<%BANNER%>

Mathematical Modeling for Multi-Fiber Reconstruction from Diffusion-Weighted Magnetic Resonance Images

Permanent Link: http://ufdc.ufl.edu/UFE0021691/00001

Material Information

Title: Mathematical Modeling for Multi-Fiber Reconstruction from Diffusion-Weighted Magnetic Resonance Images
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Jian, Bing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: deconvolution, diffusion, dtmri, dwmri, hardi, mow, nnls
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging technique that allows neural tissue architecture to be probed at a microscopic scale in vivo. By measuring quantitative data sensitive to the water molecular diffusion, DW-MRI provides valuable information for neuronal connectivity inference and brain developmental studies. The broad aim of this dissertation is to develop mathematical models and computational tools for quantifying and extracting information contained in diffusion MR images. One of the fundamental problems in DW-MRI analysis is the mathematical modeling of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In this dissertation, we present a novel mathematical model and accompanying efficient algorithms for this problem. Our model uses a continuous probability distribution over the space of symmetric positive definite matrices and is general enough to model water molecular diffusion in a variety of situations involving complex tissue geometry including single and multiple fiber bundle occurrences. We show that the diffusion MR signals and the probability distributions for positive definite matrix-valued random variables are related by the Laplace transform defined on the space of symmetric positive definite (SPD) matrices. Another interesting observation is that when the mixing distribution is parameterized by Wishart distributions, the resulting close form of Laplace transform leads to a Rigaut-type fractal expression. This Rigaut-type function exhibits the expected asymptotic power-law behavior and has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until the development of the proposed model. Furthermore, both the traditional diffusion tensor model and the multi-tensor model can be interpreted as special cases of this continuous mixture of tensors model. In tackling the challenging problem of multi-fiber reconstruction from the diffusion MR images, we further develop the mixture of Wisharts (MOW) model, as a natural parametrization of the desired tensor distribution function, to describe complex tissue structure involving multiple fiber populations. The multi-fiber reconstruction using the proposed MOW model essentially leads to an inverse problem. Computational methods for solving this inverse problem are investigated under a unified deconvolution framework which also includes several existing model-based approaches. Finally, the theoretical framework we have developed for modeling and reconstruction of diffusion weighted MRI has been tested on simulated data and real rat brain data sets. The comparisons with several competing methods empirically suggest that the proposed model combined with a non-negative least squares deconvolution method yields efficient and accurate solution for the multi-fiber reconstruction problem in the presence of intra voxel orientational heterogeneity.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Bing Jian.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Vemuri, Baba C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0021691:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021691/00001

Material Information

Title: Mathematical Modeling for Multi-Fiber Reconstruction from Diffusion-Weighted Magnetic Resonance Images
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Jian, Bing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: deconvolution, diffusion, dtmri, dwmri, hardi, mow, nnls
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging technique that allows neural tissue architecture to be probed at a microscopic scale in vivo. By measuring quantitative data sensitive to the water molecular diffusion, DW-MRI provides valuable information for neuronal connectivity inference and brain developmental studies. The broad aim of this dissertation is to develop mathematical models and computational tools for quantifying and extracting information contained in diffusion MR images. One of the fundamental problems in DW-MRI analysis is the mathematical modeling of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In this dissertation, we present a novel mathematical model and accompanying efficient algorithms for this problem. Our model uses a continuous probability distribution over the space of symmetric positive definite matrices and is general enough to model water molecular diffusion in a variety of situations involving complex tissue geometry including single and multiple fiber bundle occurrences. We show that the diffusion MR signals and the probability distributions for positive definite matrix-valued random variables are related by the Laplace transform defined on the space of symmetric positive definite (SPD) matrices. Another interesting observation is that when the mixing distribution is parameterized by Wishart distributions, the resulting close form of Laplace transform leads to a Rigaut-type fractal expression. This Rigaut-type function exhibits the expected asymptotic power-law behavior and has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until the development of the proposed model. Furthermore, both the traditional diffusion tensor model and the multi-tensor model can be interpreted as special cases of this continuous mixture of tensors model. In tackling the challenging problem of multi-fiber reconstruction from the diffusion MR images, we further develop the mixture of Wisharts (MOW) model, as a natural parametrization of the desired tensor distribution function, to describe complex tissue structure involving multiple fiber populations. The multi-fiber reconstruction using the proposed MOW model essentially leads to an inverse problem. Computational methods for solving this inverse problem are investigated under a unified deconvolution framework which also includes several existing model-based approaches. Finally, the theoretical framework we have developed for modeling and reconstruction of diffusion weighted MRI has been tested on simulated data and real rat brain data sets. The comparisons with several competing methods empirically suggest that the proposed model combined with a non-negative least squares deconvolution method yields efficient and accurate solution for the multi-fiber reconstruction problem in the presence of intra voxel orientational heterogeneity.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Bing Jian.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Vemuri, Baba C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0021691:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101108_AAAABF INGEST_TIME 2010-11-08T15:37:42Z PACKAGE UFE0021691_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 5155 DFID F20101108_AABCQD ORIGIN DEPOSITOR PATH jian_b_Page_051thm.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
93aca9bb1df92d018bb89e980e03a30b
SHA-1
20e0f4a052186d1d3333b2af2bc1f2ade75a841f
25968 F20101108_AABCPP jian_b_Page_044.QC.jpg
7f7d6c0389580b806c5d4dc9a513e3c3
5993a0fb599a9d68a022b9939b3eb91e76584010
19989 F20101108_AABCQE jian_b_Page_052.QC.jpg
80b079c77dbb4b03d75493570eb5bc89
884a84f7dcdd39b842d7b3dc9d90d50491ef412e
25031 F20101108_AABCPQ jian_b_Page_045.QC.jpg
5ebf9ae23f038a7eba471c772fb439e3
c263c7f3daac20e67592b6ae6bcfa10af30dc76c
12348 F20101108_AABCQF jian_b_Page_053.QC.jpg
65354b3c7b43495508ed0e9c4d853319
021b3ff00eb881e1ce20a296c1b64d44ee4467e3
6547 F20101108_AABCPR jian_b_Page_045thm.jpg
f21bfb079c800c7c44017422802a3424
5c10777ec56178fed7244ddde2f8deee34f88a34
4672 F20101108_AABCQG jian_b_Page_053thm.jpg
93269be170a8570184e520593d9c5fa7
a82bb13d3ef66e2fe5ea00dc2c4e0fa1f446f472
26418 F20101108_AABCPS jian_b_Page_046.QC.jpg
58656d4684cec0c7b1f4d3248425faad
35fb6c9b19e06ab25d37d0fc6335e3c6f180b1c4
16617 F20101108_AABCQH jian_b_Page_054.QC.jpg
548595d06ff6c2d17560b692aa79bf5b
b1ac63b110897d280fe07610fc7dff6efb3d7277
6755 F20101108_AABCPT jian_b_Page_046thm.jpg
0b41843b52845c1edf53f0742a5bfcb8
919395b090725381ade5f834ae6da198310247c4
26510 F20101108_AABCQI jian_b_Page_055.QC.jpg
d207e850e1b22f8f3d83dca0d2507f2e
40300c0059a69c1541bd222b64166c5af6ab8b5c
24202 F20101108_AABCPU jian_b_Page_047.QC.jpg
01f0f77d3b8e50a5eb232669b1aa9d43
214825dab2f949f52816f8ee1795f53d1f63afed
6715 F20101108_AABCQJ jian_b_Page_055thm.jpg
1e80c1d8da222d796b1343edb2c46e02
00f6eb7154af48c6cc62d4deb15c5b706348303d
6432 F20101108_AABCPV jian_b_Page_047thm.jpg
9a586a0cc92206f3f2001777a6c00773
bbc1241d35a9cd547ad3c46a4ed72b9aed2038c0
23191 F20101108_AABCQK jian_b_Page_056.QC.jpg
73c2d619a0f615ac64490e844a0ad935
8b958637e0966413229974432154e3b3278f3c84
25628 F20101108_AABCPW jian_b_Page_048.QC.jpg
2acf045cb1d2839055f3dafd24861d3b
c6400ce717c5e441f57b15bf9c621c0c6f1f5bdc
6161 F20101108_AABCQL jian_b_Page_056thm.jpg
c4208b4e031eeb479b25fb25eb3c8217
68076c15f7730f5752c9d399272fb65c8d68ae29
6482 F20101108_AABCPX jian_b_Page_048thm.jpg
129441debab810084c32635fd9859db4
ae316e7e0caaaac911effd638d98290306c85496
15972 F20101108_AABCQM jian_b_Page_057.QC.jpg
cc31c011900ec5ec166c3981085e3592
2cb34875e028cf39290ebfb5f9e6146c8c96d825
20686 F20101108_AABCPY jian_b_Page_049.QC.jpg
101152e3db94952759eae5f586e046f6
b6802a838e8979208bf9aaff813f1569d145d0ab
22088 F20101108_AABCRA jian_b_Page_068.QC.jpg
df78fb78e5d32b71a0345ad7ea02e298
e0ad44ac715857c7e201369e48c64652149885bc
4712 F20101108_AABCQN jian_b_Page_057thm.jpg
19fa6612190aa403c16678182e28281e
eb79c0e8ff9178e2c69593b99ae76d323a8e1d4c
5907 F20101108_AABCPZ jian_b_Page_049thm.jpg
2d32369e9849514086a1e69c8d3682bd
26aaad3d5d4ade2d1fb82d4d0e6f2e8daa24021d
6232 F20101108_AABCRB jian_b_Page_068thm.jpg
f552f9931d396f13daee781588b337d8
2f018b1fd6a3003d4341b196bef0b9364dbb1666
23857 F20101108_AABCQO jian_b_Page_058.QC.jpg
8d67b902442afc406d56d949ebb12d8a
55988e95636d3cc93bad62ca226c7bca4ac1a469
25536 F20101108_AABCRC jian_b_Page_069.QC.jpg
ae7cee6f4f7b6f8015e28cacdaa4d13b
dcd27eb11ef2beb7241916f5565b7e25e0a066b6
6073 F20101108_AABCQP jian_b_Page_058thm.jpg
14b3036bcae1299f0ec935b9357a3f84
c364a81eaee47ec2cba07afe8b0f3fec114a84d6
6642 F20101108_AABCRD jian_b_Page_069thm.jpg
039b86647af87a6500030e96f3d555f3
788a4cb212e1ca8a09e9138ee62f1b8cdae471da
17512 F20101108_AABCQQ jian_b_Page_059.QC.jpg
9ffc345ad7a0a0d94d8be379ac2cf673
ee95191081a800780d6e2c0529d8534cb4b06e7d
22526 F20101108_AABCRE jian_b_Page_070.QC.jpg
8d2e0adfd5dbee5ad075cbe2e4224ada
26d6af0ffa3c67ceac5d47668b0a0b2c599ffd06
5126 F20101108_AABCQR jian_b_Page_059thm.jpg
0471d7578d2c9709ecc45551248280e0
82fa7c36a30750c3f357edf6a00757a6f6bda7f1
1875 F20101108_AABBOD jian_b_Page_042.txt
b3f5e99efbae2d3056298ae9a1369a98
6887ea4402cda6bdda72ebd69672393d6e891084
6016 F20101108_AABCRF jian_b_Page_070thm.jpg
f9e98be4cf85c966629d9fa95c8bab8f
bb0caa0078f275fa2b07fcbe1bca353d7f1d0512
19102 F20101108_AABCQS jian_b_Page_060.QC.jpg
1a6bd3f18f155499b04197ca4d679044
dac78697f7acbc810f88bef0ed2e5cdd0bcf0ad2
6172 F20101108_AABBOE jian_b_Page_004thm.jpg
721b524855222fb36d586a06e159436f
7f8baa13616f5a2ff362a1f3e2bf61da64958768
27163 F20101108_AABCRG jian_b_Page_071.QC.jpg
cd1b8edc5067fda262506055a121a512
935e6a2c50ad20e94770c6b33b9e96ac75c9e162
5512 F20101108_AABCQT jian_b_Page_060thm.jpg
10871d2d82243ef5df69548a52cd952e
356eb2aa1a5a841f0d7821fb8b029ea9d847f011
1053954 F20101108_AABBOF jian_b_Page_115.tif
de38826d83714dddb407763ef927417f
cc44f26a7f99cab1fc24752e30a41ca3101d0914
6750 F20101108_AABCRH jian_b_Page_071thm.jpg
30f5c3e0c405256d145f08e49fbb3d23
4cdfcda84c6c7b35e87ce74d8ad9991a766f6b5f
17906 F20101108_AABCQU jian_b_Page_062.QC.jpg
4d688293df93e08d25fa945475861409
de3d30404035235cb8093053a9dbf0da6a22be0a
25271604 F20101108_AABBOG jian_b_Page_009.tif
11c38579a02acd5f34f665e014ae1a5e
ec4fabad5572748ce885808cd88c9259896e3d92
6563 F20101108_AABCRI jian_b_Page_072thm.jpg
a73e0b85b2b9c4d63f077b735e47b013
e612ecc85aa61a018242bf07487dd08292c29bbf
5181 F20101108_AABCQV jian_b_Page_062thm.jpg
06890ebd186ca0d96c9a17a913ce3231
25cc6b099db2756f46b5e60eda872cb249492e96
6424 F20101108_AABBOH jian_b_Page_081thm.jpg
34fa270da995683ab2a8e862afd81c4b
46c363e8fa997c13116e9482f356714f14c4755c
25792 F20101108_AABCRJ jian_b_Page_073.QC.jpg
95b647199a5b3a99da3addbb52c73e1d
ae2f1d53bd941c629d2c9944089cb5c83ebbe12c
19237 F20101108_AABCQW jian_b_Page_063.QC.jpg
e8de9a6ee60bce76e29f715d71ed529e
fae539af623eac804771fa290ac36942019b6e5e
19564 F20101108_AABBOI jian_b_Page_008.QC.jpg
524777e508e9032878027ab4537c2b74
2c95e2fae2ac4311b8d14e1af2e9974aa6a14133
28953 F20101108_AABCRK jian_b_Page_074.QC.jpg
5c4d6396fee601f1345c3653bc0ff4f4
131b5d782a9356877e86b9615d8a96d4d288db73
5636 F20101108_AABCQX jian_b_Page_063thm.jpg
e33cef218f4ab908378703e2fecca965
1c07a31cc51f98e011a7d3ff02574e9dd9630c36
53722 F20101108_AABBOJ jian_b_Page_054.jpg
6c176c357a6db7f1fa8fd7f1603a4917
a903230f3aabc94558eb111fb35b5382379ddff4
6991 F20101108_AABCRL jian_b_Page_074thm.jpg
a2760fa6ccf365093801a5fa69c0b5f8
ca9f0b11e9de28927236af7cf9e838ef39ade2a3
21111 F20101108_AABCQY jian_b_Page_065.QC.jpg
2267fdae5150a5fa3ad0890cc9268681
e4dae26ecd1848bf1a0b4d0236098a9e56f38a24
23112 F20101108_AABCSA jian_b_Page_086.QC.jpg
96b9e1a0a2e80ea59a4f66435febacb7
b6dc4fc74f3e31b420cb30d5c5d0c511ed16aa2e
2231 F20101108_AABBOK jian_b_Page_001thm.jpg
f6c473975297cef8b5a1b8c3fb223bb7
73a2af78eeb7e6fc4b1fabc97a17adf34e8afa3b
6061 F20101108_AABCRM jian_b_Page_075thm.jpg
6dd9dc580edb5b3e222f13231b7a0b54
e502cb315a2823d546176daa568c492b86427f48
22133 F20101108_AABCQZ jian_b_Page_067.QC.jpg
da2df14cb6693c680fe8740be9f45624
4300832b4d7a76e8314e113a8f842a9ffc5d96f2
6144 F20101108_AABCSB jian_b_Page_086thm.jpg
30b60a964f20fbabfb25320d6c95bb28
ffe2fa5cac5dbb1c313add62d814409327498b0f
57831 F20101108_AABBOL jian_b_Page_108.pro
69c23f467a6d96cde5be158935adf269
92f90aa01f7399c58adeb70e63e3b629a6bb86df
26500 F20101108_AABCRN jian_b_Page_076.QC.jpg
498c450dc171627203496fe3bd77af02
15f5eb0168150d51425748b04c3562c1740c5986
1454 F20101108_AABBPA jian_b_Page_062.txt
cdded041854fd9189f08e0ad2662ccc5
f123292ff99f732a927a82b266d9cea7a7d72fe2
24888 F20101108_AABCSC jian_b_Page_087.QC.jpg
360e84200cea094b674fa85cdf0c98dd
34c99c4215486c120f2dab880de7591d5a50355a
24339 F20101108_AABBOM jian_b_Page_116.QC.jpg
0ba4287565f4d331f17502410823a756
b593b4a44fd7c82033426347b4b81ac3c7531c54
6848 F20101108_AABCRO jian_b_Page_076thm.jpg
904325ba285198d225a9495a7eece8aa
bece6d93526d49e610415f8a1d9d74bff74357c2
53389 F20101108_AABBPB jian_b_Page_104.pro
5b4970b35f85809ad9c57960b336f9d9
8ed214fe917a03e160b3a80235d25f02d2b7a7a1
6648 F20101108_AABCSD jian_b_Page_087thm.jpg
0c4f54d366b882bfb06bf43aeb98a435
df6eae2feadeb775e65f46b4d745536ea0942bb8
86624 F20101108_AABBON jian_b_Page_055.jpg
11a99bd326bda076aa69af2b9b2149af
d201866855a3a049d1a46e0c8e10ddd37326d561
6932 F20101108_AABCRP jian_b_Page_077thm.jpg
ec378e219da1d6420867d6b312090684
445ddd04e257b20854ca80080921cffe8c374ea3
1730 F20101108_AABBPC jian_b_Page_085.txt
1283d62a07b07607f8ebb1c1748ade97
9ddaf82662f9cb37ee15d001e5cfd9f69cae431c
28010 F20101108_AABCSE jian_b_Page_088.QC.jpg
540e443af5beb71ed15c96a360971944
e7db8365546b911cbb16bccb1beab6e157aff7a3
2266 F20101108_AABBOO jian_b_Page_004.txt
083947cd0f25fe9a21dd2135787b610f
bbb2e7a446172fbf7bbf8c711f9dbb277f669974
6041 F20101108_AABCRQ jian_b_Page_078thm.jpg
15405444fede0df4bc4fce18cdc51255
398e043a5de4492f07db538834dfee29ee2b61d7
6847 F20101108_AABBPD jian_b_Page_053.pro
11a159ef5c952dfa819c05eb429c53f0
6316315dae600087c35bfa7da10807988c863ec1
21013 F20101108_AABCSF jian_b_Page_089.QC.jpg
aeebecdf24b219ea9101f38005262206
9e4051fc45b156f191feb92e05e2e449461b2af7
26091 F20101108_AABBOP jian_b_Page_041.QC.jpg
840e1d3406dbe7b76b6cd3977fb40d1f
ed017e408cc99967ea06814bb7b2fb86d53942ac
22209 F20101108_AABCRR jian_b_Page_079.QC.jpg
993bf66c26f070458b1eaf192376a6ab
1757c335ef64a5d08d2dd5757250d464382b9907
135556 F20101108_AABBPE jian_b_Page_113.jp2
b5eaed313a55584f44f0bb106f8441b1
6823f8a8b9b6fef90fef10e7cdb59a4a23e145ba
5841 F20101108_AABCSG jian_b_Page_089thm.jpg
708b53a8168ef73e47d8c75a0e46a4d7
eccfc094d1e3a2e4ccff2ad8212a82b2421d6b14
77746 F20101108_AABBOQ jian_b_Page_109.jpg
f2125a41b48dfa18a66eefcb2b71140d
75e7b8102a1ac126049eb00573d11780f91594b2
23147 F20101108_AABCRS jian_b_Page_080.QC.jpg
69b5a6e59537c9714f7e9d45a0883149
6bcaa4281b592e2477e84e550de50390c344d6dc
61369 F20101108_AABBPF jian_b_Page_088.pro
3b10e4effa595161712c6f4ff8dff8c3
41028f1e9af33d7a5e0ba77c5ea50af9425142ce
26069 F20101108_AABCSH jian_b_Page_090.QC.jpg
afedcfbdcc0e83645888ca5404858e8b
ed824deff1b4144689a68df41f2552528b3d44f0
49943 F20101108_AABBOR jian_b_Page_047.pro
b8018b10766205b2076a2ed9378292a5
0933826ade9670d1a1d15d83b61c22430d47abd8
5995 F20101108_AABCRT jian_b_Page_080thm.jpg
048a325a2c7a1c620210beaa6c0b4cbb
261d2148d5298d8b4771a965d742fdbac5c09f80
732632 F20101108_AABBPG jian_b_Page_036.jp2
44154b07fb827d1726dfc2498bbe877c
142d2b3f98bdb76ecfa5c25487743b8bb35ce74a
28649 F20101108_AABCSI jian_b_Page_091.QC.jpg
a63e78274ec6008ef5219b95f23c625e
dde98c65e13491b97701cb8511eb26cdf6382349
4471 F20101108_AABBOS jian_b_Page_017thm.jpg
16c45e359d4d658c9878970f34ab67fc
30fa6b09a7cbf18076e0cf6d446b515aad20c647
25238 F20101108_AABCRU jian_b_Page_081.QC.jpg
1d96bc8927bd3a504e9eebb1d9a65eeb
22c1fa1a1129b8223ef74cc5de83718287c5162a
7160 F20101108_AABCSJ jian_b_Page_091thm.jpg
c7af1dbb88f4198e62f1c38b0273a428
68c5b59bd683744a2949b74f995dde96ab16931a
56124 F20101108_AABBPH jian_b_Page_059.jpg
833f7270048ccc7dad003b3ce96658f1
c35a2b9b86d84ee9b1304d94c89b262c6579588b
293 F20101108_AABBOT jian_b_Page_053.txt
4c1fc278aca3286af0605ca5be1f695f
485be3f0f69aa23ebbdee72b2a76ae93a734dc7c
6581 F20101108_AABCRV jian_b_Page_082thm.jpg
5b525f3a60f545afb927120ac6dd50ba
373826f811ea2065e6205ea1dafa57702e60a25e
28396 F20101108_AABCSK jian_b_Page_092.QC.jpg
950d309c1955e67562b6136e2d1c4470
b67fac436cf391c7839ea29198fcaa3c4cbb3d0b
2127 F20101108_AABBPI jian_b_Page_089.txt
dac54a6535a445fe0be2c13513cf2b2f
324bf5f0f5993fcdefa6daf4fc347f047f18d091
F20101108_AABBOU jian_b_Page_067.tif
19c0cc93a20307ef86832397425d955b
b64d2548ab1c9ec0a525a237b7c4e9d4a968a25b
20908 F20101108_AABCRW jian_b_Page_083.QC.jpg
fc624952f3950a6e7e5cbd926f40c8fb
c0d0fe8a1d2a073e01dbea14fadda60a55bcddbc
6927 F20101108_AABCSL jian_b_Page_092thm.jpg
855b076a9e19c2e1134e0f960d222e07
a57e3e1b9d116569ec6b8812913189b9d366f433
6217 F20101108_AABBPJ jian_b_Page_116thm.jpg
2e7df72c3e17757d268d74b410ec9a8e
5e82569f11933f2a8940de26982159f6e4c7f358
F20101108_AABBOV jian_b_Page_027.tif
e29f7758062f719158c92b8dfc9aebc5
78930afb95fea93ebb298717199a840a661fe63a
27616 F20101108_AABCRX jian_b_Page_084.QC.jpg
c6cf068b5c3708f3ed3af13e811c6bdd
ddcda09b58827f83d39502fee895a644ab1d034e
27772 F20101108_AABCTA jian_b_Page_103.QC.jpg
dc8545bcfa246c34254fe710ba830f5b
7c586b3b5deb3da5b75f10d12ce38a0f375c053a
6192 F20101108_AABCSM jian_b_Page_093thm.jpg
bee6744f913863e909d15dd74c8079df
794575995d3408366c15fb83ca83d4bba9644419
45974 F20101108_AABBPK jian_b_Page_034.jpg
2d34bfd2367a9c8fec858577c3838e3c
67b4ac238f7d18eb39ca766d7ace8f02ecfddbb1
F20101108_AABBOW jian_b_Page_065thm.jpg
6eddbfa5224429d6e21e9161c2b3b4c8
697c2ec99ac4e9b269527fe51e1bc5fc3101a477
7011 F20101108_AABCRY jian_b_Page_084thm.jpg
5723bec35e10359e46dafc1953d8123c
0a42d8e320c81be6d5369231fa567554b2967bbb
6730 F20101108_AABCTB jian_b_Page_103thm.jpg
6a2380227d7217bca2de86b29f9a23e7
a6b35bb511ab337f34cf11f92a9ea58689dccb69
28268 F20101108_AABCSN jian_b_Page_095.QC.jpg
e581bed45b324568dae1c5bec7f8b672
e0c052f3e8659b6b30c9b9b2bc924abbf1403ddb
F20101108_AABBPL jian_b_Page_024.tif
d5e4dc201400772c070cdc65387224b5
965d1252151395c1dc30f198633115f6d3e5d52a
2356 F20101108_AABBOX jian_b_Page_092.txt
ebc11061531d04ccae44dc15c143d2e9
f5de06b8402264801d25f1d29f89f539bee17d08
5627 F20101108_AABCRZ jian_b_Page_085thm.jpg
6f50e51331725d5797a920871752e244
80364c409a7c080851c34e3ee12b71e3cd30d1da
25518 F20101108_AABCTC jian_b_Page_104.QC.jpg
3ff17fca903b1e4784bca723023e2bc2
3bacfee5ce9e7c03f85043c8f37e409172bb65e6
6937 F20101108_AABCSO jian_b_Page_095thm.jpg
e2da74540172e0cd4d4a0939805ada5b
fa1475b6d9368f0973421728c555744ee32dce71
F20101108_AABBPM jian_b_Page_056.tif
d0ff90127a0419c4b270fa68872478b1
7f2b81c97d7ee44604aaef836af5d339691e47aa
21070 F20101108_AABBOY jian_b_Page_085.QC.jpg
fcd07a26ec0c9a3806b1ae12ebe74f35
04d06565421a0bed6597527ecd44428749051343
967888 F20101108_AABBQA jian_b_Page_033.jp2
5c9b3cb69d57206e5aa8808f6bdc951d
5011f89b967fb570b2066439967f558e6c04a736
6309 F20101108_AABCTD jian_b_Page_104thm.jpg
cba4f46f034de2dc00d46111e55970b6
0d416a77704f119c8154b0997469849c8fef5632
25978 F20101108_AABCSP jian_b_Page_096.QC.jpg
a96836ff135038048bff1847d024ce1d
f51d1b5dd307e6677ba2f4683aa5f7c5c132dd21
38771 F20101108_AABBPN jian_b_Page_064.pro
91bf1e00f482698830ff347bf7c2ff84
f674836e7a4527d31a05221b9aea1ea772f0e21c
1051926 F20101108_AABBOZ jian_b_Page_082.jp2
9ce439d963834ce9aa66bff326d3c9fd
df68bc6b15771bf85730ffe11268049db0775471
1917 F20101108_AABBQB jian_b_Page_061.txt
d3fb37f0f126ff5554bed668d378e9ff
e5673883d2c81da1e3445af326edb36396f31a78
23645 F20101108_AABCTE jian_b_Page_105.QC.jpg
10fbbaf0f77409859099ddb67c75ac17
6f98e2ca022c5371070eb021d3260623da68bbb9
7018 F20101108_AABCSQ jian_b_Page_096thm.jpg
e88cdf1003d2e19a3d017bea1847ea52
7eff80e26f91d6842a79160f8c222e44625ac4f5
7789 F20101108_AABBPO jian_b_Page_117.pro
e52e379601285fcbc370e9fdbfd38dc4
e0eb9129d2fba00ec61d376c263970c0516b00da
2200 F20101108_AABBQC jian_b_Page_044.txt
31bf36c094b01f85e5373fc5fc7e2c38
1a1ce36ab084883bc7c5af2352bc582ff09222b5
6141 F20101108_AABCTF jian_b_Page_105thm.jpg
1d0f858060958fab17e7bc90feae0fcb
79507e5ccf8c3de4b7367e444ce2181e15e962c9
6494 F20101108_AABCSR jian_b_Page_097thm.jpg
9542092ec291360550102f725bc13eec
0695f6eb403dc4bb009131d04766101754c6291f
2584 F20101108_AABBPP jian_b_Page_008.txt
ae48c084a99be15dd551c7b65adbf86c
d5bb11045291a1d4af3468e5f58ade8a0ad1867b
91743 F20101108_AABBQD jian_b_Page_091.jpg
85664da34edde856bb30a9e185756039
75ca4408dc96bf2536c89b825471b7f9e8c3c07c
24089 F20101108_AABCTG jian_b_Page_106.QC.jpg
62ccfbbda066f1a45db9eac8947d154c
575de0b2540281984156b379bef8cb0d854fe10b
6694 F20101108_AABCSS jian_b_Page_098thm.jpg
292777575b5e703f6f4610d83a9cc4ae
1332cc400a9f82a0d735d780c22f6298e80b2620
81402 F20101108_AABBPQ jian_b_Page_048.jpg
de66c24f95c8bc2a53c0126f51966839
d4945f6d0c8b42b5155fba77dd1eba3785157ab7
F20101108_AABBQE jian_b_Page_103.tif
526bd26047c0e74fa0ac43ab2c194dc2
308d63b2b7ff1b692a19b2a5fd011dbbff9b9252
6447 F20101108_AABCTH jian_b_Page_106thm.jpg
04126bb94f19dfa075ffe262de32781e
a6e06b9b9ea094b45638bdce2f0fb6c14e9d4553
3930 F20101108_AABCST jian_b_Page_099thm.jpg
7a4e2eca6ed9ae9f1dd758aa2b1d1c0c
089599cdccddc42e2f91339fe95869f3bfc73736
7055 F20101108_AABBPR jian_b_Page_042thm.jpg
3e56bb39a894cac78d67dc74857d633a
5ceb5aab8f37ae7c2000f6cfe0637cfd3823e5fe
19528 F20101108_AABBQF jian_b_Page_030.QC.jpg
a72788e04df2f52cbe25ffeeed53cbbd
0f28ab6b1c85087e962992f7a07c1d0432e36a9a
25577 F20101108_AABCTI jian_b_Page_107.QC.jpg
7044c84866785cf52e81848f3b3ff9b1
71b11aa2b6b0e3e37e4ebb08a29f8c92a88c7440
25240 F20101108_AABCSU jian_b_Page_100.QC.jpg
a7bcc88e08f8e60e714f29e1ca248778
837cb38f12f5a5953e5a4d55d03f2b6ee16d584d
6222 F20101108_AABBPS jian_b_Page_094thm.jpg
93fc90dc7ea1d8ec035b1fae652b9c9a
15c2b3f99da7c840aaa9007e0e08999cbd4de761
26939 F20101108_AABBQG jian_b_Page_066.QC.jpg
679dbf55d43433385c59302afc0d011c
c129c28d0b0ddc824a71b885b52d0592d4fdad0c
6512 F20101108_AABCTJ jian_b_Page_107thm.jpg
6c9c0f3e5be337641512b8ad6ed8e01d
53b94fcf757c5f277d2e0703285067e375f8b440
6431 F20101108_AABCSV jian_b_Page_100thm.jpg
f7f2c3dfb38e5879d41318f371962f4b
fbc2e37b6fa235f7bcdedb5480bae914a22e38af
21224 F20101108_AABBPT jian_b_Page_029.QC.jpg
55c60107610ce16fc67e31937988751b
611f6a20787c179810c0f5e607781d190734e026
76738 F20101108_AABBQH jian_b_Page_108.jpg
93c95f246ecb4ad1963f476ea6a2e969
eae58b1e8d23dd30aa0204e378081dea89e9aa90
22554 F20101108_AABCTK jian_b_Page_108.QC.jpg
7831c2e0bf174cc2a0558eaa94c85d7f
ebd104eda2b95c528692906bbcbf5eb382d31d14
25804 F20101108_AABCSW jian_b_Page_101.QC.jpg
014b325fa4842ac837d6bad13205aefa
d55dd07185a590c61db75ce77e30dd593ccd2548
F20101108_AABBPU jian_b_Page_006.tif
8259f89cf7ff138fa25e231e25e1c349
b3cbd8b34e343522707764aca11b2ca0493d8763
23307 F20101108_AABBQI jian_b_Page_072.QC.jpg
0a9db062eda2da58a14a79ca51758453
9eb0ee4a329407e11fd1896f5044dd4fa3ff0dd2
6097 F20101108_AABCTL jian_b_Page_108thm.jpg
eba63103cbe339d2386ab72ae69f7aca
5eb779ff17ef70b92e39d2f4c6ae033eec364a6a
6621 F20101108_AABCSX jian_b_Page_101thm.jpg
c4a056be284497debded9e491ee55031
12a91033daca273626e00c7bee69ae0a1594e918
18694 F20101108_AABBPV jian_b_Page_038.QC.jpg
2069467a24a449a85578dffa9ada9482
b6bb730cdba2e7e14bfa45e92544a9c7fc6e2429
F20101108_AABBQJ jian_b_Page_043.tif
5a453e84ff8a269c73c2c824349c59f8
ec68259f149785eb66c06876fe5163689cb9763f
3142 F20101108_AABCUA jian_b_Page_118thm.jpg
019b51e4d0a18382857175d86f6deb63
4afc1ae633bcc25cc6b484e5e1367bcbf9b5a350
22688 F20101108_AABCTM jian_b_Page_109.QC.jpg
ac55d2836dba8f0bf4b57f942add9520
8837e70590dea2a82374acae680ee5cc9be373e1
27145 F20101108_AABCSY jian_b_Page_102.QC.jpg
4e77edd26a3448b6417dd6aaa585986d
337f855ac151ace46cd4037e1a5199c0733ea329
70388 F20101108_AABBPW jian_b_Page_067.jpg
675062211b95948dea4beda7fd9da60e
c21aa2dcf8a452db31fbdac29c380c19b5dccebb
F20101108_AABBQK jian_b_Page_063.tif
f90e2f8e8fa680ba692b760ed2140c10
e081d305a3ca0a07e800a8ae1728f2ecc31463be
135700 F20101108_AABCUB UFE0021691_00001.mets FULL
f76b71c180d0548029b61ddb402c373c
88b6d0fc4d6e86c1b6204657809328a4f21c318c
25056 F20101108_AABCTN jian_b_Page_110.QC.jpg
b2c67b401c79b14e37ab1af2fec429cf
b18aed37386c03d23de0d39124592f0a811ac8da
6802 F20101108_AABCSZ jian_b_Page_102thm.jpg
f953eab30d3792562aad30bb73178b3d
3774b27682d2c46919085696bb14ff8f3cbd6b32
15374 F20101108_AABBPX jian_b_Page_099.QC.jpg
616a8a3eef3eea5184a7e742f461b333
91191310bb0c638acf7f404c7b9f8accc124e7e3
139844 F20101108_AABBQL jian_b_Page_107.jp2
7807102f704bd17632be2c0791085d8e
d61b7392be904ead18b76c16060f2a7a9adbf02d
6390 F20101108_AABCTO jian_b_Page_110thm.jpg
739f944086ececa48965f4099a9249fe
ecef9a4fad5eb2df6e1181a9418bdae5ec9009ce
6239 F20101108_AABBPY jian_b_Page_005thm.jpg
73185ba437bb80353e8138fad7948165
5c7ba4a6fad7c08f33b5360555812b284e04501d
2419 F20101108_AABBRA jian_b_Page_088.txt
bc52e77886583a8e207308961a2ee879
c688f75212ccf889ed0e3fd10b47ee4562ecc39d
87907 F20101108_AABBQM jian_b_Page_107.jpg
21e19392ef0f497702865a7af14dd1a0
9f7a688833b73f8bbfebd159ee2d3957b629cbc0
6298 F20101108_AABCTP jian_b_Page_111thm.jpg
3aa32358ec4f767d46dec33ee1ed6627
9402c0f569ebf394f25bb898a95b0c46661b7a31
132334 F20101108_AABBPZ jian_b_Page_112.jp2
efe136f4b32d3a47cd7e4e9deb14a377
3ed98caf71724d7457fd3ad5e488f11d2b33e27d
6980 F20101108_AABBRB jian_b_Page_031thm.jpg
33e3f38d86dddf657ac36cf72108705d
a6e9b755da8f63db5becbb59eb60fe315ec76bd2
F20101108_AABBQN jian_b_Page_102.tif
b4bfa40585d35c17f69065750945be14
687307a137f0be1f916528208f9660ceb47af6ad
24385 F20101108_AABCTQ jian_b_Page_112.QC.jpg
740a89caf18833d96a88f9674e607373
42bc5cecd332b8581aae28b90bc65de5d727f515
2334 F20101108_AABBRC jian_b_Page_108.txt
693a0f5fa7cda09025e5fdb634043ae2
b256d3c08233bd10758c1dd12f72302b94511fc5
F20101108_AABBQO jian_b_Page_049.tif
29680aeb171914d7838369bcb8e71d61
cb816eed62d5bd0869c3f9835db3e551381889f7
6329 F20101108_AABCTR jian_b_Page_112thm.jpg
9cf0c09a64e89edb73a4ca1c3e9ca303
f78bdfac83411a4ca667ea42ddbfbb31944a9c6b
856414 F20101108_AABBRD jian_b_Page_027.jp2
73cff34bf29f0d08c569711a1da817d0
7c2e5c83cf0f41db0c727e87e770f9020ed5e476
27439 F20101108_AABBQP jian_b_Page_010.QC.jpg
bab858065a31a2a000f6a1f02c623b6b
d77cf0b353ac897e4c47782038052674cdd93517
24815 F20101108_AABCTS jian_b_Page_113.QC.jpg
c75139a05c780b115577219ed64aa0f0
a39b2f35e1b382f1b0a79bb98433e97c67152820
21654 F20101108_AABBRE jian_b_Page_022.QC.jpg
75bae0830d4f833a1cdfa695d97c7c99
bde1945baf0042d07c95f7e7aa83c623e3a28f17
5777 F20101108_AABBQQ jian_b_Page_064thm.jpg
cad2eaa1e2284d5fe6dcd81480676f8b
bcc2e9132c9487339dece65286a290cd363d57dd
6510 F20101108_AABCTT jian_b_Page_113thm.jpg
d554f6c7a78cc0b5b85ee16a0e834170
f9c2bfd7f556e2b24039413c50cc7afbaa41d9bc
7181 F20101108_AABBRF jian_b_Page_006.QC.jpg
7a2bc04a08975f586f4e742da17dcf20
7ba89a5384512e0fe5b20f8d8861f162af957e3a
5787 F20101108_AABBQR jian_b_Page_083thm.jpg
14bb5fe46ade8ecfabc479223c12067f
8b1256e7be9b49965da7b86d953046de0edf733b
26260 F20101108_AABCTU jian_b_Page_114.QC.jpg
3a3cae430f663b8b7f8119b917bb08d6
3112461501ce3b9bb8666b3c4f9c7f26fb2c7a64
1971 F20101108_AABBRG jian_b_Page_080.txt
f830dd69e530d1735c287385faa27b5d
b2d990a0b86859e9f1678a46c931dd0dcedc393d
2229 F20101108_AABBQS jian_b_Page_101.txt
281a82bd321f86f7d8cb5eec621dafb8
80fee246c0b5017d07c38c42c301a36124fac4be
6657 F20101108_AABCTV jian_b_Page_114thm.jpg
97fe5598349ebbdc5fb2c18402c6382d
5c65eef5e79d3709460ccf1e609f8c22a006276e
21508 F20101108_AABBRH jian_b_Page_093.QC.jpg
eca3d184d133bfa5299a7777e7f5c5a7
48a37c39369c9d2821fbe1e1d74973546092e420
24499 F20101108_AABBQT jian_b_Page_043.QC.jpg
b18b02363b560845e404438b604fce74
234216cf7a724894214b6d182a5c662a825887d4
25647 F20101108_AABCTW jian_b_Page_115.QC.jpg
95feb29bfab9b394f380ec824baee481
4c152b69fa9db2784838e524ad72fcb10ec991c3
48991 F20101108_AABBRI jian_b_Page_029.pro
d76a24acfec8b229b66bc63a6ebb6c93
244e6d1c5183b366db3eb80ef42d7a45ce298ad5
F20101108_AABBQU jian_b_Page_093.tif
474ee18a79107e0f4c4f540f61a8c8af
35ff6c648d79e0b7468c3e4890553c1552e4797c
6557 F20101108_AABCTX jian_b_Page_115thm.jpg
d7dba13662aee65c45395ca08c0ae5ef
e93896067c86c5e280a882f9c82fb7bc1387d4c7
98139 F20101108_AABBRJ jian_b_Page_100.jpg
5047a794216e67ae6b17ab0a4b82da6f
596d96f3d5004dce7765b9933e83e5edc05713d5
482 F20101108_AABBQV jian_b_Page_001.txt
20f9f441fa6a210299c1b77b9c28a0ba
4b84bc53b1f7d272cf298a4a5a8c84c9a3fe1c9d
5518 F20101108_AABCTY jian_b_Page_117.QC.jpg
3e191367c50f5c9e1e47bf6f2479fc52
a1b66c243b0de130aeeeb8f7cf28e6fae121e60b
6610 F20101108_AABBRK jian_b_Page_044thm.jpg
daca8954e07192e49991d84a20100762
e904a1c72d9d2e1aa134d3f59382ee825ca9db3b
26060 F20101108_AABBQW jian_b_Page_098.QC.jpg
3917b0cd981c94993f0712882454d992
3de39ee751f851733cd93a88e7127b31c903b074
1945 F20101108_AABCTZ jian_b_Page_117thm.jpg
b70fc2457ed63465ef046a90ef45f068
346fb36854f7ffe89fbe1c8361b3aa94d4d686ca
22959 F20101108_AABBRL jian_b_Page_061.QC.jpg
278b47f4c70e37565191c7de95c0d76d
f2a0bfb965842eb3265aa1bb9babc46b55adedc4
1679 F20101108_AABBQX jian_b_Page_038.txt
0f487c455111dd4874f1799983a3b264
4e677596f53f88e3f719697c0c8eeb0c8353b12a
56789 F20101108_AABBSA jian_b_Page_055.pro
152d12f46354244e7eff32177f2e271a
dd9c5cf94780c3418bf6a972376229c7b8fccc49
88941 F20101108_AABBRM jian_b_Page_115.jpg
08bcf9bba45a7c10903fbb00821a4691
5ca7a56a2d70d66e1d9ad7af3f6ba393821889ee
1382 F20101108_AABBQY jian_b_Page_003thm.jpg
d701a90606b13b6dff319703f9b9c999
8233c844524906d797ffa235a17ef26d9369d708
20965 F20101108_AABBSB jian_b_Page_028.QC.jpg
fa3ac18143bff48edc3641ad2b26e17e
86c4dfba76e1212dfa9595207b1306dd94020dc3
59158 F20101108_AABBRN jian_b_Page_012.jp2
3f540d651da1fc2027626fb887512923
b68941eea4bd9fbbae1babe9db52ae42ce2782a4
5939 F20101108_AABBQZ jian_b_Page_079thm.jpg
93923d3b0069082fcaf21d1ff0dbe47f
c6d45a4891b86acb3939bcdd2eb97d7e50a7202c
F20101108_AABBSC jian_b_Page_051.tif
d869b08b9d05c96f302bc6d0b7fb0815
c99533f200e6048becdf139e506076d91efa4888
84876 F20101108_AABBRO jian_b_Page_042.jpg
20409a4de96dc6da39a9f1be49c0df70
489ea20043a54da89d290ac4f6a4a2d91873a359
F20101108_AABBSD jian_b_Page_078.tif
afb900077edb156bf41dfcaa73868fbf
19e65dbb6cfa04854b0398fec5d924b228781ed1
85485 F20101108_AABBRP jian_b_Page_090.jpg
5b860fd876f7fdaf411a536f220e7544
4f1e7a0c9fb15160992cc8b03e4c941193d04061
858808 F20101108_AABBSE jian_b_Page_030.jp2
15e0084f81b9fd80f3b74a3437a9d048
0de8e6452fe3695c8604cc804d69770ec91829bc
1051986 F20101108_AABBRQ jian_b_Page_010.jp2
329d4cd8d31121cf4f42c7e1a79e5bcf
f216899b6cc1ab3aae6fd368e819b40209319334
63715 F20101108_AABBSF jian_b_Page_030.jpg
ba4e2a9a838051dbf1a1f22b1657402c
a5feb10297ca2870cf653c3b664baf5c8bfe3393
F20101108_AABBRR jian_b_Page_091.tif
08857e47ddb257ce9bccb882c7b8f5df
11a95004e9c64bc113b33f5a3f13d18cea255e7a
25733 F20101108_AABBSG jian_b_Page_082.QC.jpg
3682f3dde93219934a0dc8b669132226
748e85e9b23b77c35bdf34215cd10f26a1d85ad0
6637 F20101108_AABBRS jian_b_Page_090thm.jpg
55901f2e564ba23f360661350e2f909d
77abe92ad3b9bbb19029dbbcccb12159710c25e0
5385 F20101108_AABBSH jian_b_Page_027thm.jpg
aa9f49755c7f95e2fe4fd899f09f80c6
246493a41dba9422e326d47051d7b45c71c2570f
26591 F20101108_AABBRT jian_b_Page_042.QC.jpg
b3dd48045fe2cceb2b2eb9e4e8e443f3
d855044cf85fa505afc85478ccd3e1e3cca984aa
F20101108_AABBSI jian_b_Page_116.tif
fc39ad15d415e85b9643aa838f4bd6a9
6d1d2f14b6efd22bf9d12753a9415eb85024bf46
6400 F20101108_AABBRU jian_b_Page_073thm.jpg
389daff316aa8d13d694591abd179f6a
3f928e821e378e216e736448a2c7e6a8cddb344b
1040132 F20101108_AABBSJ jian_b_Page_086.jp2
42dcb1c4f788dba57794855970826b9b
49fee1023ace9b3bfbaa8549e777cbeeac673cc9
18724 F20101108_AABBRV jian_b_Page_023.QC.jpg
4de458cec77795430313941d81502f1f
b497882f17d04be39db5e1908904ca0af9535067
22117 F20101108_AABBSK jian_b_Page_016.QC.jpg
571c7296e810f884517e95b849db7d1f
f79cde63ad4ce2b42b77c0f72916e7b28307c5a7
78068 F20101108_AABBRW jian_b_Page_072.jpg
ee684da6406900042da03d8c781383a8
60c644415c1a5c7ebea61221f6aafee3efdbe609
2237 F20101108_AABBSL jian_b_Page_021.txt
602a5698e5caa2afe0f9e219752de58c
419b96fc5c3e4b3ced2f93836154b80438142501
2796 F20101108_AABBRX jian_b_Page_010.txt
1a2d63cc7a048d1219945c8e3c7bd213
9a6822c7fb6242b4d0b3b6c4df685a080feae800
26314 F20101108_AABBSM jian_b_Page_077.QC.jpg
c362bbfa60c79ab80efe86677cd1cfea
204a775e2d2b3dbc407652613a2748bba6648367
F20101108_AABBRY jian_b_Page_010.tif
85cfc6e5999c714f829a024c722449db
372779953983e2180d68d9b129356616d77bb57f
1051938 F20101108_AABBTA jian_b_Page_041.jp2
e79ada58fc5b1df354375a4b8bc503e1
078730bfe461f7f8940679dc060b6f967d1f9da8
87370 F20101108_AABBSN jian_b_Page_098.jpg
25d49b871bfa5c8e305466d5992ce409
5da76e264ee625b2d57792d03727641739ff8ba5
68041 F20101108_AABBRZ jian_b_Page_040.jpg
8904497152f9ee7310766d8ea47860de
e2f27576f214a106d1d5cb0dd3469d2e699bc54c
F20101108_AABBTB jian_b_Page_016.tif
c1dcbb24ad1ad4eecd1a4c29258935e4
648c8f10edf3923c65ec7e946b08366ef35b86bd
1037565 F20101108_AABBSO jian_b_Page_078.jp2
a3458b7a7aded4f5b3925c68b57e73e6
3a4e977bd8b6c8ac50ad663fd36cf9b8ff55ec3e
1779 F20101108_AABBTC jian_b_Page_049.txt
8ce7769833115e73ed512abe58fd2671
d61b07b0f75302125836df3fcb5a9661f8742d95
1051983 F20101108_AABBSP jian_b_Page_094.jp2
d5b2a206e0375feee3ebfaaef80e405b
e0be1101150eeb197f6394dedebec390468cefd7
23426 F20101108_AABBTD jian_b_Page_078.QC.jpg
515c99dca59d7ae3e0d517c8496fe3a8
5da06be36cb11622128f7e321e379d87036cefde
55680 F20101108_AABBSQ jian_b_Page_031.pro
813316cb16a2477f31b4c67e0d3ac6dc
146758176221cced6b1f874a1aa04556aafdbd5d
138860 F20101108_AABBTE jian_b_Page_115.jp2
2b267f518fa4212b2b678bc4a200e961
168a33b27d605713dfb23aaa4bc9b1604f8b6947
59325 F20101108_AABBSR jian_b_Page_062.jpg
1a4c734cc5ad158a8ffbef77aaf25844
85af322559a99dd392d47b1f1a86eef495c0bb7d
2351 F20101108_AABBTF jian_b_Page_091.txt
385cc722afedfdb959952fae75dfaa5b
1955fbc92ae7c2a162d9656b5067142c8c49538d
F20101108_AABBSS jian_b_Page_013.tif
c00e33f390170787cace7ec6514c42e3
103847fd81e6ea7828f0bc182094eb1af1562142
1051958 F20101108_AABBTG jian_b_Page_095.jp2
613b7eafb4f335e2a9441cc1e4c5e0ae
f5f7d402df83d6bdea65b8ecec5b09ddd4be471d
5612 F20101108_AABBST jian_b_Page_029thm.jpg
183d8671145d4fe336ded73573c30e82
04233e784007db030ce31980c835d1182135d573
7035 F20101108_AABBTH jian_b_Page_088thm.jpg
70ce8ecc761673f025ff4dff976d8b65
801e3711c1e8f361e9240c6bc302d5fefd31f204
F20101108_AABBSU jian_b_Page_035.tif
f3e634554cf706e10e2437e158743c7d
414cc81460f83318187b5fc0bc388aaecf6f0b47
51499 F20101108_AABBTI jian_b_Page_043.pro
af43a121664dec3b5ed458a626da451a
3cd0fbf93a07d90e68de37c71cbb37650d7b28f7
11713 F20101108_AABBSV jian_b_Page_118.QC.jpg
8d6ebab32f4f68d9b5e739ebb731a3c1
e4046fb016a17f63c34d8716c873a84cd4a69ad1
34761 F20101108_AABBTJ jian_b_Page_007.pro
bacbcfa23c71b6c5f5ed6896a7d08bfc
a4347870981487e20d50baeea038742dd5c008aa
1051985 F20101108_AABBSW jian_b_Page_058.jp2
35100e9ee75a318c5f34cf38cb82e55b
720536fa6bafbdd76eab11edb174a213bb5fa6f2
28525 F20101108_AABBTK jian_b_Page_013.jp2
8a4d2b8a2882402997a14ae7dd6fc7df
5aaf55da69b73c43d6b41661fd733052b7d08e67
53305 F20101108_AABBTL jian_b_Page_045.pro
e4796c8c07f8cec504bbea56c84119e0
ded7426df590413ef0796a570b6f198ee7969769
6353 F20101108_AABBSX jian_b_Page_026thm.jpg
65bd07d619b4f158c20ec15ab68eca40
60ff6e5f959639d0b1864bf45fb62a734667a756
24381 F20101108_AABBUA jian_b_Page_094.QC.jpg
afe058ea4d2f928be03d51e3e965ba11
98f601ecb7fe67a4e4e3b0e89e9b671d09b23a69
47616 F20101108_AABBTM jian_b_Page_058.pro
eb53766c501b89e88751dc101292c030
dc80f61144274fdcf33dcb9a2723aa76b6a0a643
1051895 F20101108_AABBSY jian_b_Page_081.jp2
1fb86a4f3754793c48c4c003e5f2b03b
42204ec3b67b29fd8758615b37adf1a61e721ad2
91 F20101108_AABBUB jian_b_Page_003.txt
063cab25502c774bae394a5e80b9a347
e5fda5dda13ef0d16b17bed4fa7cb7bd420b3c44
23998 F20101108_AABBTN jian_b_Page_011.jpg
b79658db488394e3cc54c5628231d8f5
7aa87907e5f3db0193a617f24361cc32b2fe7f4a
2190 F20101108_AABBSZ jian_b_Page_026.txt
39a8e182c728867c07408677902dbd4c
8653a98d2c43d96738818b147c50f699e5899209
26079 F20101108_AABBUC jian_b_Page_014.pro
0a4233d476f527431206f8887bd24d33
00a11a46899f7a346cd93e49e0cca3b81b480565
1430 F20101108_AABBTO jian_b_Page_059.txt
ce2f16430ac7db41969fca459b65ccf4
de1d00e3178d03846822cc88c97c0a2661b82e04
2088 F20101108_AABBUD jian_b_Page_077.txt
9e02827085de3b5c2995c081c0b2aaa4
f0e2b010a8bfd948f1fd7b6a427db77af90f12e2
6759 F20101108_AABBTP jian_b_Page_066thm.jpg
c0fc7e0fdd1c310c553e77c0f59d33ad
52a9dfdd4737cd68dae12b1fbaa044ba341aa6ed
9590 F20101108_AABBUE jian_b_Page_003.jpg
33816332deb60c2d63180771c75f7d05
ad0df437ed98d050f2e7c4826164fc0294651575
86846 F20101108_AABBTQ jian_b_Page_076.jpg
dc37472aae52b1db2fde60a838bcdd3d
70b46cbe6adc93956749d1a4a25d2f5104f5088d
1516 F20101108_AABBUF jian_b_Page_017.txt
48f9ebb446d8310a574b0b5f7fa3d8fe
7f40a2dd8c941b3254c6b0e285efb2029f98d971
F20101108_AABBTR jian_b_Page_111.tif
78ed63ed6c509c7762af97d568417bb8
46973c40e3c8b5971181a7f12143402242a16732
1051947 F20101108_AABCAA jian_b_Page_035.jp2
eea5e40def373e8a813e07c8072a1186
e7b9730361c4f3aa9a89181538f7d57891af8edc
1051939 F20101108_AABBUG jian_b_Page_055.jp2
29cb735c1c8ea6b9c8affd6084ea1d28
4d486d61d279d59de74374ad555c8844995c7390
19069 F20101108_AABBTS jian_b_Page_064.QC.jpg
10ad41635e73fa691f6c2209c12541e2
df244dafd02ad5d44f09c5915256c37e62644601
1051976 F20101108_AABCAB jian_b_Page_037.jp2
ab7239beba8b4ffd6edf54f47c00782a
b20b1feb1a273903d8a2b7ae1e329400b39289fd
23574 F20101108_AABBUH jian_b_Page_097.QC.jpg
71fbc9068f5cd8a66fdf334592f5de73
61a8cd1de730469107fe1c4cb59453798edcc197
24244 F20101108_AABBTT jian_b_Page_075.QC.jpg
952a2d64dea26c809b6c3d48fe2b294e
78bf45e34791caab24995b1b71e0f5cd67ea6037
852899 F20101108_AABCAC jian_b_Page_038.jp2
3c706bdb36b94fb3f77b30862794b60e
c1cfcf78021728bac51c26962e0e79aec07c4d1d
1051950 F20101108_AABBUI jian_b_Page_044.jp2
cef98a55c00eee36ae78646e9967dfc6
11fe3c809fa8e824153c83f39ba1783d00e558b5
105714 F20101108_AABBTU jian_b_Page_029.jp2
b48f1cae9f6ad966b3f1a12704d6f579
fb7fb131ac0bd1e81aa13d449c7fb2336fb20cce
959885 F20101108_AABCAD jian_b_Page_039.jp2
79ac7a561a8ae14701551cf9124da107
2a588547617ff3be50ab39b71a23a4d7bb3a5b14
98719 F20101108_AABBUJ jian_b_Page_028.jp2
133e027ac6d2201a3cd70a5954911fda
6f277d2107124b3925fa0ba4584e2c850c8353d3
4954 F20101108_AABBTV jian_b_Page_054thm.jpg
1be815baa6b634c58785dc6a51c800d5
2a42df32d1494b4e5297a1b260be5dcbf06d023e
950400 F20101108_AABCAE jian_b_Page_040.jp2
bd62241834986f111e2c758129688a13
ed49443758d6314969be373f458378bd69605d31
1893 F20101108_AABBUK jian_b_Page_033.txt
1fd7a68f35bea81af672966758d5acec
90dfc558dc4f74e95b9d778993fd4cd3c2cfcfdd
4965 F20101108_AABBTW jian_b_Page_052thm.jpg
3245c18437d738a74081bf4600ff2738
f13e2d9d562d042c93c94ba4d8c50dcd94301ad3
F20101108_AABCAF jian_b_Page_042.jp2
23ceae7f15dd11a4113a347098cebb92
d383305d73ee09b3612590cf27f6db0af0c7b9c6
17782 F20101108_AABBUL jian_b_Page_009.pro
c07b106831213ffe5140bdb031880477
c784cd95ef12a755e2eb91c939c2d5325c78d1d3
11771 F20101108_AABBTX jian_b_Page_006.pro
1b1e4453a2a30ab31ffdb31a773a1eff
d4d695e8ff0ec88be5b32d6b87b163371bff739b
1051909 F20101108_AABCAG jian_b_Page_043.jp2
ee34f4783ada2ecc98a8b8f40f4b1747
e57ccb63c8372391a6ab7f387a3221f9905d73c4
6466 F20101108_AABBVA jian_b_Page_035thm.jpg
576bb6d826ed5cb1d1f80875e7f56ea5
6c9f589fdaade158c806dcde2d7ac49ec7d19a35
2305 F20101108_AABBUM jian_b_Page_071.txt
7bada050980473299b449ac7e917b6c4
c00914e05bcf00b2b65184ee15f8cba7aa0e2c70
53272 F20101108_AABBTY jian_b_Page_018.pro
a4e1440e0cca94e975d776c7c8b2d8c4
0d5cdb99875937df707107d80fc205d432cfa883
1051980 F20101108_AABCAH jian_b_Page_045.jp2
78c509a17017571c249ee33e760658b6
93501995b8f0a10b36b7000289e13a66c7bea7eb
F20101108_AABBUN jian_b_Page_025.tif
2d25b8dec86919ad034bee92bc8c0f67
a31b8791df6f6208502e46cf2b5f94344c2c8002
26344 F20101108_AABBTZ jian_b_Page_034.pro
33452a1714ff962626393439fb42a0bc
cffa7526cb7438572b20f772c9011c6a1a47fae7
1051979 F20101108_AABCAI jian_b_Page_046.jp2
b5cde8efcb07db711cebc3dcf96e513b
defb876f5d0a4cf6bbd74b031ce1a69768392251
F20101108_AABBVB jian_b_Page_021.tif
fcd845fb4541b0c6d4fd950bb43a92a1
037940a3ef4f38e55788cf2657ca4f5c6305d67b
82039 F20101108_AABBUO jian_b_Page_112.jpg
96c26fb6285be61d6f353ed87fe2740d
205c1ad42dd42e12b7425eaee58829d697caf278
1051961 F20101108_AABCAJ jian_b_Page_047.jp2
0ddb287e16632c3d1e4d34ee79419bf0
b73387afe9b3998c1799c76de3b9257df520683a
F20101108_AABBVC jian_b_Page_073.tif
b2ee1d0db058a1e41ecddd98f7a80402
efb07ad662f5258b0bbda395ed9600847fa3a988
6034 F20101108_AABBUP jian_b_Page_067thm.jpg
5be8f1780c75e2877690da1210edb591
ff4d2a8f37fb13ff154e6fc6ee631643ebe646fd
1051978 F20101108_AABCAK jian_b_Page_048.jp2
075710b16a0524b53fdf1cf79bd8d2d1
8b3985ad76d4c18052a6daa06e7c31ec37b3899a
F20101108_AABBVD jian_b_Page_071.jp2
f39cfe13fcf869a4a03bce9f7b6cec84
73b6d0ff576c600ea2b3c00dbac651de5b675ced
6276 F20101108_AABBUQ jian_b_Page_061thm.jpg
51071df9af72570e1ec4c934baff7aab
cf4523d749b5806b4cb83f34aca537093c69fd2e
6149 F20101108_AABBVE jian_b_Page_109thm.jpg
3cb92aea953ece7bb59f3e7cbadd6987
a589af8577922c907f3cc7b3618481d54cdc6493
508814 F20101108_AABBUR jian_b_Page_011.jp2
9ca9169034aea0bf242900f927dd7d90
295b5c058353fb872e7a2e4f948edc8356e778d2
1051964 F20101108_AABCBA jian_b_Page_066.jp2
0aed2ded611e4c601e4af57dc5f7b974
34373cbaae9ac446dd9da1a480a59ad36e4a47d2
996417 F20101108_AABCAL jian_b_Page_049.jp2
60a44d248b633b0c58e3a7908412f712
1b5fea97aa27609b5ce82e0757c054334bb6c301
1979 F20101108_AABBVF jian_b_Page_028.txt
2e713081c4279da0a1b54d5d16214554
f8f649bfa64d75efb4a7f9232a9e5a50086be8fb
68542 F20101108_AABBUS jian_b_Page_107.pro
78f95429380cb82ea99ad0d28146de7f
e27bd975e9b9eafb99eb8b81c8de430aa6737e70
F20101108_AABCBB jian_b_Page_068.jp2
9727ba47ba38bc0ae7b21dc3a322aa58
c01d1417fa4f40579b1b7f86034a5ebdb94a4eaa
1051973 F20101108_AABCAM jian_b_Page_050.jp2
6797e113316a9266452313a335f6f8d3
f1352ca136b05567b0cce75896a1561039327b76
85134 F20101108_AABBVG jian_b_Page_111.jpg
e64e509b51005c8a5e38c22eb4833970
8d4bfbf607fd5c40573e2f631df85615dd8e850c
5709 F20101108_AABBUT jian_b_Page_022thm.jpg
8690dfb7bb9840593979fe6e4d5e0cf3
5f631fbb3d6078408326232034cd508f95b1485a
F20101108_AABCBC jian_b_Page_069.jp2
0cbd52e5ceb3ed770986c9ece6a7f0ea
251777f3c5ac551185c2d55b849d4a157a19b8c1
842757 F20101108_AABCAN jian_b_Page_051.jp2
f240b6320e6bf1d87b7e8c049ede9432
dad4156f44a6eb1e97923b87cb0dad4ebc3c044c
83326 F20101108_AABBVH jian_b_Page_097.jpg
fcfd03fbc168822e16a0ad21ab1f15eb
3213d6e7c1d5a7cbb002b1ac9b1a4b7c77e0dba0
16951 F20101108_AABBUU jian_b_Page_015.pro
4cad57cb2019ae548442f3ae235f7e3a
8c8e8155d2b58454e03d6c9816dfd7604059c36f
984955 F20101108_AABCBD jian_b_Page_070.jp2
ce5063901c6ae18f7d3695c367b912ff
6dca5984a4c6dead69c9139735ad6f244e504ffd
876933 F20101108_AABCAO jian_b_Page_052.jp2
df92a62b17931d6bb7556191d9185ee6
3e5aaaea767bb7bb46a91e744818e62239c947ec
1004699 F20101108_AABBVI jian_b_Page_067.jp2
e291b212ff59b34701f3ceba0ef55542
a7840d1f3f3a128f436141f321794714de38a7a4
24665 F20101108_AABBUV jian_b_Page_001.jpg
57fea3a92e24d405139a99e48635466d
04ed2306c589a5b90a2e49996d6323944e896f11
1051935 F20101108_AABCBE jian_b_Page_072.jp2
01568039bff55194099d8928630c30ff
4626ea1153d78b5d18525678b77bc1f074c49b3e
563636 F20101108_AABCAP jian_b_Page_053.jp2
d92a8acaaf80dc06e39148e6dd1e0b8a
6119fe5ec94086c243edd77205935db10a46185e
75374 F20101108_AABBVJ jian_b_Page_008.jpg
e556a784bbfe6d34ba5cec71109fb543
eb562b729fb03a7b75546637a8dfc0025fcfa10c
10250 F20101108_AABBUW jian_b_Page_011.pro
90256d36c94e00ee5effe742769e26a7
f850c251b6edb37d3df138105c68295641cf3feb
1051974 F20101108_AABCBF jian_b_Page_073.jp2
90d42bafc16dd8d15c6d80cf81fb0011
bc4bda42d115e94bc812a6f8a822e547d384995f
762947 F20101108_AABCAQ jian_b_Page_054.jp2
46bfa2a6b5cf50775aaaa2b3f9cc8469
30774ec8ec14f06d17db98a65798089e6c0e5d14
23831 F20101108_AABBVK jian_b_Page_035.QC.jpg
ae392c3f7df4eac0d2533f0431deae1f
1ef4ead49d6d1ce92384b29b60e69acd44e577dd
24703 F20101108_AABBUX jian_b_Page_111.QC.jpg
02be60d442d4af118de3e3cddd8621ec
59f69fd710eb07c78c8a42104ee5ef86e71b7e43
F20101108_AABCBG jian_b_Page_074.jp2
152f669680818e7c84cee4a168fef159
6eac4f72d5f7de6acf085c677df6e192d0aae9cd
69201 F20101108_AABBWA jian_b_Page_016.jpg
4a9f34e4f553041439aface2e6f73b57
bb869330dc121a5199459c2f97ac60be979ff79f
1016099 F20101108_AABCAR jian_b_Page_056.jp2
3349e983f1ff67c4ef4b2140a2ad319d
c49c3d6147ec93c1c67eb60c2d2dfeda603343c7
75256 F20101108_AABBVL jian_b_Page_058.jpg
a4127ee939be59227185d40ff1c5abf2
b8c760a68e69daf3359282e08620c5f8be956853
2345 F20101108_AABBUY jian_b_Page_006thm.jpg
926f48d3298f54af60caaa06a904de02
14c3a88deee2d6102071214a1d3f8979f2b98ede
1051917 F20101108_AABCBH jian_b_Page_075.jp2
bc8930a4bdc0b61c3554fb3c3bc4e81f
577f3d5491eeccb7592f716342af37e14fc4897e
51548 F20101108_AABBWB jian_b_Page_017.jpg
c325cb70e84f9fc078d2a4c697f0e0e9
18ebfcfd28f45c41ef58b65898c7465573a544c0
668783 F20101108_AABCAS jian_b_Page_057.jp2
6c5d6fcfe96409da2e8fba4de2a8a71c
258ac4de1a2cc346f714f01760b5695e3e8cdd90
175686 F20101108_AABBVM UFE0021691_00001.xml
60d93ae30677ac9e8277b23419f1de99
94d1ee6ed3a4d2921d5921f8064be29c3a96ac9d
34989 F20101108_AABBUZ jian_b_Page_030.pro
5a41a0baa7690b2fe5d959b6a87f9c23
3c6fde0362b666e8a40055e74f092d6ae3c340ee
F20101108_AABCBI jian_b_Page_076.jp2
7e6a4809d723d71e7b4aa80278aac70f
e79bdf8d77eea7721e8c18d0b14a57916e3feb38
767315 F20101108_AABCAT jian_b_Page_059.jp2
32d6f2af13fc0b00e510a57a6f4e411a
99e636fe2a9d9198f229e36e7371f6b4e0596e16
F20101108_AABCBJ jian_b_Page_077.jp2
745916f5df8ca46c835973e0b66781f1
0d4c738804077bfcb9ae50c552066fa54ee87388
83533 F20101108_AABBWC jian_b_Page_018.jpg
3ab3015a2414f1ccb84b962bba029f65
c550eb74cea091180a0088706c834f2b753340ec
860095 F20101108_AABCAU jian_b_Page_060.jp2
3f905deb58c19e16c4b0d723481b9dde
826cf7f37ada9331c8e3d22e98857c108234c7a0
969858 F20101108_AABCBK jian_b_Page_079.jp2
84b4569312c826bf493f844bef3578fd
8aa6ae582c59e7df5675a6beef189108113270c1
88836 F20101108_AABBWD jian_b_Page_019.jpg
aabba157217b21caf4514444ad2b2da2
a601631538e4cbc26f120b8d4dfa6d482db7deb9
1051975 F20101108_AABCAV jian_b_Page_061.jp2
f5663f4100e55153965e6c0d3807f2a9
94c0887530130eac8d739ff6728708ba7c0bf3d5
9611 F20101108_AABBVP jian_b_Page_002.jpg
187d76f5942eef0217e18c85587f712e
2211dcb84e9675f0be1e5d502177723436fd46ec
F20101108_AABCBL jian_b_Page_080.jp2
1ae716cd5797f229b98dac6e0efe115b
327a3c0a741cecb7f795399a19864fac39d55a12
83746 F20101108_AABBWE jian_b_Page_020.jpg
a0d9a6430e91b01f63c17f847c6583d9
7858bc7f5576f61fb128c128e163478f381d3210
792050 F20101108_AABCAW jian_b_Page_062.jp2
7336ac028ee013477a268606f8599390
13900f89c42198a868c1b663794d7cbe4c56c0eb
75458 F20101108_AABBVQ jian_b_Page_004.jpg
9479e7e6cafdf2f28a43e11cba234ac9
152cf46b9055df6e30fe74ba73cb5ded87bd8609
86128 F20101108_AABBWF jian_b_Page_021.jpg
733588b7957bd237197db24d7c90bdbd
acbe8fe96bb4d90c5a9cad456b6e3508f86ad9d1
899370 F20101108_AABCAX jian_b_Page_063.jp2
bf4d9e15b97ec3fea3919e6e1fd25ba7
67440270f1cf80784d082cd270591ece4c29c8a4
74261 F20101108_AABBVR jian_b_Page_005.jpg
63bb4b56cb5b4057ae452ba61eca3005
37d289e091c7f1e61d888c4279c28a5f880eb633
F20101108_AABCCA jian_b_Page_100.jp2
8edacf813e9246d4024d4ca3800aefbf
066729ab0adca5d0f139c8fe84a27b7cd6c4b611
930730 F20101108_AABCBM jian_b_Page_083.jp2
5f799a49aa725b44f7910f66ab9cd86a
93802e2107411a4ec741f20475440c7cd027cb0f
67006 F20101108_AABBWG jian_b_Page_022.jpg
56ec2c0f9452d42be702b1158ccc9281
e2c0071b8c4cce84a7fc7a234050c02080d4f230
887446 F20101108_AABCAY jian_b_Page_064.jp2
a7d9fbd774e3e43c8bc4adb96d0fb60f
5c008ca9c6602cc0dfd5e0d2fc110c34a8f2babb
22329 F20101108_AABBVS jian_b_Page_006.jpg
9c58a7330d88d36f7bffc20a5b56a621
f22de7863a0a78446537a83112d8b951faef3052
1051946 F20101108_AABCCB jian_b_Page_101.jp2
42b50194f5c8f995cf1505adfcee6d7a
60f6d9b7d50e5aee1dd44b73d7e80b40701c3b3f
F20101108_AABCBN jian_b_Page_084.jp2
19fd12436593e6f7124fe5a196787316
7e5fc6c325b0bd87cd7b8af043b3e5c75d2dafeb
60709 F20101108_AABBWH jian_b_Page_023.jpg
32150f960fb3ed0d69c7b7c24c35bf63
5a2e63f152c44d06f10c49ac19412b92ab18b763
1030353 F20101108_AABCAZ jian_b_Page_065.jp2
d7e981e5bc4f27cd78a39395efe55fe0
3acdfde6396e659f348eb99243b478e9540b9a84
81058 F20101108_AABBVT jian_b_Page_007.jpg
2267902104972b595247376230e7f85b
bb6733e582063d8786cdff2b1163e59f5a7eed7b
F20101108_AABCCC jian_b_Page_102.jp2
f05ce134ca802360f6e96e9e72092085
b9fbecab118a61bb644634a300fc480aa09cbbf1
948190 F20101108_AABCBO jian_b_Page_085.jp2
a82fb0d269b9981dd568d43b95c1c3de
984caf601529353db5f1247b73e49cb056f9bd6f
70637 F20101108_AABBWI jian_b_Page_024.jpg
795c79e4f5accf333ace7816bee4eb02
7ea7fab456f3eb7a4feca2768c075bddbc626373
33270 F20101108_AABBVU jian_b_Page_009.jpg
abe8af144f8bfe118493285d509066e6
ce571856731628145d4763fb9523486f82fc857a
1051966 F20101108_AABCCD jian_b_Page_103.jp2
f321ea510a4054f708d1daf87f686e77
d6e250e0fad553b0c9a34fe085a50aeb6bfa4969
1051965 F20101108_AABCBP jian_b_Page_087.jp2
e8ca30abf22b895754d4cc881cb47e64
1534bea248dcc492b87b8d5f7026fa1ff4a790d9
73793 F20101108_AABBWJ jian_b_Page_025.jpg
f5d94ffee0df4e79a8f13457cfb63f75
e66d4459398e8dca6b45e225491e10881d4c9599
100229 F20101108_AABBVV jian_b_Page_010.jpg
d2e57cf9deb08b91c4630e8c830f11c3
b9c90595b907d335ad1cc80b35f827e2026339d2
F20101108_AABCCE jian_b_Page_104.jp2
e70bda4de31031ed244887a9186537ea
09a71d42ec62918e6d354a335678b3b74a11a362
F20101108_AABCBQ jian_b_Page_088.jp2
03690c6241abc11f811fc9519dc8246e
5de9a8af7fe7480438dcceb6c3abbe1895d7ce6e
83075 F20101108_AABBWK jian_b_Page_026.jpg
9053c279d8a41ebd4f1f760020b1f8a6
d7316f279e89b0e6397195d82e02bb3d592c4dd8
40120 F20101108_AABBVW jian_b_Page_012.jpg
fbc8f71f74b6dd2b3deb9bfd20428ed0
9805eaf07a98d9b8f98156a56ceb164c5f666098
128157 F20101108_AABCCF jian_b_Page_105.jp2
0712c363b60dd91c22fddefa8b4c912f
b7086623fb19530ef57a112e2627314cc62587f7
81965 F20101108_AABBXA jian_b_Page_046.jpg
d71d559964191b36e4d90b1c3c963a21
d5d0d8f69375046da272aaf02f4463a2998ffc4c
959607 F20101108_AABCBR jian_b_Page_089.jp2
481a030f0986d6577d10611a83337db7
d4ffa51e4fc3bac092c8a9907a06216e0b753bf2
62355 F20101108_AABBWL jian_b_Page_027.jpg
6f318c59578c28763d10392cb8a1317b
1318725760561457e80084972b327389a0acf42a
21974 F20101108_AABBVX jian_b_Page_013.jpg
5e874dc86846ac63b8d922288baf93fd
06df82ae040421f931411ca84027b9622fddc520
129477 F20101108_AABCCG jian_b_Page_106.jp2
5083e57fdca10e23db21096d2593b5e1
b49d52f84c99a65c9d8ef427d167099cbcc932c5
77083 F20101108_AABBXB jian_b_Page_047.jpg
36ebfa10db87988b18ec160a23cff809
36e1f410e0cfb75cfd688599872a24e145914441
1051984 F20101108_AABCBS jian_b_Page_090.jp2
eeb75d96559f97306578e72d252ed111
b54f69e168a559aaf523db3c2f5e6009fc9c8437
64818 F20101108_AABBWM jian_b_Page_028.jpg
cebe2a0f6dc0498370d6e18224da3e32
e4fa47b1e16183e656c5e6ceb6efb35152b2be31
40456 F20101108_AABBVY jian_b_Page_014.jpg
9e8695e536d8a5395f43c6fa3c1d4c9c
d4132b6b07823c6a4b1f9ac6f0efe7aef0760a22
123876 F20101108_AABCCH jian_b_Page_108.jp2
6f62dc0e459cba38284433d8d41a1ba5
dfcd2d915712927fc0d5c596a56035949ce41c01
66263 F20101108_AABBXC jian_b_Page_049.jpg
03e004eb8202a90214998ca85bcd69d3
d97bd894d5f249cb0bfcbf8ec6900b3aa7674040
F20101108_AABCBT jian_b_Page_091.jp2
956e919599072cb44b13df242d2462a5
94caf38a80e01caadcf0c21f9c6fae6eb23f9373
64205 F20101108_AABBWN jian_b_Page_029.jpg
6dce3895739763ff25501fb46ec062ae
6b04a92f4674b469092fb6bad59b743f5af41eb5
29596 F20101108_AABBVZ jian_b_Page_015.jpg
a809d7946cf19fef76c47827dab72e53
a1d4e53502214fe059301f726816a724b4da4e22
123536 F20101108_AABCCI jian_b_Page_109.jp2
dc255f602ce86366c15c364e5fd84dc4
0b94fb2e7f67a41c605ab4ebeac7eb66b979c8b5
1051956 F20101108_AABCBU jian_b_Page_092.jp2
081633d49880d0921477ea666e24f6be
b38bc01328bfff00826ffc7de17df38643c2a69d
86579 F20101108_AABBWO jian_b_Page_031.jpg
8246e3483dffce7c04b31a7d7c274146
f63b12ed8f9b8f686db08dacb0a947dc0c6482a4
135220 F20101108_AABCCJ jian_b_Page_110.jp2
e150f9f30179d4c835a9d60251a2053c
604ff9fc394bebac081fc0fab4eb093a1e009721
79092 F20101108_AABBXD jian_b_Page_050.jpg
72cc69197f1166a140df66d0860581fa
b3091b731b28c5601875691bf95444e8a1f07ee3
F20101108_AABCBV jian_b_Page_093.jp2
60c7db4db7e9b1a880a52005793c2fde
99660f2cc99cc2def7a6251ebee2dad09855f45f
85439 F20101108_AABBWP jian_b_Page_032.jpg
3d704909eda04d30cbeb464bcb58fa00
03bba0cf4c2bd4a1f2a2e9d23c6ffa3ba4ea5af9
135000 F20101108_AABCCK jian_b_Page_111.jp2
47bc3871b9c14f834fe5e62a2cbe103f
b3160aa0a9f95692565566cff95d5481289c06c9
61183 F20101108_AABBXE jian_b_Page_051.jpg
c8cc0694860d2770c3bd842906839ccd
d3f62954d36e7292644708bf02d804684e3bbca7
1051913 F20101108_AABCBW jian_b_Page_096.jp2
90aedc9e27e8d551ccd87feaad09d121
b705430147a2a1a60733d899566a90dd6cd37f06
69879 F20101108_AABBWQ jian_b_Page_033.jpg
540bd1ba869ca2643bf9ccdf02065a91
551ab877589c116d5e628f0901bc33b90a70c33c
142589 F20101108_AABCCL jian_b_Page_114.jp2
e7219dbc501ba713f36d41b83a1bf692
9566ab69eade0e6d34ba2d030ff8219cb91a3d40
60380 F20101108_AABBXF jian_b_Page_052.jpg
6ed74a411da871285389a854e0510047
3aab046a0a7baed44fe976b9854d327ee0f7e0a6
F20101108_AABCBX jian_b_Page_097.jp2
1eeb587a20191b279ce0a018ce5fe9c5
1210b06116da78b0f6b7b709da9e119e42812c6b
73568 F20101108_AABBWR jian_b_Page_035.jpg
3762a1da93332a3ed1ccd721ce52b2b2
4ef2cbbfd7d051e8d0188a2e2bbbee8c88986f2b
F20101108_AABCDA jian_b_Page_017.tif
e47ab7eb7c64535e6c67b57cd2582b56
fcc53410ec6136ded017c6ab81b26dc84ddfa958
130204 F20101108_AABCCM jian_b_Page_116.jp2
0467a4e753405e4f0b12e787c21a3dce
994be5130e76106068f3c3b5b9164a025e5c52af
35188 F20101108_AABBXG jian_b_Page_053.jpg
68ee584cd86883b9a5380b783689d9e1
b1b8d9da4a9d6d283313d13fc846ad193e735969
1051903 F20101108_AABCBY jian_b_Page_098.jp2
67b5fa3c1a7b6e1d9376ca2b68cd938e
b88da3123271fcb0944a6a394dc607b04877eb3b
51447 F20101108_AABBWS jian_b_Page_036.jpg
c2c57beaf17973ee251030562800f4be
66016aa2159a4d42c6f335227c4cce6facf666c0
F20101108_AABCDB jian_b_Page_018.tif
a04b98bfb1b45d7f05e2ac88bc1ed821
8dcaaff9162f49ce7a8a1421e7ee0fd45af614b6
71272 F20101108_AABBXH jian_b_Page_056.jpg
e71b135f7c0c764be1e61c97a1a557f8
5b9e928d1a758e995243911978cc66ac9b71f31e
632246 F20101108_AABCBZ jian_b_Page_099.jp2
4b69b6dcbade30ee37bd84671870d198
6934adeeada1f5a112a453c905306e1310e4a9ed
80517 F20101108_AABBWT jian_b_Page_037.jpg
c9fc7c4bb13a3f061ff0b499076906a9
ef369688ca8086dbe733fd7b0fa559c266a11f58
F20101108_AABCDC jian_b_Page_019.tif
1eccfaf8a8e31904cd2d64a8cff3a8bf
f62268d449ef96df2ed7bfca1b9c62369ab52884
20785 F20101108_AABCCN jian_b_Page_117.jp2
2f9d08efc5e05eb296fea50f1bcee52b
84aa9cf3559f8120fe60cce33c31e4126c9c291b
51216 F20101108_AABBXI jian_b_Page_057.jpg
c47fcbf09c45da5e4693dc754e6608c7
4d80e032bd4f6da1a2d894c76a2dfc6f0d8988d8
57792 F20101108_AABBWU jian_b_Page_038.jpg
b9ea13efae80849cf8c9058be6e3c95a
339622fa3d7995ae9937aa42824cd9e49451730c
F20101108_AABCDD jian_b_Page_020.tif
9c8d706577ffac9909cb219aff9d20e6
2ff2b72f81588b82d6575113d18566e5c3e3922c
50848 F20101108_AABCCO jian_b_Page_118.jp2
deb84920dda188f4afc6e9bf8e8e519b
6480a9279d291fde00be63dd92d400d4f9436c2a
60524 F20101108_AABBXJ jian_b_Page_060.jpg
f7a1883dcffba5f8f37357c5844c0418
6f995c436800dae421f82dcad3dea845505f5d9f
67267 F20101108_AABBWV jian_b_Page_039.jpg
2ec44f1fc7a746781a58d36f721dd142
4ab9687678c3e052ce71f5cddf9c2bab4937721f
F20101108_AABCDE jian_b_Page_022.tif
d1ee7b4068b9a213b9a6d8f636b5901f
45ad7c58a65d09fc2c7eb823feddee548b304595
F20101108_AABCCP jian_b_Page_001.tif
b7be003b467b76f2e0305158e3db057a
a5412bb3bfbc8565eabc690b564014d4f79551e9
72300 F20101108_AABBXK jian_b_Page_061.jpg
dd652529350473682fbbf0a0d21b3bb4
ba5f3222bd12d040173795398d34698932d82481
88045 F20101108_AABBWW jian_b_Page_041.jpg
d00149eb13b649c558208f2c5f631c15
d06a81e19d6a9a11275d329d9d822f074acff79b
F20101108_AABCDF jian_b_Page_023.tif
78cd526d8706aa61e364c1433d8b1b01
a313915db08ba679f4fec54cfe33b4bce2b327fa
F20101108_AABCCQ jian_b_Page_002.tif
49e6032e62fce4be5e0d1c3d0d622658
8c8254505d05d4657f5ac23a56ea097bb34f7ecd
60150 F20101108_AABBXL jian_b_Page_063.jpg
a317fc5897f607da63bf62fd8eacbea5
27eea3a6debfb5ac555342d9d2ec6fc7ffede50e
78826 F20101108_AABBWX jian_b_Page_043.jpg
f562a328991ab97bf9672b1b4204aee7
a206134235e333e24010961832692d474584c19a
F20101108_AABCDG jian_b_Page_026.tif
8cf79825d95bb3176dd71e4acb9e0be9
153537793ab1ee24760e47bc2f4bd7f5aaf71c94
81670 F20101108_AABBYA jian_b_Page_081.jpg
ad5cd28b87cf299de0fd349c6c4d38ab
8423dea9aaaca02539e66a471d35078afcd0c0e0
F20101108_AABCCR jian_b_Page_003.tif
a788c02b8e1963f2cad33baebcf4e718
120ccf97fe6bcdff19749c4bf8d63ad90dc721c0
62136 F20101108_AABBXM jian_b_Page_064.jpg
8faba7f9732f9c5ff0ef0c88301c333e
08d45a532c1542f9d273b12df94c9daca2776efa
81166 F20101108_AABBWY jian_b_Page_044.jpg
3012aada5cff6515ce5c6412df9c849a
d96db4fb7d06b85bef3cb72198b50d090ef06bb4
F20101108_AABCDH jian_b_Page_028.tif
a269d0d76ad3754f7e854a1be7476392
d8814c91b0d9b93b23f35839330f67cdb9b028b4
79812 F20101108_AABBYB jian_b_Page_082.jpg
d90f3f0d2300ab8806a5b8f7b813c400
516c20298de284322c8f1f2f2c1af45a9d4dc539
F20101108_AABCCS jian_b_Page_004.tif
f3e11e5a6355065767d52777c9df0756
683a5ac764e6d305067fc038d369af8cc0f50eac
70453 F20101108_AABBXN jian_b_Page_065.jpg
bc9e9756baa15913625373141166ad7e
091445106944548a9ee4ed0d147a69952773066c
78950 F20101108_AABBWZ jian_b_Page_045.jpg
594cadbb6daef8406411122d52dc0b68
db4002f29735887a8fc51db7829c2c42e2ad78ff
F20101108_AABCDI jian_b_Page_029.tif
6761aeef88012d022053981c0283de5e
0fe63043541f9f1adc46ca676cef80734d096451
63811 F20101108_AABBYC jian_b_Page_083.jpg
a76224154e045bdcf4349f4e627d419d
9f566d2d85de92b8607ccf14a54b6aa36d66cbae
F20101108_AABCCT jian_b_Page_005.tif
3bbf83f935c082f2fd5c6fc9a4f18d26
8a7ce53b76872ac290ee24cffe59a1be027cec26
85785 F20101108_AABBXO jian_b_Page_066.jpg
2826024681d1afad24f8272737a45c3d
0a8a48092c77b60735b51267558702698a6741b5
F20101108_AABCDJ jian_b_Page_030.tif
cb24650a2bb3ee43e0646b21510c9189
cf0c3de5ea6918fdb0fac419f21b21f57350fe43
91500 F20101108_AABBYD jian_b_Page_084.jpg
4f7716ca1cb3264b5e57acecd9d2983a
4e1f51325bc299f7abe2f26752527cbe50ce3822
F20101108_AABCCU jian_b_Page_007.tif
cb7676b70fec97dfc4cbf4bd027fded3
dc5bd27c46db22e8d0e60c30cb9f757258409dbc
72189 F20101108_AABBXP jian_b_Page_068.jpg
4adc492f074257ae21fe3a8180edbfe9
5351036796bfaa804408bace02933ae308089de7
F20101108_AABCDK jian_b_Page_031.tif
9b80037e677414788018206b95623b15
cb495542efade736e6c9eb0e05c22f6377d3f072
F20101108_AABCCV jian_b_Page_008.tif
f48679fda250bb1d37df2039d353f229
679136a2a0e18fc22b0ba7efc3f2ae3fc01362a0
83319 F20101108_AABBXQ jian_b_Page_069.jpg
836054bc602c87e7ac8143e7f54bb96a
2d77600f2fe1416864affb4d258e5f9b0309f4d5
F20101108_AABCDL jian_b_Page_032.tif
d8383df701aaf1823ac39d7bb3356342
5de530b424d65f0df20b1aa6f147d3ceb9d2f158
70777 F20101108_AABBYE jian_b_Page_085.jpg
ebbeac3a09c316ffd190211516779d2d
618d99144a0089ab540756f9581848bea42df2c6
F20101108_AABCCW jian_b_Page_011.tif
7babea6b93f52ad262a1b0a3f5d3454d
fb9b30e224efe9cfb3ca98bde09164f808a46189
70303 F20101108_AABBXR jian_b_Page_070.jpg
ffe41c4a353a92f5046ae22f8bf5847f
fece435a77c041d1a3fe433b2470b7b3b25e09dd
F20101108_AABCEA jian_b_Page_050.tif
85df6d23099bb8411581e32dfdcfb788
3cc00e6d84948eb3211b4d69a5799995ffd907fc
F20101108_AABCDM jian_b_Page_033.tif
0c339dfa286647acc9a3aea3fe6d63a4
4c90fc259e6a5d8e4a0ddc185ca3dfdc42874a8f
75186 F20101108_AABBYF jian_b_Page_086.jpg
87380679d8c8d37193020a814c343363
5447aa8ffce2759ac1749839d7c61b8a938ca62a
F20101108_AABCCX jian_b_Page_012.tif
cfbda5fdc1cf40b9d27dfe04f193098c
a0d18ce11c7dd2f4409f385d6ee5e2e15663141d
88074 F20101108_AABBXS jian_b_Page_071.jpg
8f139b17ffe5b8558d13ffdd5739cc29
220cff284cad6a607db9a0f80fea43335f259b00
F20101108_AABCEB jian_b_Page_052.tif
5b8ab1e2a3154bff04b4b1e8afd7f28e
47684be0ee10c17727c5c79d05a68ab61c41c617
F20101108_AABCDN jian_b_Page_034.tif
fcc922d1fa5b7002763c4273a9e76ebc
c0884145eeb599e2e842a8b02292dc803ddd9551
81810 F20101108_AABBYG jian_b_Page_087.jpg
3db827529db3f35edb1856f22d071432
68abacdbd9e0bac5500c83ca767a8528cf943a1e
F20101108_AABCCY jian_b_Page_014.tif
68c3c9f8f8e745843abc750fee6a7e36
1885e3c685fc3afffb1e564f01037b32cb313514
82963 F20101108_AABBXT jian_b_Page_073.jpg
1844a6948f8ac4feb34cea8dd0e56323
0d1c16c4c97bda4c76781f52d73af7c5bebcd801
F20101108_AABCEC jian_b_Page_053.tif
5920848358bd7959b3c80e9a10976098
e9034fb94c7218c2826dfd440af4141421bbfdfa
91167 F20101108_AABBYH jian_b_Page_088.jpg
5f8f446ac4d70884a51c5ec5c5a810dd
26beb714470bbe1a2cd91d5e55107b31e0f3f885
F20101108_AABCCZ jian_b_Page_015.tif
db4e22ee489faec1db72e75461d48ec3
495078974879243089afd3e91da559cd8d41743e
F20101108_AABCED jian_b_Page_054.tif
8431258971995b5d68e6e6f2993ad135
1e12dc645e40f7e6e745c02962a51005058f4ce9
F20101108_AABCDO jian_b_Page_036.tif
6bf7d168c5497431808f8b72683cfe44
a543a3a3034b95dad147f707c1f0f31587a31a84
69546 F20101108_AABBYI jian_b_Page_089.jpg
c516a3fd2a50507891d557d55ca6a5ba
12a0f604408d4efcb35ecde5c7d4a8004f103638
91933 F20101108_AABBXU jian_b_Page_074.jpg
821ae052eaec1f69ba42093760d3d064
8684b8679a52aff5864b896661f90edea865c97f
F20101108_AABCEE jian_b_Page_055.tif
e01b6a12d2233ec654c637ac78fbe713
0c1ceb87e3b2358422ff6d1c6f11dc551ff643e3
F20101108_AABCDP jian_b_Page_037.tif
704825711a31fce35dd165f4b39e386e
ab167e6e35560cd550dd32cc7bd95fb2f41f21a8
89805 F20101108_AABBYJ jian_b_Page_092.jpg
1d7af786e4170bf1f8969285f25c1163
cd2766e2e6859492cef35defc26c2f87d4f429b1
78460 F20101108_AABBXV jian_b_Page_075.jpg
ea2e210279bba325cd1c7940f8136775
6203f459071a3d38e3ca4674380f7a7dfdda37a6
F20101108_AABCEF jian_b_Page_057.tif
24bcf46534a52666d8e79d6b57d90dfc
27b7a3ce70f4bbef6f4570b0e05dfc5216bd12d8
F20101108_AABCDQ jian_b_Page_038.tif
bf7a6ceceef7483ebdaa921e9d677e95
b63752e998be0a5e9a033de68943b25354eb4612
76846 F20101108_AABBYK jian_b_Page_093.jpg
cd3059bfb278475cda5252db3e07b8e2
cbae353adddc0885102c48fe397027f2d4739893
90243 F20101108_AABBXW jian_b_Page_077.jpg
3749585f48a104e783089e08ff657206
26e1bd340ae28394633afe9aeff53473da76a0b2
F20101108_AABCEG jian_b_Page_058.tif
979b78ae8ee6c8272b1e61c2cea96cf8
88ad6c4575235b7c84224dbd1e092306aa3a5871
37347 F20101108_AABBZA jian_b_Page_118.jpg
d230609dc9fd9804c3f9c76bdca8b44c
1ac0c322acff728684bc7484d4567d019f14e8ed
F20101108_AABCDR jian_b_Page_039.tif
d4400a0db32687887346c444af4f8253
0f02e5f64236afe8e6f87613b55c601e93256238
81023 F20101108_AABBYL jian_b_Page_094.jpg
bca2942f2b4b713f7f2bf86172c53941
81a11349db22b997c2460d0188cfe44037da3132
72289 F20101108_AABBXX jian_b_Page_078.jpg
7b0201e1ef32ff9991da9012f56e29b4
b2bcaeadbdb5becfc7ad85766738d4bccc3bde17
F20101108_AABCEH jian_b_Page_059.tif
ba7b022b6101323aa7909f4de8fba364
bf5425bd9924f2cd2f4ae975a521260a6a782eec
25884 F20101108_AABBZB jian_b_Page_001.jp2
bc3c34199859a1bebfbe4990c86da952
8b1845759fb8a0b8193ce1d67166cc52acdfb173
F20101108_AABCDS jian_b_Page_040.tif
d3b497e83b1f947f1f0815b8bdeb694c
c857f2d8d05c6dddc0b71fa563a268651f0bcd64
91868 F20101108_AABBYM jian_b_Page_095.jpg
353eeff8f949ce31fda6ca843ab56315
25fe503406eecb816e5c26dea46817e617a85bbe
68285 F20101108_AABBXY jian_b_Page_079.jpg
022392ba674b12662af5e632af6181b3
dedb5f054761b5c069622500e8d45d6c0470db3b
F20101108_AABCEI jian_b_Page_060.tif
9c4d563b8de663afeb8d6b4a94c68d2f
38cb2991c8f494e29b3d0bdb4fd3112abcfedc40
4703 F20101108_AABBZC jian_b_Page_002.jp2
accce1bc0673135d83906ef4a5cdd43c
1d2a206be7ea81a1a9898a5e14a6385a444e6192
F20101108_AABCDT jian_b_Page_041.tif
29457b6e3d1e4e1b7727f938424448dc
08d300cd2531630a8557adb16a641ff6f010aaaa
96444 F20101108_AABBYN jian_b_Page_096.jpg
ad4e946ee1237d2421e893a9c4d6e762
9579216520fe24d9a536c6e79a4497912c8e8572
73950 F20101108_AABBXZ jian_b_Page_080.jpg
022d67c1e9f71102ae582908c83d2557
4037856393ba2747b23051459c39acd8ff64441d
F20101108_AABCEJ jian_b_Page_061.tif
10a1a02e4ff857d95b48e27e65489c30
183c9b3f188d6a3a033f7f4f309e5261a4e5a0ae
5073 F20101108_AABBZD jian_b_Page_003.jp2
ed76cc32389353b57731290388af1f45
6ca25a0e6316f37debddbd0997f53845b0127969
F20101108_AABCDU jian_b_Page_042.tif
e919e4e1bccedeb4779e728385b7e139
d5f7d90af768576ca901f5784873b778a6d07b47
48888 F20101108_AABBYO jian_b_Page_099.jpg
236147899d878eb25a924e14e6ff8016
e289c1a784f8b580579a653dedfc35a0539878ce
F20101108_AABCEK jian_b_Page_062.tif
71b8b8691f68e58627459bc860f943a5
2f0de35eed2f51cc0cc0c4ed8fce6b3b4aa77cc6
117347 F20101108_AABBZE jian_b_Page_004.jp2
4a805a3374e1de18e72eed01c1268785
b2a0c404a12d5d8ae5cf7d872d30cc3c563c58d5
F20101108_AABCDV jian_b_Page_044.tif
b62b30c40d622defddb90061bbf3c154
79e7bd9922a2a8b2eff5932e65617a5cad42c823
84806 F20101108_AABBYP jian_b_Page_101.jpg
a39a97143e849cadbef165835545f590
43f4de814865b0fcae3994bed64f5b59edcbbde5
F20101108_AABCEL jian_b_Page_064.tif
9e99f789ab71318ddcd7220e6c4d3b4c
647830a8bec656d71ec38df6194cf2e8609b8148
F20101108_AABCDW jian_b_Page_045.tif
72fc199c6e2146024973f970e97a6d0b
9c95f67df5abc0fb10a510ad351a5c4b869dea8d
86439 F20101108_AABBYQ jian_b_Page_102.jpg
1a620eff424e17aebc4c9fb7c47201af
c43475ae992e61cb6e21ffd45dba2bd906f19362
F20101108_AABCEM jian_b_Page_065.tif
4add86eb9eedf7b28db112b6e776e4bb
009d7bd7d8a27c0de6025f9b1ec5abe01ad37de6
113487 F20101108_AABBZF jian_b_Page_005.jp2
4e65a44d683dd19feac73852e68b7d1d
8ac948a7e4e1ae9b53c0d46ba808ad896cee1ac0
F20101108_AABCDX jian_b_Page_046.tif
f203dfeb29e2c8d85bec0e1d55461605
14c6c35041f6e092a73b42bdf6244f8dd1857a23
89303 F20101108_AABBYR jian_b_Page_103.jpg
99dc80b2494510743dca578ce55d1705
009b46a6759b3b2e9aead467a576ba779c13c404
F20101108_AABCFA jian_b_Page_082.tif
071b4ab0df2ddf8a95dc377b897c2321
0a220e4f3fc8df5f80b1d7e57b2a3b242a9e2e95
F20101108_AABCEN jian_b_Page_066.tif
52dffd9c4392210bf71fc4a0172b8f74
fb8314c023f3844c7f905e0cc01410423df124c2
28462 F20101108_AABBZG jian_b_Page_006.jp2
2e0c8fd44fccafd6f866a46df4d8772e
fe06536452ae4f276094cbccf71ec96be20d7786
F20101108_AABCDY jian_b_Page_047.tif
131dfc1c503464cf4bbe5f5617782180
e6f529fdbbd504b9884871b2d34d189db8be8397
83248 F20101108_AABBYS jian_b_Page_104.jpg
168e91843809c172912ac021585a9e35
d4422f6c0e3937a1354b3c7c66397e34ed889807
F20101108_AABCFB jian_b_Page_083.tif
494271dc53f4c4ac181e29b591ddb8d4
83227aaae3fc20f0ca51233920df96715814e8d1
F20101108_AABCEO jian_b_Page_068.tif
aeebf93857a7a08f5f7a79649c746e84
d2110bdfd597a1cccf1522a0f68512a02b37e228
F20101108_AABBZH jian_b_Page_007.jp2
f72be30470a2c13eadae76cdde398ea5
dbb558823a52db6bce3be16429165cee5dae297a
F20101108_AABCDZ jian_b_Page_048.tif
6552a243c7bca6d04a8027bf0e211076
0183780acb067bc2f58b9983ac748fe06da65b7a
80711 F20101108_AABBYT jian_b_Page_105.jpg
baabd5a30e8d1d73b424ac7e3734b1e7
0254331a645a803e85b9c13bdce4aca2b129b8f3
F20101108_AABCFC jian_b_Page_084.tif
ef3bdd52b9d68621d3bf3743ea51cc97
eb5011079edc6da26822d817366139bffc20ee9a
1051969 F20101108_AABBZI jian_b_Page_008.jp2
1ac8ffbca5d6598effd151c3dd0ad089
de816d327de29b230f2e204f316eb50934b826f7
83668 F20101108_AABBYU jian_b_Page_106.jpg
b21ed25317675b4bb255f7d8cfbd297b
28be48c9ed28ec0f9df0d6c11b17f6247a21d27d
F20101108_AABCFD jian_b_Page_085.tif
1e1f000c497778dd1e5f3a4e06b560da
ab513f5b2121afcbbe755f89406de931af3cdbb8
F20101108_AABCEP jian_b_Page_069.tif
c9c5a3bd1933b2eab512d26c2a376b0f
d45dd9b069472c40d5e8f022f38df61e82a7ea3b
787161 F20101108_AABBZJ jian_b_Page_009.jp2
a583e41a250c03e2fee79f8aa4c4c407
a34958e19677873f5beebf4385cd6c7ab29f1d3d
86500 F20101108_AABBYV jian_b_Page_110.jpg
92bb89134400fde16a8dfaff1a3660b7
2672034aca4ac5a307a2f87f28e5ddf8cc53d93b
F20101108_AABCFE jian_b_Page_086.tif
13a3a2d764e1a2d68218c1ed8a1a7846
86e44d99ee1703e5143a4cdd684d60ce9b0e736e
F20101108_AABCEQ jian_b_Page_070.tif
02e85a042cdaeadad5bc85553c7d3584
f7c2b63633215824ac538f2b4e528abdaa420195
60933 F20101108_AABBZK jian_b_Page_014.jp2
0c214a2acb450ba1914fe7cfad0b9d9d
da4f12425145952ebb4608e5cb564f8bd84daa93
86497 F20101108_AABBYW jian_b_Page_113.jpg
574505b0d7dd3a1bf34fbf2e0e669281
4fa635466b74eabeb63399b36eb8d20c9bb7a51d
F20101108_AABCFF jian_b_Page_087.tif
ec906689325c8ce735476489e28b1a54
03c6f6939b8fc7711c3d2028d509b7703e6516cf
41503 F20101108_AABBZL jian_b_Page_015.jp2
f1112c4aa1560aa068d8bdf6bbb520e2
ab479099794e3980ea7e661ba884a11370a40869
91058 F20101108_AABBYX jian_b_Page_114.jpg
57781d1018e30d2171ad254239cf42dd
b269f9bbecfbe34d12e36a121349e6c899c26954
F20101108_AABCFG jian_b_Page_088.tif
69fbfd55f0243943771cdd09218d73aa
ca6e08c092d3520679aa8afe7c25946ade99e062
F20101108_AABCER jian_b_Page_071.tif
c35cbcffd07189b060ad6b64eb9903f1
e8703b91d4b7f9e949020a6796fcc2ec3bb9eb56
110126 F20101108_AABBZM jian_b_Page_016.jp2
0a69669db17e2284120465d834b8c401
53c79d85cd03abf20bfc4b3c11a0ed845b390611
85787 F20101108_AABBYY jian_b_Page_116.jpg
d279bf436cd4b923916ac7ee85511ddf
4a63a0ba44cb5752102230572337e267fef6529b
F20101108_AABCFH jian_b_Page_089.tif
70299afbcbae93f33eede8319987eef4
ab2d190198ca77601ee264ea1d1af3ee6799c785
F20101108_AABCES jian_b_Page_072.tif
295e336bf5cd44852d37637a738e38a1
589404d3f481565e8f5f173d968af1e9896bf55a
81255 F20101108_AABBZN jian_b_Page_017.jp2
ae86d3023880190d5c6df198af2bc8c2
ce0228ce78eb815836e107e7f96951fec17a62bf
17613 F20101108_AABBYZ jian_b_Page_117.jpg
2930c5c2ab45e1c922ec1d42e49ec645
a516f32fcab2ae37fbfd8dd61b2699834d98f21a
F20101108_AABCFI jian_b_Page_090.tif
4742e1843d03e94cef96540e0fb20618
270bc192108f089224b1df86ba7504aa5d31098f
F20101108_AABCET jian_b_Page_074.tif
b20ff157986e370eba37845418c766b6
4cc71ef25a87d3cb109f9e17a50cb13214509f41
F20101108_AABBZO jian_b_Page_018.jp2
98921c7a75510a596bcf939d75c4c0ca
2a728a14f2b28d9b1beccea9713c65710bdf608e
F20101108_AABCFJ jian_b_Page_092.tif
44e7f90c8652d6ecd1c0901c5030cdce
98715e888698b4d123efe57b528f2416d28cdb15
F20101108_AABCEU jian_b_Page_075.tif
892dc392cbf6f9b55a548fed2bf3a7d8
b57835845dff7123163086f9db6ce601bcfdd07f
F20101108_AABBZP jian_b_Page_019.jp2
ce9a40a9fc7e05b61fa353589ba7f463
8b8e453ba188f2ab27718dcb5966ea9343710ccb
F20101108_AABCFK jian_b_Page_094.tif
f17605dcdb18048ccde639e7f2f783c4
0cf9b14f11612b7cd3d20ca3eb6d74d05e1e48e5
F20101108_AABCEV jian_b_Page_076.tif
d52a9035d5fef3da59b7291daa2b2d37
9e4701bf47ed292f374d4cda26314e4daf42e10d
1051916 F20101108_AABBZQ jian_b_Page_020.jp2
d65aded92227a4c3f26d696ab569726c
bc4aa50eb9490d7ba6f23d379fd3a1d68cb3fdb8
F20101108_AABCFL jian_b_Page_095.tif
547032caeceed7adfc016ad569e5810c
99db55e7adad0e412507a1dc2a81a244e9f817a5
F20101108_AABCEW jian_b_Page_077.tif
f0a99c14c3961b34261c5bce10f12bbb
38bb518c4a038af6889697e5a0d3fba13ce4ea47
F20101108_AABBZR jian_b_Page_021.jp2
cee2d27a1bb74db500ae11add8315ee1
5259d5e10ddfebf3c44dc451adb83626bbe55b50
F20101108_AABCGA jian_b_Page_113.tif
62ae11e4fec59dcc501fc713e2fccef9
e6056de70cd3cb2159c1639793995f6ec3808ac3
F20101108_AABCFM jian_b_Page_096.tif
a50e680e2ea2f3dbddaf0822c19950cc
85cd57f436540df119ff7847b82c66a13928b05d
F20101108_AABCEX jian_b_Page_079.tif
bd650627cec5e0e311cfe637f1e5190c
d8512e997055e51b60095e17f9fc1245feeced19
106248 F20101108_AABBZS jian_b_Page_022.jp2
ba733fdf693a162c4498e5e49be382da
37a439e678fc53eaa4c8edbc1d618beeb37afae1
F20101108_AABCGB jian_b_Page_114.tif
5252ab6b1093c47d9848b31828735f5e
0587b307182bac5b2534727f1645e2871d04bd95
F20101108_AABCFN jian_b_Page_097.tif
d16aaad4e7153c7f90a9efe6fa206cc0
567da637a8734ed2682bfee1f99cf6bd6de73211
F20101108_AABCEY jian_b_Page_080.tif
12a4055b516c2ceefba7f631fdb142f8
cb0428aa76e959736a9219af8e5f730e29856cc8
837600 F20101108_AABBZT jian_b_Page_023.jp2
f54a50081201f9745493a0ffc92d7ee5
638924f2c56d3658c8ee668aab87bca5222a13b0
F20101108_AABCGC jian_b_Page_117.tif
ef26efa46866a3816a72d6bd56bddc61
c58073b3c84627775a0ffe0e35c4b5cc57797dfc
F20101108_AABCFO jian_b_Page_098.tif
be3fd1cf61b8b6802f241bc99da83195
3ddfef31e1a8d7433b96162fd846599dce7fa577
F20101108_AABCEZ jian_b_Page_081.tif
a04557653507d84f11d9fd7205783896
cf3fffff695d44ee59c0a784fe824d50bfefbfd7
971923 F20101108_AABBZU jian_b_Page_024.jp2
fdbb3be3f3fcb5dbd54f24fac1550696
a88a8dd947d6d60e1fe03cabea076c63d151f196
F20101108_AABCGD jian_b_Page_118.tif
cc72a6f4aeaf0653846c575a399fc20f
ca315bea55c6d7fc58b2dd7df2297e3435a2e915
F20101108_AABCFP jian_b_Page_099.tif
11d9ca404244a0e49164bdb37365f806
8a108d6dac55d4a87f4e95c288ae06d98ad8d790
1051962 F20101108_AABBZV jian_b_Page_025.jp2
fb3e239cf223b5de572ccaecb60173ab
a87476d3a673034568674953ee674f16e76c5ef0
8678 F20101108_AABCGE jian_b_Page_001.pro
48d3d99544841f933c72289eb917db37
feea8c1ede1f31439f784e83a8319859477f0da1
F20101108_AABBZW jian_b_Page_026.jp2
1190d6adfea8b0176d34abbf4ef2817b
836861bf85ac8b826d4e5eab77865b1ff78edc48
672 F20101108_AABCGF jian_b_Page_002.pro
988e2052a58a9c96ede738b880b31858
56bc87a79436fa8b9937b92c50ff137b62873737
F20101108_AABCFQ jian_b_Page_100.tif
9d4e35e86efb816a15f3972b30f6055a
eb525270b8bfaebb9b721c6a36d21f0516942c45
1051954 F20101108_AABBZX jian_b_Page_031.jp2
b95a7e64f85aa70aa1aa422739e19ff4
862d84922f8b051b690dc4839398e4574520e0dd
854 F20101108_AABCGG jian_b_Page_003.pro
a173b759e4db60200d96cb32fb00891d
2e91ad97b72eef3965659d76c107823d1e875ec5
F20101108_AABCFR jian_b_Page_101.tif
192d2958dc3ecf760559acf5002603ac
a3b4ed75a9d99414f94edfdce135e612e9c48773
F20101108_AABBZY jian_b_Page_032.jp2
c17063020a32445c62b5b5869f185cdc
857516e085109d3ec17551dbd9288ddea7c52048
56806 F20101108_AABCGH jian_b_Page_004.pro
09d2ba401d3d690144069d48856be0f8
4490c8ccb4285bea57d1a40627f8ec83880dcd5e
F20101108_AABCFS jian_b_Page_104.tif
d3dd467ed4ece6c0f23183bf82b00788
df85e6ed67bb57d0112c1d3ef5525e56daacd70f
612643 F20101108_AABBZZ jian_b_Page_034.jp2
b45cd8a3367f2ab5349e478aaf373dbe
c11dd508be42af5f30c666b61e8322daff46c7de
55006 F20101108_AABCGI jian_b_Page_005.pro
d79e5b8a326e6f0461ceb14f010c6f89
a488908b37d56da3a825b12ecd0d210b7320edd4
F20101108_AABCFT jian_b_Page_105.tif
e00fbc110f5c0566ede7680d03e727e7
6af1f66a31f3ed0f292ad42c341428268f145269
57075 F20101108_AABCGJ jian_b_Page_008.pro
2237b05cc782e800e9e0fc3682d5a6e3
b3c57882c49ce1abadcd9ee6dd753a015b257700
F20101108_AABCFU jian_b_Page_106.tif
bb9c642868b2909a8cbe444cd97cc48d
ea5b044bb415a2d5269d454653f6c9341e015be8
68016 F20101108_AABCGK jian_b_Page_010.pro
bec052bcdc62817081262fcda9176e2e
431c0459e567b207554504ea397b7d0d0da995eb
F20101108_AABCFV jian_b_Page_107.tif
94277176ccf39ba06365e0953afdbb82
e06719936b6f063962fb5ed34e50dd5da389455f
24536 F20101108_AABCGL jian_b_Page_012.pro
d0f368049d3e6e56af2dd1f7e915fd12
640eafa9dd9f828048acf9c649d295699f245cc0
F20101108_AABCFW jian_b_Page_108.tif
8a683eb4c53962b60d1b466f5143e8e1
9a7fd8e27547fd3f7bae92c3a29d7793a9ab7fa8
11042 F20101108_AABCGM jian_b_Page_013.pro
99f46d88a74bc89241141bccacffc531
125921f875653969fdbb72708e075978c16243b2
F20101108_AABCFX jian_b_Page_109.tif
4657b4cdad4eb849e22d0cf8fa161a29
a4932cc19fb34892bb63cfed0c654938f83c7fc6
42452 F20101108_AABCHA jian_b_Page_033.pro
3d0994468ddd62303b5ccb7f7e0a8575
e9992688a98a22adcfc8cf7c24700e087078f684
51537 F20101108_AABCGN jian_b_Page_016.pro
9bc37b64959b9697bb4e86fd449dd812
b33be885a612824b076a78700024a098ac0e29d5
F20101108_AABCFY jian_b_Page_110.tif
9b2385291a58c4cd4894ddeef6229f8c
1d470b9604fb01f6c51a89b2c4616f9693386e2a
44378 F20101108_AABCHB jian_b_Page_035.pro
48d7052dd3eecfdb644e9938218965a4
f142a08913326ea9b4dbce41f646859a0734a349
37972 F20101108_AABCGO jian_b_Page_017.pro
318dff208b13c48c69766650d98cddcc
2e74c40ac57330a3c203f38d992fc50112de4d3e
F20101108_AABCFZ jian_b_Page_112.tif
c3884b624431777bfed29a485b213eca
f0997ab7342c7f26a194c20ac29434ab7ddfeadb
30867 F20101108_AABCHC jian_b_Page_036.pro
3424ae412f5c5bcb39a15ae66b0de972
137855ae39d8792fd4a75be0f4ec466cb7d28279
60145 F20101108_AABCGP jian_b_Page_019.pro
df36b9e6813220aba68dfbd9ff6f8d4e
8418014c4ffbb7ca606b9359cbcab0146106e07d
54457 F20101108_AABCHD jian_b_Page_037.pro
7281c25886cde1b60ded0366ac86ea4d
365f879fac897e1b69d6974514187bb36d5c06cd
56334 F20101108_AABCGQ jian_b_Page_020.pro
6438dabedca2df65bf3f7f8993cc003e
479e9d6397eed932dfea293db3d8e8b3d0db8e7a
36513 F20101108_AABCHE jian_b_Page_038.pro
a6c4416f7c51bbe12fb61903cf90c7c1
a1dc6eebf8546335e4de645be207a16260318917
41333 F20101108_AABCHF jian_b_Page_039.pro
96fc082ef58df2686ead5bd856196dd8
c691528d1ac7e9f16de391e265234c395a43cf0b
56727 F20101108_AABCGR jian_b_Page_021.pro
134b03bb5a15b69e554161b0a62368f3
d3f5b5c7d1732fc4b7d8586ca6bbc1500782a47a
42532 F20101108_AABCHG jian_b_Page_040.pro
1a04f1bc50a281663086af5d99389031
ecaeaa2dc4bf482e2fb20e555a58f77585cb4d87
49759 F20101108_AABCGS jian_b_Page_022.pro
e8e579ffcd45482ee67654e0b01355c5
406265a5907a60ea1da39464957f655a766a5937
54867 F20101108_AABCHH jian_b_Page_041.pro
5ea1f0f880778dc0636f35e6532af075
5adfd022e2b97987aea629a5424b679c06a6c9ae
37061 F20101108_AABCGT jian_b_Page_023.pro
b8a17ececb72cecc08678029d4dd16be
babcab625df49268b194087aa0f45d7c14811599
46633 F20101108_AABCHI jian_b_Page_042.pro
45c95ef53fcd228bc0fcb78f82216ef9
3b374b81b285203721a52920678c9e61d178ea27
43293 F20101108_AABCGU jian_b_Page_024.pro
d5ea0e3c92980ff775f02565a9f17450
baf0b7f9e9f70396cf836499fce060c0bfdab226
52118 F20101108_AABCHJ jian_b_Page_044.pro
dd7cf6c549750ccc1ef5a3ae7c8c2aad
67779a1047fec50025aea6a215d1e9fede9e3243
49257 F20101108_AABCGV jian_b_Page_025.pro
9623b447e9100b5b629e7eb6326741b3
2c8f280803861815a16828744ce7aef6aaab9475
51612 F20101108_AABCHK jian_b_Page_046.pro
8b08418a33e465ae119182b9ba9e0489
13d37cc72dfb7d66f933ff7de14dc314173bf9c2
53117 F20101108_AABCGW jian_b_Page_026.pro
129d10d0503d232fda07617edb1e528f
2319c4ee6bac6742b375bfa7b7cec583c017f5f1
50970 F20101108_AABCHL jian_b_Page_048.pro
6e7ad41f37f4f1c0b2f1fdf89f56dffd
e6b43ea253f04f109c2ae5892616d2a6a38c358f
39298 F20101108_AABCGX jian_b_Page_027.pro
5da1c598de79f5ea64797e317c33a652
39aa32f75ba936a35034281bd8f43a11d382f9ca
43485 F20101108_AABCIA jian_b_Page_067.pro
b1493bac84a544025e950664a94f383d
0bb8f044ade7e9264f1d832b67bb1a1ba3e50640
40623 F20101108_AABCHM jian_b_Page_049.pro
20275a2d96cafd4cf30d3e267be9d06c
2d392b3789655af269fa0e0f218bf78b60e3c0b5
47023 F20101108_AABCGY jian_b_Page_028.pro
6a5308e8a6f6a0089df8225250baf8a6
28c50b5fc92d42e1db134410d598527abde1c0d0
48684 F20101108_AABCIB jian_b_Page_068.pro
8b9af8a3c23bea9953849b2000b20e6e
4e0877bb5eaf7e76f37243d868289b30a4614a31
47087 F20101108_AABCHN jian_b_Page_050.pro
d41194fe54d1c911e6fe645d16f2b402
853aa6076c98823249193c898c1e1ee8f9aa90a9
58085 F20101108_AABCGZ jian_b_Page_032.pro
1ead726c6fdbf6eccf263518d8fc5fe6
e526a15465ebbdf375c129fe13b3348dd25dcd0e
40401 F20101108_AABCIC jian_b_Page_069.pro
21295285ab193bbbef668bc397fbc50a
e6d674dbf2db72befb17d6a33ce3e206983f003d
32525 F20101108_AABCHO jian_b_Page_051.pro
5482f5bd3845970a9df33f2eb3bafcbb
97add7adaef49b8abfcce298cf04d903a4e6925b
40569 F20101108_AABCID jian_b_Page_070.pro
d6c101a48baa58c10eb938193cfb8ff8
86e370962bda6ba12f472e1c84a40c12f2484f5e
40429 F20101108_AABCHP jian_b_Page_052.pro
bda53b988e85feeeb653403ed4c6003a
13a5843e208b0020cddcb1ef9990764865805a31
57601 F20101108_AABCIE jian_b_Page_071.pro
bcf4034471a2dcadad1d28c8261818b6
39ac853c60e5322fc2956261922f9917eb63dd4d
31959 F20101108_AABCHQ jian_b_Page_054.pro
1b74e83d87999bc53b038b059fcc5619
c6ceaf61789ebae397786b154070a12b7b2057b5
48844 F20101108_AABCIF jian_b_Page_072.pro
f8446bf79ab08e6dc7b18dc0dcb56850
d9e7e149ab35e60610bbbe8d1f14f412064615d8
45455 F20101108_AABCHR jian_b_Page_056.pro
769b106cbc69d47d41fca8cd994d81ae
18ea3b66f8ea6241bbe5f3e14bb58407d1c2ba70
54489 F20101108_AABCIG jian_b_Page_073.pro
ae25bab54f5826d10889d7fd85687665
a52ce947b17a89d0b8c9103ce4e84c2d3d4f6e6c
60895 F20101108_AABCIH jian_b_Page_074.pro
24fb178dad1cec2d8b97f27d82ed4c1d
96d7206718e483bbba6209575b4e11cee2dc88cb
29617 F20101108_AABCHS jian_b_Page_057.pro
8f6177ae19b4802d605e58df86aeaf21
c741675768759eb7a3ef37aa00f3af67d86c9248
48630 F20101108_AABCII jian_b_Page_075.pro
5fdc19ae8fd4c2a732e74a05d1d50c7f
3959d56cad6d94eeec2a270cc262e576761176cb
29477 F20101108_AABCHT jian_b_Page_059.pro
eeb109d42d39cf78e777306f7c979673
79f66db860236637d6d0802b8233f9b53e6c12e3
57366 F20101108_AABCIJ jian_b_Page_076.pro
21bf2b7769ea05521e757e3ccb220ade
44e04ff7042731e844c8ae731c1fb47fe6aadc91
35053 F20101108_AABCHU jian_b_Page_060.pro
eb5f3c4fc04c2471ddfb914e3e129a4d
7379733a058f3dfff0a0446c4a2b0a458c71e4cb
52791 F20101108_AABCIK jian_b_Page_077.pro
209d2416eb519642da8a0e1355d0c810
c6e96737403df12c8d3149073fad33c1a26624d3
47561 F20101108_AABCHV jian_b_Page_061.pro
a177e013da6a8abe40221ad6383dcb19
9ce30181151e4e0b2d9cb0adfe93033245447365
47076 F20101108_AABCIL jian_b_Page_078.pro
e6272b0f71a078e818417baaddd0aea4
17b66f50786ac734bd4a4509d532ae3beff71a4f
33940 F20101108_AABCHW jian_b_Page_062.pro
c3ba561378ded1229d2ccba86da98f63
bff6a24c9e9d1e3a9e4c7f51e0bd2ccf8e816799
52641 F20101108_AABCJA jian_b_Page_094.pro
72ec50cf4b478adba0c86d8a129141c9
ccaeb5da0ed09d4af2eb5f25eeea0bded1101977
42762 F20101108_AABCIM jian_b_Page_079.pro
d2f5c411ffb942477d305bbcb7110c89
b27626e4a81e14e49e2bf8332bf9fbf19d40942d
30564 F20101108_AABCHX jian_b_Page_063.pro
cef3e44db112908c60d3eae61fae61ce
349c4073820fd9e1aab545bb77b07f141730eb6e
59786 F20101108_AABCJB jian_b_Page_095.pro
16970d85cb65c31d00e2ed1ef5340886
4d476bd17d54ec322cec3a8d82837c613d995c42
47618 F20101108_AABCIN jian_b_Page_080.pro
6b6417d4983cac7239629eec80bd5fd9
7605b6d041fc2a9fb71db07be508ab003bfeafc1
36364 F20101108_AABCHY jian_b_Page_065.pro
b8bbf62151a7b49794489f44f926b4f8
92012e6e5292f0965761feacb289b13bffd71e36
25339 F20101108_AABCJC jian_b_Page_096.pro
59a6ee31e1a731d3828ddfe5a73869c7
d324044ba8ee32d61cd5d9c0a70e986ac80013b3
51320 F20101108_AABCIO jian_b_Page_081.pro
b7e21c50d9e6b03ecf7bf0227bc05014
db9da20a2c723dc665341d1cd0dbd06e9e72ac21
56966 F20101108_AABCHZ jian_b_Page_066.pro
77eef991fdb4d245144ad618be2daba9
d9db5c4d46bd376f677e652c4545e6294e0e0a0a
16322 F20101108_AABCJD jian_b_Page_097.pro
14c2b4965c62c2df154278c4659b95b4
d6c8c6fb21d7629ebbb0ce90661774beab80a595
53507 F20101108_AABCIP jian_b_Page_082.pro
7f8f942a1ceb63133dd966c825fa05a5
99389ed28298be5668d19c28d5545cb97c2caf01
55277 F20101108_AABCJE jian_b_Page_098.pro
d31e17c97049fc8f7c7234e8d63d8f1e
cf6e153aa84f1a0751a014c8010004754da6f749
42052 F20101108_AABCIQ jian_b_Page_083.pro
5b2650dce8c255d394d35d1d50bd4d8c
761d5b5a0af548ee297249a904e16051445dd218
28792 F20101108_AABCJF jian_b_Page_099.pro
173c7d5cb1d348f81bf40dcee35ba9a3
3048a76c1a984990dee280b5f61de47f242015b1
60143 F20101108_AABCIR jian_b_Page_084.pro
a0afdcc67c43703cb9c9277aaf33893e
25cae2591ad59f22f8f7478b31054658a9855d0b
31161 F20101108_AABCJG jian_b_Page_100.pro
5a347d3caea59445283ccb6ec5c41f47
f173993881c22852d092f1e424ce303c512825b4
42959 F20101108_AABCIS jian_b_Page_085.pro
81cdfac05abd05cf2f37024ee708acec
72d0197515086fbeedb7848a68b89e9425d3db74
54070 F20101108_AABCJH jian_b_Page_101.pro
8ac7541e818e178ee791e00c38d9dbeb
30f91797df0e9562d724d9a79787775ea7cb6d98
56492 F20101108_AABCJI jian_b_Page_102.pro
f4abc077b7272d2107252e3b36965b04
2dd9dec9b2bc15130369c0bfda1c6e3c44d90706
45024 F20101108_AABCIT jian_b_Page_086.pro
3a16e0ea34c41e33f9f6664f2156afb1
09aab3530975a18cc772d456fcc49eff40c5164e
57402 F20101108_AABCJJ jian_b_Page_103.pro
68d50151390343be023ec80f2af65ddb
c4d02f248f40050313a4b813b074e096fe9cb11f
47985 F20101108_AABCIU jian_b_Page_087.pro
e1ea1ed97b58c0f4af5888e2dc06eca6
48c9ced5cdc6950d1630d1c4398affe3fc1b7fb0
61068 F20101108_AABCJK jian_b_Page_105.pro
9c1ab0c047194fa1993a60475080922c
74afefff7db203a3df33b6604b1b3c95c1bbbe11
F20101108_AABCIV jian_b_Page_089.pro
ba5c6eaf6bc848f462143d86afdb6f26
5350aa5ee633b7d81ebeddcf4a4c72c4e9428fbe
62517 F20101108_AABCJL jian_b_Page_106.pro
c59ae5ea6d7634140972e7405bf35f51
c30751f1b9047610a16621964d53c450c860ad99
54741 F20101108_AABCIW jian_b_Page_090.pro
46287a1c03a806b77b5ae35148fe81e1
5a1b4cc9285d4fee1a5930d37f581ca2e51f0d79
58131 F20101108_AABCJM jian_b_Page_109.pro
a5ccc7fe10e6ab520697b7e04cdc730f
e26d2b573f448a13129375922301d00a957c0b84
59947 F20101108_AABCIX jian_b_Page_091.pro
8aaf8c38e12a500b1111fbb17d3c3a34
09ccfd4d6f9b376283d14a4aa1d79256097de249
420 F20101108_AABCKA jian_b_Page_011.txt
cb6ebce877ea424b5eb0adf2ae1dd69e
30bdfb3b702092698a55b82102d8987daeb24201
66752 F20101108_AABCJN jian_b_Page_110.pro
2a7800eeb52d47587d43708da29610da
f86141912114a7d8065f10ffd1d23e63e84e2f70
59197 F20101108_AABCIY jian_b_Page_092.pro
c7b22c34a3ce7d9d102e1342e84cbf66
dc2e5b991899ed0e74e9b81da5121f01b6e038a0
1046 F20101108_AABCKB jian_b_Page_012.txt
d45bf54fd60cbbf018ce473ef3fa3d10
abb0fb3cdcbaeb226f2350a8e513b8d339c1998a
65914 F20101108_AABCJO jian_b_Page_111.pro
673fe002b9beb0109f34dac63b9e99ef
10023f902f0c4606822f1a47fd11cc55424c2b5e
31478 F20101108_AABCIZ jian_b_Page_093.pro
5bda37145f2674b28f48ada565b0f17d
54a002272bb8ec8f0930ed4af5a1262e140e7b12
487 F20101108_AABCKC jian_b_Page_013.txt
dcb8707884102615bdaf3e2dc9cb9909
fff58e32308b2851ed524b8b0e2ee596fd5e5b4b
1124 F20101108_AABCKD jian_b_Page_014.txt
a5bee5375be6a030cae24cc7b4abc645
20fa765e31f3052fc1005aa5882cfc5b88597c4b
62819 F20101108_AABCJP jian_b_Page_112.pro
98e041cfd8fe523218cbc1a49c840114
3589fe6327ea26b466a18833d7e23a2c739e2fe7
701 F20101108_AABCKE jian_b_Page_015.txt
225e854a0bb121310fd74fda63cfbf19
cd7d471b643bd743d766e1ebc24c6fd2cf41d024
65991 F20101108_AABCJQ jian_b_Page_113.pro
66ca53d6bc3cbc1f7f5e57ca182f91b3
4ad47c267aac478d96599d5a7b814a14b360df57
2204 F20101108_AABCKF jian_b_Page_016.txt
a40c5f632f770471b40e4a5b5df4cf47
47ca1115bd0c6587a3d0fabd1d0d51d681e781ad
70968 F20101108_AABCJR jian_b_Page_114.pro
8fffeb8c2c05c57adf68b821aae63076
b26b21675042632293f880d859bec767d25da4ce
2212 F20101108_AABCKG jian_b_Page_018.txt
23e8848741c30657b0c41a80e697f152
14b8dd62e9cc87446d718b533b75e511e2c25e0a
65785 F20101108_AABCJS jian_b_Page_115.pro
f35176a4cbc710041430b302ebcd432a
39f09e753ce57d6920abf9d17f0914d1ac4e05ad
2394 F20101108_AABCKH jian_b_Page_019.txt
cfee2aa591215e61c292f02a051c626f
e5b3e19346c0262584b0599c0765a9739956f5e4
62951 F20101108_AABCJT jian_b_Page_116.pro
0631d2e685d77aed42180c38555c325e
4eef125f4010f8b8ba1b4f277b3c3f53de4662c3
2263 F20101108_AABCKI jian_b_Page_020.txt
aa03fa14f6c260309bedc5c6670b83ef
1f8e88054e544a579d1b69cff53f00b5e851ce6e
2049 F20101108_AABCKJ jian_b_Page_022.txt
9baed7ecf3efbad082bd7df7aefda5f4
e31bf8d929b53a99b4be5454bc23f767aa88930c
22559 F20101108_AABCJU jian_b_Page_118.pro
b91b15535933dae4e26081be1da57582
5ba7b3975462828c531de19f8090d7814b46f4ad
1816 F20101108_AABCKK jian_b_Page_023.txt
e7fa5f31ef93d23fee206093c172d9d8
6b7c8bac3910e49431767cf97988cf1473528b91
80 F20101108_AABCJV jian_b_Page_002.txt
a78ea60ccebb83e8bd34f4b9c77b5ec3
7ee9b08fd808945395524d2812435a7d4782989b
1993 F20101108_AABCKL jian_b_Page_024.txt
5a7da11a6023e45095794c9f314ef874
30d7ed00b4bc24d966248250f5484e5a82079d00
2174 F20101108_AABCJW jian_b_Page_005.txt
c41775ddb4de1cae5f767bb6ffa67ed7
44c088bcf98fc1cd987a76285bbe0d356ccd7636
2132 F20101108_AABCLA jian_b_Page_045.txt
f86e08cf9f8552d9edd5c4cf1fbedbaa
54c3cc05b6688943b76800c843a53b489c6c1327
2168 F20101108_AABCKM jian_b_Page_025.txt
c7da18ff0f70ceb62f518104d537191d
55df14ea0b94fbe8e883ea0b1533e61490664c70
477 F20101108_AABCJX jian_b_Page_006.txt
11884805dcde3df12160cada4862dde6
7b4e2fc683fd7ff4b126245bb7f460201f24af00
2047 F20101108_AABCLB jian_b_Page_046.txt
672dbd3cc6934347735fda5d4f7712f6
8e662cb1b1c5a375b77a516fd775dda8f40a6db9
1716 F20101108_AABCKN jian_b_Page_027.txt
1fa63ad26f104c2764661300f0ead750
fa811a39fc962be15ea8854e5a07e10e6c21bcce
1462 F20101108_AABCJY jian_b_Page_007.txt
4be3bd50d7b14d49434f2bad8bcf7716
2f9ddbd180a9920d0634141706063bc7c5f809a6
2148 F20101108_AABCLC jian_b_Page_047.txt
f5650ad2b1f9b529b0d875a1d3701cb3
a38a5350f3c3766b771586f942ba32b89e196f8d
2023 F20101108_AABCKO jian_b_Page_029.txt
e0e83bbea58035168ddddc11fd29995d
df859abe0667e138b4895e5ceb40ba75407b95a2
743 F20101108_AABCJZ jian_b_Page_009.txt
290240bb10243f13480259599c919367
d46904508fd566ba3c8f1b0519644f9627c69ff5
2067 F20101108_AABCLD jian_b_Page_048.txt
4ce80eba1772fbfbbf61a1837d55ba20
94afbf1c48527c0261ba09101f2d39cc9938d525
1722 F20101108_AABCKP jian_b_Page_030.txt
10725f979529cd71adbb5c4235ee6548
2bea2649acac0f8ff1324c0bbb3c0499c837c309
1997 F20101108_AABCLE jian_b_Page_050.txt
ad2023ee27a04c2ae7d59900b8d3844b
e46b2fc741ba9fd5da054fb0b9ea78f147af4810
2217 F20101108_AABCKQ jian_b_Page_031.txt
804f075626dbb23e685790724b5bd7a5
3bb5be9d8713ccdb2510a1c1703a969c4741367e
1518 F20101108_AABCLF jian_b_Page_051.txt
4c0b10f12ab1dbf082d32671b29c0d1b
255d6cf27534cb8d43482d09c8ae1b6a5143d61c
2326 F20101108_AABCKR jian_b_Page_032.txt
76c39efcf00d98327751c6676d66cac7
e83b8b7c94144184880ffd76272b450ab947c278
1650 F20101108_AABCLG jian_b_Page_052.txt
f27b212ff220665a921f5d7d11d0d645
9198649affcfded3b8a5daed5a835612bd712907
1110 F20101108_AABCKS jian_b_Page_034.txt
8dde98dc1100bda71628de0563a237da
ac49e79e843bdb786f3111ba1099e34e0ecae3e2
1495 F20101108_AABCLH jian_b_Page_054.txt
5ed4285d3045f438f2970f8e82a05527
1ef4e6615c99c953b5ad7a01487849bc3dfbf2a9
1879 F20101108_AABCKT jian_b_Page_035.txt
d630b0e7e660d9da29452019f425c4db
d77a67448ea0231846c4b32cd80a13eae08abccd
2248 F20101108_AABCLI jian_b_Page_055.txt
9e3eb14813a2f12a41117bdce534af9d
1bcbf07dd0359dcf2ad1ba2603c12246cf53145f
1492 F20101108_AABCKU jian_b_Page_036.txt
fcb01658d6458696ce0b7b02b1b6bba4
2fd9095476d26fb167e7acaf1ee155a6e39bc767
1929 F20101108_AABCLJ jian_b_Page_056.txt
c6197cc0058d1dd45188a55def2ddafd
f4dca0ee6c793619b6d0556c5b681fda6d0a7e70
1389 F20101108_AABCLK jian_b_Page_057.txt
ee973c759d1013213c654952b37bf28f
9d6c8096f6cb76ee2213746c1f48d1a94667387a
2219 F20101108_AABCKV jian_b_Page_037.txt
263d500ef6d64f67adc21ba8aac2222f
f6854350efcf844d90035dff7c548e1cbec079da
1985 F20101108_AABCLL jian_b_Page_058.txt
9b11db2989743277601393c1bd9bbd07
9a2376598780dd973155277416c413b35d99a0ce
1918 F20101108_AABCKW jian_b_Page_039.txt
0cbc0420e39da2f72a1d415cef111e5a
2ae7e2982f8566926b4154b4034bd3f1d4939bcc
1514 F20101108_AABCLM jian_b_Page_060.txt
1a1baff241f7312fe23f9d6646fbea62
77e683f307589ad4555823fda6f8bb538fdc8963
1791 F20101108_AABCKX jian_b_Page_040.txt
f5d48f1d9c28cc409a0afb78409c9065
e98f09c824510f29aa5b7ee2a706f06adbd77837
2009 F20101108_AABCMA jian_b_Page_078.txt
cc0eb35d7f3fcd8a6dfae4140370dcf0
f8f7eaa78bffac38e28401428d657094d63ddaf6
1294 F20101108_AABCLN jian_b_Page_063.txt
cb8fc88b0be6a0f89dafacac65ad0ffc
a645028b33617c8c9ff413434a50b76ffd9447fe
2483 F20101108_AABCKY jian_b_Page_041.txt
b0e20be87149d9e9486670dbb7b07a2f
9211436b0480ba340a3a73014978b1a7d4358b3c
1905 F20101108_AABCMB jian_b_Page_079.txt
9b3679ec6db2080723fe61655508cab1
e49193e1499f19d6a1418a0fff3fd970319527e5
1586 F20101108_AABCLO jian_b_Page_064.txt
078608bfc7df6f70dca94d3deb6dcd8c
1e61bff347a0848dd85fa0c5701a0cfd6d8352ef
2156 F20101108_AABCKZ jian_b_Page_043.txt
bb7593a0a03e9792f624fb054b35ee37
fd6494317376eb19f69f89de9bea8954d8373808
2074 F20101108_AABCMC jian_b_Page_081.txt
f741a718ddc0cba1c566e7e16f2526f7
902cf185043c5c7ce217b7414a38ad241627427f
1627 F20101108_AABCLP jian_b_Page_065.txt
deb8c1cdae6f9e49c9a2286bc807ea49
6ccb5eda9253b8b534498091f5722976d2728f92
2186 F20101108_AABCMD jian_b_Page_082.txt
392dd3cc87c33aeafec4cdb4f161e6bd
ef36d77dd21ae060226c055d9e2da40429ad4187
F20101108_AABCLQ jian_b_Page_066.txt
8b189a1cc1f98696e6df1e689aee784a
eaaa59f93f9eee20e99e2d83d29e5faa2b248c4a
1937 F20101108_AABCME jian_b_Page_083.txt
8a803bbc2277f591921dc14cbf1ce45e
19ac5397110aeb5506958588d10974c2c5a01773
1974 F20101108_AABCLR jian_b_Page_067.txt
2002f6c36f3c4eceb2c20024fc5eb09c
31ae61313a79040ef9c0d64fec58037e75191861
2381 F20101108_AABCMF jian_b_Page_084.txt
2b243d556b232696d5c32d2561ebc75e
81c254ebca628d8f080926b9d312e2e6956d44c7
2058 F20101108_AABCLS jian_b_Page_068.txt
509c7c8797ee21af049cd2234f6582e2
da27c85385be09f21a008bd9b30205454edb4ecb
2028 F20101108_AABCMG jian_b_Page_086.txt
05df1c20f47eef37423e79643ea8d5c4
7a6144136de40176826d688acbac07d88ce4904c
1707 F20101108_AABCLT jian_b_Page_069.txt
faf273390fcb5107e4fe4d7f3cbdfabc
5d2a5e4d235e83be603f95d86745029d9f6b7a39
1964 F20101108_AABCMH jian_b_Page_087.txt
5c50969369ed7481d00a02ded0f241e5
8f48a66520d6601cc0a5b1b8fda146972aabbdb4
1794 F20101108_AABCLU jian_b_Page_070.txt
e2ffbc5089ab8d103bb514af507df5c7
2c499ecece0f8a211915b1e5eaa12423723dfe74
2206 F20101108_AABCMI jian_b_Page_090.txt
5efd302096e505da42dedab7fa0c40c5
a485c090f7565baa0b87fb387a01371cb2cd5773
2163 F20101108_AABCLV jian_b_Page_072.txt
a74e848e45ff55a4f9f0c09dc1f348b8
6269299b8e8063edf15b453339b117ffe0d7a66c
1361 F20101108_AABCMJ jian_b_Page_093.txt
521488e86e18b7fe2c12c6cc757f51bb
4df26cf9ccdb47e1dead99be9fd5eb331cefd4a8
2218 F20101108_AABCMK jian_b_Page_094.txt
545224090c44ae07aa7a67825565daf1
fb9a734bfab88b72414c23659c8955d6e31b3f6f
2194 F20101108_AABCLW jian_b_Page_073.txt
5db3a92e65cf7907361a132da8f6b5f7
41ab555e10c9b08db399474db529db4e680813d3
F20101108_AABCML jian_b_Page_095.txt
38c230f7b6ae59610b6a2c941e4d89b1
46c20e83b4c7ee8488627a3b3a21aadcd22df11c
2427 F20101108_AABCLX jian_b_Page_074.txt
fe77c5d60ab19de2b93a65616c4c5339
402df847bd55135dee360c8035ff6ba0bfc36770
2548 F20101108_AABCNA jian_b_Page_112.txt
340559a8431491573d1388989e4b72a9
b0d16c14e4943b7a8b1f09837d62e563d1197206
1070 F20101108_AABCMM jian_b_Page_096.txt
4bc19c5d5d57a4e5058e02513942c3de
59ac62c6e03d2eef7118ebe01d290cf34e8a1c9f
1990 F20101108_AABCLY jian_b_Page_075.txt
219e4d72936356d4166b43a22ee25ce8
3a5b8db6cd836abe57364b737a87a8c0baa048fe
2675 F20101108_AABCNB jian_b_Page_113.txt
90750a2fbeb4935a19b5718288a1ab48
12cb3fe4470986f842482f3142bbeca24abdeb6a
773 F20101108_AABCMN jian_b_Page_097.txt
217a1d059ad93250a7040346758a52e4
14fa97f55177a5c4fc213a45b03754b301e00154
2321 F20101108_AABCLZ jian_b_Page_076.txt
2023f762261b91a8f402a40245004f88
fad38bff7cbed02157d0bc647919d769fea01e54
2857 F20101108_AABCNC jian_b_Page_114.txt
866fd29d4365d013cb81d3ccbb93615c
e516d7338a994c4652c83b7c632ebf8877cb5133
2278 F20101108_AABCMO jian_b_Page_098.txt
9d5be9ca67f0a3e457f69b312c8df0f7
6a1a4143b11dfda416e817fff598bd4c4f2293c1
2647 F20101108_AABCND jian_b_Page_115.txt
2a261f3ff04ec69fd98b9ef01df0ba00
52324063c9ee7893510e1c828ada20a002c95e70
1149 F20101108_AABCMP jian_b_Page_099.txt
ea8fbeda4f759d842345a880239ef662
47500091698ead3b620350d05464ba8309cf1d28
2543 F20101108_AABCNE jian_b_Page_116.txt
f67044c09a5d8374d2729124f8eb8c29
937b78d9137c4c800919b5ec9ed1c79b60ac35dc
1353 F20101108_AABCMQ jian_b_Page_100.txt
2af45281a05a5814aae5aecbfda6c7f8
0fc4ed1dbca1ee4b0721f0936d3f3bb1d762bc97
319 F20101108_AABCNF jian_b_Page_117.txt
7cb26e1e186654c09e6c253d662e96d0
c753490ece76ea411bfdf47c492ba319f2bf91d2
2254 F20101108_AABCMR jian_b_Page_102.txt
800c20903e540109d00f77529fb2c359
6b6e00833e941e4554fe9620f58ae828f2483e13
931 F20101108_AABCNG jian_b_Page_118.txt
382a95cde0e3c928b4065a2f539d5026
9d7cd2334ee5031e1eb689b6c59b698d6f8ad931
2258 F20101108_AABCMS jian_b_Page_103.txt
90dd853c4798e313a09f54a880d06ac8
157cf347c5530a9545c0fb36ce62000271320295
3309989 F20101108_AABCNH jian_b.pdf
57106309c01abbd617c0183c25cbd814
20a0b7197fc810c84eb28a2c99331374f4829a53
2115 F20101108_AABCMT jian_b_Page_104.txt
80a5c21c7a7e997ba0a06dba9ae77a2c
4e4717780cd56a2521f6ccdb8cc25a5143723ae4
7509 F20101108_AABCNI jian_b_Page_001.QC.jpg
aef7414e489cb486dea22e204dcb5830
fbb6a117c45d25ac2d400ba19a4ab5ebae27c534
2465 F20101108_AABCMU jian_b_Page_105.txt
40ef3a59c8cde8a5965dec9a2f592e71
d6226b91c6f05a071c0444e6a27a39fb1e0bcef5
3114 F20101108_AABCNJ jian_b_Page_002.QC.jpg
d84a761438e6d7bdfa608585ec25919d
57aa8b7f6a84a94af18124331faeb35c06076cbe
2505 F20101108_AABCMV jian_b_Page_106.txt
8003d623c225b7eb65a4eb4813b2a6be
131d9a4cceaebc567b9cb3688c08cee8510f1225
1315 F20101108_AABCNK jian_b_Page_002thm.jpg
66f62cfe97b2199a9f2313d79de22ec5
947ec0a06b61a21c4a1a8ff6dc454a645839f92f
2744 F20101108_AABCMW jian_b_Page_107.txt
c6532a9dfa10fdf05c8fdae408e03353
b446a6b52d8183c0b2e96aad2c83b87a3874040d
3070 F20101108_AABCNL jian_b_Page_003.QC.jpg
22d428dfde1a0d568df3b1bd0ce03b10
03971146fbe1e6d533eb7a81a26099f69fffd482
13652 F20101108_AABCOA jian_b_Page_014.QC.jpg
e5ade30cb84828ccd5aad9304970ebe3
3427e42055a56921115b220bd4b5dfbfe67ce718
24615 F20101108_AABCNM jian_b_Page_004.QC.jpg
9d345316cd28d5037f60922fd2d3e71f
306624bfbeb412238ffec492946279e6f543d727
2337 F20101108_AABCMX jian_b_Page_109.txt
113f0aa3b2700e93ea373cf59d833a2c
0bf30db6a2822b2a56ab636ba16bf7fd6a2597f3
3787 F20101108_AABCOB jian_b_Page_014thm.jpg
9cdc92d3846759df4a431302e3adebb1
18ffcdf638d3bad32f512264fc9819e76613c53f
23719 F20101108_AABCNN jian_b_Page_005.QC.jpg
435d3f78c889988d5fd72cd11268d1c5
830976bf89a2aa98c80d5ee3c0e2535394c6f50b
F20101108_AABCMY jian_b_Page_110.txt
6ce6345d103b8f27980ded5efdf700de
017b72549af9c39ba0f0ad1bb35edd0189c03057
9145 F20101108_AABCOC jian_b_Page_015.QC.jpg
af44edb5c3ffc3a8f42f4c72aa565fa7
609d30c3aa5996e963c8c1656e21503cac2fab45
22233 F20101108_AABCNO jian_b_Page_007.QC.jpg
e76b66e3af9c0b2cd1e04dc902d51720
f10315fc5c8cd10f29ed7675f8efddee9f1bde0d
2637 F20101108_AABCMZ jian_b_Page_111.txt
d00ec15e4ed9e86f7825a6e2bdbb2c34
c4b0d638c86dfcdd5b24b572dd06d0a1846ecdd1
3068 F20101108_AABCOD jian_b_Page_015thm.jpg
35e02ac9bfe94349add09c9f45434e07
63297d7e439e42c2fc753294998a62e4e487e7e3
5388 F20101108_AABCNP jian_b_Page_007thm.jpg
f380862af0dcd00efb23ae112d1aef25
e37075df674705c5a93c97b1593e135948e3e5e3
5786 F20101108_AABCOE jian_b_Page_016thm.jpg
ff40f9d0cd758965445f1a95a53773ef
dbb5bfd2bd444c172b755b5f7c9df433781d22bc
5036 F20101108_AABCNQ jian_b_Page_008thm.jpg
fc25bd72d71c4ee0e22be0efbbad29b0
1a456e8787d855c4597304e54b418aa415aeec64
17100 F20101108_AABCOF jian_b_Page_017.QC.jpg
1baca7aa236858fd3885cdcb33566f49
aabf94c79d55885c773fb404e3fa257c97190a34
9648 F20101108_AABCNR jian_b_Page_009.QC.jpg
20fbb16089939ac72e26a9643c682566
3959de69730dd0d41cd0bef79fd710eb3eac17df
25832 F20101108_AABCOG jian_b_Page_018.QC.jpg
37141dc6bb9869e02e7836764286024e
6920e8ee12f9a555e7df44e8a7f439aeefa0b194
2783 F20101108_AABCNS jian_b_Page_009thm.jpg
cfbccf24bf921beb72bf78b6da542b30
621d558b8346d6847a04d28aca3ee95fdb36f435
6404 F20101108_AABCOH jian_b_Page_018thm.jpg
e5dcecfbda201ba4a74f1851959fc567
2ad416c5fd9184fee2b6197f6ea08cf35bed1125
6919 F20101108_AABCNT jian_b_Page_010thm.jpg
f0be429f1bf35fb2170e319bb0e08692
8b25860c6719deb775d76143a77c90095734cd4e
28384 F20101108_AABCOI jian_b_Page_019.QC.jpg
35e054a9804b6a9d5a205e1a0504301a
d8126eb737842733d8e66335eb928c40c682b9ce
F20101108_AABCNU jian_b_Page_011.QC.jpg
6446408c5b88f228a82a967df47dff84
e63c71cb5b0919741d730e54d7793a4fbda57a84
7057 F20101108_AABCOJ jian_b_Page_019thm.jpg
1ec1f3e49acc192fcb58b41baa18b03b
9542ab1aae5ba5cef257ceef2e8c233fff458883
2089 F20101108_AABCNV jian_b_Page_011thm.jpg
8faa9458e06b80fecd2929e773a1901f
7a948fa3360f4f4ded809110237fac9cdbfa4d40
26235 F20101108_AABCOK jian_b_Page_020.QC.jpg
ffad6811b887f5865d685d9799db5f6e
3431f63b93dc2b7f7cb46b9207bb2e12bbac6d57
13179 F20101108_AABCNW jian_b_Page_012.QC.jpg
11a9e2ae0296d570ecceac9cbbf8fc93
4e05cbefcc949b5a6a00aac532b76b75c54270f1
6565 F20101108_AABCOL jian_b_Page_020thm.jpg
66445eb004fe7e90bc55c5189ba898b1
e04fe9eda2554257e247a01aa2de226e87846522
3554 F20101108_AABCNX jian_b_Page_012thm.jpg
03d2f20444c318876cae3c2c2c2d2939
c4a17f2b6764f2c1b9476014dfd468338afa2225
22266 F20101108_AABCPA jian_b_Page_033.QC.jpg
654219991925644ce9e47b02d63fe411
e74068d6d836c3c63f82d67e74e849600ddb0418
27045 F20101108_AABCOM jian_b_Page_021.QC.jpg
e38a457c3e06e841d787cf083a32af6e
59d5c6832a36beebc280b1f4e224c7f187548307
7290 F20101108_AABCNY jian_b_Page_013.QC.jpg
c0c10c8a742732f934810c8ad4fd1309
896c810685c945a4a062cce872c9e8ac3207232d
5893 F20101108_AABCPB jian_b_Page_033thm.jpg
0adc5a18bc1e9aaab183f124d106f7f4
a5e0ab379c8e7f811cf9d92adce393f278778c84
6773 F20101108_AABCON jian_b_Page_021thm.jpg
d6154005023d0cb1721562f8aa886974
631ad4619c8feef303ed63cafd91d6ba8f709d96
2313 F20101108_AABCNZ jian_b_Page_013thm.jpg
73690d4575116a893336d5e1cf6ae20b
e376f8dc7b970aa273dc52a7de187b6ad4a419ab
14568 F20101108_AABCPC jian_b_Page_034.QC.jpg
bc362c00494d71dfec36d38c8f9f9de4
4a9096e6bb4619896db998621a81f3fbc01d0004
5415 F20101108_AABCOO jian_b_Page_023thm.jpg
bf26fee783d3e338fef9ef8f1aee6c67
bb8b36c909c1457ff04cf7d87b9b333296fd33c9
4126 F20101108_AABCPD jian_b_Page_034thm.jpg
8e7bfcdff4817a3fc0c4a826325e4e7f
0b21f069a48b9dc37c496c40723e526759a1d3ba
21376 F20101108_AABCOP jian_b_Page_024.QC.jpg
e896b52d0eb88dd66ef60cc9cbd9ac06
7a701d7402c59d85b22ba8ac620601d8849c1f97
16225 F20101108_AABCPE jian_b_Page_036.QC.jpg
d3a645b1aa1aa45e35c20bd72924c61f
35bd3a7e0c3f6046980c6e10c00ae44ee8f714b7
5900 F20101108_AABCOQ jian_b_Page_024thm.jpg
1166cedebc82136c113b66fad57cfd72
98c489199a85f916f2687064ad8e9b49f709f530
5037 F20101108_AABCPF jian_b_Page_036thm.jpg
2fa7b2d8c06acfd3ebbf98cc876042b5
982e02b21512b3784c5fec0b19b34fda6db5b15c
22950 F20101108_AABCOR jian_b_Page_025.QC.jpg
91f68e3fe4a5a126529b8221255624c1
e7b7bcb9d79de95e36c2b4de3757134f6f7921b7
25008 F20101108_AABCPG jian_b_Page_037.QC.jpg
fe85c32f6d6046c2fad1fb1ba70f07db
f4fa1b3b5e1d2c7cefaf5fb4739fe62cf9a0c1bd
6090 F20101108_AABCOS jian_b_Page_025thm.jpg
35ba679f3a152fba49b8b38629e535c4
4cce3dbe1a1d372e17d3597667837d3339dcc5bb
6659 F20101108_AABCPH jian_b_Page_037thm.jpg
34bdee4e04402401b203fbb9fe1a2ded
8180e4a834463342d3a97208fcf2414d275c3196
26012 F20101108_AABCOT jian_b_Page_026.QC.jpg
333b1982456ceab3176b3f876fbcf31a
4fbea2f01e089f011a4162322bc0df47c1ca928c
5581 F20101108_AABCPI jian_b_Page_038thm.jpg
4e2201540fbd4c2bf2bb68842f4d3a75
d587122e9795b75f9ee9c029387dadaf70d0f787
19985 F20101108_AABCOU jian_b_Page_027.QC.jpg
ceb28b60211d9179ec380bb4b84526a4
00a4fed0555c8cf5d67a2829568c4f406dbdcf35
20984 F20101108_AABCPJ jian_b_Page_039.QC.jpg
50de3f9e974b4204445aeefe4859784d
1dad27812b03338756819d3debb8032ef58257d3
5608 F20101108_AABCOV jian_b_Page_028thm.jpg
9d70143f267f325768ef8270f4bae3e7
7254b11939dc5fbaa8688940dfe328b329f9d98c
5881 F20101108_AABCPK jian_b_Page_039thm.jpg
09502dff0bea9834de8bd3765c201ce5
7b7ad507c1cb6d512ae7c438727aca9209957005
5315 F20101108_AABCOW jian_b_Page_030thm.jpg
c78e966813f864ca09ad473f4199681f
7af2f76ed9778a0db14857c6fd16234fc6d4b1ba
21682 F20101108_AABCPL jian_b_Page_040.QC.jpg
84aea3c14cb784fcc6e41c9216aa31a7
0e9d4e0dd0dd2a35f84a8a69f0cc5ed1da0c7176
27099 F20101108_AABCOX jian_b_Page_031.QC.jpg
e47a6d7a37396ddf55efa1b47ede8077
6e46b378dc31e9cbe768e3a04021330983ed09b4
24019 F20101108_AABCQA jian_b_Page_050.QC.jpg
a6c68a40d40ddb098b3a5c627f23575b
6035b012464bf2435472452a71321f2ebeb5cf5f
5847 F20101108_AABCPM jian_b_Page_040thm.jpg
6db2b2ba34c711d714a7baa7a3e8d0d9
1f7365edb68491ac1137bc5284a0a8acbd125022
27349 F20101108_AABCOY jian_b_Page_032.QC.jpg
631c4687907801ba0a96628c7694e5a7
92c3786c45fba267b3c117c593458e33363b2783
F20101108_AABCQB jian_b_Page_050thm.jpg
c92961c3f9ba0b00d89d1eb530499859
ed596d07e23b261484cf3a7952218d2385ae2e1b
6801 F20101108_AABCPN jian_b_Page_041thm.jpg
2e3bb8c8fcb9655ae23ee6f4ca8c1151
4ab1f0ae0e331806cae9fc85361ea23e86f9e829
18462 F20101108_AABCQC jian_b_Page_051.QC.jpg
30ab40d76ece40caf4da5faa35f2d144
7e8c7312a9fbb201b1924645270ef1a1d6beb180
6645 F20101108_AABCPO jian_b_Page_043thm.jpg
75d783f90f3ad76d58548ba2a5cb65b1
88b65645ad9cbc0bb2683a1f8f045d858e7d587f
6941 F20101108_AABCOZ jian_b_Page_032thm.jpg
3bf2859fc09076f11138abf7d726f1c8
d4297af2954d1b0a8e8313c970a59b39a44ca085







MATHEMATICAL MODELING FOR MULTI-FIBER RECONSTRUCTION FROM
DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES


















By
BING JIAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009

































S2009 Bing Jian



































To my parents and Geri










ACKNOWLEDGMENTS

This dissertation would not have been possible without the help and support front

many people. First and foremost, I would like to express my ininense gratitude to my

advisor, Dr. Baha Venturi, who introduced me to the field of medical image analysis

and assisted me in each step throughout my PhD study. The research described in this

dissertation was mostly horn front the lively meetings and intense discussions with Dr.

Venturi. Without his patient guidance and strongest support, I could never have reached

this important point in my life. I feel so fortunate to have him as a good friend and kind

mentor.

I would also like to thank my coninittee nienters, Dr. Sartaj Sahni, Dr. Anand

Rangaredi .Il Dr. Arunava Banerjee, and Dr. Siriphong L li.--111,.'.1-l I'!;I1.1 for being

very generous with their time and knowledge and ahr-l-:- heing supportive during the

completion of my PhD study. The five years I have spent in the University of Florida

is definitely one of the most nienorable experiences in my life. I am very grateful to all

the professors who have taught me many interesting and rewarding courses in computer

science, niathentatics and statistics. Especially, I would like to thank Dr. Baha Venturi,

Dr. Siriphong L li.--111,.'.1-l I'!;I1.1 Dr. .Jay Gopalakrishnan, Dr. Arunava Banerjee, Dr.

Yunnmei C'I. in~ Dr. .Jeffrey Ho, Dr. Tint Davis, Dr. Xianfeng Gu, Dr. David Groisser,

Dr. Sergei Shabanov, Dr. Brett Presnell, and Dr. Richard ?-. i.--us! Ia for all the time and

energy they have invested into my education and research. Nevertheless, any errors in this

dissertation are solely my own.

Most of my doctoral research activities were carried out in the Center for Vision,

Graphics and Medical Imaging (CVGMI) where I have learned a lot from Dr. Baha

Venturi, Dr. Anand Rangaredi .Il Dr. Arunava B lia. j. Dr. .Jeffrey Ho and my fellow

students during weekly seminars and daily interaction. I thank former and current

CVGMI lah nienters: Zhizhou Wang, Eric Spellnman, Timothy McGraw, Hongyu Guo, .Jie

Zhang, Fei Wang, Nicholas Lord, Santhosh K~odipaka, Adrian Peter, Angfelos Barnipoutis,










O'Neil Smith, Ajit Rajwade, Ozlent Subakan, Ritwik K~umar, Ting C'I. >.~ Guang C'I. 19_

Meizhu Liu, and Yuchen Xie, for making our CVGMI lah an intellectually thriving and

socially enjoi-l l,h environment. I consider it a big fortune of me to have the opportunity

to know these wonderful persons. I also want to take this opportunity to thank all my

friends in the CISE department, CVGMI SIAM Gators, ITSTC-ITF alumni, and many old

classmates.

The diffusion AIRI data used in this dissertation were provided by the McE~night

Brain Institute at the University of Florida. Thanks go to Dr. Thomas Mareci, Dr.

Paul Carney, and other nienters in their groups for the collaboration on this aspect. In

particular, I would like to thank Dr. Evren Ojzerslan for helping me better understand

the diffusion AIR imaging and interpret the key niathentatical model developed in this

dissertation. Evren also kindly provided me some simulated data used in the experiments

and the IDL visualization program to render the surfaces paranletrized by spherical

harmonics.

Across the campus of the University of Florida, there are many warnthearted staff

nienters whose excellent work and services make the study at ITF very pleasant. They

are the wonderful staff at the CISE department, the International Center, the libraries,

and many unsung heroes who deserve my deep thanks. Outside the ITF, I want to thank

Dr. Behzad Dariush at the Honda Research Institute ITSA for giving me the opportunity

to work with him as an intern in suniner 2006 and for his continued help. I would also

like to extend thanks to Dr. Arun K~rishnan, Dr. Sean Zhou, Dr. Gerardo Hermosillo, Dr.

Yoshihisa Shinagawa, Dr. Jiannming Liang, and many other colleagues, for their kind help

and support during my early career in the Computer Aided Diagnosis group at Siemens

Medical Solutions ITSA.

Finally, I would like to express my deepest gratitude to nly parents in Chan!~ I and my

girlfriend Geri, for their unwavering love, tremendous support, and firm belief in me. This

dissertation is dedicated to them.










The research for this dissertation was financially supported by NIH grants R01-

EB007082 and R01-NS42075 to Dr. Baha Vemuri. I also have received travel grants from

the CISE department, the Graduate Student Council at the University of Florida, and

the IEEE Signal Processing Society. I gratefully acknowledge the permission granted by

IEEE, Elsevier, and Springer for me to reuse materials from my prior publications in this

dissertation.










TABLE OF CONTENTS


page


ACK(NOWLED GMENTS


LIST OF TABLES.


LIST OF FIGURES

LIST OF ABBREVIATIONS

LIST OF SYMBOLS ....


ABSTRACT

CHAPTER


1 INTRODUCTION


Motivation.
Alain Contributions.
Outline .


2 DIFFUSION AIR FUNDAMENTALS BACKGROUND REVIEW

2.1 The Basics of Diffusion Physics
2.2 The Basics of Nuclear Magnetic Resonance ..........
2.2.1 Dynamics of Nuclear Spins.
2.2.2 Magnetic Resonance, Relaxations, and Bloch Equations .. ..
2.3 Measuring Diffusion using NMR .. ........ ..
2.3.1 Spin Echlo anld Diffusion Effects in NMRl. ........
2.3.2 The Fourier Transform Relationship .. ......

:3 DIFFUSION AIR MODELING AND DIFFUSION PROPAGATOR RECON-
STRITCTION -CLASSICS AND THE STATE OF THE ART ......

:3.1 From the Bloch-Torrey Equation to the S' i-1: I1-Tanner Equation
:3.2 Diffusion Tensor Imaging .
:3.2.1 Tensorial Steil1: I1-Tanner Equation
:3.2.2 Estimation of the Diffusion Tensor.
:3.2.3 Fiber Orientation and Anisotropy Measures Derived from the Diffu-


sion Tensor .
Problems of the Diffusion Tensor Imaging.
Angular Resolution Diffusion Imaging (HARDI).
Modeling Diffusivity Profiles.
:3.3.1.1 Spherical harmonics series.
:3.3.1.2 Generalized diffusion tensor imaging
:3.3.1.3 The limitation of ADC profile.


:3.2.4
:3.3 High
:3.3.1











:3.3.2 Multi-Conipartniental Models .. .. 45
:3.3.3 Deconvolution Approaches ..... .. . 46
:3.3.4 Model-independent Q-Space Imaging Approaches .. .. .. .. 46
:3.3.4.1 Diffusion spectrum imagfingf . . 47
:3.3.4.2 Q-hall imaging . ...... .. .. 48
:3.3.4.3 Diffusion orientation transform ... .. . .. 50
:3.4 Conclusion ......... .. .. 51

4 METHODS .... ._ ... .. 54

4.1 Some Mathematics on ~P, ......... ... .. 56
4.1.1 Measure and Integration on ~P, ..... .. . 56
4.1.2 The Laplace Transform on ~P, .... .. .. . 57
4.1.3 Wishart and Matrix-Variate Ganina Distributions .. .. .. .. 60
4.2 The Expected AIR Signal from Wishart Distributed Tensors .. .. .. 6:3
4.3 Methods for Multi-Fiber Reconstruction ... ... .. 68
4.3.1 The Mixture of Wisharts Model .... .. .. 68
4.3.2 A Unified Deconvolution Framework .... .. .. .. 70
4.4 Computational Issues ......... .... 74
4.4.1 Regularization and Stability . ..... .. 75
4.4.2 Nonnegativity and Sparsity Constraints .. .. .. .. 81
4.4.2.1 L1 nminintization methods .... .. .. 81
4.4.2.2 Non-negative least squares (NNLS) ... .. . .. 82

5 EXPERIMENTAL RESITLTS ......... ... .. 86

5.1 Simulations ......... .. .. 86
5.2 Real Data Experiments ......... .. .. 92

6 DISCUSSION AND CONCLUSIONS . ..... .. .. 101

6.1 Suninary ......... .. .. 101
6.2 Open Problems ......... . .. .. 102
6.2.1 Nonparanletric Inverse Laplace Transform ... .. .. 102
6.2.2 Adaptive Sparse Dictionary Learning .... .. .. 10:3
6.2.3 Sub-voxel Fiber Bundles Classification ... .. .. 104

REFERENCES .. .......... ........... 105

BIOGRAPHICAL SK(ETCH ..... ._. . .. 118










LIST OF TABLES


Table page

3-1 A list of diffusion anisotropic measures that can be derived from the eigfenvalues
of the diffusion tensor. (D) = trace(D)/3 = (At + X2 + 3) 3 is known as the
mean di~f: .It;, which indicates the average diffusivity over all directions. .. 41

4-1 A list of previously published fiber reconstruction methods expressed in the pro-
posed unified deconvolution framework. See text for meaning of symbols. Re-
produced from with permission. @[2007] IEEE. .. .. .. 72

5-1 Imaging parameters used for the optic chiasm dataset. ... .. .. 94

5-2 Imaging parameters used for the rat brain dataset. ... .. .. .. 98










LIST OF FIGURES


Figure page

2-1 The diffusion propagators for cases as the free diffusion are probability density
functions of Gaussian distributions. The diffusion coefficient and diffusion ten-
sor are closely related to the random displacement of particles. .. .. .. 25

3-1 An illustration of subvoxel fiber configurations arising from the intro-voxel ori-
entation heterogeneity (IVOH). ......... ... .. 42

3-2 From the diffusion data to orientation distribution function (ODF) via the spher-
ical random transform with q-ball imaging (QBI). ... ... .. 49

3-3 A schematic illustration of the diffusion orientation transform (DOT). .. .. 51

3-4 Various quantitative profiles derived from diffusion weighted signals simulated
from 1-fiber, 2-fiber, and 3-fiber geometries. ..... .. . 53

4-1 Plots of density functions of gamma distribution Y4,1 W.r.t the non-invariant
measure and scale-invariant measure. . ..... .. 63

4-2 The Wishart distributed tensors lead to a Rigaut-type signal decay. Reproduced
with permission from [75] @[2007] Elsevier. ..... .. . 65

4-3 Sphere tessellations using an icosahedron subdivision model with different itera-
tion numbers. ......... ... .. 69

4-4 The ill-conditioning problem in the spherical deconvolution approaches. Repro-
duced from [73] with permission. @2007 IEEE. .. .. 77

5-1 HARDI simulations of 1-, 2-, and 3-fibers (b = 1500s/mm2) VISualized in Q-ball
ODF surfaces. Reproduced with permission from [73] @2007 IEEE. .. .. .. 87

5-2 Results of w on 1-fiber HARDI simulation data using different deconvolution
methods. Reproduced from [73] with permission. @2007 IEEE. .. .. .. .. 89

5-3 Mean and standard deviation of (a) angular correlation coefficient and (b) er-
ror angles for the two-fiber simulation. Reproduced from [73] with permission.
@2007 IEEE. ........ . ... . 93

5-4 The statistics of angular correlation coefficients and error angles for the 2-fiber
simulation. Reproduced from [77] with permission. @2009 Springer. .. .. .. 93

5-5 Probability maps computed using (a) damped least squares with GCV; (b) Min-
Li with quadratic constraints (e = 1) initialized from (a); (c) damped least squares
with fixed regularization parameter (a~ = 0.6);( d) non-negative least squares
from a rat optic chiasm data set overlaid on axially oriented GA [114] maps.
Reproduced from [73] with permission. @2007 IEEE. .. .. .. 96









5-6 Probability surfaces computed from a rat optic chiasm image using (a) QBI-
ODF, (b) DOT, (c) MOW+Tikhonov regularization, and (d) MOW+NNLS.
Reproduced from [77] with permission. @2009 Springer. .. .. .. 97

5-7 Probability maps of coronally oriented GA images of a control and an epileptic
hippocampus. Reproduced with permission from [75] @2007 Elsevier. .. .. 100









LIST OF ABBREVIATIONS


ADC

CNS

DLS

DOT

DSI

DTI (DT-MRI)

DWI (DW-MRI)
EAP

FA

FOD

FT (FFT)
GA

GCV

GDTI

HARDI

IVOH

MLE

MRI

MOG

MOW

NNLS

NMR

ODF


apparent diffusion coefficient, or simply diffusivity, in units of m2 -1

central nervous system

damped least squares
diffusion orientation transform

diffusion spectrum imaging

diffusion tensor (magnetic resonance) imaging

diffusion weighted (magnetic resonance) imaging

ensemble average propagator

fractional anisotropy

fiber orientation distribution

(fast) Fourier transform

generalized anisotropy

generalized cross validation

generalized diffusion tensor imaging

high angular resolution diffusion imaging

intravoxel orientational heterogeneity

maximum likelihood estimate

magnetic resonance imaging
mixture of Gaussian distributions

mixture of Wishart distributions

non-negative least squares

nuclear magnetic resonance

orientation distribution function










PAS

PDF

PGSE

RBF

RF

SH (SHS, SHT)
SNR

SPD

SVD

TE

TR

QBI

QSI


persistent angular structure

spin displacement probability density function, probability profile

pulsed gradient spin echo
radial basis functions

radio frequency

spherical harmonics (series, transform)

signal-to-noise ratio

symmetric positive definite

singular value decomposition
echo time

repetition time

q-ball imaging

q-space imaging









LIST OF SYMBOLS


b-value, b = 22G( /) taeB

B-matrix, B m bggT

duration of diffusion gradient

Dirac delta function

time between diffusion gradients

(apparent) diffusion coefficient, or diffusivity

diffusivity profile as a function of directions g

(apparent) diffusion tensor

a prolate tensor with eigfenvalues At > X2 = 3 and dominant eigfenvec-
tor v

Fourier transform

inverse Fourier transform

gyromagnetic ratio

a Wishart (matrix-variate Gamma) distribution over ?P, with shape

parameter p > 0 and scale parameter C E 7P,

diffusion gradient

magnitude of diffusion gradient, G = |G|

direction of diffusion gradient, g = G/|G|

initial spin density

diffusion propagator giving the probability of a particle traveling from

position z to zt in the diffusion time t, cPDF

ensemble average propagator, PDF

probability profile, P(r) := P(r, t = -r)

the manifold of 3 x3 symmetric positive definite matrices


b

B



6(-)


D

D(g)
D

Dy












G

G





P(z/|x, t)



P(r, t)

P(r)










q displacement reciprocal vector, or wave vector, q = 7~6G



{qi}=1 a diffusion imaging scheme with NV different wave vectors sampled in q-space

r relative displacement

R3" 3-dimensional Euclidean space

S2 2-dimensional unit sphere in R3"

S(q) spin echo signal, diffusion MR signal

So spin echo signal in the absence of an applied diffusion gradient

S(q)/So diffusion MR signal attenuation

-r effective diffusion time -r = a 6/3

v a unit vector.

{v~K~ k Sphere tessellation containing K sample directions.

YEm spherical harmonics of degree I and order m

w w = {wkKIS a vector, of nonnegtive weights. i,.e., > 0.

(-) ensemble average









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MATHEMATICAL MODELING FOR MULTI-FIBER RECONSTRUCTION FROM
DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES

By

Bing Jian

August 2009

C'I ny~: Baba C. Vemuri
Major: Computer Engineering

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging

technique that allows neural tissue architecture to be probed at a microscopic scale in

vivo. By measuring quantitative data sensitive to the water molecular diffusion, DW-MRI

provides valuable information for neuronal connectivity inference and brain developmental

studies. The broad aim of this dissertation is to develop mathematical models and

computational tools for quantifying and extracting information contained in diffusion MR

images.

One of the fundamental problems in DW-MRI analysis is the mathematical modeling

of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In

this dissertation, we present a novel mathematical model and ....c.. I .nvII~ing efficient

algorithms for this problem. Our model uses a continuous probability distribution over

the space of symmetric positive definite matrices and is general enough to model water

molecular diffusion in a variety of situations involving complex tissue geometry including

single and multiple fiber bundle occurrences. We show that the diffusion MR signals

and the probability distributions for positive definite matrix-valued random variables

are related by the Laplace transform defined on the space of symmetric positive definite

(SPD) matrices. Another interesting observation is that when the mixing distribution

is parameterized by Wishart distributions, the resulting close form of Laplace transform

leads to a Rigaut-type fractal expression. This Rigaut-type function exhibits the expected










.I-i-up!lletic power-law behavior and has been phenomenologically used in the past to

explain the AIR signal decay but never with a rigorous niathentatical justification until the

development of the proposed model. Furthermore, both the traditional diffusion tensor

model and the niulti-tensor model can he interpreted as special cases of this continuous

mixture of tensors model.

In tackling the challenging problem of niulti-fiber reconstruction front the diffusion

AIR images, we further develop the mixture of Wisharts (\lOW) model, as a natural

paranietrization of the desired tensor distribution function, to describe complex tissue

structure involving multiple fiber populations. The niulti-fiber reconstruction using the

proposed MOW model essentially leads to an inverse problem. Computational methods

for solving this inverse problem are investigated under a unified deconvolution framework

which also includes several existing model-based approaches. Finally, the theoretical

framework we have developed for modeling and reconstruction of diffusion weighted AIRI

has been tested on simulated data and real rat brain data sets. The comparisons with

several competing methods enipirically -II---- -1 that the proposed model combined with

a non-negative least squares deconvolution method yields efficient and accurate solution

for the niulti-fiber reconstruction problem in the presence of intra voxel orientational

heterogeneity.









CHAPTER 1
INTRODUCTION

1.1 Motivation

The desire to discover the .Inr I~linin and functionality of the human body, especially

the brain structure and central nervous system (CNS), has been one of the driving forces

behind efforts to develop sophisticated medical imaging technologies. In the mid 1940s, a

breakthrough achievement was made involving the discovery of nuclear magnetic resonance

(NMR). This eventually led to the invention of magnetic resonance imaging (jl RI) whose

imaging capability due to the spatial localization of NMR signal was first demonstrated

and implemented in the 1970s. Since then, MR imaging has advanced tremendously

and become an indispensable diagnostic tools in modern medicine that is used everyday

for clinical and research applications. Built on technologies enabling the fast acquisition

time and high image quality, MR continues to pIIli .a important role in the diagnosis and

treatment of a number of diseases associated with the brain, heart, liver, and other organs

in the human body. Furthermore, "the ability to use MR imaging to noninvasively probe

the individual regions of the brain that control vision, sensation, motor function, memory,

language, and other processes has made this an extremely valuable modality to those

engaged in virtually any kind of brain-related research.[128]"

Among the many types of MRI modalities, diffusion Weighted MRI (Diffusion MRI,

DW-MRI or DWI) is a unique MRI technique that permits in-vivo measurement of the

diffusion of water molecules within tissue samples being imaged [88]. By exploiting the

sensitivity of the MR signal to the random motion of water molecules, diffusion MRI is

able to quantify different water diffusion characteristics in tissue samples locally. Because

these diffusion characteristics may be substantially altered by diseases, neurologic disor-

ders, and during neurodevelopment and aging, diffusion MRI is now recognized as a very



1 See [40] for a beautifully written history of NMR and MRI.










important clinical tool for brain-related diseases. For example, it has been successfully

applied to the evaluation of early ischemic stages of the brain [108]. Furthermore, the

directional dependence of water diffusion in fibrous tissues, like muscle [39] and white-

matter in the brain [107], provides an indirect but powerful means to probe the anisotropic

microstructure of these tissues. Like the idea of reconstruction the map of higfh--li in

a geographical region from the direction and frequency information of vehicle traffics in

that area, the motion of water molecules in neuro tissues can he potentially used to draw

inference about neuronal connections between different regions of the central nervous

system. Because of this powerful potential, diffusion AIRI has a distinct position in the

field of neuroscience research.

The broad aim of this dissertation is to develop mathematical models and computa-

tional tools for quantifying and extracting information contained in diffusion AIR images.

As opposed to a straightforward qualitative study, the process of producing accurate

quantitative results necessarily involves substantially more time and effort. However,

"the benefits of quantification are that fundamental research into biological changes in

diseases, and their response to potential treatments, can proceed in a more satisfactory

way. Problems of hias, reproducibility and interpretation are substantially reduced. [141]"

1.2 Main Contributions

The most significant original contributions of this dissertation are summarized

below. First, we present a novel mathematical model for diffusion-attenuated AIR signal

which involves a continuous probability distribution over the space of symmetric positive

definite matrices. This model is general enough to model water molecular diffusion in

a variety of situations involving complex tissue geometry including single and multiple

fiber bundle occurrences. We make the interesting observation that the diffusion AIR

signals and probability distributions for positive definite matrix-valued random variables

are related by Laplace transform defined on the space of symmetric positive definite

(SPD) matrices. We further show that in the case of Wishart distributions or mixtures










of Wishart distributions, a closed form expression for the Laplace transform exists and

can he used to derive a Rigaut-type .I-i-mptotic fractal law for the AIR signal decay

behavior which has been observed in the past [84] hut never with a rigorous mathematical

justification until now. This theoretical result depicts surprising consistency with the

experimental observations. Moreover, Diffusion Tensor Imaging (DTI), currently the most

widely used technique, can he viewed as a limiting case of the proposed model.

The measurements from diffusion AIRI provide unique clues for extracting orientation

information of brain white matter fibers and can he potentially used to infer the brain

connectivity in vivo using tractography techniques. Nowced .va- the widely used DTI tech-

nique fails to extract multiple fiber orientations in regions with complex microstructure.

In order to overcome this limitation of DTI, a vali'. iv of reconstruction algorithms have

been introduced in the recent past. One of the key ingredients in several model-based

approaches is deconvolution operation which is presented in a unified deconvolution

framework in this work. Additionally, some important computational issues in solving

the deconvolution problem that are not addressed adequately in previous studies are

described in detail here. Further, we investigate several deconvolution schemes towards

achieving stable, sparse, and accurate solution. Experimental results on both simulated

and real data are presented. The empirical comparisons -II---- -1 that non-negative least

squares method is the technique of choice for the multi-fiber reconstruction problem in the

presence of intra voxel orientational heterogeneity.

1.3 Outline

This dissertation is organized as follows:

ChI Ilpter 2 provides the background knowledge for understanding models and methods

used in diffusion magnetic resonance imaging. The first two sections of this chapter

briefly review hasic concepts in diffusion physics and nuclear magnetic resonance (NMR),

respectively. Then the key principles for measuring the diffusion phenomena using NAIR,










together with the fundamental relationship between the measured NMR signal and the

statistical properties of molecular diffusion, are presented.

('!s Ilter 3 reviews existing models and methods used in diffusion magnetic resonance

imaging. The first section in this chapter presents the classical Bloch-Torrey equation

and the Stei-l: I1-Tanner equation which are the foundations of the diffusion AIR imaging

modeling. The second section in this chapter introduces the now widely used diffusion

tensor imaging (DTI) method. The third section in this chapter is dedicated to the more

recent high angular resolution diffusion imaging (HARDI) methods with a focus on the

multi-fiber reconstruction techniques.

The first two sections in C'! Ilpter 4 present the technical details of the proposed

continuous tensor distribution model with a brief introduction to Laplace transforms

and Wishart distributions on the manifold of symmetric positive definite matrices. This

part builds the mathematical foundation of the proposed research. The last two sections

in C'!s Ilter 4 focus on the multiple fiber reconstruction from diffusion AIRI using the

proposed mixture of Wisharts model. In this chapter the proposed reconstruction method

and several existing approaches in literature put into a unified deconvolution framework.

Additionally, some important computational issues in solving the deconvolution problem

that are not addressed adequately in previous studies are described in detail here. Further,

we investigate several deconvolution schemes towards achieving stable, sparse, and

accurate solutions.

Experimental results on both simulations and real data are presented in ('! .pter

5. The comparisons with other approaches demonstrate the merits of the proposed

continuous tensor model together with the deconvolution reconstruction framework solved

using non-negative least squares method for the multi-fiber reconstruction problem in the

presence of intra voxel orientational heterogeneity. Finally in ('! .pter 6 we summarize

the main contributions of this dissertation and discuss a few open problems for further

research.










CHAPTER 2
DIFFUSION AIR FUNDAMENTALS BACKGROUND REVIEW

This chapter provides the background knowledge for understanding models and

methods used in diffusion magnetic resonance imaging. The first two sections of this

chapter briefly review hasic concepts in diffusion physics and nuclear magnetic resonance

(NMR), respectively. Then the key principles for measuring the diffusion phenomena using

NAIR, together with the fundamental relationship between the measured NMR signal and

the statistical properties of molecular diffusion, are presented.

2.1 The Basics of Diffusion Physics

On a macroscopic level, diffusion results from the microscopic random thermal

agitation of the particles in a medium. This phenomenon of random thermal agitation is

ubiquitous and known as "Brownian motion", named after the botanist Robert Brown

who first discovered and described it in 1827. In the context of this study, diffusion

refers specifically to the perpetual random translational motion of water molecules in any

part of a human or animal .Inr Irh ow.J Diffusion of water molecules in biological tissues,

observed as a macroscopic manifestation of Brownian motion, highly depends on many

factors including restrictions due to cell membranes, and a variety of microstructure

properties. The ability to measure the diffusion process of water molecules in tissues and

the understanding of how it is affected by these factors are extremely useful for studying

the biological microstructure.

It is natural to model the random motion of particles using a probabilistic framework.

A starting point of studying diffusion processes is the so-called self-diffusion propagator.

Generally, the propagator P,(z/|x, t) denotes the conditional probability density that a

particle initially located at position coordinate z moves to a position in a volume element

dz at zl after a time interval t.









The classical treatment of the diffusion propagator is to describe it as the Green's

function for the diffusion equation via the Fick's law


vls~= V D(VP,(2/|x, t)) (2-1)


subject to the initial condition


P,(z/|x, 0)) = 6(xt 2) (2-2)


and proper boundary conditions. In (2-1), D is the molecular self-diffusion tensor, a

physical quantity describing the diffusion processes in a more compact form.

In the case of unrestricted self-diffusion, also known as free diffusion, the boundary

condition of (2-1) is simply P,(z/|x, t) 0 as z/ c o which yields

1 (2t 2)TD-1(2t 2)
Psx/xt)=exp(- )(2-3)
zdet(D)(4xt) 4t

combined with the initial condition (2-2).

An interesting observation is that Ps in (2-3) depends only on the displacement r

but not the initial position 2, which reflects the Markov property of Brownian motion

statistics. Another important observation is that the self-diffusion tensor D is related to

the time dependence of covariance matrix (rrT) by


D = (rr T). (2-4)

On the basis of the central limit theorem, the above results may be derived either from a

formal Markov's method as in [33] or from an elementary random walk model as presented

in [55, pp. 325].
In an isotropic medium, the diffusion tensor D is proportional to an identity matrix.

Accordingly, the diffusion propagator (2-3) simplifies to

1 |xt |21
P(z/|x, t) = exp(- )(2-5)
(4xDL) 4Dt









where D is the self-diffusion coefficient given by the diagonal of the diffusion tensor D. In

this case, the corresponding diffusion equation via Fick's laws is


=DV2ps(z1/|x t) (2-6)

and we obtain

D = trace(D)/3 = ~(rTr) (2-7)

which agrees with the well known Einstein relation [51].

Because it is unrealistic to measure the random motion of a single particle, we shall

take an in,~ in!!. --ave I, ii view to depict the macroscopic behavior of a large number of

molecules on a statistical basis. Starting from the self-diffusion propagator, one can define

the total probability, W(zt, t) of finding a particle at position z/ at time t as


W~m/ t) W~, 0)s~x|x, )dz(2-8)


where W(2,0O) is just the initial particle density p(z). Using the notation of the displace-

ment vector, we can modify (2-8) and define the so-called ensemble average l".-y..;yl.>r

(EAP), also known as the displacement ly .-?..:?-9i;, distribution function (PDF), by inte-

grating the diffusion propagator over all initial positions [29, 79]


P~ ) P~x+rlz, t)p(z)dz (2-9)


where p(z) is the initial particle density. As we shall see later, this ensemble averaged

propagator which cumulates all the microscopic contributions distributed in a voxel at the

macroscopic scale may be measured directly by NMR. This is the fundamental principle

exploited in the diffusion MR and it will be the topic investigated later in this chapter.

In cases such as the free diffusion where P,(z + rlz, t) is independent of the initial

position 2, the ensemble average propagator is also a Gaussian

1 rTD-lr
P(r, t) =exp(- ). (2-10)
d et(D 4x) 41










The diffusion propagators for the free diffusion as well as the corresponding diffusion

coefficient and the diffusion tensor are summarized in Fig. 2-1.

Diffusion tensor =-DP)1rD^ r
SP(r, t) = exp(- )
D = (rr T dtD(4) 4t

isotropic isotropic

Diffusion coefficient =D~V2P 1 |7| T
,, P(r, t) = exp-
D = (r Tr) 4Dtt

Figure 2-1. The diffusion propagators for cases as the free diffusion are probability density
functions of Gaussian distributions. The diffusion coefficient and diffusion
tensor are closely related to the random displacement of particles.


It is worth noting that the diffusion tensor framework and the Gaussian propagators

described above are derived from the free diffusion which only represents a very limited

class of diffusion phenomena. Commonly observed are a vast range of phenomena, such

as restriction, heterogeneity, anomalous diffusion, finite boundary permeability. Generally,

the diffusion equations are imposed with nontrivial boundary conditions depending on

the nature of confining geometries and other physical properties. Solutions to diffusion

equations in a nr1I~1ii of simple geometries are available in [44]. It has been shown that

if the diffusion boundaries are closed, unrelaxing, and completely impermeable, then in

the long run (long time of diffusion), the propagator will assume the shape of the space

accessible to particles [29, 150]. Even though the limiting conditions are rarely satisfied in

biological applications, the diffusion-structure relations nevertheless provide useful clues

for probing microstructures, which has fostered a vast promising research field.

2.2 The Basics of Nuclear Magnetic Resonance

In this section, we present the basic physical principles of nuclear magnetic resonance

(NMR). The main references we consult are [29, 92, 94, 133].










2.2.1 Dynamics of Nuclear Spins

The fundamental concept of NMR is the spin aq,:(;,lar momentum, an intrinsic prop-

erty possessed by certain atomic nuclei. In atomic physics, the spin angular momentum p

of an atomic nucleus is quantized by the nuclear spin quantum number I according to the

formula

2x1p| ( (2-11)

where & is the Planck's constant. The spins possessed by an atomic nucleus are combined

by the spins of the protons and the neutrons inside the nucleus. Since the spins of protons

and neutrons may interact in various configurations, e.g., parallel or anti-parallel to each

other, the value of nucleus spin quantum number I is chosen as the one in the lowest

energy nuclear state, also called the iI,. ;,,../ state nuclear spin. Depending on the nuclear

structure, the number I can only be an integer, half-integer, or zero with the following

rules[92]: (1) if the numbers of protons and neutrons are both even, the ground state

nuclear spin I is zero; (2) if the numbers of protons and neutrons are both odd, the

ground state nuclear spin I is a positive integer; (3) if the total number of protons and

neutrons is odd, then I is given by a half-integer. Note that the nucleus of 1H, which is

most abundant in nature and human body, contains a single proton, hence its nucleus

quantum spin number I is 1/2; while the nuclei 12C 160, and 56Fe all have a ground state

spin I = 0. A complete listing of nuclear isotopes and their NMR properties may be found

at http: //www web elements com.

Another intrinsic property possessed by fundamental particles is the magnetic

moment. The spin angular momentum p and the magnetic moment p are proportional to

each other as given by

I- = 7p (2-12)

where y is called the ~i;, tion 7,: It.: H: ratio, a characteristic value of a particular atomic

nucleus. According to the classical mechanical model, when a nucleus with spin angular

momentum is subject to an external magnetic field, the interaction between the magnetic









moment p and the field Bo will generate a torque force L trying to align the two:


L = dp/dt = p x Bo.


The outcome of this torque force is the precessional motion of the nucleus, in which the

magnetic moment vector p rotates around the direction of the external magnetic field

Bo in addition to the spinning motion of the nucleus around its own axis. The equation

describing the precession is


dp/ldt = ydp/dt = yL = wox p- (2-13)


where wo = -7Bo is the so-called Larmor /, ~i;, ,: ;t of precession. Note that the minus

sign in (2-13) implies that the rotation obeys the left-hand rule.

According to the quantum mechanical model, the nuclear magnetic moment p can

only have 2I + 1 orientations in a magnetic field Bo, corresponding to 2I + 1 energy levels:


E = -p Bo. (2-14)


Here the minus sign in (2-14) indicates that the magnetic energy is lowest when p is

parallel to the Bo field while the magnetic energy is highest when p is anti-parallel

(spin-down) to the external field Bo. The difference between the two energy levels is

proportional to the strength of the applied field Bo:


AE = yBo. (2-15)


With I = 1/2, the 1H nucleus can either align with (spin-up) or against (spin-down) the

applied field, corresponding to low and high energy states respectively. When an ensemble

of nuclei are immersed in an external magnetic field, the distribution of the allowed

orientations is described by Boltzmann statistics. The resulting macroscopic magnetization









vector M~ can be defined as the vector sum of all the microscopic magnetic moments


M = pi (2-16)
i= 1

where ps is the individual magnetic moment of the i-th nuclear spin, and NV is the total

number of spins. At equilibrium, two macroscopic effects can be observed: (1) the overall

transverse component of M~, i.e., the component perpendicular to the applied magnetic

field direction, is zero because of the random phases introduced by the processing magnetic

moments, (2) more nuclei relax into the parallel orientation (lower energy state) than the

antiparallel orientation (high energy state), resulting in a bulk magnetization vector M~.

The direction of this bulk magnetization vector M~ is aligned with the applied magnetic

field direction, and its strength depends on the proton 1 1

2.2.2 Magnetic Resonance, Relaxations, and Bloch Equations

The principle of NMR is that the alignment of nuclear spins can be perturbed by

applying a circularly polarized magnetic field B1, called a radio fr .;, e.'.;i excitation

pulse, which is rotating about Bo at the Larmour frequency wo. A typical B1 field is

perpendicular to Bo and its generation takes the following form:


Bl(t) = Bl(t)e-i("Ot+4) (2-17)


where 4 is the initial phase angle and can be assumed to be zero as it has no significant

effect on the excitation result. The excitation property of an RF pulse is specified by the

shape and duration of the envelop function Bl(t).Two most widely used RF pulses are the

r Litty(;J.;<, pulse and the since pulse.




1 The term RF pulse is so called because wo/2xr is normally in the frequency range
of radio waves. The field usually lasts for a few microseconds or milliseconds. Also, the
strength of the field B1 is much weaker than the static magnetic field Bo.









On the microscopic level, the energy absorbed by nuclei during the RF excitation

makes some nuclear spins jump from the lower energy state to the higher energy state and

hence changes the overall magnetization vector M~. Although the microscopic behaviors

of individual spin magnetic moments are quantized, the observed bulk magnetization as a

macroscopic manifestation may be treated classically. The excitation induced by B1 tips

the bulk magnetization vector M~ away from the direction of Bo by a spiral precession

around Bo described by the following equation


dM/ldt = wox M~ = ylM x Bo. (2-18)


Note that the processing magnetization creates a periodically changing magnetic field in

the transverse plane. This effect may be detected and measured by a closely placed wire

coil according to Faradovl~i's law of induction. The resulting signal then can be exploited in

NMR spectroscopy and magnetic resonance imaging.

At the time when the RF pulse is turned off, M~ makes an angle a, called flip nl

with the static field Bo. In general, the flip angle a~ depends on the strength By (t) and the

duration -r of the applied RF:



~o (2-19)
= B1-r (for a constant B1 amplitude).

Because the amplitude of the alternating voltage induced in the receiver coil is propor-

tional to the transverse magnetization component, a 900 pulse is commonly used as an

excitation pulse as it produces a maximum transverse magfnetization component equal to

the equilibrium magfnetization.

Once the RF pulse is removed, the nuclear spins will tend to restore the equilibrium

energy levels distribution by releasing excess energy into the surroundings. This process is

called relaxation and has two-fold effects: (1) the transverse magnetization component My,

decays to zero exponentially, which is referred as spin-spin relaxation, (2) the longitudinal









magnetization component 14 returns to the equilibrium value ii1, gradually, which is

referred as spin-lattice relaxation.

These two types of relaxations are phenomenologically described by the following

equations


dt T~
1 ~(2-20)

dt T2
with solutions


(2-21)
It) = 1Av(0) exp(-t/T2)

where the parameters Ti and T2 are known as lory~ll.:;J~.:..al relaxation time and the trans-

verse relaxation time, respectively. Note that both Ti and T2 depend on the molecular

environment and thus can be used to characterize different samples.

In the rotating reference frame where the x-y axis keeps rotating with the precession

frequency wo, the phenomenological descriptions of spin-lattice and spin-spin relaxation

can be combined together into a system known as the Bloch equations [27]

dM~(t)
= TM(t) x B(t) R(M~(t) MI,) (2-22)

where B(t) includes both the static field and the RF pulse, i.e., B(t) = Bo + Bl(t), and R

is the relaxation matrix:


R = 0 /T2 0(2-23)



and the vector M.i = (0, 0, il,)T. The Bloch equations provide a valuable reference

in describing macroscopic phenomena in NMR imaging. Note that if there were no

relaxation, i.e., both Ti and T2 were approaching infinity, the Bloch equations simplify

to Eq. (2-18), as expected. With the relaxation effect after the single RF pulse B1,









the subsequent NMR signal induced by the (. 1;11 Ir;1). decaying magnetization in free

precession is then detected in the time domain. It is therefore known as the free induction

I/ .,ru (FID). The measured signals are conveniently represented by complex numbers

where the real part and the imaginary part correspond to the x-direction and the y-

direction in the rotating frame respectively. By Fourier transformation the signals may be

represented in the frequency domain. For a detailed discussion, the reader is referred to

[29].

It is important to understand that the Bloch equations (2-22) only describe the

dynamics of the magnetization under the effects of spin-lattice and spin-spin relaxation in

an NMR experiment. By introducing a diffusion term, Torrey modified the original Bloch

equations to reflect the effects of molecular diffusion. The modified equations are known

as the Bloch-Torrey equations [142] and plI li- a fundamental role in modeling the diffusion

imaging data that we will discuss later.

2.3 Measuring Diffusion using NMR

2.3.1 Spin Echo and Diffusion Effects in NMR

One assumption of the Bloch equations is the homogeneity of the magnetic field Bo

in the absence of B1, which is never true in practice. The variations in the local magnetic

field contribute to the gradual reduction of the transverse magfnetization in addition to

the spin-spin relaxation. Consequently, May decays more rapidly than the exponential

decay May = il.*:-tl/ predicted by the Bloch equations. However, if a 1800 pulse is

applied at time t following the initial 900 pulse, then the additional decay due to the field

inhomogeneity can be reversed at time t after the application of that 1800 pulse. This

phenomenon of the signal restoration is called spin echo. The time between the initial 900

pulse and the echo formation is called TE (echo time) which is twice the time interval

between the two RF pulses.

Another factor affecting the signal reduction is the random thermal motion of the

spins, or the molecular diffusion at the macroscopic level. Simply speaking, the molecular










diffusion plus spatial variation of the magnetic field will induce a phase distribution in the

transverse plane and hence result in signal attenuation. This signal attenuation is closely

related to the amplitude of the spin displacements caused by the diffusion. Intuitively,

the faster the diffusion, the more random the phases distributed, and a larger signal

attenuation will be observed. This is the exact idea used in the classical S' i-. H:.-Tanner

pulse-gradient spin-echo (PGSE) experiment [138], the first NMR experiment specifically

designed for quantitatively measuring diffusion in a sample.

In the standard PGSE pulse sequence for diffusion in.! I_;oy a pair of two identical

field gradients are placed before and after the 180" pulse, to perform the "diffusion

encodingt In the remainder of this dissertation, these two gradients are denoted by G

with strength G = |G| and direction g = G/G. Each gradient pulse will last a time 6, and

the pair is separated by a time a between the leading edges of the two gradient pulses.

Because that the 180" pulse only cancels the phase shift of spins induced hv the first field

gradient, the final detected signal will take into account the phase shift of spins induced

by the molecular diffusion during the time period a which separates the two gradient

pulses. Note that for a complete AIR image formation the applied diffusion gradients must

he combined with a sequence of .slice .selection, S,. it;,. m e; encoding, and phase encoding for

spatial localization of AIR signals. However, the details of these so-called k-space sampling

techniques are beyond the scope of this dissertation and we refer the reader to [94].

2.3.2 The Fourier Transform Relationship

Let 1 be the time when the first gradient is applied, the phase shift #1 of the spin

transverse magnetization induced by this gradient pulse with duration 6 is given by


I1 = ]* GTzdt = q*6GTz (2-24)


where ]* is the gyro-magnetic ratio and z is the spin position supposed to be constant

under the narrow pulse assumption (6
gradient pulse is applied, suppose the spin moves to a different position a + r, then the net









phase-shift induced by this pair of gradients will be:


S= y6GTr. (2-25)

Note that if spins were stationary, i.e. r = 0, a perfectly refocused echo will occur.

The NMR signal measurement is proportional to the total transverse magfnetization

My, from a very large number of spins. Consider the millimetric resolution provided by
NMR devices and the micro-metric scale of the molecular diffusion, it is reasonable to

neglect the inter-voxel diffusion effects on a voxel scale. Let E(G, a) denote the amplitude
of the "echo signal", then it can be expressed as the ensemble average


E(G, a) =(exp(i )) = (exp(iyGGTr)). (2-26)

It is important to note that here E(G, a) is 1...)~ II!. .1. ii so that E(0, a) = 1, in other

words, E(G, a) should be considered as the signal decay induced only by the molecular

diffusion but excluding the attenuation of the echo due to the T2 T68XatiiOn.

The net phase distribution that weights the ensemble average of individual phase term

exp(iyGGTT) iS the probability for a spin to travel from z to z + r during time a. Note
that this probability is precisely p(z)P,(z + r)|x, a) where p(z) is the spin density and
Ps (x + | t) is the, diffusion, propagator westudied~ in the,,,;, prviussetio.Inetngti

probability into (2-26) and using the substitution zt = z + r, we obtain

E(G, A) = psP~ ~, x~gG~t-x)md

= ( p(ms: D~~)i)Psl(/7 + ~r~x~sc) epi)Trt-x) (2-27)

= P(r, A\) exp)(6igbGl))dr

where P(r, t) is exactly the ensemble average propagator (EAP) defined in Eq. (2-9).

By introducing a displacement reciprocal vector, q, defined as


q = 6G, (2-28)
2xr









we can now rewrite (2-27) in a more so-----~ -lHi.- form


E(q, a) = ~ ,A)epi7rT~ -1[P(r, A). (2-29)


As clearly expressed in (2-29), there is a simple Fourier reciprocal relationship between the

spin echo signal E(q, a) and the EAP P(r, a). This Fourier relationship is fundamental

in the direct reconstruction of the EAP P(r, a). The space of all possible q vectors is

called q-space. By varying either of diffusion gradient G or the gradient duration 6, the

signal can he measured at many sampled points in q-space and the EAP P(r, a) can he

obtained by taking an inverse Fourier transform of (2-29)


P(r, a) = FT[E(q, a)] = F[S(q)/So]. (2-30)


Employing Eq. (2-30) to calculate the EAP is called "q--p. .. analysis[29, 42]. Since

in practice, the diffusion time a is treated as an experimental constant, the notation

P(r) = P(r, a), as the displacement probability distribution function (PDF) defined on

RW", is the central mathematical quantity being studied in this dissertation.









CHAPTER 3
DIFFUSION MR MODELING AND DIFFUSION PROPAGATOR RECONSTRUCTION
-CLASSICS AND THE STATE OF THE ART

In this chapter, existing models and methods used in diffusion magnetic resonance

imaging are reviewed. The first section presents the classical Bloch-Torrey equation and

the Stei-l: I1-Tanner equation which are the foundations of the diffusion MR imaging

modeling. The second section introduces the now widely used diffusion tensor imaging

(DTI) method. The third section is dedicated to the more recent high angular resolution

diffusion imaging (HARDI) methods with a focus on the multi-fiber reconstruction

techniques. For published review articles on these topics, we refer the reader to [2, 89,

104].

3.1 From the Bloch-Torrey Equation to the Stejskal-Tanner Equation

For free diffusion in an isotropic medium, combining Eqs. (2-5) and (2-27) yields


S(q)/So = exp (-(y6G)2DA). (3-1)

There are two problems associated with this simple model. First, in commercial MRI

units, long gradient durations 6 are required to produce observable dephasing effects,

hence diffusion occurring during the application of the gradient pulses may not be ignored.

Second, it does not take into account the effects induced by other additional gradient

pulses, including imaging pulses and background residual gradients [24]. To address these

issues, we resort to the following Bloch-Torrey equation




which adds a free diffusion term to the original Bloch equations (2-22). Detailed math-

ematical derivations [24, 113, 150] show that in infinite and homogeneous media the

solution to (3-2) with a spin-echo sequence is given by


dL,(t) = !1,(0) exp(- k(u) TDk(u)dzL) (3-3)









where

k(t) = 7 G(u)du. (3-4)

Note that the sign of G in Eq. 3-4 has to be inverted for all gradient pulses following the

refocusing 1800 RF pulse.

Furthermore, for an isotropic medium, the signal decay at echo time TE in a spin-

echo experiment is


,(E)= ,() xp-D k(t)Tk(t~dt). (3-5)
12,ti.) IIY(I)C*0/TET

By substituting the so-called ,9 II! 1.0~ factor", b-value, defined as in [88]

0/TE


into (3-3), we obtain a simpler expression for the signal attenuation


14,(TE) = 14,(0) exp(-bD) (3-7)

which is strictly valid for diffusion in unrestricted, homogenous, and isotropic media. Note

that the b-value is a useful quantity that characterizes the sensitivity of NMR sequences to

diffusion [24].
For the most widely used PGSE sequence [138], an exact solution to Eq. (3-5) exists

and the corresponding signal attenuation is given by

S(q) = So exp(-72 2G2(A 6/3)D) = So exp(-bD) (3-8)

where q = 276G and

b=4xr2 q2( 2/) y2G2 _9).

The time constant (A 6/3) in Eq. (3-8) is therefore known as the effective diffusion time

where the 6/3 correction accounts for the diffusion that occurs during the application of

gradient pulses.









It is important to note that the S' i- 1-1 .-Tanner equation (:38) is derived from

the Bloch-Torrey equation (:32) which is a microscopic, free-diffusion physical model;

while the observed diffusion AIR signal measured at a millimetric voxel is an ensembled

contribution from all the microscopic displacement distribution of the water molecules

in this voxel, as shown in (2-27). To reflect and partially bridge the gap between these

two scales, one can consider replacing the physical diffusion coefficient, D, with its global,

statistical counterpart, Do>,, termed the apparent diffusion coefficient (ADC) [88], in the

Steil1: I1-Tanner equation, which now reads


S(q) = So exp(-bDo;;). (:310)


Although the concept of ADC requires some intra-voxel homogeneity assumption, ADC

has been largely used in the literature since it was introduced. [87]

The Steil1: I1-Tanner equation (:38) also assumes free diffusion as no boundary

conditions were imposed. However, the water molecular diffusion in tissues is obviously

both hindered and restricted. This fact inevitably leads to a deviation of the observed

signal from the monoexponential behavior implied by Eq. (:310).

More sophisticated models, such as the hiexponential model[:38, 111], the Laplace

transform model [164], the Rigaut-type .I-i-mptotic fractal expression [129, 1:30], and the

stretched exponential model [22], have been investigated in literature to depict the non-

exponential signal decay. A brief discussion on these models with comments can he found

in [11:3, Ch.2].

3.2 Diffusion Tensor Imaging

The scalar Steil1: I1-Tanner equation (:38) depicts the dependence of the diffusion AIR

signal on the gradient strength, however it can not describe the diffusion anisotropy which

is observed in fibrous biological tissues, for example, in muscles [:39], the spinal cord [108],

and the brain white matter [:37]. Diffusion tensor AIRI (DT-AIRI or DTI), introduced by

Basser et al. [17, 18], provides a relatively simple way of quantifying diffusion anisotropy









as well as extracting fiber directions locally from multidirectional diffusion MRI data.

The present section provides a brief description of DTI, more complete treatments can be
found in recent review articles on DTI, for example, [25, 83, 120].

3.2.1 Tensorial Stejskal-Tanner Equation

In Section 3.1, the solution to the Bloch-Torrey equation with an anisotropic diffusion

tensor term (3-2) for free diffusion in homogeneous media measured by a spin-echo

sequence is given by


i,(t) = !I,(0) exp(- k(u)TDk(u)dzL) (3-11)

where

k~t) 7 G~~da.(3-12)

Introducing the matrix

B =k(t~k(t)Tdt (3-13)
0/~TE
which is referred to as the B-matrix [18], we obtain the tensorial Stei-l: I1-Tanner relation:


S(q) = So exp (-trace(BD)) (3-14)

which is the foundation of the diffusion tensor imaging (DTI). Note that to be meaningful

at the voxel scale, the D in (3-14) should be considered as an apparent diffusion tensor

Day, different from the original microscopical diffusion tensor in (3-2) based on the same
argument provided for apparent diffusion coefficient (ADC) in Section 3.1.

Let g be the unit vector representing the direction of the diffusion gradient, i.e.,
G = Gg, the B-matrix can be quite accurately approximated by a pure outer product of

the gradient direction scaled by the b-value, i.e.,

B mbggT (315)

where b = y2 2G2( 6 3) aS defined in Eq. (3-9).









Defining the apparent diffusion coefficient D(g) along a certain direction g


D(g) = gTDg (3-16)

reveals the connection between the two versions of Steil1: I1-Tanner equation as follows

S(G) = S(Gg) = So exp(-bD(g)) = So exp(-bgTDg) or
(3-17)
S(q) = So exp(-74xr2 TDq)

where -r = a 6/3 is the effective diffusion time and q = 276G is the displacement

reciprocal vector.

Substituting S(q)/So from (3-17) into the Fourier relationship (2-29) leads to the

Gaussian propagator
1 -rTD-lr
P(r, a) =exp[ ]. (3-18)
|/D|(4xA)3) "
The comparison of (3-18) and the previous result in (2-3) demonstrates the consistency of

the Steil1: I1-Tanner equation and the diffusion propagator formalism [14].

3.2.2 Estimation of the Diffusion Tensor

Taking the natural logarithm at both sides of Eq. (3-14) yields the simple relation


trace(BD) log(So) = log(S(q)). (3-19)


The left-hand side of Eq. (3-19) is linear with respect to D and log(So), therefore, the

tensor components are obtained by solving a linear system of equations formed by stacking

(3-19) with a set of measurements S(q) and corresponding B-matrices.

It is worth noting that the linear system approach is equivalent to maximum likeli-

hood estimate (j!1.11) when assuming the Gaussian noise-model on the log-transformed

signals. However, in practice, the more realistic noise model should be Rayleigh dis-

tributed rather than Gaussian distributed. Consequently, the Gaussian MLE will be biased

if high q-values are used since Gaussians are no longer good approximation to Rayleigh










distributions at the tails. For more detailed discussion on modeling noise in diffusion 1\RI

data, see [1:36].

Since there are six independent elements in diffusion tensor, image acquisitions

from at least six linearly independent directions, together with a reference image So, are

required. Usually, a more reliable measurement of the diffusion tensor requires more than

six directions. Even though a diffusion tensor should be positive definite theoretically, the

positivity of the estimated diffusion tensor may be destroyed by the signal noise, therefore,

one might consider methods that can preserve the positivity of the estimated tensors, for

example, [158].

3.2.3 Fiber Orientation and Anisotropy Measures Derived from the Diffusion
Tensor

Represented by a :3 x :3 symmetric matrix, the diffusion tensor can he decomposed

into:

D = &QAQT (:320)

where Q = (ele283) is an orthogonal matrix of eigenvectors and A = diag(Ay, X~, 3S) is a

diagonal matrix of real eigenvalues ordered by At > Xa2 X3-

Intuitively, the fiber orientation can he estimated by taking the directions along the

peaks of the probability profile given in Eq. (:318). Finding the peaks of the probability

profile in Eq. (:318) is equivalent to finding the minima of the quadratic form: rTD-lr

for r E S2. It iS easy to show that TTD-lr has a uique minimum when r takes the

direction of the dominant eigfenvector, el, provided that X1 > X2 3.XS In this case the

ellipsoids generated from the isosurfaces:


rTD-lr = C


also align to this dominant direction, hence they are widely used to visualize the estimated

diffusion tensors.









Table 3-1. A list of diffusion anisotropic measures that can be derived from the eigfenvalues
of the diffusion tensor. (D) = trace(D)/3 = (At + X2 + 3) 3 is known as the
mean di~ffr.: .It;, which indicates the average diffusivity over all directions.
Name Definition

Relative Anisotropy (RA) C ~(As (D))

Frac~tional Anisotfropy~~ (F) 2 1(s- D)
Volume Ratio X1 2 3
(D) 3
Prolateness Metric X1-X
3 (D)
Oblateness Metric 2(Ag-X3)
3 (D)
Sphereness Metric


While the largest eigenvalue and its corresponding eigenvector of the diffusion tensor

describe the quantity and direction of the principal diffusion, its eigenvalues are also

exploited to derive some scalar measures that are meaningful for studying the nature of

diffusion anisotropy exhibited in the tissue of interest. Table 3-1 lists several anisotropy

metrics that have been studied in the literature [15, 162].

Note that all metrics defined in Table 3-1 are dimensionless and rotation-invariant.

Among these metrics, the fractional anisotropy (FA) has been most widely used because

it is relatively insensitive to noise [15] and does not require the sorting of the eigenvalues.

Additionally, FA is automatically normalized to the unit interval, FA takes the value of 0

when the diffusion tensor is totally isotropic, i.e., At = X2 3 X, Whereas it takes the value

of 1 when the diffusion tensor is extremely anisotropic, i.e., X2 = 3 = 0.

Anisotropy metrics are quite useful in quantitatively assessing the orientational

coherence of the diffusion compartments within a voxel [120]. For example, the high FA

values are typically associated with strongly aligned fibers such as axons in white matter,

while the FA values are expected to be relatively low in regions of fiber intersections or in

dense tissues where diffusion is restricted equally in all directions.

3.2.4 Problems of the Diffusion Tensor Imaging

As previously mentioned, the diffusion tensor framework can only describe a very

limited class of diffusion phenomena. Although promising results have been achieved using









DTI to study regions of the brain and spinal cord with significant white-matter coherence

and to map anatomical connections in the central nervous system [19, 41, 105, 106, 156]

Sthe major drawback of diffusion tensor MRI is that it can only reveal a single fiber

orientation per voxel and fails in regions where fiber populations cross, kiss, splay, branch

or twist as shown in illustrated in Fig. 3-1. In a recent study [21], it was estimated that

this intra-voxel orientational heterogeneity (IVOH) problem affects one third of white

matter voxels. In those regions, tractography applications based on the diffusion tensor

model may result in artefactual reconstructions of pathi- .va~ [19]. In recent years, high

angular resolution diffusion imaging (HARDI) techniques have been able to address the

challenges inherent in diffusion tensor imaging and will be the topic of the next sections.













(a) kissing fibers (b) crossing fibers (c) splaying fibers

Figure 3-1. An illustration of subvoxel fiber configurations arising from the intro-voxel
orientation heterogeneity (IVOH).


3.3 High Angular Resolution Diffusion Imaging (HARDI)

Despite the promising results achieved by the diffusion tensor it! s. ./). the diffusion

tensor model is known to be inadequate for resolving complex neural architectures,

particularly in regions with complicated intra-voxel fiber patterns [4, 56, 152, 154].

This limitation of the diffusion tensor model has stimulated exploration of both more

demanding image acquisition strategies and more sophisticated reconstruction methods.

Tuch et al. [151] developed a clinically feasible approach in which apparent diffusion

coefficients are measured along many directions distributed almost isotropically on a










spherical shell in the diffusion wave vector space. Since then, many methods that augment

the angular resolution of the diffusion model have been used in the literature and are

commonly referred to as high r,:ll;larr resolution diffusion :l,,nryl.e..y (HARDI).

3.3.1 Modeling Diffusivity Profiles

In the original HARDI method [151], with diffusion gradients applied along many

directions, the diffusivity profile is calculated by using the scalar version of the Steil1: I1-

Tanner equation (:38) along each direction but does not assume any particular model. In

diffusion tensor in.! I_;by the diffusivity profile is assumed to take the form of Eq. (:316)

which is intrinsically a quadratic model. Since it has been shown that the diffusivity

function exhibits complex local geometry in voxels with orientational heterogeneity

(IVOH) [152, 154] and the diffusion tensor model is inadequate in such situations, various

higher order models have been proposed to approximate the underlying diffusivity profile.

3.3.1.1 Spherical harmonics series

Fr-ank [56] introduced the use of spherical harmonics (SH) series [60] to model the

diffusivity profile
00 1

l=0 z= -1

In (:321) the coefficients airn can he calculated using the spherical harmonics transform

(SHT)

at,>= Dg)*>gd. (:322)

The spherical harmonics series (SHS) in (:321) is truncated so that only the most sig-

nificant terms are included in the expansion. Furthermore this SHS should only include

even-order spherical harmonics due to the positivity (D(g) > 0) and antipodal symmetry

(D(g) = D(-g)) of diffusivity profile. Instead of directly applying SHT, Alexander et al.

[4] used linear regression to estimate air, and also -II_0-r-- -1. a hypothesis testing method

to determine up to which order the SHS should be truncated. If the highest order of SHS

is L, then the number of aine for all even I from 0 to L is (L + 1)(L + 2)/2, which reduces










to 6 when L = 2. Descoteaux et al. [47] emploi- a the real-valued spherical harmonics basis

and proposed to regularize spherical harmonics coefficients using the Laplace-Beltrami

operator.

3.3.1.2 Generalized diffusion tensor imaging

Starting with an extension to the Bloch-Torrey equation, Ojzarslan and Mareci [114]

proposed to use Cartesian tensors of rank higher than 2 to model the measured diffusion

coefficients
33 3
D)(g) = o?- D9ixi..izixi 9 (3-23)
ii=0 iz=0 i3=0
where Dixi,...i, are the components of the Cartesian, rank-1 tensor. To ensure the positivity

and antipodal symmetry of D(g), I must be an even number and Diiz...il have to be

realized as a totally symmetric tensor which contains (l+ 1)(l+ 2)/2 independent elements.

In their work [114], the correspondence between the coefficients in the SHS and Cartesian

tensors of higher ranks was derived as well, which reveals the equivalence between these

two approaches. To generalize the anisotropy measures for DTI, Ojzarslan et al. [117]

also proposed scalar measures in terms of variance or entropy derived from the higher

order tensor coefficients. Recently, Barmpoutis et al. [10] represent a 4th-order tensor as

a homogeneous polynomial of degree 4 in 3 variables, the so-called ternary quartic, and

then impose the positivity of the 4th-order tensors in the estimation from diffusion MRI

data based on the Hilbert's theorem which states that any non-negative ternary quartic

can be expressed as a sum of squares of three quadratic forms. Barmpoutis et al. [12]

further introduced a novel parametrization of the 4th-order tensors for the simultaneous

estimation and regularization of the 4th-order field while preserving the positivity of the

estimated tensors.

It is worth noting that another formulation of generalized diffusion tensor imaging

was independently proposed by [96, 97] where the diffusion process is quantitatively

characterized by a series of diffusion tensors with increasing orders. Similarly, diffusion









kurtosis imaging (DK(I) has also been proposed to characterize the non-Gaussian property

of water diffusion by a so-called diffusion kurtosis '~~-~ is- [71, 86, 98].

3.3.1.3 The limitation of ADC profile

Although the measured diffusivity file can be used to indicate the complexity of

the fiber structure within the voxel, it is important to point out that the maxima of

the diffusivity profile may not correspond to the underlying distinct fiber orientations.

von dem Hagen and Henkelman [154] observed that the peak of the ADC profile measured

from a voxel containing two perpendicular fibers occurs at an angle in the middle of the

two fibers but not in the direction of either. Similar observation was also reported by Tuch

et al. [152], Zhan and Yang [165] in vivo. Due to this fact, the diffusivity profile can not

be used directly for extracting fiber orientations, and one might still need to investigate

the average diffusion propagator by taking the Fourier transform of the signal attenuation

implied by the diffusivity profile as done in [115, 119].

3.3.2 M ult i- C ompart me ntal M models

A direct extension of the DTI model proposed by Tuch et al. [152] assumes that the

diffusion propagator takes a form of Gaussian mixture densities. Under this assumption,

the signal can be modeled as a finite mixture of Gaussians as well:


Scq) = So my~ exp(-bgTDyg) (3-24)


where my is the apparent volume fraction of the compartment with diffusion tensor Dj.

Behrens et al. [20] introduced a simple partial volume model where the diffusion sig-

nal is expressed as the combination of an infinitely anisotropic component and an isotropic

component. A B li-. -1 Ia inference is then used to estimate the model parameters. This

partial volume model was further extended in [21, 67] to allow the estimation of multiple

fiber orientations. However, both extensions require complicated solution techniques to

address the model selection problem properly, for example, the Markov C'I I!1, Monte Carlo










(\!C\!lC) eI~! 1i--; used in [67] and the automatic relevance determination (ARD) used in

[21].

A slightly more complex multi-compartment model, called composite and hin-

dered restricted model of diffusion (CHAR MED) was described in [7, 8]. Similar to the

multi-Gaussian model, CHARMED also interprets the signal as a weighted sum of the

contributions from a highly restricted compartment and a hindered compartment. Note

that the diffusion in the hindered compartment is approximated by a Gaussian while the

diffusion in the restricted compartment is described by a Neuman's model for restricted

diffusion in a cylinder 110 .

3.3.3 Deconvolution Approaches

1 To avoid determining the number of components in the modeling stage and possible

instabilities associated with the fitting of these models, Tournier et al. [14:3] emploi-, I1 the

spherical deconvolution method, assuming a distribution, rather than a discrete number,

of fiber orientations. Under this assumption, the diffusion AIR signal is the convolution

of a fiber orientation distribution (FOD), which is a real-valued function on the unit

sphere, with some kernel function representing the response derived from a single fiber. A

number of spherical deconvolution based approaches have followed [:3, 5, 145] with different

choices of FOD parameterizations, deconvolution kernels and regularization schemes.

More detailed discussion on the deconvolution approach is presented in OsI Ilpter 4 of this

dissertation.

3.3.4 Model-independent Q-Space Imaging Approaches

In contrast to previous approaches which assume diffusion tensor model or multi-

compartmental models, the so-called q-space :lr.:,:ny: (QSI) technique directly employs

the Fourier relation between the diffusion measurements in q-space and the probability

profile in displacement space without invoking any assumption on the underlying diffusion



1 This subsection is reprinted with permission from [7:3]









process. Originally, the q-space imaging principle was used to vield the one-dimensional

density autocorrelation function in real space from time-scale echo attenuation data in

structural imaging of inanimate mate ~ 1 .h [.$, 42]. After the the emerging of the high

angular resolution diffusion imaging (HARDI) [151], several in vivo q-space imaging

techniques have been recently proposed to approximate the ensemble averaged diffusion

propagator in :3D displacement space by performing the full :3D Fourier transform on

q-space measurements using different sampling schemes.

3.3.4.1 Diffusion spectrum imaging

In diffusion spectrum imaging (DSI) [159, 160], the diffusion signal is obtained by

sampling a dense three-dimensional Cartesian lattice in q-space with a very high number

of gradient directions and different h-values. Then the full displacement probability

density function (PDF) of the diffusion, termed diffusion .spectrum in [159, 160] for each

voxel is reconstructed directly based on the Fourier transform relation (2-29)

S(q)/So = P(r)ol iexp'i2rq~ i-
(:325)
P(r) = .F[|S(q)|/So].

It has been shown in [150, 160] that in an isolated system with time invariant (i.e. homo-

geneous) diffusion properties the Fourier transform of the EAP in the stationary state is

real and positive. Thus the signal magnitude information is sufficient to reconstruct the

EAP due to a homogeneous diffusion process. For the purpose of determining the fiber ori-

entations and better visualization, the resulting PDF is usually reduced to an orientation

density function (ODF) by a weighted radial projection:


ODF(u) = (u~T3_26)

with |u| = 1, ru = r, and Z is a normalization constant[160].

Note that in order to make q-space imaging feasible in vivo, DSI adapts the twice-

refocused balanced echo (TRBE) sequence to reduce the eddy- current-induced artifacts









created hv the 180" refusing pulses of the PGSE[127]. In addition, effectively constant

gradients are emploi- II in the TR BE experiment [150]. This modification violates the

narrow pulse condition assumed in the conventional PGSE experiment, and actually

measures the probability of a spin departing from its mean position over time 0 to TE/2

to its mean position over time TE/2 to TE. According to Tuch [150], this so-called

apparent center-of-mass (CO1\) propagator still preserves the orientational structure

of the originally desired diffusion propagator. Recently, diffusion spectrum images have

been successfully acquired on both human and small animal subjects [62, 95, 150]. These

experiments have demonstrated the ability of DSI to resolve complex tissue microstructure

such as intravoxel fiber crossing and divergence; However, the heavy sampling burden of

DSI still makes the acquisition time-intensive and limits the wide spread application of

DSI.

3.3.4.2 Q-ball imaging

The ODF reconstruction by using radial projection in DSI captures the salient

angular contrast of the diffusion function but discards all of the radial information

contained in the diffusion function. Inspired by this observation, Tuch [149, 150, 153]

proposed a model-independent sampling and reconstruction scheme termed q-hall imaging

(QBI). QBI samples the diffusion signal S(q) only on a single sphere and then directly

reconstructs a function defined on the unit sphere by taking the spherical Radon transform

of the diffusion signal 2

The spherical Radon transform, also known as Funk-Radon transform, sends a

spherical function to another spherical function hv the following integration


(R [f]) (u)= f (x)6(uTx)dx (3-27)




2 QBI can also be extended to multiple shell HARDI data. K~hachaturian et al. [82]
describes a two-shell acquisition scheme to improve the sampling efficiency and signal-to-
noise ratio of QBI.









where |u| = 1. Tuch [150] showed that the spherical Radon transform of the diffusion data

sampled on the sphere, (R [S])(u), closely resembles the ODF(u) obtained by the radial

projection of the PDF

ODF(u) = ~ u .(3-28)

Recent studies have expressed QBI's Funk-Radon transform in a spherical harmonic

basis [5, 48, 66]


ODF,(0, ~)= (-)/ ~(,)(3-29)
even I m=-l
where

sim = S(, V)IMm(0,1 4)in Od~idc (3-30)

is calculated from the spherical harmonics transform of the diffusion signal. In addition


difusin sgnl S0, -spherical radon transform -qbl D

difsimn = Jn S(04) (, 4)- sinl Odd >(1)



Figure 3-2. From the diffusion data to orientation distribution function (ODF) via the
spherical random transform with q-ball imaging (QBI).


to the spherical harmonics, other mathematical models for representing spherical functions

or distributions have also been used to express and compute the q-ball ODF, including the

mixture of von-M~ises[102], the mixture of Watson densities[126], spherical wavelets[112],

and spherical ridgelets[103].

The QBI reconstruction has advantages of being efficient and model-independent,

which made it a popular high angular resolution reconstruction scheme in recent works

[30, 122, 123, 148], however, the end result obtained by the radial projection of the three-

dimensional displacement PDF via a line integral is a convolution of the real probability

values with a 0-order Bessel function [149] but not the probability values themselves. This









convolution gives rise to an undesirable "contamination" of the probability along one

direction with probabilities from other directions and induces spectral broadening of the

diffusion peaks. To address this problem, recently alternative approaches attempting to

yield a more accurate approximation of the ODF using the spherical radon transform have

been investigated[1, 147].

3.3.4.3 Diffusion orientation transform

The diffusion orientation transform (DOT), introduced in [116, 119], is a robust and

fast model-independent method. The key mathematical tool used in DOT is the Rayleigh

expansion (plane-wave expansion) which expands a plane wave in terms of the product of

derivatives of spherical Bessel functions and spherical harmonics:


exp(iigr) = xid~rM(/8)E~/r)(331)
1,m

where ji(-) is the 1-th order spherical Bessel function and YEm is the spherical harmonic

function. Inserting the Rayleigh expansion in the Fourier transform relation (2-29), we

obtain

J m~*4~d S"3

where

Iz~~p) 4x q2 (2qr| ) exp(- 4xr22tD(p)) (3-33)

and D(p) is the diffusivity profile (angular distribution of apparent diffusivities) that can

be obtained from the Stei-l: I1-Tanner expression. Note that the integral in Eq. (3-33)

can be evaluated analytically by using the formulas derived in Ojzarslan et al. [119]. With

these powerful tools, DOT is able to transform the diffusivity profiles into probability

profiles either directly or parametrically in terms of a spherical harmonic series (Figure

3-3). The estimated probability function from DOT is also "impure" in the sense that

the end result is the true probability values convolved with a function which cannot be

specified analytically. It is worth noting that much of the blurring in the DOT is due to















Mono- or Multi-exponential
-I I-I .I-Tanner Equation


Transform


Figure 3-3. A schematic illustration of the diffusion orientation transform (DOT).


the monoexponential decay assumption of the MR signal, hence using the extension of the

transform to multiexponential attenuation as described in [119] can alleviate the blurring,

however, it in turn would necessitate collecting data on several spherical shells in the

q-space.

3.4 Conclusion

To conclude this chapter, we simulate diffusion weighted MR signals using a restricted

diffusion model [137] and compute different quantitative profiles described above. The

signal simulation is based on the exact form of the MR signal attenuation derived from the

diffusion propagator for particles diffusing inside cylindrical boundaries [137] which can be

considered as a simplified model for diffusion inside real neural tissues.

For diffusion within a cylinder of length I and radius r, the signal attenuation with

diffusion coefficient D obtained by Soderman and Jansson [137] is as follows:


E(q, 8, a) = Kmr Sn22)
n=0 k=1 m=0 [(12r/T1) (2;7qlr cos 8)2]

x[1 (-1)" cos(2;ql cos 8)] [J (2;qr sin 8)]2
12 [am -, (2xrqpsinO)2] 2~(~, m 9
x~~~k ex -2 7 D










In Eq.(:3-34), 8 is the angle between the direction of the magnetic field gradient and

the symmetry axis of the cylinder, a is the time separation, Jr~ is the mth order Bessel

function, Gkm, is the Akuz solution of the equation J,,(ca) = 0 with the convention colo = 0,

and K,an's are constants defined by K,an = 21,,>.21-,,> where 1A is the indicator function

on a set A.

The gradient directions were chosen to point toward 81 vertices sampled on a unit

hemisphere from the second-order icosahedral tessellation. The orientations in our

1-, 2- and :$-fiber configurations are specified by the azimuthal angles of $1 = :300,

2a = {200, 100"} and #:3 = {200, 750, 1350} respectively. Polar angles for all fibers were

taken to be 8 = 900, so that a view from the x axis will clearly depict the individual fiber

orientations. as illustrated in Fig. 3-4.

Three .straightforward observations can be maade from this Agure: (1) the di~fu~son

tensor model is not able to cheeracterize the IV/OH; (2) the di~f;,. .it;, 14.e..01. does not ;;.:. I1

the correct Aber orientations; (3) the diffusion y-e~..;yl.r'l~l derived profile~s are calpuble of

resolving comp~lex: tissue microstructure~s .such
the profiles presented in the right three columns correspond to the targeted results of three

model-independent techniques: DOT, QBI, and DSI, respectively



















SThe actual results of these three profiles were computed from the method proposed in
this dissertation. See details in later chapters.





Figure 3-4. Various quantitative profiles derived front diffusion weighted signals simulated
from 1-fiber, 2-fiber, and 3-fiber geometries.


Simulated
Fiblers


Traditional
DTI


Diffusivity
Profile


Plr = ro)


SP(r)dr


SP(r)r dr


)tC









CHAPTER 4
METHODS

The fundamental formula in the diffusion tensor imaging (DTI) is the tensorial

Stei-l: I1-Tanner relation [18]:


S(q) = So exp (-trace(BD)) (4-1)

It is derived as the solution to the Bloch-Torrey equation with an anisotropic diffusion

tensor term (3-2) for free diffusion in homogeneous media


i,(t) = !I,(0) exp(- k(u) TDk(u)dzL) (4-2)

where

k(t) = 7 G(u)dia (4-3)

and

B =k(t~k(t)Tdt. (4-4)

As discussed before, the Bloch-Torrey equation only describes the microscopic

phenomena and the measured diffusion MR signal at the voxel level is an macroscopic

quantity. Hence Eq. (4-1) is only valid by assuming the homogeneity of diffusion tensors

in a voxel. And the diffusion tensor estimated from the Eq. (4-1) is called the apparent

diffusion tensor.

In this work, we model the measured diffusion MR signal in a voxel using the ensem-

ble sum of the individual transverse magnetization vectors:




where each individual 14, is modeled by Eq. (4-2) with its own diffusion tensor. Suppose

the diffusion tensors in a voxel are distributed according to a density function f(D), then

combining (4-1) and (4-5), we propose the following tensor distribution model for the









diffusion weighted MR signal:


S(q) =So ex~'.p (-trace(BD)) f (D)dD (4-6)


where ?P, denotes the manifold of n x n symmetric positive-definite matrices, and

by default, refers to the the manifold of 3 x 3 symmetric positive-definite matrices

throughout this work. The key postulation in the proposed model (4-6) is that each voxel

is associated with an underlying probability distribution defined on the space of diffusion

tensors. Clearly, Eq. (4-6) is a more general form of multi-compartmental models and

simplifies to the diffusion tensor model when the underlying probability measure is the

Dirac measure.

In this chapter, we will first review the necessary mathematics for studying the

integration on ~P,. Interestingly, Eq. (4-6) is exactly the Laplace transform of a prob-

ability distribution on ?P, whose formal definition is given later in this chapter. Since

diffusion tensors are used to depict the time dependence of the covariance matrices of

random molecular displacements, it is natural to choose this distribution as the Wishart

distribution [163] on which we also present a short discussion in this chapter. The signal

decay associated with a Wishart-distributed random tensors is no longer a Gaussian, but

a Rigaut-type .I-i-~!lp)tic fractal expression given by the closed form Laplace transform

for Wishart distributions. The technical details of this closed form Laplace transform will

be derived later in this chapter. This Rigaut-type .I-i-mptotic fractal expression has been

used in previously published literature [84] to explain the MR signal decay phenomenolog-

ically. To the best of our knowledge, it is our statistical model that first gives a rigorous

mathematical justification of this Rigaut-type .I-i-mptotic fractal expression. Furthermore,

our formulation is readily extended to a mixture of Wishart distributions to tackle the

multi-fiber reconstruction problem. In fact, DTI and the multi-compartmental models

are limiting cases of our method when the tensor distribution is chosen to be a Dirac

distribution or a mixture of Dirac distributions. In the last section of this chapter, out









continuous tensor distribution model is then put into a unified deconvolution framework

[73], and several deconvolution schemes are further designed and investigated to achieve

stable, sparse and accurate solutions.

4.1 Some Mathematics on ?P,

4.1.1 Measure and Integration on ~P,

To define integration on ~P,, which is not a vector space, we need to introduce some

fundamental facts about the geometry and the measure on ~P,. Consider the general linear

group G = GL, of nonl-sing~ular Ix n rLeal mlatric~es andu define lthe auction of geG c on

Ye ?, by

Y Y [g] = gTYg. (4-7)

It is easy to show that G acts transitively on ?P, according to the action [g] defined in

(4-7), whichl imlies that~ ?, is a homogUenou~s spacet of ~lthe general Ilinear group GL,. (e

[131, 139] for the definition and meaning of homogenous spaces.)

In the following, we will also see that ?P, has a GL,-invariant measure (volume

element). By GL,-invariant measure (volume element), we basically means that if dp, is

a G~L,-invariant volume element on ~P,, then dp,(Y) = dp,(Y [g]) should hold for any

g E GL,. Since for any Y, WE ~P,, there is ag E GL, such that W = Y [g], then we

should have dp,(Y) = dp,(W) for any Y, We p, which is a quite useful property for

doing integration on ~P,.

As a convex cone embedded in the space of symmetric matrices, ?P, has a natural

induced Lebesgue measure defined as the direct product of the Lebesgue measures over the

independent elements of the matrix variable, that is, for Y = (yij) E pn,


dY =dy (4-8)


where dayi is the Lebesgue measure on RW. Though dY is commonly used for integrals

involving functions of matrix argument, it is not a GL,-invariant volume element on ?P,










which can be shown by the fact that


Y = aX 4 dY = an(n+1)/2dX (4-9)


where a is a scalar quantity.

To find the relation between a GL,-invariant measure and the dY, we first consider

the Jacobian of the mapping induced by the group action with respect to dY.

Theorem 1. [101, p.82],{189, p.19] Let J(g) be the Jacobian of the I'trry.::ll


Y H Y[g] = gT~g


with respect to dY for g e GL,, i.e. J(g) = |dW/dY | for Y e Sym, and W = Y[ g].

Then

J(g) = | det(g)|i" (4-10)



Theorem 2. [189, p.18] Let dp, be the measure on ?, I.lb:

n-E1
dpl = dpl(Y) = (det Y)- z dY. (4-11)


The&IUn dp, is th G~L,-invariant volume element on ~P,.

Note that a special case of this measure when n = 1 is just the scale-invariant

dp-(x) = (1/x)dx = d log x for x e (0, 00).

4.1.2 The Laplace Transform on ?P,

In this section we first give the definition of the Laplace transform on pn which pIIl we

a central role in our model, and then we briefly discuss two kinds of matrix argument

special functions that are used later for describing our model.

For definition of Laplace transforms on ~P,, we follow the notations in [139, p.41].









Definition 1. [189, p.41] The Lap~lace I,,r,,,4,rm of f : ~P, C,@ denoted by 2f, at the

/ iiiili. matrix: Z E Cox" is 1, Ji, .1 by:


ff (Z) f i(Y) ex -rnaceYZ)]dY (4-12)

where dY = dayi 1 < i < j < n.

An equivalent definition of the Laplace transform for matrix-variate functions is also

given in [101, p.255].

The Laplace transform converges in the right half plane, Re(Z) > Xo, for a sufficiently

nice function f, where Re(Z) denotes the real part of Z and A > B means A Be ~P,.

Theorem 3. [65, pp.479-480]. The inversion formula for this Lap~lace I,,r,:,,,,rm is:


(2i-~+)2ff(Z) exp [trace(YZ)] dZ f (YfrY r (4-13)
ne=xo 0, otherwise.

Here dZ = n dzzy and the :,.J Iyeal is over ;; ./ i~ii. matrices Z with ix~ed real part.

If f is the density function of some probability measure FT on ?P, with respect to the

dominating measure dY, i.e. dPT(Y) = f(Y)dY, then Eq. (4-12) also defines the Laplace

transform of the probability measure FT on ?P, which is denoted by MFT. Note that the

L/aplace tr t-fr, tsI, can also be 7. fr...l ~ by replacing the dY with the invariant volume ele-

ment dp,(Y) discussed in the preceding section and I,;.;;.9. y1 f (Y) to f (Y) (det Y) il" )

accordingly. In the rest of this work, the Lap~lace i,r,,,, 4,>rm & of a matrix-variate function

1~ :;,.. on ~P, will be interpreted as integration with respect to the invariant measure dp,

unless il.. 11/l~ stated.

In order to understand the property of the model being discuss later, we digress for

moment to study two important matrix argument special functions on ~P,, namely, the

power function and the gamma function.

The most basic special function on ?P, is a generalization of the complex power

function y*~, y7~E P1 = RW+, s EC and is defined as follows:









Definition 2. [189, p.89] The power function p,(Y) for Ye EP,, and a = (sl, .. s) E



ps(Y = (et Y)" ,(4-14)
j=1
where Yi is the jth leading principal minor of Y.

Proposition 1. [189, p.89] If Ye E and U is an upper t,..attyul;,lrr matrix;, then

ps (Y[U]) = p,(Y)p,(I[U]).

The proof for this proposition is given in [139] where other interesting properties of

power functions are also discussed, for example, power functions are eigfenfunctions of

invariant differential operators on ~P,.

The ordinary gamma function of a complex number z with positive real part is

defined by:

F(z) = tz- etdt= to (-u)dlog(t) = 2(tz)(1) (4-15)

which can be viewed as the Laplace transform of a power function evaluated at the

identity. Similarly, the multivariate Il.::::::;.; function for ~P,, denoted by F,(s), is defined

by:

Definition 3. [189]


En~) P(Y)ex(-race(Y))dyl(Y), (4-16)




Re( si) > (j )2 ,.,n.


In fact, the multivariate gamma function En(s) can be expressed as a product of

ordinary gamma functions 01 = 0 as given in the following theorem:

Theorem 4. [61, pp.19/, [189, pp.41]



j= 1 i= j











Let 2(p,)(Z) be the Laplace transform of the power function p,(Y), note that

the multivariate gamma function F,(s) is defined as (p,) (I). Herz [65] proved that

M(p,)(Z) is absolutely convergent for any complex symmetric matrix Z with Re(Z) > 0
and leads to the following quite useful identity:

Theorem 5. [61, pp.19/, [189, pp.41], [101, pp.254]


2 (ps)(Z) =~ p. Y) e~xp(-tmrrac(YZ7))d,() (Z )u() (4-18)

for Re(Z) > 0 and Re(C" s;) > (j 1)/2, j=1,.. n.
4.1.3 Wishart and Matrix-Variate Gamma Distributions

In Theorem 5, let n = 1, a = k > 0, Z = 0 > 0, the fumetion


f (Y)= Y"e~le (Y > 0) (4-19)
Okf (k)

integrates to 1 with respect to the invariant measure Y- dY and actually gives the density
ftmetion of a gamma distribution of a shape parameter k and scale parameter 8. For

general n, let s = (0,. ., p) and Z = E-l > 0, then we have p,(Y) = p,,, !,) (Y) = (det Y)"
and

ii, (etC)P,()I (det Y)Y exp(-trace (YE- ))dp(Y) = 1 (4-20)

where



Hence, the ordinary gamma distribution can be naturally generalized to the matrix variate

case as follows [91]:

Definition 4. {91]l For EE E ailnd for p i~n AZ = { 1, "-1 } U (" ,0 ther W~ishart

distribution y,,c with scale parameter E and shape parameter p is [n 1s


dy,,r (Y) = F,(p)-l (det Y)p-(n+1)/2 (det E) exp(-trace(E- Y)) dY. (4-22)









It is easy to see from the above definition that the matrix variate gamma distribution

is closely related to the Wishart distribution, a probably more well known name, in

multivariate statistics [6, 61, 109].

Definition 5. [61] A random matrix: Ye E is said to have the (central) Wishart

distribution W,(p, E) with scale matrix: E and p degrees of freedom, a < p, if the joint

distribution of the entries of Y has the following 1 ,.'-.11/ function with respect to the

L/ebesgue measure dY.


f (Y) = c (det Y)(p-a-1)/2 (det E)-p/2 exp -taeE ),(-3


with E E ?P, and c = 2-up'/20a(p/2)-]

Note that the correspondence between the two notations (4-22) and (4-23) is simply

given by yp/2,2E = W,(p, E). In the rest of this dissertation, we will keep using the

notation y,,c but will refer to Wishart distribution and matrix-variate gamma distribution

interchangeably.

The Wishart distribution is one of the most important probability distribution

families for nonnegative-definite matrix-valued random variables. It has been typically

used for describing the covariance matrix of multivariate normal samples in multivariate

statistics [109]. Its importance in the study of multivariate statistics can be seen from the

following property:

Theorem~ ~ ~~I 6. {61, p.8,19 p.88 L i be a random variable in R", for i = 1, 2, .. ., n,

where n < p. And suppose that X1,...,X, are I,,;,linall;i independent and distributed

according to NV(0, ); that is, normal with mean 0 and covariance matrix: EE ?P, Let

X = (X1,...X,) E Rnxp and Y = XXT. Then with ] <..7.11,*/,*///l 086, Y iS in ~Pn Gnd iS

distributed I. I;,.../ by (428).

As a natural generalization of the gamma distribution, the Wishart distribution

preserves the following two important properties of the gamma distribution [61, 91, 101]:









Theorem 7. [61] The Lap~lace I,,r, J~rm of the (generalized) Il.::::::;.; distribution y,, is


exp(-trac( 8Y)) dyl:(Y) (det(I, + OE))-" (4-24)

where (8 + E-l) E ~P,.

Theorem 8. [61] Let Y be a random variable (matrixc) with a (matrix;-variate) Il.::::::;.;

distribution y,,, then its expected value Eax C(Y) is pE.

However, it should be pointed out that the expected value pE does not yield the max-

imum value of the density function in (4-22) or (4-23). For instance, the mode of gamma

density function expressed in (4-22) occurs at (p 1)E. Interestingly, if the dominating

measure is chosen to be the GL,-invariant measure, dp(Y) = (det Y)-(n+1)/2dY, then the

corresponding density function, which is


dy,rc(Y) = F,(p)-l (det Y)" (det E)-" exp(-trace(E- Y)) dp (4-25)

does reach its maximum at the expected point pE.

Proof. Let X = YE-1, then

dy,rc d(det X)" exp(-trace(X))
dX dX

=p(dlet X)"X- exp(-tIrace(X)) (dlet X)"exp(-trac~e(X))I (426i)

=(det X)" exp(- trace(X) )(pX I)

When X = YE-1 = pl, the above derivative becomes zero which implies that the density

function reaches maximum when Y = pE. O

The difference between the density functions of gamma/Wishart distribution with

respect to the two different carrier measures is illustrated in Figure 4-1.









density functions of y4


density functions of Y1


Figure 4-1. Plots of density functions of gamma distribution Y4,1 W.r.t the non-invariant
measure and scale-invariant measure. The x-axis in the right figure is on a log
scale. Clearly, the expected value 4 corresponds to the peak of the density
function w.r.t. invariant measure but not for the case of the non-invariant
measure .


4.2 The Expected MR Signal from Wishart Distributed Tensors

Substituting the general probability distribution in Eq. (4-6) by the Wishart (matrix-

variate gamma) distribution dy,,s, we obtain


S(q)/So = |1, + BE|--


(4-27)


by applying the closed-form Laplace transform of Wishart distribution stated in Theorem
7. If the B-matrix is approximated by B = by g T, then we further have


S(q)/So = |1, +BE|-"= (1 + (bgT S)-p


(4-28)


1 + (bgTCg), we first prove the following


To establish the second equality, |1, + BE|


useful results on the Schur

Theorem 9. [124] Let P

Then


complement in linear algebra and theory of matrices.

AB" ~ ~;----r


(1) if |A| / 0 then |P| = |A| |D CA-1IB.

(2) if |D| / 0 then |P| = |A BD-1C| |D|.









Note that in Theorem 9, the matrix D CA-1B (or A BD-1C) is called the Schur

complement of A (or D) in P [166].

Let PI = Then 1 + by TCg is the Schur complement of I in P, and we

have


det P= det Idet(1 + bgTCS STC


In an analogous manner, I + b(Eg)gT is the Schur complement of 1 in P which yields


det P = det(I + b(Eg)gT)det@1).


Combining the above two equations and noting that B = by gT, we have


|1, + BE|= |1, + EB|= |1, + bCggT|= 1 +(bgTC)

Note that the well known identity |1,2 + BE| = |1,, + EB| can also be derived by applying

Theorem (9) on the Schur complements of -e

Consider the family of Wishart distributions 7;>z and let the expected value he

denoted by D = pE. In this case, Eq. (4-28) takes the form:


S(q) So (1 +(bg TDglp)-"


(4-29)


This is a familiar Rigaut-type2 .Iinji**lle'~ fractal expression [129]. The important point

is that this expression implies a signal decay characterized by a power-law in the large-|q|,

hence largfe-b region exhibiting .I- inia.,l'tic behavior. This is the expected .I-i-in!llani s

behavior for the 1\R signal attenuation in porous media [135]. Note that, although this



2 The phrase "Rigaut-type" is used to distinguish this function from Rigaut's own for-
mula [129] function. Although slightly different, the Rigaut-type function shares many of
the desirable properties of Rigaut's own function such as concavity and the .I-i-inia.,'tic
linearity in the log-log plots.









form of a signal attenuation curve had been phenomenologically fitted to the diffusion-

weighted MR data before [84], to the best of our knowledge, the proposed Wishart
distribution model is the first rigorous derivation of the Rigaut-type expression that was

used to explain the MR signal behavior as a function of b-value. Therefore, this derivation

could be useful in understanding the apparent fractal-like behavior of the neural tissue in

diffusion-weighted MR experiments [84, 118].

Here a question naturally arises. How to choose the parameter p? In our model,

we choose p to be 2 based on the following theoretical consideration. The .-i-mptotic

form of the signal attenuation due to diffusion/scattering from inhomogeneous materials

obeys the well known Debye-Porod law [46] which states that is a power-law. And in
three-dimensional porous media the signal decays as E(q) ~ q-4. Since the p-value is

defined to be the exponent of the b-value in Eq. (4-29) and b is proportional to q2, the

most meaningful choice should be p = 2.


Rigaut-type
asymptotic fractal_
Gaussian








expected sigrmnal;
intermediate surns
all 10000 tensors


OO


Figure 4-2. The Wishart distributed tensors lead to a Rigaut-type signal decay.
Reproduced with permission from [75] @[2007] Elsevier.


6i5


0 .1 00 0


0.0100o



0. 0010



0. 0001

0.0


0. 10

Y r(in-1')









The relation (4-29) can be empirically validated by the following simulation. From a

Wishart distribution with p = 2, we first draw a random sample that contains a random

rank-2 tensors {D1,...,D,} and then simulate the corresponding multi-exponential signal

decay using a discrete mixture of tensors:


E.(q) =S(q)/So = exp (-bgTDig). (4-30)
i= 1

For each sample size n, we plot the signal decay curve by fixing the direction of diffusion

gradient q and increasing the strength q = |q|. The relation between signal decay behavior

and the sample size is illustrated in Figure 4-2. The left extreme dotted curve depicts

the signal decay from a mono-exponential model, where the diffusion tensor is taken to

be the expected value of the Wishart distribution. The right extreme solid curve is the

Rigaut-type decay derived from (4-29). Note that the tail of the solid curve is linear

indicating the power-law behavior. The dotted curves between these two extremes exhibit

the decay for random samples of increasing size but smaller than 10,000. The dashed curve

uses a random sample of size 10,000 and is almost identical to the expected Rigaut-type

function. As shown in Figure 4-2, a single tensor gives a Gaussian decay, and the sum

of a few Gaussians also produces a curve whose tail is Gaussian-like, but as the number

of tensors increases, the attenuation curve converges to a Rigaut-type .-i-mptotic fractal

curve with desired linear tail and the expected slope in the double logarithmic plot.

If p is an integer, then a well known result is that the gamma distribution y,,

also describes the distribution of the sum of p independent exponentially distributed

random variables with parameter a. It follows from the central limit theorem that if p

(not necessarily an integer) is large, the gamma distribution y,~, can be approximated

by the normal distribution with mean pa and variance pa2. More precisely, the gamma

distribution converges to a normal distribution when p goes to infinity. A similar behavior

is exhibited by the Wishart distribution. Note that when p tends to infinity, we have


S(q) iSo exp(-bgTg ( 1









which implies that the mono-exponential model can be viewed as a limiting case (p

00) of our model. Furthermore, the Taylor expansion of (4-29) in the long wavelength or

low-q regime leads to a quadratic decay:


S(q)/So = 1 4xr2 TDq(A 6/3) + O(q4) _432)


which can also be derived from the diffusion tensor imaging model [14]. Therefore, Eq.

(4-29) can be seen as a generalization of Eq. (3-17). By the linearity of the Laplace

transform, the bi-exponential and multi-exponential models can be derived from the

Laplace transform of the discrete mixture of Wishart distributions as well.

When a Wishart distribution used as the mixing distribution in Eq. (4-6) with a

pre-specified p parameter, Eq. (4-28) can be rewritten as


S(q)So)- ,,, 43


Let K be the number of diffusion measurements acquired at each voxel, the above

equation can be further expressed in a matrix form:

1 \

(S2)- B, -- 2Bz C,, 1
(4-34)


(SK)- Bz Ex -- 2Bzz / Ezz 1

where Bij and Egy are the six components of the matrices B and E, respectively. The lin-

ear system formulated in (4-34) clearly -II_0-1-;- -a linear regression method for estimating

diffusion tensors from diffusion-weighted images, which is very similar to the traditional

diffusion tensor estimation methods [16]. Note that in this model the final estimation

of diffusion tensor D is given the expected value of the Wishart distribution y,,4, i.e.

D = pE.

It is worth noting that ]rn Ilw alternative methods which involve nonlinear optimiza-

tion and enforce the positivity constraint on the diffusion tensor, as in [34, 158], can be










applied to the direct nonlinear problem:


mmi (S(q) So(1 + trace(BE))")2 (4-35)

formulated from the Wishart model proposed here. Similarly, the resulting diffusion

tensor field can also be analyzed by numerous existing diffusion tensor image analysis

methods [161].

4.3 Methods for Multi-Fiber Reconstruction

4.3.1 The Mixture of Wisharts Model

The density of a simple Wishart distribution as a function of diffusion tensors reaches

one single diffusion maximum at its expected value, therefore, a single Wishart model

can not resolve the intra-voxel orientational heterogeneity. Clearly, the Laplace transform

relation between the MR signal and the probability distributions on ?P, naturally leads

to an inverse problem: to recover a distribution on ?P, that best explains the observed

diffusion signal. The difficulties entailed in inverting the Laplace transform are well known

(see [54] for "cogent reasons for the general sense of dread most mathematicians feel about

inverting the Laplace it~ lI!-1its~! ), especially for noisy data in high dimensional space,

which is the case in our application domain. In order to make the problem tractable, the

following simplifying assumptions are made:

First, just as a discrete mixture of tensors model can be adapted, so can a discrete

mixture of Wisharts model where the mixing distribution in Eq. (4-6) is given as


f (D) =~ 7, (D) (4-36)
i= 1

where r; .: (D) is the density function of the matrix-variate gamma distribution

dr; Note that in this model the set of (pi, Es) is treated as a discretization of the 7-

dimensional parametric family of Wishart distributions. Hence, the number of components

in the mixture, NV, only reflects the resolution of this discretization and should not be

interpreted as the expected number of fiber bundles.










Secondly, it is further assumed that all the pi take the same value, pi = p = 2, which

is a reasonable assumption based on the analogy between Eq. (4-29) and the Debye-

Porod law of diffraction [135] in 3D space. Since the fibers have an approximate axial

symmetry, it also makes sense to assume that the two smaller eigenvalues of diffusion

tensors are equal. In practice, the eigfenvalues of Di = p~i are fixed to specified values

(Ai, A2, 3S) = (1.5, 0.4, 0.4)p2m MSCOnSIStent With the values commonly observed in white-

matter tracts [152]. Due to this rotational symmetry, the discretization of ?P, forming the

mixture of Wisharts is reduced to a spherical tessellation (Figure 4-3). Accordingly, the

prominent eigenvector of each Es can be taken from the unit vectors uniformly distributed

on the unit sphere. Because of the antipodal symmetry, the sampling is actually performed

on the projective plane, i.e. Only half of a normal spherical tessellation is used.













(a) 92 vertices (b) 162 vertices (c) 252 vertices

Figure 4-3. Sphere tessellations using an icosahedron subdivision model with different
iteration numbers.


Note that all the above assumptions leave us with the weights w = (I,) as the

unknowns to be estimated. Given K measurements with qj, the signal model equation:


Scq) = SoC it -(1 + trace(BEi))-P. (4-37)
i= 1

leads to a linear system Aw = s, where s = (S(q)/So) contains the normalized measure-

ments, A is the matrix with Aji = (1 + trace(Bj~i))- and w, = (I, ) is the weight vector









to be estimated. The details of solving Eq. 4-37 will be the central topic of the upcoming
sections .

After obtaining the vector w, the diffusion weighted MR signal attenuation model

E(q) = S(q)/So can be analytically expressed using the general form as in Eq. (4-43).
As a result, the displacement probability P(r) can then be computed by the Fourier

transform P(r) = f(S(q)/So) exp(-iq r) dq where r is the displacement vector. The

P(r) function defined above describes the probability for water molecules to move a fixed
distance and has been emploi- II in the DOT method [119].

Note that when the mixture of Wisharts model is used, the resulting P(r) can be

approximated as a mixture of oriented Gaussians

P(r, -) =~ exprl(-qTUqr) dF(D) exp(-iq r) dq

exr"p(-qT) qr)ep(-iq r)dq dF(D)
1 (4-38)

eC xp(Ta-r Dr/47) d(D
i' -1~l~


where Di = p~i are the expected values of r;

In this case, many of the quantities produced by other methods including the radial

integral of P(r) in QBI [149] and the integral of P(r)|r|2 in DSI [160] are analytically
computable. This fact provides us the opportunity to understand these quantities and
evaluate their performances in resolving complicated local fiber geometries.
4.3.2 A Unified Deconvolution Framework

Interestingly but not surprisingly, the proposed mixture of Wisharts model (Eq. 4-37)
can be cast into a general deconvolution framework

S~q)/o = ~ q, ) f x~dx(4-39)

that unifies several model-dependent HARDI reconstruction approaches in literature.









The signal in Eq. (4-39) is expressed as the convolution of a probability density

function f(x) and a kernel function R(q, x). The integration is over a manifold MZ/, each

point, x, of which contains information indicating fiber orientation and anisotropy. The

convolution kernel, R(q, x) : RW3 X iZ/1 HWmodels how the signal receives response from

a single fiber. In order to handle the intra-voxel orientational heterogeneity, the volume

fractions represented by a continuous function f(x) : MRR I models the distribution of

fiber bundles. Hence, the deconvolution problem is to estimate f(x), given the specified

R(q, x) and measurements S(q)/So.

A common approach in practice is to represent f(x) as a linear combination of NV

basis functions: f(x) = E wyfj(x). The choice of convolution kernels and basis functions

often depend on the underlying manifold MZ/. A simple example is to let MZ/ be the unit

sphere (or more precisely, the projective plane due to the antipodal symmetry), which

leads to the spherical deconvolution problem [3, 143]. Several other approaches start from

the manifold of diffusion tensors, but again reduce to the spherical deconvolution problem

since only rotationally symmetric tensors are considered. [5, 125, 143].

Following [5, 125, 143], we choose the standard diffusion tensor kernel in our model.

However, it is the Wishart basis function that distinguishes our method from these related

methods. It is worth noting that the Wishart basis reduces to the Dirac function on ?P,

when p = 00 and thus leads to the tensor basis function method [125] as well as to the

FORECAST method [5], which both estimate fiber orientations using the continuous

axially symmetric tensors and hence resemble our method very closely.

Table 4-1 summarizes some of the existing multi-fiber reconstruction methods in the

above described unified deconvolution framework. The first approach listed in Table 4-1

corresponds to the well known diffusion tensor imaging method Basser et al. [18] where the

mixing distribution in the convolution integral is taken to be a single Dirac distribution


S(q)/BSo~ = ep(-qTDq)6(DD l^)dD exp(-qT ^) (4 40)









Table 4-1. A list of previously published fiber reconstruction methods expressed in the
proposed unified deconvolution framework. See text for meaning of symbols.
Reproduced from with permission. @[2007] IEEE.

Method MZ/ x f (x) R(q, x) unknown
Basser et al. [18] ?,n D 6(D; D) exp(-qTDq) D
Tuch et al. [152] ~P, D I,, JC(D; Dk) exp(-qTDq) { Tournier et al. [143, 146] S2 U Clm~m exp(-(U q)2) (Clm
Anderson [5] S2 Uv Clm~m exp(-qTDyq) {czm)
Alexander [3] S2 U, ,i, e? ex(( ) t
Jian et al. [75] pn D Ramirez-Manzanares et al. [125] S2 U t23 k exp(-qTDyq) w={<*}


where ?P, is the manifold of 3 x3 positive definite matrices, 6(x; xo) denotes the Dirac delta

function centered at point xo and the problem is to estimate the unknown D that best fits

the diffusion tensor model.

Instead of the single Dirac delta function used in the diffusion tensor model, all other

methods listed in Table 4-1 express the fiber orientation distribution f(x) as a linear

comblination of K basis functions f(x) C~= I E to fk(x). For. example, the second approach

listed in Table 4-1 corresponds to the multi-tensor model [152]
K K
S(q)/So = exp(-qTDq) C (D; Dk)dD expl(-qTDkf) 1

In the original multi-tensor model proposed by [152] the objective is to find a set of

Kt tensors Dk and corresponding fractions no by performing conventional nonlinear

optimization algorithms. Even though the eigfenvalues of Dk are Set tO Specified values

to favor physiological solutions, it is usually desirable to fix the number K to be a small

number, 2 or 3, to ensure the stability of the nonlinear optimization, which in turn

introduces a model selection problem.

Due to its close relation to the diffusion physics, the diffusion tensor response kernel


R(q, D) = exp(-qTDq) (4-42)









is still widely used in a number of recently proposed reconstruction methods. However,

unlike the traditional diffusion tensor model and multi-tensor model where K is set to a

small number (1,2,or 3), these extended diffusion tensor based models tend to let K be a

relatively large number in order to gain a quite good coverage of the underlying manifold

MZ/. In this sense, the number K only depends on the resolution of the discretization

on the manifold MZ/ and should not be interpreted as the number of expected fiber

populations.

In order to avoid the difficulty of directly working on the manifold of diffusion tensors

~P,, many models take the similar assumptions used by our model. Some models [75, 125],

only consider a set of K rotationally symmetric tensors, DI, with specified eigenvalues ,

A = { At > a2 =3 }, but varying dominant eigenvectors, {vl, .. ., vK } obtained from a

sphere tessellation. For this reason, the underlying domain of deconvolution is reduced

to the sphere from the manifold of diffusion tensor, ~P,. It is also worth noting that the

response kernel exp(-(v q)2) used in [3, 143] is just the special case of the exp(-qTDyq)

when X2 = 3 = 0. However, it is the Wishart basis function that distinguishes our method

from these related methods since Yp,Dy aS a COntinuOUS distribution on ?P, implicitly takes

contributions from the entire ?P, even there are only K discrete Dy. It is also interesting

to note that the Wishart basiS p,~Dyx reduces to the Dirac function on 7P, when p 00o and

thus the models in [5, 125] can be treated as special cases of our model.

Finally, except for the single diffusion tensor model and the multi-tensor model

[152] where both the volume fraction I, and the parameter of basis functions (Dk) are

unknown, all the other methods lead to a problem that can be expressed in the general

form of

Aw = s + rl, (4-43)

where, on the righthand side, s is a column vector containing NV multi-directional mea-

surements, {si} = {S(qi)/So}, and rl represents the noise, while on the lefthand side,










A = {aik) is an 1VX K matrix given by aik, = MI RGi, x) k(x)dx and w is the un-

known weight vector. Note that the integral to compute the entries of A may have an

analytical solution as in [75] or needs to be numerically approximated as in [3], depending

on the choices of convolution kernels and basis functions. But once the response kernel

R(q, x) and the basis function are specified, the matrix A can be fully computed (or

approximated) and only w, a column vector containing K unknown coefficients, remains

to be estimated. It is worth noting that a further spherical harmonics (SH) expansion on

w in [5, 143, 146] leads to a parametric deconvolution of w in terms of SH coefficients.

However, it is straightforward to change this parametric deconvolution back to the direct

nonparametric deconvolution of w. In our opinion, it is more natural and easier to inter-

pret and handle the weight vector than the SH coefficients vector. For this reason, we will

only discuss the nonparametric deconvolution of w, in this work.

4.4 Computational Issues

In the terminology of signal representation theory, the NVx K matrix A in Eq. (4-43)

as a collection of vectors ai E R"N, i = 1... K is usually referred as a "J..i.o ,:.re;, and the

column vectors (ai) are called "rlia;,, ". The deconvolution problem is then equivalent to

representing the signal s = Ci f as a linear combination of atoms in the dictionary A.

If we require all the weights to be non-negative, then the linear combination is restricted

to a conic combination. In addition, a parsimonious representation for signal is preferred if

only a few atoms are able to produce the signal s.

In the normal clinical setting, the number of diffusion MR image acquisitions, K, is

rarely greater than 100. On the other hand, a high resolution tessellation with NV > 100 is

usually preferred for an accurate reconstruction. This situation yields a II rectangular

system matrix A in (4-43) and in this case the dictionary A is called overcomp~lete. An

under-determined system of equations may have infinitely many solutions in the least

squares sense. To make things even worse, most deconvolution models used in literature

result in extremely ill-conditioned linear systems whose standard least squares solutions










may be -r I---- l in-Oy unstable. This disturbing consequence of the numerical stability issue

can he a real concern as was shown in [5].

Since the weights w = {m, } K1 correspond to volume fractions, they are expected to

be non-negative. Negative weights are not physically meaningful and should be penalized

by adding a regularization term or excluded by imposing an explicit non-negativity

constraint. In addition, it is reasonable to assume that most white matter voxels only

contain contributions from relatively few fiber bundles. Therefore, apart from a few

significant peaks, we expect w has a sparse support, i.e. most entries of w are expected

to be zero (or very small). It is important to note that the sparsity property also has

advantages in the optimization process required to find the maxima of the water molecule

displacement probability function, which represent the fiber orientations in the multi-fiber

reconstruction problem.

4.4.1 Regularization and Stability

The first question one needs to consider before solving the deconvolution problem

is how to measure the goodness of a solution, in our case, the discrepancy between Aw

and s. Commonly used measures include the L2 norm, the L1 norm, or more generally,

an L, norm for 1 < p < 00. If all the elements of A,w,s are assumed to be nonnegative,

then, some information divergence e.g. K~ullback-Leibler (K(L) divergence can also be used

to measure the distance between Aw and s after appropriate normalization. Under the

assumption that the measurement errors are independent and follow an identical normal

distribution, the maximum likelihood estimate of w naturally leads to the L2 Ilorm as a

goodness measure and is equivalent to the corresponding least squares (LSQ) problem [26]

that minimizes the residual sum of squares


(PI) min ||Aw s||2 4 )









The solution of (P1) in the least squares sense is given by w, = A+s where A+ is the

pseudo-inverse of the A given by


A+ = (ATA)-1AT ifA has full colunin rank (-5

SAT(AAT -1 ifA has full row rank. 45

Direct methods are adequate here since in our application the size of the linear system in

Eq. (4-44) is not that large to require iterative methods. 1\oreover, since the matrix A is

independent of the spatial location, the pseudoinverse is only computed once, and hence

the computational burden is light. Despite its simplicity and efficiency, a direct solution

to the linear system is highly susceptibility to noise, especially when A is ill-conditioned,

which is usually the case in our application.

To illustrate this ill-conditioning phenomena, we study two different convolution

models under the unified framework of Eq. (4-39). The first model involves the use of

radial basis functions and a Gaussian kernel with a degenerated rank-1 covariance matrix,

as in [:3]. The second model uses a standard diffusion tensor kernel weighted by a mixture

of Wishart distributions as discussed in [75]. The reason for choosing these two models

for the purpose of comparison is due to the fact that the delta function basis as used in

[5, 125, 14:3] is actually a special case of the Wishart basis when p c o. The further use

of spherical harmonics expansion on w in [5] is equivalent to a parametric deconvolution of

w which is different front the nonparanletric deconvolution of w as discussed in this work.

Consider an example situation wherein we are given 81 gradient directions chosen to

point toward the vertices sampled on a unit hemisphere front a second-order icosahedron

tessellation of the sphere. To fully specify the matrix A, we still need to specify its column

space which is dependent on the number of discretization points on the sphere, i.e., the

discretization resolution on the sphere. We will illustrate the ill-conditioning with two

values of resolution (column size) namely, 81 and :321. Thus, the matrix A will be of size

(81, 81) in one case and (81, 321) in another. As seen from Figure 4-4, both the models









x17
2x 0


12x 0

10~ P



6t

--size(A): 81 x 821


E




o


-a-size(A): 81 x 81
|+size(A): 81 x 321


20 40 60 80 100 ,0 100 200 300
a in the radial basis functions p in the Wishart distributions

The linear system constructed in (4-43) is often extremely ill-conditioned with
very high condition number. The plot on the left shows the profile of condition
numbers when the A matrices are constructed from the radial basis function
and the tensor kernel model as in [3]. The plot on the right shows the case
with a standard diffusion tensor kernel weighted by a mixture of Wisharts. In
both scenarios there are K = 81 diffusion gradient directions. Two tessellation
schemes of different resolution levels (NV = 81 and NV = 321) are considered for
each model. Reproduced from [73] with permission. @[2007] IEEE.


Figure 4-4.


result in extreme ill-conditioning of matrix A. The plot on the left shows the case where

the A matrices are constructed from the radial basis function model. Note the condition

numbers are of the order of le + 06 and steadily increase with increasing o-, the value

of radial basis function parameter. The plot on the right shows the case where the A

matrices are constructed from the mixture of Wisharts model [75]. The condition numbers

in this case are an order of magnitude less than those for the radial basis function model

and quickly converge to an upper limit. Note that when the parameter p in Wishart model

goes to infinity, the resulting system converges to the one obtained in [5, 125, 143].

II I.!y methods aiming at reliable multi-fiber reconstruction in the presence of noise

have been emploi-v I including low-pass filtering [143], minimum entropy [144], and the

maximum entropy spherical deconvolution [3]. Recently, Tournier et al. [145] proposed

to combine the spherical deconvolution with Tikhonov ,ell;,lar,.:~r..rHn, a very popular

technique for solving ill-conditioned problems [140]. In the general framework of Tikhonov









regularization, the original problem in Eq. (4-43) is reformulated as finding the estimate
w that minimizes

(P2) min ||Aw s||2 + I~l2 _46)

where T is a suitably chosen regularization operator that imposes a constrain function

||Tw||2 On W), and a~ is a non-negative scaling factor that controls the balance between the

residual term and the penalization term. In the statistical literature, the method using

Tikhonove regularization is also known as ridge regression.

Typically the matrix operator T is chosen as a discrete approximation to some

derivative operator if smoothness is the desired property of the underlying solution. In

our case, we take T to be the identity operator, I, which leads to a zeroth order Tikhonov

regularization, giving preference to solutions with smaller manitudes. K~awata et al. [81]

described an application of the zeroth order Tikhonov regularized least-squares method to

3-D reconstruction of optical microscope tomography data. They also pointed out that the

Wiener-Helstrom filtering [64] or the minimum mean-square error (1111510;) deconvolution

[80] can also be derived from the zeroth order Tikhonov regularization by letting the

signal-to-noise ratio be a~ and assuming that the signal and the noise are uncorrelated

stochastic processes. Minimization of the objective function in Eq. (4-46) with T = I

yields the following normal equation:


ATs = (ATA + a~TTT~ 47


with an explicit solution given by


w = (ATA + a~TT -1ATs 48


The so-called damped least squares (DLS) method[157], which is equivalent to the

zeroth order Tikhonov regularization technique, is emploi- II in [75] to reconstruct the

mixture of Wisharts representing a continuous distribution over diffusion tensors, where









the solution is given by


w = (ATA + alI)- ATS
(4-49)
=AT(AAT + I-18

In equation (4-49), (ATA + alI)-1AT (or AT(AAT + ~)-1) iS called the damped least

squares inverse of A and a~ is called the damping factor. The damped least sqaures inverse

can be expressed in terms of the truncated Singular Value Decomposition (SVD) [58] as:


(ATA + l)- ATi a cGi UiiT _50)

where r is the rank of A, ai's are singular values of A, and vi(ui)'s are the associated

left-(right-) singular vectors of A.

The non-negative damping factor a~ controls a trade-off between accuracy and

stability. The determination of a suitable a~ to prevent the solution from the under-

regularization or the over-regularization is often am rl i difficulty in the use of Tikhonov

regularization. A similar damping factor appears in the well known and widely used

Levenberg-Marquardt algorithm for solving nonlinear least squares problems and can be

adjusted at each iteration.

Recently, a Damped Least Squares method was used to regularize the fiber orientation

distribution [132], where the optimal damping factor a~ is determined by minimizing the

Generalized Cross Validation (GCV) criterion [155]. The underlying statistical model

used in the GCV method is that the components of s are subject to random errors of zero

mean and covariance matrix a2I, Where a may or may not be known. Let A,+ denote the

damped least squares inverse of A with damping factor a~. The predicted values of s from

the solution in (4-49) is then


li = Aw = AA,+s = Pos (4-51)


where the symmetric matrix P, = A(ATA + alI)-1AT iS called the f'illa. 0..: matrix;.









When a2 is known, it was -II_t-r-- -1.. in Craven and Wahba [45] that a~ should be

chosen as the minimizer of an unbiased estimate of the expected true mean square error

l1 81 2 a2 (4-52)


where ni is the number of rows in A. If a2 is unknown, then the so-called generalized

cross-validation (GCV) function given by


m||s s|| /[trace(I P,)]2 45:3)


may be used to estimate the proper value of c0. The main reason of using this GCV

function is in that the minimizer of (4-53) is .I-i-ingdo' tically the same as the minimizer of

(4-52) when nz is large [59].

The GCV criterion is a one-dimensional function of the regfularization parameter

(damping factor c0) and can he computed from the SVD of A and Pa~. 1\ore efficient

numerical methods based on the bidiagonalization of 4 [5:3, 6:3, 68] have also been devel-

oped; see [26] for more discussions on the minimization of GCV function. However, GCV

does not guarantee the real optimal solution due to the theoretical limits of the GCV

method, as pointed out in [155], "the theory jll-r;fi-;in-4 the use of GCV is an .I-i-ini!!lle'~

one. Good results cannot he expected for very small sample sizes when there is not enough

information in the data to separate signal from in~ ss--

Additionally, it is important to note that the regularization scheme discussed here is

performed voxel by voxel, which is quite different from some other regularization schemes

in the context of smoothing diffusion tensor field [4:3, 121] where a spatial coherence

term is usually added in the minimization problem. The presented unified deconvolution

framework could be combined with such spatial regularization schemes as well (e.g, [125]).

However, the discussion of spatial regularization schemes is not the main focus of this

work.










4.4.2 Nonnegativity and Sparsity Constraints

4.4.2.1 L1 minimization methods

The so-called Lo norm is the most straightforward measure of sparsity by counting

the number of nonzero entries. However, the naive exhaustive search for the sparsest

solution with smallest Lo norm is known to be NP-hard. A recent strand of research has

established a number of interesting results, both theoretical and experimental, on stable

and efficient recovery of sparse overcomplete representations in the presence of noise (see

[32, 49] and references therein). One such important result is that, under appropriate

conditions on A and s, minimizing L1 norm of the solution often recovers the sparsest

solution. This phenomenon is referred to as LI/Lo equivalence. It is shown in [50], that

when the dictionary A has a property of mutual incoherence (defined in [50]) and that

when the ideal noiseless signal s has a sufficiently sparse representation with respect to

A, minimizing the L1 norm of the solution, (|| w ||1 = Ci i' I ) often recovers the sparsest

solution and is locally stable, i.e., under the addition of small amounts of noise, the result

has an error which is at worst proportional to the input noise level. Unfortunately, it

turns out that our systems consistently have very high mutual coherence values, in the

range of 0.95 to 1, which makes these nice theoretical hounds inapplicable to our problem.

To investigate the performance of L1 minimization based methods in the context of our

problem, we experimentally tested the L1-MAGIC package a collection of MATLAB

routines for solving the convex optimization problems central to sparse signal recovery

[31]. Among the several programs implemented in L1-MAGIC package, we are particularly

interested in the following two programs:

*Min-L1 with equality constraints. The program can he mathematically stated as


(P3) min ||w||I1 subject to Aw = s. (4-54)




1 http://www.11l-magic.org










This program is also known as the ba~si~s pursuit [:36] and the solution is the vector

with smallest L1 norm (||w||I1 = Ci 'II |) that explains the observation s. Candes

et al. [:31] have shown that if a sufficiently sparse wi exists such that Awi = .s, then

this technique will find it and when s, A, w have real-valued entries. Note (IS) can

he recast as an linear-programming (LP) problem.

*Min-L1 with quadratic constraints. In this formulation, one finds the vector

with minimum L1 norm that comes close to explaining the measurements s, i.e.,


(P4) min I~11 subject to ||Aw s||2 < e (4-55)


where e is a user specified parameter. Candes et al. [:31] have shown that if a

sufficiently sparse wi exists such that s = Aw + e, for some small error term ||e||2

e, then a solution w, to (P4) Will be clOSe to wi in the sense that || w wil || < C ,
where, TC is asmll ,,cnsant It has1, also~,,, been shown, that,,, (P,,, eTCRtR

second order cone programming (SOCP) problem.

The experimental results for these two programs are reported in Chapter 5. Other

programs implemented in L1-MAGIC but not investigated here include minimum L1 error

approximation, minimum total variation (TV) with equality constraints and minimum TV

with quadratic constraints, etc.

4.4.2.2 Non-negative least squares (NNLS)

A direct solver may produce a solution with many negative-valued components, which

is not physically meaningful. For example, the zeroth order Tikhonov regularization is

able to suppress the large spurious negative spikes in the weight vector w as expected,

however, there are still many negative entries in the resulting w, vector. Note that the

above L1 minimization methods (P3) and (P4) do not explicitly enforce the nonnegativity

constraints either. It has been shown in image reconstruction literature [80, 81] that

constraining the solution to the nonnegative space could drastically reduce the ambiguity

of the solution and hence improve the final reconstruction results. It has been recently









reported in [146] that a spherical deconvolution technique imposed with a non-negativity

constrain on the estimated FOD does not require a low-pass filtering which was previously

----I _-r. .1 in [143].

The least squares problem subject to a nonnegativity constraint is formulated as


(Ps) min ||Aw s||2 Subject to w > 0. (4-56)

The non-negative least squares (NNLS) problem is an important special case of least

squares problems with linear inequality constraints (LSI) [85, Ch. 23]:


min ||Ax b||2 Subject to CTx > d. (4-57)

The LSI problem is essentially a quadratic programming problem that minimizes a

concave quadratic function in a linearly bounded convex feasible hyperspace. Let ce denote

the i-th column vector of the matrix C. The separating hyperplane defined by the i-th

constraint is {x : cTx = di}. Note that the vector ce is orthogonal to this separating

hyperplane and points to the feasible halfspace {x : cTx > di}. Given a feasible point x, it

may exactly sit on some separating hyperplanes, the indices of these hyperplanes form the

set 8 (equality) and the complementary set of 8 is called S (slack). The optimal solution

Ai to the LSI problem should meet the following K~uhn-Tucker conditions [99]:

jri > 0 if ie E i.e. cN X= di
(4-58)
jri = 0 if i E S, i.e. TcN > di

where jr defined by Cjr = AT(AAi b) is the dual vector for this problem.

Proof. [85, Ch. 23] Let g = AT(AAi b) denote the gradient vector of the quadratic cost

function ~(x) = ||IAx b||2 at ~i. The K~uhn-Tucker conditions in (4-58) lead to


fii (4 59)
iLES









which implies that the negative gradient -g at the solution is expressed as a nonnegative

(ysi > 0) linear combination of the outward pointing normals -ci to the constraint

hyperplanes on which ~i lies (ie E ).

Consider an arbitrary perturbation n of 1i, which yields a new value of the cost

function: #(Ai + u) = #(~i) + UTg + ||AU||2/2. Note that for &i + n to remain feasible,

U~cs > 0 for i e must hold, hence a~g = Ci, eg ye c > 0 and it follows that

~(A + u) > ~(i) for any feasible perturbation n of ~i. O

Based on the K~uhn-Tucker conditions in (4-58), Lawson and Hanson [85, Ch. 23]

developed an active set strategy [57] to find the optimal solution to the NNLS problem.

In this classic algorithm, a series of least squares problems without constraints are solved

sequentially according to the tentative status of an active set p, which contains only

positive components and is initialized to an empty set. A positive component is added

to P in the main loop, followed by a possible deletion of some components from P in the

inner loop. On termination, x will be the solution vector and y will the dual vector, both

satisfying the K~uhn-Tucker conditions in (4-58) with C = I and d = 0.

An interesting observation is that the active set strategy emploi- II in this algorithm

tends to find a sparse solution if there exists one, even though the sparsity constraint

is not explicitly imposed. This behavior was also reported in many other applications

[28, 69, 78, 93]. Furthermore, even though the algorithm finds the solution to (P5) it-

eratively, it has been proved in [85] that the iteration alr-ws- converges and terminates

in a finite number of steps. Hence, this algorithm does not require the tuning of cutoff

parameters and the output is not sensitive to the initial guess. The number of iterations

to reach the full convergence, as expected, depends on the amount of noise in the measure-

ments. However, a fairly satisfactory solution can often be achieved well before the full

convergence, since the solution gets improved smoothly with iteration.

Since the number of the insertion/deletion operations is proportional to the size of

the solution vector, an active set algorithm can be slow for large scale problems. In the










Algorithm 1: The active set algorithm for NNLS [85, Ch. 23]
Input : AE RmX", bE Rm
Output: Ai > 0 such that ^< = arg min||IAx b||2
1 begin
2 x := 0; y := AT(b Ax)
3 Z:= {1,2,..., n}; p:= 0
4 while Z / 0 and maxiez yi > 0 do
.5 3 = arg maxiez yi
B Move index j from Z to P
7 Solve the least squares || Cj,? zyAj b|| where Aj is the column j of A
s zj =0Ofor j EZ
9 while min(zy) < 0 do
to a = miniap Xi/(Xi ze)
11 x = x + a~(z x)
12 Find index j in P such that xj = 0
13 MOVe index j from P to Z
14 Solve the least squares || C ,~ zy~ A b||
1s zj = 0 for j EZ

17 y = AT(b Ax)
Is end


context of our reconstruction problem, the size of the matrix is relatively small, at most

hundreds by hundreds, which makes the active set method still a reasonable choice over

some other algorithms for large scale problems [35]. However, in real reconstruction of

volume data, we do have to solve the NNLS problem voxelwise. Unlike the unconstrained

least squares and regularized least squares where the pseudoinverse or the damped inverse

can be computed only once and reused for multiple right-hand sides, the active set method

usually has to solve different sequences of subproblems in its inner loop. To alleviate this

problem, a recent variant of NNLS algorithm [23] is able to avoid the many unnecessary

recomputations by rearranging the calculations in the standard active set method on the

basis of combinatorial reasoning. This so-called fast combinatorial NNLS (FC-NNLS) has

been tested in our experiments and proved much faster than the standard NNLS algorithm

in the real MR volume data.









CHAPTER 5
EXPERIMENTAL RESULTS

In this chapter, we present experimental results on both simulations and real data to

validate the proposed model and methods we have investigated in Chapter 4. Portions in

this chapter are reprinted, with permission, from [73, 75, 77].

5.1 Simulations

In order to compare the performance of the deconvolution methods described in

the previous chapter, a series of experiments were first performed on simulated data.

Throughout this chapter, the signals were simulated by using the exact form of the MR

signal attenuation derived from the diffusion propagator for particles diffusing inside

cylindrical boundaries [137] which can be considered as a simplified model for diffusion
inside real neural tissues.

For diffusion within a cylinder of length I and radius r, the signal attenuation with

diffusion coefficient D obtained by Soderman and Jansson [137] is as follows:

OC 0 002Kamr2(2?iqr)4Sin2(20)a m
E(q, 8, A)=22
n=0 k=1 m=0 [(nr/1)a a(2r cos 8)"2
x[1 (-1)" cos(2;ql cos 8)] [J (2;qr sin 8)]2(51
12 [am, 2xqpsin)!2] 2a: m 2



In Eq.(5-1), 8 is the angle between the direction of the magnetic field gradient and

the symmetry axis of the cylinder, a is the time separation, Jm is the mth order Bessel

function, akm, is the kth solution of the equation J (a) = 0 with the convention a~lo = 0,

and Kam's are constants defined by Kam = 21">o21m>o where 1A iS the indicator function

on a set A.

As in [119, 154], in our simulations we used the following parameters: 1 = 5 mm,

r = 5 p-m, D = 2.02 x 10-3 mm2 S, a = 20.8 ms, 6 = 2.4 ms, b = 1500 s/mm2 and

the infinite series were truncated at n = 1000 and k, m = 10. To simulate 2- and 3-fiber






















Figure 5-1. HARDI simulations of 1-, 2-, and :$-fibers (b = 1500s/nnn2) ViSualized in
Q-hall ODF surfaces using [5, (21)] with spherical harmonics expansion
terminated up to I = 8. The orientation configurations used in the simulation:
azimuthal angles 01 = :300 2 = { 200, 1000 }, 3 = { 200, 750, 1:350 }; polar angles
are all 900. Reproduced with permission from [7:3] @[2007] IEEE.


geometries, we simply used an additive model based on the Eq.(5-1) by mixing signals

simulated from two or three cylinders with known orientations.

The gradient directions were chosen to point toward 81 vertices sampled on a unit

hemisphere from the second-order icosahedral tessellation. The orientations in our

1-, 2- and :$-fiber configurations are specified by the azimuthal angles of $1 = :300,

2a = {200, 100"} and #:3 = {200, 750, 1350} respectively. Polar angles for all fibers were

taken to be 8 = 900, so that a view from the x axis will clearly depict the individual fiber

orientations. Figure 5-1 shows the corresponding orientation distribution function profiles

computed from a model-independent q-hall ODF formula derived in [5, Eq. (21)] with

spherical harmonics expansion up to I = 8.

For the experiments using simulation data, the deconvolution methods described in

the previous chapter were all implemented using MATLAB as follows:


The pseudo-inverse method w = pinv(A)s as in (P1) where piny is MATLAB's
built-in function for computing pseudo inverse of a matrix;
The damped least squares method as a solver of the Tikhonov regularization
problem (P2) aS Well aS the GCV criterion were implemented using the MATLAB
regularization toolbox developed by Hansen [6:3];
The two different L1 minimization methods (?3) ndil (P4) were imp~lll~lemete using
the L1-MAGIC MATLAB package;










*The nonnegative least squares method was implemented by the the 1\ATLAB
optimization toolbox function, Isquonneg.

The first experiment was done on the 1-fiber HARDI simulation data shown in Figfure

5-1 without noise. The deconvolution problem to be solved was formulated with the

matrix A being constructed by using the Wishart model with p = oc and the tessellation

containing :321 unit vectors sampled from a hemisphere. Since the given signal, on the

righthand side, is 81 dimensional, the matrix A is of size 81 x :321 and the unknown of this

under-determined system is a :321 dimensional weight vector.

Figure 5-2 plots the results of w obtained from these methods. The initial guess

for w was set to the zero vector for all the iterative methods (P3), (P4) and (P5). A

qualitative impression of these methods is clearly indicated from Figure 5-2. The least

squares solution to (P1) contains a large number of negative weights and has a relatively

large magnitude. A zeroth-order Tikhonov regfularization is able to reduce the magnitude

significantly but does not help achieve the sparsity and nonnegativity. By minimizing the

L1 norm with equality or quadratic constraints, (P3) yields a relative sparse solution, but

the magnitude and the negative values are not well controlled. The result obtained by (P4)

has better sparsity and nonnegativity. Evidently, the best result is produced by solving

(P5) using the nonnegative linear least squares. Among the :321 components, only two are

significant spikes, both of which lie in the neighborhood of true fiber orientation (:$Oo 90o).

It is important to note that the true fiber orientations do not necessarily occur at the

maxima of the discrete w vector. Although all of these different results of w actually lead

to very similar displacement probability functions P(r) after taking the Fourier transform,

a sparse positive representation of w is much more desirable as it provides very good

initial guess for estimating the extrema of P(r) which correspond to the fiber orientations.

A more realistic comparison on noisy 2-fiber simulation profiles is presented in Figure

5-1. The noisy profiles were created by adding Rician-distributed noise with increasing

noise levels (o- = .01,.02,...,.10). And for each noise level, we generated 100 random











(P1) Pseudoinverse solution 0.2 p3) Min-L1
(Pg) Tikhonov regularization
0.15 with equality constraints

001



-1 -0.05 -4
0 100 200 300 0 100 200 300 0 100 200 300

1.5 0.15
--(31.7, 90.0)
(P4) with t = 0.1 (P4) with t = 0.5 0.8
1~ 0.1
0.61 (P5)
0.51 0.05~11 Non-negative least squares
0.4
0 =- -- 1- C / II p 'I 0.2
(23.8, 90.0)
-0.5 -0.05 0
0 100 200 300 0 100 200 300 0 100 200 300

Figure 5-2. Results of w on 1-fiber HARDI simulation data using different deconvolution
methods. The x-axis shows the indices of w while the y-axis presents the
corresponding numerical values of w. The matrix A is of size 81 x 321 and is
built from the tensor kernel model and Wishart basis with
p c o, A = {0.0015, 0.0004, 0.0004}. Reproduced from [73] with permission.
@ [2007] IEEE.


samples. Then, we formulate the problem as in Eq. (4-43) using the 321 tessellation

vectors and the Wishart model with p= 2. In this experiment, the following numerical

methods were tested to recover the weight vector w:

(a) the pseudo-inverse method as in (P1);

(b) the damped least squares as in (P2) (the damping factor is empirically chosen to be

So- which gives quite satisfactory results);

(c) the damped least squares method with the damping factor determined using the

GCV criterion;

(d) the Min-L1 norm with equality constraints as in (P3);

(e) the Min-L1 norm with quadratic constraints as in (P4) (e = 1.0) ;

(f) the nonnegative least squares as in (Ps)


2









In all these methods, the deconvolution formulation is derived from the mixture of

Wisharts model [75]. The GCV solution was used as the initial guess in both the methods

(d) and (e). For the nonnegative least squares (NNLS) approach, the initial guess for w
was ahr-l- .- set to the zero vector.

As discussed in OsI Ilpter 4, once the deconvolution problem formulated from the

mixture of Wisharts model is solved, quantities including the probability displacement

function restricted on a sphere, P(|r| = r), as in DOT [119], radial integral of P(r), as in

QBI [149], and the integral of P(r)T2 aS in DSI [160] are analytically computable. In this

experiment, the f P(r)T2dT iS chosen as the quantity to be compared and all the resulting

functions are represented by spherical harmonics expansions terminated after order 1 = 8.

First, to assess the noise resistance of these methods, for each noise level, we cal-

culated the similarity between the resulting probability profile and the corresponding

noiseless profile using the angular correlation coefficient formula [5, Eq. (71)]


TA"= =" (5-2)
[E m=- -U1 |wm2 1/2 1 1Im=-1l 2 1/2

where {wm}) and {<.. } are the spherical harmonic coefficients of the two functions to be

compared. The range of the angular correlation coefficient is from 0 to 1 where 1 implies

identical orientational profiles.

We also estimated the fiber orientations of each system by numerically finding the

maxima of the spherical functions with a QuI I-;-N. i.--ton numerical optimization algorithm

and computed the deviation angles between the estimated and the true fiber orientations.

As expected, the results of method (a), the pseudo-inverse method (P1), degrades quickly

as the noise level increases. It has also been observed that method (e), Min-L1 with

quadratic constraints (P4), produces much worse results than the other four methods

(b,c,d,f). In general, the results of method (d), Min-L1 with equality constraints (P3), arT

very close to the results of method (c), the Damped Least Squares method with damping










factor determined by GCV, due to the fact that method (d) starts with the solution of

method (c).

Figure 5-3 shows the mean and standard deviation of the angular correlations and

the deviation angles in the two-fiber experiment, respectively. In these two figures, we

do not include the results of the pseudo-inverse method (P1) and the results of Min-L1

with quadratic constraints (P4) since both methods are extremely sensitive to the noise

according to our observations. The angular correlations obtained using the four methods

shown in Figure 5-:3 are all very high (> 0.9). While the GCV and Min-L1 with equality

constraints are slightly worse with relatively larger standard deviations. It is also observed

that the angular correlations produced by the NNLS method are very close to 1 and have

the smallest standard deviation. The DLS method with the damping factor empirically

chosen to be So- also exhibits quite satisfactory stability. In terms of the accuracy of the

fiber orientation estimation, it is also clear from Figure 5-3 that the NNLS method gives

the most accurate results consistently. While there is no significant difference between the

other three methods. Note that in all the cases, Min-L1 methods with both equality and

quadratic constraints were initialized with the solution returned by the GCV method and

the NNLS method alv-a-l- started with a zero vector.

Finally, as a conclusion to our experiments on the simulated data, a quantitative

comparison was performed among the proposed method mixture of Wisharts (\l OW)

model and the two model-free methods, namely, the Q-hall ODF [149] and the DOT

[119]. All the resulting P(r) surfaces were represented by spherical harmonics coefficients

up to order 1 = 6 and the Q-hall ODF is computed using the formula in [5, Eq. (21)].

First, to gain a global assessment of these methods in terms of stability, we calculated

the similarity between each noisy P(r) and the corresponding noiseless P(r) using the

angular correlation coefficient formula given in [5, Eq. (71)]. The angular correlation

ranges from 0 to 1 where 1 implies identical probability profiles. Then, we estimated the

fiber orientations of each system by finding the maxima of the probability surfaces with










a CMI I-;-N E-ton numerical optimization algorithm and computed the deviation angles

between the estimated and the true fiber orientations. Figure 5-4 shows the mean and

standard deviation of the angular correlation coefficients, and error angles, respectively,

for the two-fiber simulation. Note that among the three methods examined, only MOW

results in small error angles and high correlation coefficients in presence of relatively

large noise. This trend also holds for the 1-fiber and the 3-fiber simulations. This can be

explained by noting that NNLS is able to locate the sparse spikes quite accurately even in

the presence of a lot of noise.

5.2 Real Data Experiments

The aforementioned reconstruction methods were tested on two real datasets provided

to us by researchers with the McE~night Brain Institute at the University of Florida. The

first dataset was acquired from a perfusion-fixed excised rat optic chiasm at 14.1 Telsa

using a Bruker Avance imaging system (Bruker NMR Instruments, Billerica, MA) with a

diffusion-weighted spin echo pulse sequence. Note that the rat optic chiasm is well suited

for the validation of the fiber reconstruction results because of its distinct myelinated

structure with both parallel and decussating (crossing) optic nerve fibers.

The imaging parameters used for the optic chiasm dataset are shown in Table 5-1. In

this optic chiasm dataset, there are 46 images acquired from 46 gradient directions at a

b-value of 1250s/mm2 and 6 additional images acquired at b a Os/mm2. Echo time and

repetition time were 23ms and 0.5s respectively; a and 6 values were set to 12.4ms and

1.2ms respectively; bandwidth was set to 35kHz; signal average was 10; a volume size of

128 x 128 x 5 and a resolution of 33.6 x 33.6 x 200pm3 WaS used. The optic chiasm images

were downsampled to 67.2 x 67.2 x 200pm3 TOSolution before the subsequent computation.

Two sets of experiments were performed on this optic chiasm dataset. The first

experiment was designed to compare the four different numerical methods discussed in

Section 4.4, namely, (a) damped least squares with GCV, (b) Min-L1 with quadratic

constraints, (c) damped least squares with a fixed regularization parameter, and (d)













-- I- IP2)Damped Least Squares (DLS)
IP2)DLSIGCV
-i (P3)M nLlwth equa ty
~ IPS)Nonnegatue Least Squares


O

S098~
















O


0.8


0.0 0



Figure 5-4.


std. dev. of noise std. dev. of noise

(a) (b)
Mean and standard deviation of (a) angular correlation coefficient and (b)
error angles for the two-fiber simulation. At each noise level, the four methods
compared here are (1) the DLS method as in (P2) (the damping factor is
empirically chosen to be So-); (2) the DLS method with the damping factor
determined using the GCV criterion; (3) the Min-L1 norm with equality
constraints as in (P3) initializedd with the solution obtained by DLS+GCV;
(4) the NNLS method as in (Ps). The di;11lai- II values are averaged over the
two fibers. Reproduced from [73] with permission. @[2007] IEEE.


-- DOT
QBI
*MOW


-.DOT
-QBI
*MOW
0.05
std. dev. of noise


0.05
std. dev. of noise


The plots on the left and on the right show the statistics of angular correlation
coefficients and error angles for the 2-fiber simulation, respectively. The
di1 pl i- 4I values for error angles are averaged over the two fiber orientations.
Reproduced from [77] with permission. @[2009] Springer.









Table 5-1. Imaging parameters used for the optic chiasm dataset.
Imaging parameters used for the optic chiasm dataset
magnetic field strength 14.1 Telsa
gradient directions 46 at high b-value, 6 at low b-value
high b-value 1250s/mm2
low b-value a Os/mm2
echo time (TE) 23ms
repetition time (TR) 0.5s
gradient pulse separation (A) 12.4ms
gradient pulse duration (5) 1.2ms
bandwidth 35kHZ
signal averaging 10
volume size 128 x 128 x 5
voxel size 33.6 x 33.6 x 200p~m3


non-negative least squares (NNLS). The resulting displacement probability profiles on a

region of interest are shown in Figure 5-5. For each method, the corresponding So image

is also shown in the upper left corner of each panel as a reference. For the sake of clarity,

we excluded every other voxel and overlaid the probability surfaces on the generalized

anisotropy (GA) maps [117]. GA is the variance of normalized diffusivity function. Higher

values of GA (brighter regions) indicate higher anisotropy.

As can be seen from Figure 5-5, the fiber-crossingf in the optic chiasm region is not

identifiable in either results obtained from the method using GCV (a) or the results

obtained from the method of Min-L1 with quadratic constraints (b), while both the

DLS method (c) and the NNLS method (d) are able to demonstrate the distinct fiber

orientations in the central region of the optic chiasm where ipsilateral m--- linated axons

from the two optic nerves cross and form the contralateral optic tracts. The failure of the

GCV method is due to the fact that the damping factors estimated by the GCV method

may not ak- -li- be optimal. The Min-L1 method with quadratic constraints only yields

satisfactory results in regions mostly populated with single fiber, which is consistent to

the previous observation on the 2-fiber simulation. It is also evident from Figure 5-5 that

compared to other three numerical methods, the NNLS scheme yields significantly sharper

displacement probability profiles.










Similar to the experiments on the simulated data, we also compare the proposed mix-

ture of Wisharts (\lOW) model with two model-free methods, namely, the QBI-ODF[149]

and the DOT [119]. The rest of this paragraph is reprinted from [77] with permission and

is copyrighted by Springer. Figure 5-6 shows the reconstruction results generated using

four different methods, namely, (a) QBI-ODF, (b) DOT, (c) MOW+Tikhonov regular-

ization, and (d) MOW+NNLS, on the same region of interest shown in Figure 5-5. The

fast combinatorial NNLS method [23] is used here as the NNLS solver. The computation

time for this region of interest containing 1024 voxels is less than 0.5 second for all four

methods on an Intel Core Duo 2.16 GHz CPU while the standard NNLS takes about 8

seconds. As seen in the Figure 5-6, the fiber-crossingfs in the optic chiasm region cannot

he identified by using the QBI-ODF method. Note that both the DOT method and the

MOW method with two different schemes are able to demonstrate the distinct fiber ori-

entations in the central region of the optic chiasm. However, it is evident from the figure

that compared to all other solutions, the MOW technique in conjunction with the NNLS

scheme yields significantly sharper displacement probability surfaces. This is particularly

horne out in the optic chiasm, in the center of each panel. The probability surfaces in the

QBI and DOT models are blurred, in part, because both vield a corrupted P(r) rather

than the actual displacement probability surfaces.The corrupting factor for the QBI is a

zeroth order Bessel function, for the DOT method it is a function that does not have an

analytic form. This corruption affects the accuracy of the reconstructed fiber orientations

as evidenced in the simulated data case where the ground truth was known. Note that

validating the precise angle of the fiber crossing in this real data set is non-trivial as it will

need a histology stack to be created and then fiber directions estimated from this stack to

be validated against those obtained from the DW-AIRI data.

To investigate the capability of diffusion weighted imaging in revealing the effects in

local tissue caused by diseases or neurologic disorders, further experiments were carried


















optic nervp- /)yj t;* & M// j
//////###1/// optic
&~s~b~9#J//QH~+$ ChiaSm

~"ff 4 optic tract
~--u~~idM/98~3ft

(a) Inisj J sis 3






optic nervp- / / 1' / # #207//

~u~p#p/)######ly~ ChiaSm


,~d~~U~9~ aft oWLptic tract


p ,,,,,,,/B ,, cmas

opti nr opictrc
(b) I st M I/I



opicneve/ / 4 / < *4/p


we .,47,ogyE. cmiasm

=ws+tit ptic tract


Probability maps computed using (a) damped least squares with GCV; (b)
Min-L1 with quadratic constraints (e = 1) initialized from (a);(c) damped least
squares with fixed regularization parameter (a~ = 0.6); ( d) non-negative least
squares from a rat optic chiasm data set overlaid on axially oriented GA [114]
maps. The decussations of myelinated axons from the two optic nerves at the
center of the optic chiasm are readily apparent. In all the plates, the
corresponding reference (So) image is shown in the upper left corner.
Reproduced from [73] with permission. @[2007] IEEE.


Figure 5-5.





















optic nerve ))J'di3///// L

&///s,~~uodgyj optic
JJ.~~~~C~~IeS ddjgJjIty cniasm



(a) QBI3J optic tract



opt Qicnene s //i fe ///


*** &// ess tge op4 c
*** ~44 &/O#$$# nam



R Iaic nev optic tract d
Figue 5-. Prbabiity surface coptdfr


optcne'e 7 944///

----**>>>Ms $$l cmasm







optic nerve ,)//b+/+/////




Nd MSOW optictrc



m~ ~ ~ a rat optic chas iae sig a


QBI-ODF, (b) DOT, (c) MOW+Tikhonov regularization, and (d)
MOW+NNLS. Note the decussation of row;linated axons from the two optic
nerves at the center of the optic chiasm. Reproduced from [77] with
permission. @ [2009] Springer.









out on two data sets collected from a pair of epileptic/normal rat brains. The rest of this

section is reprinted from [75] with permission and is copyrighted by Elsevier.

Table 5-2. Imaging parameters used for the rat brain dataset.
Imaging parameters used for the rat brain dataset
magnetic field strength 17.6 Telsa
gradient directions 46 at high b-value, 6 at low b-value
high b-value 1250s/mm2
low b-value 100s/mm2
echo time (TE) 28ms
repetition time (TR) 1.4s
gradient pulse separation (A) 17.5ms
gradient pulse duration (5) 1.5ms
bandwidth 750M~HZ
signal averaging 20 for high b-value and 5 for low b-value
volume size 200 x 100 x 32
voxel size 150 x 150 x 300p~m3


The imaging parameters used for the rat brain dataset are shown in Table 5-2. The

multiple-slice diffusion weighted image data were measured at 750 MHz using a 17.6 Tesla,

89 mm bore magnet with Bruker Avance console (Bruker NMR Instruments, Billerica,

MA). A spin-echo, pulsed-field-gradient sequence was used for data acquisition with a

repetition time of 1400 ms and an echo time of 28 ms. The diffusion weighted gradient

pulses were 1.5 ms long and separated by 17.5 ms. A total of 32 slices, with a thickness

of 0.3 mm, were measured with an orientation parallel to the long-axis of the brain (slices

progressed in the dorsal-ventral direction). These slices have a field-of-view 30 mm x 15

mm in a matrix of 200 x 100. The diffusion weighted images were interpolated to a matrix

of 400 x 200 for each slice. Each image was measured with 2 diffusion weigfhtingfs: 100

and 1250s/mm2. Diffusion-weighted images with 100s/mm2 Were measured in 6 gradient

directions determined by the vertices of an icosahedron in one of the hemispheres. The

images with a diffusion-weighting of 1250s/mm2 Were measured in 46 gradient-directions ,

which are determined by the tessellation of the icosahedron on the same hemisphere. The

100s/mm2 images were acquired with 20 signal averages and the 1250s/mm2 images were

acquired with 5 signal averages in a total measurement time of approximately 14 hours.










Figure 5-7 shows the displacement probabilities calculated front excised coronal

rat brain 1\RI data in a, (a) control and (b) an epileptic rat. The hippocanipus and

entorhinal cortex is expanded and depicts the orientations of the highly anisotropic and

coherent fibers. Note voxels with crossing orientations located in the dentate gyrus (dg)

and entorhinal cortex (ec). The region superior to CA1 represent the stratum lacunosuni-

moleculare and statunt radiatunt. Note that in the control hippocanipus, the molecular

1 .,-< c and stratum radiatunt fiber orientations paralleled the apical dendrites of granule

cells and pyramidal neurons respectively. In the epileptic hippocanipus, the CA1 subfield

pyramidal cell 1 .,-< c is notably lost relative to the control. The architecture of the dentate

gyrus is also notably altered with more evidence of crossing fibers. Future investigations

employing this method should improve our understanding of normal and pathologically

altered neum~ .Il 1Inyl~: in regions of complex fiber architecture such as the hippocanipus

and entorhinal cortex.












































22


s-- DG


S~tS$SSSA
dd ~iirid





Hilu


Figure 5-7.


Probability maps of coronally oriented GA images of a control and an epileptic
hippocampuus. Upper left corner shows the corresponding reference (SO) images
where the rectangle regions enclose the hippocampi. In the control
hippocampus, the molecular 1 ;r and stratum radiatum fiber orientations
paralleled the apical dendrites of granule cells and pyramidal neurons
respectively, whereas in the stratum lacunosum, molecular orientations
paralleled Schaffer collaterals from CA1 neurons. In the epileptic
hippocampus, the overall architecture is notably altered; the CA1 subfield is
lost, while an increase in crossing fibers can he seen in the hilus and dentate
gyrus (dg). Increased crossing fibers can also be seen in the entorhinal cortex
(ec). Fiber density within the statum lacunosum molecular and statum
radiale is also notably reduced, although fiber orientation remains unaltered.
Reproduced with permission from [75] @[2007] Elsevier.


.: ::. CA3
SSSrI Go
W~Iilus1


Oddestd//dd

d o as SSlllllltfijdS
IU~~1Y ~ Al1ddstaiddsra
**###ekkthdd 445St-
(l#ba~theb66 San--r~r __


(b) epileptic









CHAPTER 6
DISCUSSION AND CONCLUSIONS

6.1 Summary

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging

technique that allows neural tissue architecture to be probed at a microscopic scale

in vivo. By producing quantitative measurements of on water molecular motion, DW-

MRI can be processed to map the fiber paths in the brain white matter. This valuable

information can be further exploited for neuronal connectivity inference and brain

developmental studies [87].

In this dissertation, a novel mathematical model for the diffusion weighted MR

signal attenuation is presented. The key postulation of the proposed model is that at

each voxel the diffusion of water molecules is characterized by a continuous mixture of

diffusion tensors. An interesting observation based on this continuous tensor distribution

model is that the MR signal attenuation can be expressed as the Laplace transform of the

associated tensor probability distribution function. It has also been show that when the

mixing distribution is parameterized by Wishart distributions, the resulting close form of

Laplace transform leads to a Rigaut-type fractal expression. This Rigaut-type function

exhibits the expected .-i-mptotic power-law behavior and has been phenomenologically

used in the past to explain the MR signal decay but never with a rigorous mathematical

justification until the development of the proposed model [74-76]. It is easy to show

that both the traditional diffusion tensor model and the multi-tensor model are limiting

cases of this continuous mixture of tensors model. Additionally, in the long wavelength or

low-q regime the proposed model leads to a quadratic decay which is consistent with the

traditional diffusion tensor model [14].

Another contribution of this work is on tackling the challenging problem of multi-fiber

reconstruction from the diffusion MR images. The mixture of Wisharts model, as a natu-

ral parametrization of the desired tensor distribution function, is used to describe complex










tissue structure involving multiple fiber populations. The multi-fiber reconstruction is

then formulated in a unified deconvolution framework [72, 73] which also includes some

other previously published approaches in this field. One central topic of this work is to

investigate several different deconvolution methods in the context of multi-fiber recon-

struction in diffusion AIRI. Experiments on both simulations and real data have shown

the favorable results from the proposed mixture of Wisharts (\lOW) model in conjunction

with the nonnegative least squares (NNLS) method, comparing with competing models

and methods from literature.

6.2 Open Problems

In this dissertation, we have developed a novel mathematical model which depicts the

diffusion AIR signal attenuation reasonably well, supported by both theoretical analysis

and experimental results. Regrettably but fortunately, there remain a number of open

problems .

6.2.1 Nonparametric Inverse Laplace Transform

The key postulation of the proposed mathematical model is that at each voxel the

diffusion of water molecules is characterized by a continuous mixture of diffusion tensors.

Based on this assumption, we also reveal the Laplace transform relationship between the

signal attenuation and the underlying tensor probability distribution. In our approach,

this tensor distribution is modeled by wishart distributions or mixture of Wisharts.

This choice is mainly justified by the existing closed form Laplace transform of Wishart

distributions which also yields the expected Rigaut-type fractal expression. Though it

is tempting and maybe more convincing to directly invert the Laplace transform in a

nonparametric way, one has to recognize the difficulties entailed in inverting Laplace

transform in the high dimension from the noisy data (see [54] for "cogent reasons for the

general sense of dread most mathematicians feel about inverting the Laplace ti Ion-1. .i ts ).

Recent work by Leow et al. [90] is very similar to our model in that a tensor distri-

bution function (TDF) is associated to each voxel and the inverse problem there is also










to recover this tensor distribution function that best explains the observed diffusion AIR

signal. In [90], the domain of the TDF is a four-paranleter space which is reduced front the

original 6-dintensional tensor space by assuming that fiber tracts are cylindrical. Then a

gradient descent method is used to solve for an optimal TDF that nminintizes the difference

between the observed signal and the predicted signal in the least-square sense. The current

mixture of Wisharts (\lOW) model eniploi-. .1in this work actually reduces to a spherical

deconvolution model by assuming the cylinder syninetry and fixing the eigenvalues of the

tensor parameter in each Wishart components. These simplifying assumptions lead to a

linear system and hence can he solved efficiently. An alternative but much more compli-

cated deconvolution method would be a B li-o Io approach where all the parameters in

the MOW model including the tensor eigfenvalues and the weights can he imposed with

suitable prior distributions.

6.2.2 Adaptive Sparse Dictionary Learning

We have shown that several existing nmulti-fiber reconstruction models can he ex-

pressed in a unified deconvolution framework model and reformulated into the problem of

seeking a sparse and positive solution to a linear system [7:3]. In most existing approaches

that lie in this unified deconvolution framework, the system matrix, A, or the so-called

dictionary in the context of signal representation theory, is fixed depending on the choices

of the convolution kernel, the paranietrization of the volume fraction function, and the

discretization scheme. The exception is a recent work proposed in [1:3] where the shape of

the convolution kernel function is estimated simultaneously with the rest of the unknowns

of the model. However, the basis function of this adaptive kernel is still fixed as a spline

function. A significant extension to [1:3, 7:3] would be a supervised learning hased approach

that infers a learned dictionary front the training data instead of a pre-defined dictionary

as in [52, 100] and furthermore allows a sparse encoding of the diffusion signal using this

learned dictionary.









6.2.3 Sub-voxel Fiber Bundles Classification

It has been widely recognized that the diffusion tensor model does not perform

well in regions containing intra-voxel orientational heterogeneity. This limitation of the

diffusion tensor model has prompted the development of many multi-fiber reconstruction

methods, including those reviewed in Section 3.3, such as higher order tensor it., 1_;ng [7 14],

diffusion spectrum ill. 1_;hly [7 60], q-hall ill. 1_;ity [7 49], probabilistic PAS-AIRI[70], spherical

deconvolution[14:3], diffusion orientation transform[119], and the mixture of Wisharts

model [75] proposed in this work. However, to the best of our knowledge, none of these

approaches is able to differentiate between complex configurations of intra-voxel fiber

bundles, e~g fibers crossing, kissing, bending or twisting within a voxel. In other words,

these methods still suffer from topological ambiguity, which can not he easily resolved by

only estimating fiber orientations.

We believe that the ability of successfully labelling voxels into distinct sub-voxel

fiber bundle configurations would be crucial in the next generation of fiber tractography

algorithms. Recently, UCniie7lj~ v et al. [1:34] proposed a curve inference method that uses

differential geometric estimates in a local neighborhood to differentiate the structures

of fanning and curving fiber bundles. Barmpoutis et al. [11] presented a novel method

for estimating a field of .l-i-mmetric spherical functions, dubbed tractosemas, which can

he used to model .I-i-mmetries such as pl w-ing fibers. The mixture of Wisharts model

proposed in this work has been shown in [9] as an excellent generative model for depicting

the diffusion AIR signal attenuation. It would be an interesting future research topic to

investigate if the combination of this generative model and some discriminative models can

he potentially used to distinguish different sub-voxel fiber bundle categories and improve

the performance of the fiber pathway reconstruction.









REFERENCES


[1] I. Aganj, C. Lenglet, and G. Sapiro. ODF reconstruction in q-ball imaging with
solid angle consideration. In IEEE International Symp~osium on Biomedical Imaging
(ISBI), 2009.

[2] D. C. Alexander. Multiple-fibre reconstruction algorithms for diffusion MRI. Proc.
N. Y. Acad. Sci., 1064:113-133, 2005.

[3] D. C. Alexander. Maximum entropy spherical deconvolution for diffusion MRI. In
Information Processing in M~edical Imaging, pages 76-87, 2005.

[4] D. C. Alexander, G. J. Barker, and S. R. Arridge. Detection and modeling of non-
Gaussian apparent diffusion coefficient profiles in human brain data. M~agn. Reson.
M~ed., 48(2):331-340, 2002.

[5] A. W. Anderson. Measurement of fiber orientation distributions using high angular
resolution diffusion imaging. M~agn. Reson. M~ed., 54(5):1194-1206, 2005.

[6] T. W. Anderson. An Introduction to M~ultivariate Statistical A,:al;,.: John Wiley
and Sons, 1958.

[7] Y. Assaf and P. J. Basser. Composite hindered and restricted model of diffusion
(CHARMED) mr imaging of the human brain. Neurolmage, 27(1):48-58, 2005.

[8] Y. Assaf, R. Z. Freidlin, G. K(. Rohde, and P. J. Basser. New modeling and exper-
imental framework to characterize hindered and restricted water diffusion in brain
white matter. M~agn. Reson. M~ed., 52(5):965-978, 2004.

[9] A. Barmpoutis and B. C. Vemuri. Information theoretic methods for diffusion-
weighted mri analysis. In Emerging Trends in V/isual Conre,Hay,.:: LIX Fall Collo-
quium, pages 327-346, 2008.

[10] A. Barmpoutis, B. Jian, B. C. Vemuri, and T. M. Shepherd. Symmetric positive 4th
order tensors and their estimation from diffusion weighted MRI. In N. K~arssemeijer
and B. P. F. Lelieveldt, editors, IPM~I, volume 4584 of Lecture Notes in Comp~uter
Science, pages 308-319. Springer, 2007.

[11] A. Barmpoutis, B. C. Vemuri, D. Howland, and J. R. Forder. Extracting tractosemas
from a displacement probability field for tractography in DW-MRI. In M~ICCAI (1),
pages 9-16, 2008.

[12] A. Barmpoutis, M. S. Hwang, D. Howland, J. R. Forder, and B. C. Vemuri. Reg-
ularized positive-definite fourth-order tensor field estimation from DW-MRI.
Neurolmage, 45(1 sup.1):153-162, March 2009.

[13] A. Barmpoutis, B. Jian, and B. C. Vemuri. Adaptive kernels for multi-fiber
reconstruction. In IPM~I, 2009.










[14] P. J. Passer. Relationships between diffusion tensor and q-space MRI. M~agn. Reson.
M~ed., 47(2):392-397, 2002.

[15] P. J. Passer and C. Pierpaoli. Microstructural and physiological features of tissues
elucidated by quantitative-diffusion-tensor MRI. J. M~agn. Reson. B, 111(3):209-219,
1996.

[16] P. J. Passer and C. Pierpaoli. A simplied method to measure the diffusion tensor
from seven MR images. M~agn. Reson. M~ed., 39:928-934, 1998.

[17] P. J. Passer, J. Mattiello, and D. Le Bihan. MR diffusion tensor spectroscopy and
imaging. B.:'rine;, J., 66:259-267, 1994.

[18] P. J. Passer, J. Mattiello, and D. Le~ihan. Estimation of the effective self-diffusion
tensor from the NMR spin echo. J. Magn. Reson. B, 103:247-254, 1994.

[19] P. J. Passer, S. Pajevic, C. Peierpaoli, D. J, and A. Aldroubi. In vivo fiber tractogra
phy using DT-MRI data. M~agn. Reson. M~ed., 44(4):625-632, October 2000.

[20] T. Behrens, M. Woolrich, M. Jenkinson, H. Johansen-Berg, R. Nunes, S. Clare,
P. Matthews, J. Brady, and S. Smith. C'I I) Il:terization and propagation of un-
certainty in diffusion-weighted mr imaging. M~agn. Reson. M~ed., 50(2):1077-1088,
2003.

[21] T. Behrens, H. Johansen-Berg, S. Jbabdi, M. Rushworth, and M. Woolrich. Proba-
bilistic tractographny with multiple fibre orientations: What can we gain? Neurolm-
age, 34: 144-155, 2007.

[22] K(. M. Bennett, K(. M. Schmainda, R. Tong, D. B. Rowe, H. Lu, and J. S. Hyde.
C'I I) Il:terization of continuously distributed cortical water diffusion rates with a
stretched-exponential model. M~agn. Reson. M~ed., 50(4):727-734, 2003.

[23] M. H. V. Benthem and M. R. K~eenan. Fast algorithm for the solution of large-scale
non-negativity-constrained least squares problems. Journal of C'I,; ;..;;;., Irics, 18(10):
441-450, 2004.

[ 24] D. L. Bihan, editor. Diffusion and Perfusion M~agnetic Resonance Imaging, Appllica-
tions to Functional M~RI. Raven Press, 1995.

[25] D. L. Bihan and P. V. Zijl. From the diffusion coefficient to the diffusion tensor.
NM~R Biomd., 15(7-8):431-434, 2002.

[26] A. Bjorck. Theoretical Numerical M~ethods for Least Squares Problems. SIAM, 1996.

[27] F. Bloch. Nuclear induction. Phys. Rev., 70:460-474, 1946.

[28] D. S. Briggs. High [.:ll, 1.:1; Deconvolution of M~ .J.~ ,,rl 1; Resolved Sources. PhD
thesis, New Mexico Institute of Mining and Technology, 1995. Ch.4.










[29] P. T. Callaghan. Principles of Nuclear M~agnetic Resonance M11:- '* ***"rI;, Clarendon
Press, Oxford, 1991.

[30] J. S. Campbell, K(. Siddigi, V. V. Rymar, A. F. Sadikot, and G. B. Pike. Flow-based
fiber tracking with diffusion tensor and q-ball data: Validation and comparison to
principal diffusion direction techniques. Neurolmage, 27(4):725-736, 2005.

[31] E. J. Candes, J. Romberg, and T. Tao. Stable signal recovery from incomplete and
inaccurate measurements. Communications on Pure and Applied M~athematics, 59(8):
1207-1223, 2006.
[32] E. J. Candies, J. K(. Romberg, and T. Tao. Rous unetit rnipe:eatsg
nal reconstruction from highly incomplete frequency information. IEEE Transactions
on In:I~ for tal.>n The <;, 52(2):489-509, 2006.

[33] S. ('I! .1..11~ I-ekhar. Stochastic problems in physics and astronomy. Reviews of
Modern Ph;;-.. 15(1):1-89, 1943.

[34] C. ('1.. 1 I'hotel, Tschumperli6, R. Deriche, and O. Faugeras. Regularizing flows for
constrained matrix-valued images. Journal of M~athematical Imaging and V/ision, 20
(1-2):147-162, 2004.

[35] D. C'll, is and R. J. Plemmons. Nonnegativity constraints in numerical analysis. In
A. Bultheel and R. Cools, editors, The Proceedings of the Symp~osium on the Birth of
Numerical A,:ale;,: World Scientific Press, 2007.

[36] S. C'h.~ is, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit.
SIAM~ Journal on Scienti~fic Conr,,l;/. .9l 20(1):33-61, 1998.

[37] T. C!. II. i.- I t, A. Brunberg, and J. Pipe. Anisotropic diffusion in human white
matter: demonstration with MR techniques in vivo. EiRl.:; J. J..;,i 177:401-405, 1990.

[38] C. L. Chin, F. W. Wehrli, S. N. Hwang, M. Takahashi, and D. B. Hackney. Biexpo-
nential diffusion attenuation in the rat spinal cord: computer simulations based on
anatomic images of axonal architecture. M~agn. Reson. M~ed., 47:455-460, 2002.

[39] G. G. Cleveland, D. C. ('!, a ss, C. F. Hazlewood, and H. E. Rorschach. Nuclear
magnetic resonance measurement of skeletal muscle: anisotropy of the diffusion
coefficient of the intracellular water. B.:\ rIte,;, J, 16(9):1043-1053, 1976.

[40] R. Conlan. A life-saving window on the mind and body: The development of
magnetic resonance imaging. In B. ;;. .t.J D:..:. r e; ; The Path from Research to
Human BeniI~~ National Al I1. iny: of Sciences, March 2001.

[41] T. E. Conturo, N. F. Lori, T. S. Cull, E. Akbudak, A. Z. Snyder, J. S. Shimony,
R. C. McE~instry, H. Burton, and M. E. Raichle. Tracking neuronal fiber patly wei~ in
the living human brain. Proc Natl Acad Sci, 96:10422-10427, 1999.










[42] D. G. Cory and A. N. Garroway. Measurement of translational displacement
probabilities by NMR: an indicator of compartmentation. M~agn. Reson. M~ed., 14:
435-444, 1990.

[43] O. Coulon, D. C. Alexander, and S. R. Arridge. Diffusion tensor magnetic resonance
image regularization. Medical Image A,:tale;,.: 8(1):47-67, 2004.

[44] J. Crank. Mathematics of Diffusion. Oxford University Press, 2nd edition edition,
1980.

[45] P. Craven and G. Wahba. Smoothing noisy data with spline functions. Numer.
M~ath., (31):377-403, 1979.

[46] P. Debye, H. R. Anderson, and H. Brumberger. Scattering by an inhomogeneous
solid. ii. the correlation function and its application. J. Appl. Phys., 28:679-683,
1957.

[47] M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche. Apparent diffusion
coefficients from high angular resolution diffusion imaging: Estimation and applica-
tions. M~agn. Reson. M~ed., 56(2):395-410, 2006.

[48] M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche. Regularized, fast and
robust analytical q-ball imaging. M~agn. Reson. M~ed., 58:497-510, 2007.

[49] D. L. Donoho and J. Tanner. Thresholds for the recovery of sparse solutions via 11
minimization. In 40th Annual Conference on Information Sciences and S 1.~ii
pages 202-206, 2006.

[50] D. L. Donoho, M. Elad, and V. N. Temlyakov. Stable recovery of sparse overcom-
plete representations in the presence of noise. IEEE Transactions on Information
Ti,. ;, 52(1):6-18, 2006.

[51] A. Einstein. Investigations on the Theory of the Brownian M~ovement. Dover, 1956.
(Collection of Einstein's 5 papers on Brownina motion translated from the German).

[52] M. Elad and M. Aharon. Image denoising via sparse and redundant representations
over learned dictionaries. IEEE Transactions on Image Processing, 54:3736C3745,
2006.

[53] L. Elden. A note on the computation of the generalized cross-validation function for
ill-conditioned least squares problems. BIT, (24):467-472, 1984.

[54] C. L. Epstein and J. C. Schotland. The bad truth about Laplace's transform. Siam
Review, 50(3):504-520, 2008.

[55] W. Feller. An Introduction to P,~rol~.:l..7.;i Theory and Its Applications, V/olume I.
Wiley, 1973.










[56] L. Frank. C'I I) Il:terization of anisotropy in high angular resolution diffusion
weighted MRI. M~agn. Reson. M~ed., 47(6):1083-1099, 2002.

[57] P. Gill, W. Murray, and M. Wright. Practical Op~timization. Academic Press,
London, New York, 1981.

[58] G. H. Golub and C. F. Van Loan. Matrix: Comp~utations. The Johns Hopkins
University Press, 1996.

[59] G. H. Golub, M. T. Heath, and G. Wahba. Generalized cross-validation as a method
for choosing a good ridge parameter. Technometrics, (21):215-223, 1979.

[60] H. Groemer. Geometric Applications of Fourier Series and Spherical Harmonics,
volume 61 of Er:. ;;. 1.>pedia of M~athematics and Its Applications. Cambridge
University Press, 1996.

[61] A. K(. Gupta and D. K(. N I, I.r. Matrix: variate distributions, volume 579 of M~athe-
matics and Its Applications. C'!. 11pin .1' & Hall, 2000.

[62] P. Hagmann, T. G. Reese, W.-Y. I. T ngr R. Meuli, J.-P. Thiran, and V. J.
Wedeen. Diffusion spectrum imaging tractography in complex cerebral white matter:
an investigation of the centrum semiovale. In ISM~RM, page 623, 2004.

[63] P. C. Hansen. Regularization Tools: A Matlab package for analysis and solution of
discrete ill-posed problems. Numerical Algorithms, 6:1-35, 1994.

[64] C. W. Helstrom. Image restoration by the method of least squares. J. Op~t. Soc. Am.
A, 57:293-303, 1967.

[65] C. S. Herz. Bessel functions of matrix argument. The Annals of M~athematics, 61(3):
474-523, 1955.

[66] C. P. Hess, P. Mukherjee, E. T. Han, D. Xu, and D. B. Vigneron. Q-ball recon-
struction of multimodal fiber orientations using the spherical harmonic basis. M~agn.
Reson. M~ed., 56(1):104-117, 2006.

[67] T. Hosey, G. William, and R. Ansorge. Inference of multiple fiber orientations in
high angular resolution diffusion imaging. M~agn. Reson. M~ed., 54:1480-1489, 2005.

[68] M. F. Hutchinson and F. R. de Hoog. Smoothing noisy data with spline functions.
Numer. M~ath., (47):99-106, 1985.

[69] D. L. James and C. D. Twigg. Skinning mesh animations. AC'f~ Trans. Graph., 24
(3):399-407, 2005.

[70] K(. M. Jansons and D. C. Alexander. Persistent angular structure: new insights from
diffusion MRI data. Inverse Problems, 19:1031-1046, 2003.










[71] J. H. Jensen, J. A. Helpern, A. Ramani, H. Liu, and K(. K!1:; Diffusional
kurtosis imaging: the quantification of non-gaussian water diffusion by means of
magnetic resonance imaging. Iaugn. Reson. Afed., 5:3(6):14:32-1440, 2005.

[72] B. Jian and B. C. Vemuri. Multi-fiber reconstruction from diffusion AIRI using
mixture of Wisharts and sparse deconvolution. In Infortuation Processing in Afedical
Imaging, 2007.

[7:3] B. Jian and B. C. Vemuri. A unified computational framework for deconvolution to
reconstruct multiple fibers from diffusion weighted AIRI. IEEE Trans. Afed. Imaging,
26(11):1464-1471, 2007.

[74] B. Jian, B. C. Vemuri, E. Ojzarslan, P. Carney, and T. Alareci. A continuous
mixture of tensors model for diffusion-weighted AIR signal reconstruction. In IEEE
International Symp~osium on Biomedical Imaging, pages 772-775, 2007.

[75] B. Jian, B. C. Vemuri, E. Ojzarslan, P. Carney, and T. Alareci. A novel tensor
distribution model for the diffusion weighted AIR signal. Neurolmatge, :37(1):164-176,
2007.

[76] B. Jian, B. C. Vemuri, E. Ojzarslan, P. R. Carney, and T. H. Alareci. A novel tensor
distribution model for the diffusion weighted AIR signal. In ISAIRM, 2007.

[77] B. Jian, B. C. Vemuri, and E. Ojzarslan. A mixture of wisharts (\lOW) model for
multifiber reconstruction. In J. Weickert and D. Laidlaw, editors, V/isualixation and
Processing of Tensor Fields Advances and Perspectives. Springer, Berlin, 2009.

[78] B. Johansson, T. Elfying, V. K~ozlov, Y. Censor, P.-E. Forssiin, and G. Granlund.
The application of an oblique-projected landweher method to a model of supervised
learning. Afrathematical and C'omputer M~odelling., 4:3:892-909, 2006.

[79] J. K~arger and W. Heink. The propagator representation of molecular transport in
microporous crystallites. J. Iaugn. Reson., 51:1-7, 198:3.

[80] S. K~awata and O. Nalcioglu. Constrained iterative reconstruction by the conjugate
gradient method. IEEE Trans. Afed. Imaging, 4:65-71, 1985.

[81] S. K~awata, O. Nakamura, and S. Minami. Optical microscope tomography. I.
support constraint. J. Op~t. Soc. Am. A, 4(1):292-297, 1987.

[82] 31. H. K~hachaturian, J. J. Wisco, and D. S. Tuch. Boosting the sampling efficiency
of q-hall imaging using multiple wavevector fusion. Iaugn. Reson. Afed., 57(2):
289-296, 2007.

[8:3] P. B. K~ingsley. Introduction to diffusion tensor imaging mathematics: Part III.
tensor calculation, noise, simulations, and optimization. Concepts in Iafetnetic
Resonance. Part A, 28A:155-179, 2006.










[84] M. Koipf, R. Metzler, O. Haferkamp, and T. F. Nonnenmacher. NMR studies of
anomalous diffusion in biological tissues: Experimental observation of Liivy stable
processes. In G. A. Losa, D. Merlini, T. F. Nonnenmacher, and E. R. Weibel,
editors, Fractals in B..J.-it ~;, and M~edicine, volume 2, pages 354-364. Birkhiluser,
Basel, 1998.

[85] C. Lawson and R. J. Hanson. Solving Least Squares Problems. Prentice-Hall, 1974.

[86] M. Lazar, J. H. Jensen, L. Xuan, and J. A. Helpern. Estimation of the orientation
distribution function from diffusional kurtosis imaging. M~agn. Reson. M~ed., 60(4):
774-781, 2008.

[87] D. Le Bihan. Looking into the functional architecture of the brain with diffusion
MRI. Nature Reviews Neuroscience, 4(6):469-480, 2003.

[88] D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, and M. Laval-
Jeantet. MR imaging of intravoxel incoherent motions: Application to diffusion and
perfusion in neurologfic disorders. EiR~l..J.-it ~;, 161:401-407, 1986.

[89] C. Lenglet, J. S. W. Campbell, M. Descoteaux, G. Haro, P. CI .<.d-iiev, D. Wasser-
mann, A. Anwander, R. Deriche, G. B. Pike, G. Sapiro, K(. Siddigi, and P. Thomson.
Mathematical methods for diffusion MRI processing. N. ;,, .:l.Un.;p. 45(1 (Supplement
1)):S111-S122, 2009.

[90] A. D. Leow, S. Zhu, L. Zhan, K(. McMahon, G. I. de Zubicaray, M. Meredith, M. J.
Wright, A. W. Toga, and P. M. Thompson. The tensor distribution function. M~agn.
Reson. M~ed., 61(1):205-214, 2009.

[91] G. Letac and H. Massam. Quadratic and inverse regressions for Wishart distribu-
tions. The Annals of Statistics, 26(2):573-595, 1998.

[92] M. H. Levitt. Sp~in D:;min: Basics of Nuclear M~agnetic Resonance. John Wiley &
Sons, 2001.

[93] L. Li and T. P. Speed. Deconvolution of sparse positive spikes. Journal of Comp~uta-
tional and Grap~hical Statistics, 13(4):853-870, 2004.

[94] Z.-P. Liang and P. C. Lauterbur. Principles of M~agnetic Resonance Imaging : A
S. 0, al Processing Persp~ective. IEEE Press Series in Biomedical Engineering, 1999.9

[95] C.-P. Lin, V. J. Wedeen, J.-H. Clo! n!, C. Yao, and W.-Y. I. Tseng. Validation of
diffusion spectrum magnetic resonance imaging with manganese-enhanced rat optic
tracts and ex vivo phantoms. Neurolmage, 19(3):482-495, 2003.

[96] C. Liu, R. Bammer, and M. E. Moseley. Generalized diffusion tensor imaging
(GDTI): A method for characterizing and. imaging diffusion anisotropy caused by
non-gaussian diffusion. Isr J C'I,; ;;, 43(1-2):145-154, 2003.










[97] C. Liu, R. Bammer, B. Acar, and M. E. Moseley. C'I I) Il:terizing non-gaussian
diffusion by using generalized diffusion tensors. M~agn. Reson. M~ed., 51(5):924-937,
2004.

[98] H. Lu, J. H. Jensen, A. Ramani, and J. A. Helpern. Three-dimensional character-
ization of non-gaussian water diffusion in humans using diffusion kurtosis imaging.
NM~R Biomd., 19(2):236-247, 2006.

[99] D. G. Luenberger. Linear and Nonlinear P,~~,rl..yiense,: Addison-Wesley Publishing
Company, 1984.

[100] J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman. Supervised dictio-
nary learning. In Advances in Neural Information Processing S ;,1.i 2 1, page
1033C1040, 2008.

[101] A. M. Mathai. Jacobians of M~atrix: T,.rt.-f.rmations and Functions of M~atrix:
Argument. World Scientific, Singapore, 1997.

[102] T. E. McGraw, B. C. Vemuri, R. Yezierski, and T. H. Mareci. Von Mises-Fisher mix-
ture model of the diffusion ODF. In IEEE International Symp~osium on Biomedical
Imaging (ISBI), pages 65-68, 2006.

[103] O. V. Michailovich and Y. Rathi. On approximation of orientation distributions
by means of spherical ridgelets. In IEEE International Symp~osium on Biomedical
Imaging (ISBI), pages 939-942, 2008.

[104] L. Minati and W. P. Weglarz. Physical foundations, models, and methods of
diffusion magnetic resonance imaging of the brain: A review. Concepts in M~agnetic
Resonance, Part A, 30A:278-307, 2007.

[105] S. Mori and P. C. M. van Zijl. Fiber tracking: principles and strategies a technical
review. NM~R Biomd., 15(7-8):468-480, 2002.

[106] S. Mori, B. J. Crain, V. P. C'I II 1:.., and P. C. M. van Zijl. Three-dimensional
tracking of axonal projections in the brain by magnetic resonance imaging. Ann
Neurol, 45:265-269, 1999.

[107] M. E. Moseley, Y. Cohen, J. K~ucharczyk, J. Mintorovitch, H. S. Asgari, M. F. Wend-
land, J. Tsuruda, and D. Norman. Diffusion-weighted MR imaging of anisotropic
water diffusion in cat central nervous system. Eirl.;J..J.it~;, 176(2):439-445, 1990.

[108] M. E. Moseley, Y. Cohen, J. Mintorovitch, L. Chileuitt, H. Shimizu, J. K~ucharczyk,
M. F. Wendland, and P. R. Weinstein. Early detection of regional cerebral ischemia
in cats: comparison of diffusion and T2-Weighted MRI and spectroscopy. M~agn Reson
M~ed, 14:330-346, 1990.

[109] R. J. Murihead. Aspects of multivariate statistical 'i,,,. ,;, John Wiley & Sons, 1982.










[110] C. H. Neuman. Spin echo of spins diffusing in a bounded medium. Journal of
C'I, ;;; ... .; Ph;,; -:. 60 (11):4508-4511, 1974.

[111] T. Niendorf, R. M. Dijkhuizen, D. G. Norris, M. van Lookeren Campagne, and
K(. NW,-,]- ,\-. Biexponential diffusion attenuation in various states of brain tissue:
implications for diffusion-weighted it., I_;ay M~agn. Reson. M~ed., 36(6):846-857,
1996.

[112] S. C. Olhede and B. J. Whitcher. A statistical framework to characterise microstruc-
ture in high angular resolution diffusion imaging. In IEEE International Symp~osium
on Biomedical Imaging (ISBI), pages 899-902, 2008.

[ 113] E. Ojzarslan. Developments in diffusion-weighted M~RI with applications to neural
tissue. PhD thesis, University of Florida, 2004.

[114] E. Ojzarslan and T. H. Mareci. Generalized diffusion tensor imaging and analytical
relationships between diffusion tensor imaging and high angular resolution diffusion
imaging. M~agn. Reson. M~ed., 50(5):955-965, 2003.

[115] E. Ojzarslan, B. C. Vemuri, and T. Mareci. Fiber orientation mapping using
generalized diffusion tensor imaging. In International Symp~osium on Biomedical
Imaging: From Nano to M~acro, pages 1036-1038, 2004.

[116] E. Ojzarslan, T. M. Shepherd, B. C. Vemuri, S. J. Blackband, and T. H. Mareci. Fast
orientation mapping from HARDI. In J. S. Duncan and G. Gerigf, editors, M~ICCAI,
volume 3749 of Lecture Notes in Computer Science, pages 156-163. Springfer, 2005.
ISBN 3-540-29327-2.

[117] E. Ojzarslan, B. C. Vemuri, and T. H. Mareci. Generalized scalar measures for
diffusion MRI using trace, variance, and entropy. M~agn. Reson. M~ed., 53(4):866-876,
2005.

[118] E. Ojzarslan, P. J. Basser, T. M. Shepherd, P. E. Thelwall, B. C. Vemuri, and S. J.
Blackband. Observation of anomalous diffusion in excised tissue by characterizing
the diffusion-time dependence of the MR signal. J M~agn Reson, 183(2):315-323,
2006.

[119] E. Ojzarslan, T. M. Shepherd, B. C. Vemuri, S. J. Blackband, and T. H. Mareci.
Resolution of complex tissue microarchitecture using the diffusion orientation
transform (DOT). Neurolmage, 31:1086-1103, 2006.

[120] G. J. M. Parker. Analysis of MR diffusion weighted images. The British Journal of
FiRl.:; J. J..;,i 77:S176-S185, 2004.

[121] O. Pasternak, N. Sochen, and Y. Assaf. PDE based estimation and regularization of
multiple diffusion tensor fields. In J. Weickert and H. Hagfen, editors, V/isualization
and Image Processing of Tensor Fields. Springer, Berlin, 2005.










[122] 31. Perrin, C. Poupon, B. Rieul, A. Constantinesco, J.-F. Alangin, and D. L. Bihan.
Validation of Q-Ball imaging with a diffusion fibre-crossingf phantom on a clinical
scanner. Philo~s Tan~s R Soc Lond B Biol Sci., :360(1467):881-891, 2005.

[12:3] 31. Perrin, Y. Cointepas, A. Cachia, C. Poupon, B. Thirion, D. Riviibre, P. Cathier,
V. E. K~ouby, A. Constantinesco, D. L. Bihan, and J.-F. Alangin. Connectivity-based
parcellation of the cortical mantle using q-hall diffusion imaging. International
Journal of Biomedical Imaging, 8 (3) :1-10, 2008.

[124] V. Prasolov. Problems and theorems in linear rll'l', In. Translations of Mathematical
Monographs. American Mathematical Society, 1994.

[125] A. Ramirez-Manzanares, 31. Rivera, B. C. Vemuri, P. R. Carney, and T. H. Alareci.
Diffusion basis functions decomposition for estimating white matter intravoxel fiber
geometry. IEEE Tan~s. on Afedical hanging, 26(8):1091-1102, 2007.

[126] Y. Rathi, O. V. Michailovich, S. Bouix, and 31. E. Shenton. Directional functions for
orientation distribution estimation. In IEEE International Symp~osium on Biomedical
Damaging (15BI), pages 927-9:30, 2008.

[127] T. Reese, O. Heid, R. Weisskoff, and V. Wedeen. Reduction of eddy-current-induced
distortion in diffusion AIRI using a twice-refocused spin echo. Iaugn. Reson. Afed., 49
(1):177-182, 200:3.

[128] S. J. Riderer. AIR imagings: Its development and the recent Nobel prize. EiRl.:;J..J..I,i
2:31(:3):628-6:31, 2004.

[129] J.-P. Rigaut. An empirical formulation relating boundary lengths to resolution in
specimens showing 'non-ideally fractal' dimensions. J Microse, 13:3:41-54, 1984.

[1:30] J. P. Rigaut, D. Schoevaert-Brossault A. 31. Downs, and G. Landini. Asymptotic
fractals. In G. A. Losa, D. Merlini, T. F. Nonnenmacher, and E. R. Weibel, editors,
Fmectal~s in B..~~I J..;,i and M~edicine, volume 2, pages 7:385. Birkhauser, Basel, 1998.

[1:31] T. Sakai. Riemaunnican geometry, volume 149 of Tmansdetions of M~athematiall
Afonogra~h~s. American Mathematical Society, 1996.

[1:32] K(. E. Sakaie and 31. J. Lowe. An objective method for regularization of fiber
orientation distribution derived from diffusion-weighted AIRI. Neurolmerge, :34:
169-176, 2007.

[1:33] 31. F. Santarelli. Basic physics of mr signal and image generation. In L. Landini,
V. Positano, and 31. F. Santarelli, editors, Advanced charge Processing in Ahagnetic
Resonance Imaging. Taylor & Francis, 2005.

[1:34] P. CI7T71-ii.'v, J. S. W. Cambell, 31. Descoteaux, R. Deriche, G. B. Pike, and K(. Sid-
digi. Labeling of ambiguous subvoxel fibre bundle configurations in high angular
resolution diffusion AIRI. Neurobmatge, 41:58-68, 2008.










[135] P. N. Sen, M. D. Hiilimann, and T. M. de Swiet. Debye-Porod law of diffraction for


[136] J. Sijbers. S~ll,..rl and noise estimation from M~agnetic Resonance Images. PhD
thesis, University of Antwerp, M1 li- 1998.

[137] O. Soderman and B. Jonsson. Restricted diffusion in cylindirical geometry. J. M~agn.
Reson. B, 117(1):94-97, 1995.

[138] E. O. Steil1: I1 and J. E. Tanner. Spin diffusion measurements: Spin echoes in the
presence of a time-dependentfield gradient. J. C'I,; ;; Phys., 42(2):288-292, 1965.

[139] A. Terras. Harmonic ri,.71;,; on -;n,a Iiii ..i~ spaces and applications. Springer, 1985.

[140] A. Tikhonov and V. Arsenin. Solutions of Ill-posed Problems. Winston & Sons, 1977.

[141] P. Tofts, editor. Quantitative M~RI of the Brain: M~easuring C'lar,:l. Caused by
Disease. Willey, 2003.

[142] H. Torrey. Bloch equations with diffusion terms. Phys. Rev., 104:563565, 1956.

[143] J.-D. Tournier, F. Calamante, D. G. Gadian, and A. Connelly. Direct estimation
of the fiber orientation density function from diffusion-weighted MRI data using
spherical deconvolution. Neurolmage, 23(3):1176-1185, 2004.

[144] J.-D. Tournier, F. Calamente, and A. Connelly. Improved characterization of
crossing fibres: optimisation of spherical deconvolution parameters using a minimum
entropy principle. In Proceedings of the ISM~RM 18th Scienti~fc M~eeting and
Exhibition, Miama, Florida, 2005.

[145] J.-D. Tournier, F. Calamente, and A. Connelly. Improved characterisation of
crossing fibres: spherical deconvolution combined with Tikhonov regfularization.
In Proceedings of the ISM~RM 14th Scienti~fc M~eeting and Ex~hibition, Seattle,
Washington, 2006.

[146] J.-D. Tournier, F. Calamante, and A. Connelly. Robust determination of the fibre
orientation distribution in diffusion MRI: Non-negativity constrained super-resolved
spherical deconvolution. Neurolmage, 35:1459-1472, 2007.

[147] A. Tristan-Vega, C.-F. Westin, and S. Ai I-F l 1. .1.1. Estimation of fiber orien-
tation probability density functions in high angular resolution diffusion imaging.
Neurolmage, 47(2):638-650, 2009.

[148] D. Tuch, J. Wisco, M. K~hachaturian, L. Ekstrom, R. K~otter, and W. Vanduffel.
Q-ball imaging of macaque white matter architecture. Philos Trans R Soc Lond B
Biol Sci., 360(1457):869-879, May 2005.

[149] D. S. Tuch. Q-ball imaging. M~agn. Reson. M~ed., 52(6):1358-1372, 2004.

[150] D. S. Tuch. Diffusion M~RI of Complex; Tissue Structure. PhD thesis, MIT, 2002.










[151] D. S. Tuch, R. M. Weisskoff, J. W. Belliveau, and V. J. Wedeen. High angular
resolution diffusion imaging of the human brain. In Proc. of the 7th ISM~RM, page
321, Philadelphia, 1999.

[152] D. S. Tuch, T. G. Reese, M. R. Wiegell, N. Makris, J. W. Belliveau, and V. J.
Wedeen. High angular resolution diffusion imaging reveals intravoxel white matter
fiber heterogeneity. M~agn. Reson. M~ed., 48(4):577-582, 2002.

[153] D. S. Tuch, T. G. Reese, M. R. Wiegell, and V. J. Wedeen. Diffusion MRI of
complex neural architecture. Neuron, 40:885-895, 2003.

[154] E. von dem Hagen and R. Henkelman. Orientational diffusion reflects fiber structure
within a voxel. M~agn. Reson. M~ed., 48(3):454-459, 2002.

[155] G. Wahba. Sp~line M~odels for Observational Data. SIAM, 1990.

[156] S. Wakana, H. Jiang, L. M. T I, I.--Poetscher, P. C. M. van Zijl, and S. Mori. Fiber
tract-based atlas of human white matter anatomy. Eirl.:J.J.-it~;, 230(1):77-87, 2004.

[157] C. W. Wampler. Manipulator inverse kinematic solution based on damped least-
squares solutions. IEEE Trans. S;,liii Man and C;,lo ,, t.:. 1. 16(1):93-101,
1986.

[158] Z. Wang, B. C. Vemuri, Y. C!. li, and T. H. Mareci. A constrained variational
principle for direct estimation and smoothing of the diffusion tensor field from
complex DWI. IEEE Trans. M~ed. Imaging, 23(8):930-939, 2004.

[159] V. J. Wedeen, T. Reese, D. S. Tuch, M. R. Weigl, J.-G. Dou, R. Weiskoff, and
D. C!. -1. i. Mapping fiber orientation spectra in cerebral white matter with fourier
transform diffusion mri. In Proc. of the 8th ISM~RM, page 82, Denver, 2000.

[160] V. J. Wedeen, P. Hagmann, W.-Y. I. Tseng, T. G. Reese, and R. M. Weisskoff.
Mapping complex tissue architecture with diffusion spectrum magnetic resonance
imaging. M~agn. Reson. M~ed., 54(6):1377-1386, 2005.

[161] J. Weickert and H.Hagen, editors. Visualization and Processing of Tensor Fields.
Springer, 2005.

[162] M. R. Wiegell, H. B. W. Larsson, and V. J. Wedeen. Fiber crossing in human brain
depicted with diffusion tensor MR imaging. EiRl.:; J. J..;,i 217:897-903, 2000.

[163] J. Wishart. The generalized product moment distribution in samples from a normal
multivariate population. Biometrika, 20:32-52, 1928.

[164] D. A. Yablonskiy, G. L. Bretthorst, and J. J. Ackerman. Statistical model for
diffusion attenuated MR signal. M~agn. Reson. M~ed., 50(4):664-669, 2003.









[165] W. Zhan and Y. Yang. How accurately can the diffusion profiles indicate multiple
fiber orientations? A study on general fiber crossings in diffusion MRI. Journal of
Magnetic Resonance, 183(2):193-202, 2006.

[166] F. Zhang. The Schur Complement and Its Applications. Springer, 2005.









BIOGRAPHICAL SKETCH

Bing Jian was born in XinYu, JiangXi, CH.I. I After he completed his undergraduate

studies in computer science at the University of Science and Technology of CluI, I in July

1999, he chose to continue his studies at the same university, which earned him a master's

degree in electrical engineering in July 2002. He then joined the Ph.D. program of the

Department of Computer and Information Science and Engineering at the University of

Florida in August 2002. After his first year in the Ph.D. program, he started working

in the field of computer vision and medical image analysis. His research interests cover

a wide range of topics from computer vision and machine learning to medical imaging

and computer aided diagnosis. He hopes to make contributions to our society through

innovative research on where these disciplines meet.