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Shape Analysis via Unified Segmentation, Smoothing, and Registration of Riemannian Structures

Permanent Link: http://ufdc.ufl.edu/UFE0021640/00001

Material Information

Title: Shape Analysis via Unified Segmentation, Smoothing, and Registration of Riemannian Structures
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Lord, Nicholas Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: asymmetry, deformation, geometry, hippocampus, intrinsic, level, multiple, mumford, nonrigid, parcellation, reconstruction, registration, riemannian, segmentation, set, shah, simultaneous, smoothing, surface, variational
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: Locality is an important but oft-ignored aspect of shape asymmetry quantification, and segmentation is one method by which to make the locality of an analysis explicit. We have thus formulated an approach to surface registration and shape comparison which features an integrated segmentation component for the purpose of simultaneously identifying and separating regions by their evolving deformation characteristics. In the first instantiation, we achieve this effect through an adaptation of the Chan-Vese approach for image segmentation to the problem of segmenting the Riemannian structures of the very surfaces comprising the domain of the segmentation. We have successfully used the method's output on hippocampal pairs in an epilepsy classification problem, demonstrating improvement over global measures. Noting that a Chan-Vese-based approach to simultaneous segmentation and registration is inherently limited, we have also developed a unified approach to segmentation, smoothing, and nonrigid registration of images via extension of the Mumford-Shah functional, devised in such a way as to be applicable symmetrically and consistently to multiple (two or more) inputs. To conclude, we propose an extension of this unified framework (dubbed 'USSR') to the previously considered problem of 2D surface shape analysis (asymmetry quantification and localization), conferring the benefits of unbiasedness, consistency, multiple input processing, and nontrivial data field reconstruction.
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 Nicholas Andrew Lord.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Vemuri, Baba C.
Local: Co-adviser: Ho, Jeffrey.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021640:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021640/00001

Material Information

Title: Shape Analysis via Unified Segmentation, Smoothing, and Registration of Riemannian Structures
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Lord, Nicholas Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: asymmetry, deformation, geometry, hippocampus, intrinsic, level, multiple, mumford, nonrigid, parcellation, reconstruction, registration, riemannian, segmentation, set, shah, simultaneous, smoothing, surface, variational
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: Locality is an important but oft-ignored aspect of shape asymmetry quantification, and segmentation is one method by which to make the locality of an analysis explicit. We have thus formulated an approach to surface registration and shape comparison which features an integrated segmentation component for the purpose of simultaneously identifying and separating regions by their evolving deformation characteristics. In the first instantiation, we achieve this effect through an adaptation of the Chan-Vese approach for image segmentation to the problem of segmenting the Riemannian structures of the very surfaces comprising the domain of the segmentation. We have successfully used the method's output on hippocampal pairs in an epilepsy classification problem, demonstrating improvement over global measures. Noting that a Chan-Vese-based approach to simultaneous segmentation and registration is inherently limited, we have also developed a unified approach to segmentation, smoothing, and nonrigid registration of images via extension of the Mumford-Shah functional, devised in such a way as to be applicable symmetrically and consistently to multiple (two or more) inputs. To conclude, we propose an extension of this unified framework (dubbed 'USSR') to the previously considered problem of 2D surface shape analysis (asymmetry quantification and localization), conferring the benefits of unbiasedness, consistency, multiple input processing, and nontrivial data field reconstruction.
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 Nicholas Andrew Lord.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Vemuri, Baba C.
Local: Co-adviser: Ho, Jeffrey.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021640:00001


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SHAPE ANALYSIS VIA UNIFIED SEGMENTATION, SMOOTHING, AND
REGISTRATION OF RIEMANNIAN STRUCTURES


















By
NICHOLAS A. LORD


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

2007



































S2007 Nicholas A. Lord



































for more than I asked for,

for more than I deserved,

for more than I can rep ai










ACKENOWLED GMENTS


There's nothing original about striking a modest tone in an acknowledgment section,

but, seriously, I did nearly nothing. I was the fortunate beneficiary of a constant stream of

support, guidance, and inspiration, without which the probability of my getting through

the past four and a half years would have been zero. Thanks are in order to, at bare

minimum, the following:

Denise, who took the lead in seducing me into changing my master's degree program
application into the one that resulted in all of this. What was to be a handful of
additional classes for a substantial increase in salary instead became a miniature lifetime
of self doubt and sporadic insanity. Given the chance to do it again, I'd gladly let her lead
me .ll lii once more.

The University of Florida, the Department of Computer and Information Sciences and
Engineering, and the Alumni Fellowship Fund, for giving me the opportunity to complete
this degree, and with fine treatment throughout. It was good to be home for a while (and
I stand by my anti-northeastern rants now more than ever).

My advisor Dr. Vemuri, who gave me a spot in one of the best computer vision labs in
the world and held me to its standards. Or, more accurately, held me to its standards
after waiting through two years of my getting my stuff together. The experience was
mostly brutal, as with almost anything worth doing. I can safely recommend him to all
masochists interested in the field: if my experience is indicative, the world will be your
oyster once it's over. I suppose I can finally retire that voodoo doll.
The rest of the members of my dissertation committee, for their various contributions: my
external member, Dr. Hager, for donating his time; Jeff, for working closely with me on
all of my papers, for being the source of much of what was inspired about them, and for
ahr-l- .- finding a way to make rigorous explanations of mathematical arcanities digestible;
Anand and Arunava, for being the most intelligent, engaging, and inspirational lecturers
I've ever had the pleasure to sit with, and for making Indians less confusing to me.

Mom and Dad, who were put on this earth to deflect the bullet when I attempt to shoot
myself in the foot. Every time my neglect gets me into trouble, I call the same people
to bail me out of it. I'm sure I'll have to kick the habit eventually, but I wouldn't have
been able to walk this far without that crutch. Beyond the last-minute rides, calls, and
shipments, whatever I needed to make my life more comfortable, a laptop, a car, whatever,
it either literally or figuratively showed up on my doorstep. For telling me that everything
was going to be alright, and making sure that everything really was. Love you both,
pardon the slip into second person.

Nate, for being the coolest brother possible, and K~ristan, for sharing the wealth.










Gra and Pa, who have lived their lives under the odd impression that I am bright, from
the d 7i-a of Paddingfton and mini-golf until now, and without whom C'I!s -I~i x-it I- wouldn't be


Grandma Pauline and Larry, for ensuring that a sometimes-destitute graduate student
was able to do the travelling that he both wanted and needed to do. It's a lot more fun
without the scraping, and few have the luxury of avoiding it.

My entire extended family (in the Third World sense: those who know, know), including
the above, for being there, and never doubting.

The many people whom I've called labmates over these years, for being an absurdly
brilliant crop. When I joined, it was with the hope that I would finally be able to consider
myself as being surrounded by peers: I overshot and wound up with my superiors instead.
Few things are more humbling than considering what some of them have already managed
to do with their young lives. Special thanks is again due to Santhosh for his assistance in
the statistical analysis portion of my first journal paper, without which I might still be
working on my first journal paper.

The entire DeVicente clan, for taking me in in the midst of that little transitional period
of mine, and resisting the temptation to invoke the Baker Act. Starcraft II will be out
before long, if anyone's down.

Ranee and his family, for a desperately needed trip out of Dodge, and Michele, for a
desperately needed female ear. But I do hope that bit about the hyphenated surname was
a joke.

K~ofi, for an introduction to the San Francisco Bay Area which bordered on professional.
If that engineering thing of his doesn't pan out, he has a bright career as a tour guide
ahead of him. Also, for the thirteen-odd years prior, over which I was exposed to so much
I!. Il1l1s competition that I am now unable to mistake the pathological v-1I .i iv for anything
other than what it is.

Josh and Sakar, for being quite hairy, and for continuing the one long conversation that
they and I have been having for some time now.
Barth, for wasting nearly every wo~~rl-day of every workweek talking to me on AIM. (By
symmetry, I must not have gotten too much done either.) Also, for falling asleep at
stoplights, screening Secretary for his mother, and in general serving as a reminder of how
good it is that Germany lost the Second World War.
Dan and Tiffany, for that spot on the air mattress in the living room.

Anna, for a lovely packing job.

Lisa, who took painfully awful singing and gently, patiently sculpted it into mildly
less painfully awful singing. I envied her job until I realized that it must require the
constitution of a saint. (Bonus points for liking the timbre of Serj's voice.)

Si Gung ('I! I1, Poi for bringing Wah Lum to Florida, Sifu Dale for bringing it to
Gainesville, Si Hing Austin for running it, and Si Hing Matt for keeping it going as










long as he did. Particular thanks to those last two for going well beyond what they were
required to do or compensated for, and further thanks to that last one for getting my jokes
and sending them back. I owe the strength of both my body and mind to the school and
its teachers. Those front outside have no way of understanding what I'ni talking about,
and I have no desire to try to explain it to them.

All of the musicians, writers, and filninakers who've made my life worth living, and all of
the women who've done the same. I believe Vomnegut said a few things about this topic
before he left us.

M :-:- ,~: for everything. There's too much for me to even try. To be discussed over scones.











TABLE OF CONTENTS


page

ACK(NOWLEDGMENTS ......... . . 4

LIST OF FIGURES ......... .. . 9

ABSTRACT ......... ..... . 11

CHAPTER

1 PROPOSAL SITAINARY . ...... ... .. 12

1.1 Local Surface Shape Asyninetry Analysis .... . .. 12
1.2 ITSSR: Unified Segmentation, Smoothing, and Registration .. .. .. .. 12
1.3 Application of ITSSR Framework to Shape Analysis ... .. .. 1:3

2 LOCAL SITRFACE SHAPE ASYMMETRY ANALYSIS ... .. .. 15

2.1 Background: Hippocanipal Shape All ll-k- ... .. .. 15
2.2 Background: Simultaneous Registration and Segmentation .. .. .. .. 16
2.3 Application of Registration and Segmentation to Shape Analysis .. .. 17
2.4 Derivation of Framework for Closed 2D Surfaces .. .. .. .. .. 18
2.4.1 Practical Differences Between FFF and SFF as Matching Criterion 2:3
2.5 System Energy Functional ......... ... 25
2.6 Inmplenientation Details ... .. .. . .. 27
2.6.1 Hippocanipal Surface Data Acquisition and Formatting .. .. .. 27
2.6.2 Discrete Representation of the Deformation Map and the Segmentation
Curve................. ..... 28
2.6.3 Sparsity of the Hessian ........ .. .. 29
2.7 Results ......... . .. . :30
2.7.1 Output Visualization ......... .. :30
2.7.2 Classification Analysis ........ ... .. :32

:3 ITSSR: UNIFIED SMOOTHING, SEGMENTATION, AND REGISTRATION
(FOR MULTIPLE IMAGES) ......... ... :39

:3.1 Motivation for Unification of Image Procesin Ortin......... 39
:3.2 Derivation of Framework for Flat Image Sets ... .. .. .. 41
:3.3 Derivation of System Euler-Lagrange Equations .. .. .. 44
:3.4 Computational Framework Details ...... .. . 46
:3.5 Results on Image Sets ......... .. .. 48

4 APPLICATION OF ITSSR FRAMEWORK( TO SHAPE ANALYSIS .. .. 5:3

4.1 Motivation for Extending ITSSR to Shape Analysis Problem .. .. .. 5:3
4.2 Modification of Variational Principle ..... .. .. 54
4.2.1 Movement of Computation to Shared Canonical Domain .. .. 54











4.2.2 Functional Form ......... ... .. 56
4.2.3 Asyninetry Measure ........ ... .. 57
4.2.4 Final System Variational Principle ... . .. 58
4.3 Extension of Muniford-Shah to Case of Weighted Integral .. .. .. .. 59
4.3.1 Necessity ......... .. .. 59
4.3.2 Derivation ......... . .. 60
4.4 Ch1 I.ng ; in Inmplenientation . ...... ... .. 62
4.4.1 Alteration of Map Representation and Hessian Structure .. .. 6:3
4.4.2 Switch front CI .I.-Vese to Muniford-Shah Segmentation .. .. .. 65
4.4.2.1 Inmplenientation Details for Muniford-Shah Minintization .65
4.4.2.2 Inclusion of Periodicity and Metric Correction Operator .67

5 CONCLUDING REMARK(S ....._._. .. .. 69

REFERENCES ..... .._._.. ............. 71

BIOGRAPHICAL SK(ETCH .. ..... .. 74










LIST OF FIGURES

Figure page

2-1 Illustration of shape analysis framework. ...... .. 19

2-2 A pair of cylindrical surfaces with identical "Gaussian bumps" of opposite orientation.
The extrinsic orientation information is discarded in using the FFF as the matching
criterion: in this framework, the two surfaces are considered identical. .. .. 24

2-3 A pair of cylindrical surfaces which are differentiated only by surface noise. Under
the (identity) map which would correspond the noise-free versions, the SFFs
correspond poorly. ......... . 25

2-4 Hessian sparsity structure. ......... . 29

2-5 Synthetic results: case of cylinders with surface distortion. .. .. .. 32

2-6 Segmentation of distortion between hippocampi of LATL set member. .. .. 33

2-7 Segmentation of distortion between hippocampi of RATL set member. .. .. 34

2-8 Segmentation of distortion between hippocampi of control set member. (The
smaller of the two segmented regions has been chosen for enlargement.) .. .. 35

2-9 Epilepsy classification results: optimal test accuracies over all classifiers shown
in bold. ......... ..... . 37

3-1 Nature of interdependence between segmentation, smoothing, and registration. .40

3-2 Unbiasedness and compatibility can be guaranteed by performing all computations
on a canonical domain D. ......... .. .. 42

3-3 Processing of variations on a hand position. Each column represents the evolution
for a single input image in the trio, with the original input on top, and the final
smooth reconstruction at bottom. The evolving segmentation and registration
are depicted through the red curve and blue grid, respectively. Note the success
in corresponding the fingers between all images. ... .... .. 50

3-4 Processing of various openings of the hand, organized as in Figure 3-3. Note
the identification of the fingers in the first column despite their near closure in
the input image, under heavy noise. This owes to the "implicit atlasingt inherent
in the method. .. ... . .. 51

3-5 Identification of the ventricles of the brains of three different patients (in respective
rows) in cross-sectional MRI scans. Note the successful identification in the case
of the second input, despite the relative fineness of the structure. The final smoothed
images are di11l-p II in the last column. .... .. .. 52










3-6 Di~plw-~ and analysis of inter-image registrations for brain MRI dataset di;11lai- II
in Fig. 3-5. .. ... . .. 52

4-1 The reformulated version of the local .I- i- illia~ nIy an~ lli--;- proposal: All K input
surfaces are corresponded through the K maps fi between their parametric domains
Pi and shared canonical domain D. Using the metric information of the gi, we
can define an energy function over D which can be simultaneously segmented
via y, which in turn appears on each Si as yi through fi, segmenting the surfaces
themselves based on group .I-i-mmetry structure. ... .. . .. 55

4-2 The structure of the argument map vector, used as input to the optimization
method and with respect to which the system Hessian is defined (shown transposed
into a row vector). The a coordinates of the K maps (K segments with mn elements
each) are concatenated first, followed by the variable v coordinates likewise (K
segments of m(n 2) elements), for a vector of total length 2Km(n 1). .. 64

5-1 Your author .. ... . .. 74









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

SHAPE ANALYSIS VIA UNIFIED SEGMENTATION, SMOOTHING, AND
REGISTRATION OF RIEMANNIAN STRUCTURES

By

Nicholas A. Lord

December 2007

('I! I!1-: Baba C. Vemuri
Co(ll. I!1-: Jeffrey Ho
Major: Computer Engineering

Locality is an important but oft-ignored aspect of shape .I-i-mmetry quantification,

and segmentation is one method by which to make the locality of an analysis explicit.

We have thus formulated an approach to surface registration and shape comparison

which features an integrated segmentation component for the purpose of simultaneously

identifying and separating regions by their evolving deformation characteristics. In

the first instantiation, we achieve this effect through an adaptation of the ('I! .Il-Vese

approach for image segmentation to the problem of segfmentingf the Riemannian structures

of the very surfaces comprising the domain of the segmentation. We have successfully

used the method's output on hippocampal pairs in an epilepsy classification problem,

demonstrating improvement over global measures. Noting that a ('I! .Il-Viese-based

approach to simultaneous segmentation and registration is inherently limited, we have

also developed a unified approach to segmentation, smoothing, and nonrigfid registration

of images via extension of the Mumford-Shah functional, devised in such a way as to

be applicable symmetrically and consistently to multiple (two or more) inputs. To

conclude, we propose an extension of this unified framework (dubbed "USSR") to the

previously considered problem of 2D surface shape analysis (.I- i- ini n. i ry quantification

and localization), conferring the benefits of unbiasedness, consistency, multiple input

processing, and nontrivial data field reconstruction.










CHAPTER 1
PROPOSAL SITAINARY

1.1 Local Surface Shape Asymmetry Analysis

The problem of quantitative shape analysis of two-dintensional simple closed surfaces

appears in many applications, and is as such a well-studied problem. However, we

note that while such applications could oftentintes benefit front characterizing shape

differences on the basis of their location in addition to their extent, this is not the

general practice. Moreover, among those more sophisticated of recent proposals which

do explicitly determine shape disparity functions rather than global measures, there

still lacks a framework for ready interpretation of this data. The provision of such a

framework could advance the state of applications reliant on shape analysis, particularly

those seeking to use the data to perform classification. We have successfully proposed

and intpleniented one such framework, and applied it to a set of point clouds representing

the hippocanipal surfaces of patients belonging to either of two epileptic classes or a

control group. Driven by the clinical observation that .I-i-all...~ r -y between the two halves

of a given patient's hippocanipus appears to differ between epileptics and controls, and

given that this variation is generally described in terms of relative local enlargement and

shrinkage, we devised a scheme based on simultaneous registration and segmentation of

the Rieniannian surface structures themselves. The registration component ensures that

the surfaces are placed into sensible correspondence, while the C'I .I.-Vese (CV)-- i-1 i

segmentation component identifies surface subregions which are particularly disparate

under this correspondence. Using information front both the match disparity and the

segmentation thereof, we were able to successfully differentiate patient class niembers via a

fully automated system.

1.2 USSR: Unified Segmentation, Smoothing, and Registration

The issue of simultaneous segmentation and registration itself separately hears

further consideration. As before, the segmentation scheme utilized in the first iteration










of the epilepsy classifier was based on the CV framework. In essence, this means that its

resultant segmentation corresponds to the optimal piecewise-constant reconstruction of

the field being segmented (the Riemannian surface structure in the above case, image

intensities traditionally As such, CV can he viewed as a scheme which not only

segments a field but simultaneously reconstructs it. When coupled with a registration

evolution, a system is then produced which simultaneously performs segmentation,

registration, and reconstruction of its input. However, the piecewise constancy inherent

in C'I .Il-Vese limits the segmentation partially, and the reconstruction severely. This

relates directly to the fact that the framework is simply a special (limiting) case of the

minimization of the general Mumford-Shah (\!8) functional for image reconstruction.

We have thus proposed a general framework for and implemented a specific instantiation

of a scheme for simultaneous segmentation, smoothing, and (nonrigid) registration of

two-dimensional image suites via extension of the MS functional. Our implementation

surpasses any comparable techniques known to us by virtue of possessing all of the

following properties at once: (1) meaningful mutual assistance between all three image

processing suboperations with nontrivial results in all, (2) applicability to input sets of size

greater than two, (3) symmetry/lack of hias with respect to input arguments (including

sets of size greater than two), and (4) lack of requirement of prior knowledge of region

statistics, count, or shape. Results on sample image suites demonstrate successful mutual

assistance of the three image processing operations under the simultaneous framework, by

addressing cases under which the operations fail in isolation. (We henceforth refer to this

method as "USSR", for "Unified Segmentation, Smoothing, and Registration".)

1.3 Application of USSR Framework to Shape Analysis

In the final instantiation, we apply the USSR method (an extended MS framework)

to the shape .I-- iin!~i n.I y analysis problem in a manner analogous to our adaptation

of the CV scheme. Even a cursory glance will reveal that the implementation of such

an extension is more involved than in the special case of CV, but it carries with it the










following benefits: (1) the fact that 1\S involves a degree of freedom more than CV allows

for the possibility of optimizing the free parameter (establishing the balance between data

fidelity and smoothness of the reconstruction) for the application at hand, (2) the added

arguntent-wise syninetry of the USSR framework adds a necessary consistency, and the

multiple input capability allows for the possibility of batch processing in the absence of

an explicit reference shape (using the example of the hippocanipus dataset, one might

wish to examine deviations between the left hippocanipi of many patients, rather than

the left-to-right .I-i-int...~ I1y of each patient: for a structure such as the hippocanipus, it

might he especially prudent to avoid explicitly defining a notion of nornialcy of shape

through an atlas), and (3) theoretically, this is an interesting development: a proposal for

simultaneous segmentation, nonrigfid registration, and piecewise-smooth reconstruction

of data defined over 2-nlanifolds is to our knowledge unprecedented in the literature, and

the fact that the data in question is in fact the Rieniannian structure of the manifold

itself makes it more so; even in its current instantiation, our algorithm has ininediate

applications beyond comparing hippocanipi.









CHAPTER 2
LOCAL SURFACE SHAPE ASYMMETRY ANALYSIS (APPLIED TO HIPPOCAMPI)

2.1 Background: Hippocampal Shape Analysis

The temporal lobe's role in memory and learning makes it a natural point of

focus in the study of neurodegenerative disease progression. In particular, the medical

imaging literature abounds with attempts to identify and correlate abnormalities of the

hippocampus with Alzheimer's disease, schizophrenia, and epilepsy. While early studies

typically characterized hippocampal shape in terms of such simple global measures

as volume, length, and surface area, it was shown as early as [30] that analysis of

regional .I-i-mmetries could improve disease classification capability. Several methods

for fine-grained regional hippocampal shape analysis have since been -II__- -r. II Gerig

et al. [12] included a medial shape representation with age and drug treatment data in

an exploratory statistical analysis of the hippocampus's link to schizophrenia. Shen et

al. [21] conducted a statistical analysis based on the spherical harmonic ('SPHARM')

representation method. Styner et al. [23] tested the power of a SPHARM-based

medial representation to separate monozygotic from dizygotic twins through lateral

ventricular structure, and schizophrenics from normals through hippocampal and

hippocamplu-- I....vgdalan structures. Davies et al. [9] devised a minimum description

length framework for statistical shape modeling and extracted modes of variation

between normal and schizophrenic populations. Bouix et al. [1] emploi-. I medial

surfaces in a local width analysis. Using a viscous fluid flow model, Csernansky et al.

[8] computed diffeomorphic maps of patient hippocampi onto a reference, producing a

dense inward/outward deformation field over each hippocampal surface. The surface

itself was additionally manually segmented to allow for the regional comparison of the

deformation fields as part of an attempt to separate healthy individuals from those

exhibiting dementia of the Alzheimer's type. All reported results have indicated the

benefit of incorporating regional information into the analysis. This is unsurprising, as










simple scalar measures of entire structures necessarily discard a wealth of information

concerning the full characterization of those structures. However, the obtained statistical

results are preliminary, and shape analysis results can actually appear contradictory

across different studies (for instance, Styner notes in [24] the contrast between the primary

abnormality localization in the hippocampal tail found in that work and the localization

in the head reported in [7]). Furthermore, while statistically significant differences of

hippocampal shape have been identified between diseased and normal sample populations,

reliable classification of a sizeable number of patients with respect to those categories has

not occurred previously.

2.2 Background: Simultaneous Registration and Segmentation

The first scheme for simultaneous segmentation and registration known to us

was given by Yezzi et al. [:34], who introduced the idea of explicit interdependence

of segmentation and registration: registration methods can make use of the feature

detection inherent in segmentation, while segmentation (particularly in the case of noisy

or incomplete/occluded data) can utilize the redundancy provided by correctly registered

images of the same structures to segment said structures more robustly jointly than

individually. Their proposal involved the unification of mutual information(\!1)-hased

flat 2D rigid image registration with a level set implementation of a piecewise constant

((CI .Il-Vese) segmentation scheme through a variational principle. Wyatt et al. [:32]

accomplished much the same thing, solving instead a maximum a posteriori (j1\! P)

problem in a 1\arkov random field (illtF) framework. Xiaohua et al. [:33] extended

this to the nonrigid registration case, and Unal et al. [26] addressed the same problem

using coupled partial differential equations (PDEs). In [29], F. Wang et al. presented

a simultaneous nonrigid registration and segmentation technique for multi-modal data

sets. Richard and Cohen [19] proposed a variational framework for combining region

segmentation and registration (matching) using free boundary conditions. Jin et al. [1:3]

simult aneously evolved a surface segment ation and a radi ance-dis continuity-detecting










segmentation of the evolving surface itself. Young et al. [35] combined joint segmentation

and (scaled rigid) registration with morphing active contours for segmentation of groups of

CT images. Pohl et al. [17] produced an expectation nmaxintization (EM)-hased algorithm

for simultaneous affine registration and subcortical segmentation of AIRIs using a labeled

cortical atlas.

2.3 Application of Registration and Segmentation to Shape Analysis

The need to identify and possibly isolate subregions of interest on the hippocanipal

surface -II--- -is segmentation (henceforth synonymous with 'parcellation'), and the need

to compare hippocanipal surfaces between and within (i.e. detection of ..-iin..l!l! i 1y

between halves of the same structure) patients represents an inherent requirement

for registration. A unified segmentation and registration scheme is thus a natural

approach to the problem at hand. As is to be seen, the applicability contes through

viewing the Rieniannian surface structures of the input surfaces themselves as the

data to be operated upon. Our scheme then rests on the fundamental assumptions

that: (1) the deformity of homologous anatomical structures can he quantified as the

deviation from isonietry of the deformation nmap between their surfaces, and (2) the

evolution of the global correspondence must allow for a partial disconnection between the

normal and abnormal regions, as these are by definition not expected to exhibit the same

deformation patterns. As such, surface segmentation rests at the heart of our approach,

which can he viewed both as a registration tool and an .I-i-int...~ r -y localizer. For the

topologically spherical surfaces considered here, our intrinsic approach leads naturally

to an elegant 2D parametric representation of both the segmentation and registration.

Our use of Rieniannian surface structure in the matching criterion is similar in spirit

to work done by Y. Wang et al. (e.g. [31]), in which the authors presented a technique

to nonrigidly register surfaces using a 2D parametricc) diffeomorphic nmap constructed

front Rieniannian surface structure information. The integrated manifold segmentation

is enough to distinguish our approach front this work, but additionally, our definition of










system energy in terms of piecewise isonietry (i.e. using first fundamental forms in the

matching criterion) inherently balances the consideration that the nmap should accurately

match Rieniannian surface characteristics with the idea that the nmap should incur as

little deformation as possible in doing so. As such we eliminate the need for additional

regularization of the nmap gradient. In fact, the real novelty of the algorithm is more

fundamental: this is the first proposal for simultaneous nonrigfid surface registration

and segmentation wherein the segmentation is driven by the evolving registration nmap

itself. (Though, we do note that there exist works of similar spirit. For example, see

the "defornlotion" concept of Soatto and Yezzi [22] for separating physical motion from

surface deformation in registration of correspondent regions.)

2.4 Derivation of Framework for Closed 2D Surfaces

To begin the explanation of our framework, we first turn attention to the already-studied

problem of 2D image pair segmentation and nonrigid registration. A solution to this

problem (regardless as to how one obtains it) clearly takes the form of (1) a differentiable

one-to-one and onto mapping function which shares its domain with one of the images

and its range with the other, and (2) a segnienting curve which lies within one image

domain and is carried into the other through the mapping function. With images,

there is a natural and obvious basis for computing this solution: the image intensity

values. Intensity agreement represents the matching criterion by which algorithms such

as mutual information nmaxintization estimate pairwise image registration. Statistics

derived from intensity values form the driving force of region-based segmentation methods

(in C'I .Il-Vese [3], for instance, the statistic is the mean of each curve-defined region).

Further, differential qualities (e.g. gradient, Laplacian) of the intensity profiles can prove

useful in identifying edges, corners, and other useful features. When conducting pairwise

segmentation and registration of shapes rather than images, however, the appropriate

measures are perhaps less obvious. It remains to us to define them.

















Y1


I
;T~=~S_;f~~
_i- 111
r


laru ~ "Iz1
--L
fSY -~-I t~E~'- ~~
; I-rcL .7. =fL~ r~ i
~._Lu~'t ~- L~-~~l;t:~F~-.
-~ -~~
"Tr-e ---- ct
~;-'2~-ct c c


Figure 2-1.


Illustration of shape analysis framework: The input to the algorithm is a pair
of homologous shapes S1 and S2. Use of appropriate boundary conditions
allows us to associate the shapes with flat rectangular patches P1 and P2
respectively. These patches, now considered as parametric domains, store all
Riemannian metric information gl and g2 of their respective surfaces in terms
of the first fundamental form matrices G1 and G2. The metric information is
then matched between P1 and P2 through a variational principle which drives
both a homeomorphic map f of P1 onto P2 (diffeomorphic except possibly on
the curves ]*) and a segmenting curve y1 in P1 which is carried by f into P2 aS
y2. This process registers and segments the surfaces S1 and S2 via the
parametric correspondence. Note that the map f between the parametric
domains can he visualized as the deformation of a regular grid representing the
left parametric domain P1, and that the segmentation interiors (red) and
exteriors (blue) can he represented on S1 through shading.


Before definingf a matching criterion, we first show how it is possible to cast the

shape registration and segmentation problem into a computational framework similar

to the image framework described above (save for the matching criterion itself) through

appropriate parametrization. To accomplish this, one need note only the following simple

topological facts: (1) the surfaces of hippocampi can he thought of as 2D Riemannian

manifolds embedded in IR3, (2) these manifolds are topologically equivalent to the sphere

(closed and genus zero), (3) the exclusion of a pair of poles from the sphere produces










a cylindrical topology, and (4) any surface topologically equivalent to the cylinder can

he parametrized by a class of functions whose common domain is a single rectangular

patch with periodic boundary conditions at a pair of opposite sides. Note that this last

fact directly implies that any function defined on the surface in question can he fully

represented as a function over a closed rectangular domain with the stated boundary

conditions. In particular, a smooth map between two hippocampi can he represented as

a smooth map (deformation field) between two parametric rectangular domains satisfying

the boundary conditions. Through this simple and natural representation of the surfaces,

our solution format becomes quite similar to that of the 2D image segmentation and

registration problem in that all of the computations are performed on and between

rectangular domains. In contrast, when computation takes place in the embedding (3D)

space before being restricted to an approximation to the surface within that space, one

cannot draw nearly so direct a parallel (though one may take this approach if one wishes:

see for example [4]). The approach can he visualized in the manner depicted in Fig. 2-1.

Clearly, the matching criterion which we define in place of image intensity must

communicate shape information about the surfaces being registered and segmented.

We now take the opportunity to define the first and second fundamental forms of a

parametrized surface, and subsequently motivate their consideration:


F F F = X z,-Xi X z, -X ,, (2-1)





SFF = NXuvz N-Xv (2-2)


where X is the 3D surface map, u and v are the map parameters (equivalently, the

coordinate system of the parametric domain), NV is the surface normal (defined as
x,,xx), and, subs~ripts dennote differentiation.
Ix,,XXV/ II UUlr UIV UI\I/II*UI
But a well-known theorem due to Bonnet (as stated in [10]) reads as follows:









Theorem 1. Let E, F, G, e, f, g be di~ferentiable functions, I. I;,.. l in an open set V C RW2

with E > 0 and G > 0. Assume that the given functions ;-od:-fy for, I,,allhi the Gauss and
Manardi~r /(~]-Coazi equations~I and- tha- EG( F2 > 0. Then, for every q E V there exists

a neighborhood U E V of q and a di~feomorp~hism x : U x(U) C RW3 SUCh ltha the

,n 1,pil. r ;; it..: x(U) C R3" haS E, F, G and e, f, g as coefficients of the first and second

fundamental forms, -i 1. :; l;, Furthermore, if U is connected and if

x : U ic( U) cR 3

is another di~feomorp~hism c;H-litr;ing the same conditions, then there exists a translation T

and a proper linear oril,,.~I ~,.,,l .rt.-f.>,rmation p in R3" SUCh thtf X = T o po X.

(NOTE: In this statement of the theorem, the form coefficients E, F, G, e, f, g
correspond elementwise to the quantities in the definitions of Eqns. 2-1and 2-2 as

follows:)

FFF =




SF~f g9

Though Thm. 1 may appear arcane in its given form, a brief explanation should

illuminate its application to the problem in question. The set V in the theorem represents

a patch of a two-dimensional parametric domain (such as P1 and P2 in Fig. 2-1). E > 0

and G > 0 are necessary conditions for those quantities to be valid coefficients in the

FFF of any surface parametrized on V, which is obvious from the definition in Eqn. 2-1.

EG F2 > 0, the requirement that the FFF matrix be positive definite, is necessary for

map regularity, which is in turn a necessary condition for diffeomorphism, the only class

of maps with which we are here concerned. (Since z/EG -F2 can in fact be interpreted










as the size of an area element on the surface corresponding to a parametric patch for,

the requirement of FFF positive definiteness everywhere dictates that any subset of the

parametric domain with positive area must map to a subregion of the parametrized surface

with nonzero area: this is one way of obtaining a rudimentary intuition for the notion of

regularity. We will make use of this relationship in the implementation as well.) These

stipulations all exist simply to ensure that E, F, and G are possible FFF coefficients

of a surface parametrized hv a diffeomorphism defined on V. The theorem then goes

on to mention the Gauss and 1\ainardi-Codazzi equations (known as the 'compatibility

equations' of surface theory): without taking the clarification too far tangentially, we can

ii that satisfaction of these equations means that the combined information content

of the FFF and SFF is self-consistent.2 This completes the stipulation portion of the

theorem: essentially, E, F, and G must he suitable FFF coefficients, and e, f, and y must

not contradict them as SFF coefficients.

What the theorem then tells us is that if these conditions are satisfied, then V is

everywhere locally parametrizing some regular surface through a diffeomorphic map

.r with FFF and SFF specified by E, F, G, e, f, and g. In fact, not only does such a

surface exist, it is determined up to rigid motion, and as suchit is unique in the .sense of

.shap~e. Therefore, if we consider only maps .r which are diffeomorphic over the entirety of

their (connected) domains, then for given topology we can state that the FFF and SFF

contain all possible shape information. These tensor fields are thus a natural choice for the

matching criterion sought.




1 Consult reference [11 Cle! 2-5 and 2-8, for a detailed discussion of why this is so.

2 (Note that this implicitly broaches the fact that the shape information of the two
forms is not entirely independent. For instance, the Gaussian curvature, the product of the
minimum and maximum curvatures, a.k.a. principal curvatures, can he expressed either in
terms of both the FFF and SFF coefficients, or, remarkably, in terms of FFF coefficients
alone. This is the content of Gauss's Theorema Egregfium, or Remarkable Theorem. Thus,
there are SFF matrices which cannot coexist with given FFF matrices.)










There are different r- :--s to conceptually distinguish the two quantities. For one,

the geometric information contained in the FFF is of first order, while that of the

SFF is second order in nature. Alternately, the FFF encodes the intrinsic geometry of

the parameterized surface while the SFF encodes the extrinsic geometry, or, the FFF

encodes local length information, the SFF surface curvature (the mean curvature, an

extrinsic characteristic, is used as a matching criterion in [31]). Because of their lower

differential order (and thus greater relative stability) and intrinsic nature, we choose to

base our approach on comparison between the FFF tensors of the two surface parametric

domains. Given topological equivalence, FFF equivalence everywhere through a bijective

deformation map defines that map as a global isometry: this indicates that the extent of

failure to match this characteristic serves well as a measure of shape dissimilarity.

2.4.1 Practical Differences Between FFF and SFF as Matching Criterion

We have established that consistent FFF and SFF fields are sufficient to specify the

shape of a surface of given topology. Unsurprisingly, their information is non-redundant,

i.e., one form does not fully specify the other. Yet we have chosen to use only the FFF

tensor fields in the matching criterion, meaning that we are ignoring some aspect of shape

in a shape analysis algorithm. This fact bears consideration: we will illuminate what our

perspective retains and what it discards, and justify our decision.

The sort of information contained in the SFF (and thus discarded in omitting it from

the matching criterion) can be visualized through pairs of surfaces with SFF fields which

differ under parametrizations which match their FFF fields exactly. Fig.reffig:inandout

di pl .--s such a case. In this example, it is easy to see that there is an isometric mapping

between the two surfaces (and thus zero energy in the sense of deviation from isometry),

and what that isometric mapping is. However, under said mapping, the SFF fields differ

over the portion of the domain parametrizing the two protrusions (inverted relative

to one another): all curvatures are equal but opposite in sign. Viewed sitting in its

embedding space (R3"), this fact is obvious, but to the proverbial "bug on the sul I .










it is imperceptible: in the sense of measures, the surfaces are indistinguishable, which is

precisely what defines isonietry. All such distinction is represented in the SFF fields, which

include the surface nornials (a concept inherently tied to the embedding space).















Figure 2-2. A pair of cylindrical surfaces with identical "Gaussian hunips" of opposite
orientation. The extrinsic orientation information is discarded in using the
FFF as the matching criterion: in this framework, the two surfaces are
considered identical.


However, the quality of the match criterion is application-dependent For instance,

when considering the case of hippocanipus pairs, we seek to identify regions of localized

growth or shrinkage (in line with the clinical motivation). The sort of shape difference

represented in Fig. 2-2 does not represent such a case. Further, an example of the

drawback of SFF (essentially, curvature) matching is illustrated in Fig. 2-3.

As regards this application and many others, any difference between the two surfaces

is attributable to high-frequency noise in the point cloud data: that is, there is no

meaningful difference between the surfaces. However, the trivial isometric nmap (identity)

between the noise-free versions of the surfaces will by no means provide a low-nmagnitude

difference in their second fundamental forms under that mapping. In fact, infinitesimal

perturbations in surface data are theoretically capable of producing surface curvatures

of arbitrarily large magnitudes at the perturbation locations. The problem with using

the curvature structure of a surface as an identifying characteristic is the fact that it

is inherently of high order: matching maps through their second derivatives cannot he
























Figure 2-3. A pair of cylindrical surfaces which are differentiated only by surface noise.
Under the (identity) map which would correspond the noise-free versions, the
SFFs correspond poorly.


expected to be stable, and, worse still, in many cases cannot be expected to be physically

meaningful even with numerical considerations cast aside. This fact is especially true for

the inputs considered herein, which explains our omission of this aspect of the surface

characterization.

2.5 System Energy Functional

We now formalize the problem and fully specify its solution as follows: Let S1 and

S2 be two surfaces in IR3. The Euclidean metric in IR3 induces Riemannian metrics

gl and g2 on S1 and S2, TOSpectively. The goal is to segment regions in S1 and their

corresponding regions in S2 While mapping those correspondents so as to optimally match

their metric structures. Specifically, the algorithm must compute a homeomorphism f

between S1 and S2 and a set of closed curves yl on S1 and y2 in S2. The restriction of

f to the complement Sl\yl is a diffeomorphism between Sl\yl and S2~a 72- 1, n

and VI, V, denote the collections of open components topologicall discs) of Sl\yl and

S2 72, TOSpectively, then f maps Ui diffeomorphically onto 1K for each i while matching the

Riemannian structures of each Ui and 1K = f(Ui).

We solve the simultaneous segmentation and registration problem outlined above

using a variational framework. The energy functional 8 is defined as a functional of a pair










(f, Y1), where yl is a set of closed curves on S1 and f : S1 S2 is a homeomorphism

which is C" on Sl\y Let f*g2 denote the pull-back metric on S1. S(f, yl) is then

defined as







/3 |e es w| dA + 1ton |e cout,1|d 23

where

S= |f *g2 91| ,1 (2-4)


e .", ema= u (2-5)
~In d,4 1 I, dA 4

Si,z and Sla,,t are the regions of S1 inside and outside of ]*1 respectively, and c0,7 are

positive weighting constants.

The quantity e defined in Eqn. 2-4 provides, at each point on S1, a measure of

similarity between the Riemannian structures gl and g2 as they correspond under the

current estimate of f. As such it represents the local deviation from isometry, and its

integral over the domain (we have adopted the convention of using P1, the parametric

domain of S1) provides a global measure of how far the given deformation f is from an

isometry. This is the first term in the system energy represented in Eqn. 2-3. In local

parametricc) coordinates, it is given by the distance between the two matrices G1 and



|.7 G2. GI| ,, (2-6)

where .7 is the Jacobian of f when f is expressed from the local coordinate system of

P1 onto that of P2, and G1 and G2 are 2 x 2 positive definite matrices expressing the

metrics gl and g2 in the coordinate systems of P1 and P2 Tespectively, as in Eqn. 2-1.

The second term in 8 restricts the length of the segfmentingf curve, so as to achieve the










minimal necessary segmentation (shorter length and/or fewer open components) under

the other energy constrains. Finally, the third and fourth terms place a homogeneity

constraint on e within each connected component defined by the segmenting curve. This

corresponds to our notion that the curve should partition the surfaces in such a way as

to isolate regions exhibiting one sort of deformation characteristic from those exhibiting

another. Note that these terms are defined in terms of both f and ]*1, and as such the

minimization of their energy will drive both segmentation and registration. This coupling

represents the fundamental difference between a simultaneous framework and one in which

the segmentation follows the registration (in series).

2.6 Implementation Details

The minimization of the system energy defined by Eqn. 2-3 is achieved by alternating

between the deformation map and segmentation estimation processes. When optimization

steps are constrained to be sufficiently small, this alternation approximates minimization

of 8 simultaneously with respect to both f and ]*1. The Euler-Lagrange equation for

Eqn. 2-3 can he derived through standard calculus of variations. The equation for f

contains a fourth-order expression involving f and its derivatives. Because of the high

differential order of this analytic expression, we instead choose to minimize the functional

8 with respect to f directly through a constrained optimization process (more detail in

Sees. 2.6.2 and 2.6.3). Minimization with respect to yl, on the other hand, is implemented

via a level set segmentation as described by C'I I1. and Vese in [:3]. The only difference

between this module of our method and that described therein is our generalization of the

process to genus zero 2-manifolds, which entails respecting nonuniform surface length and

area elements in accordance with the metric information.

2.6.1 Hippocampal Surface Data Acquisition and Formatting

To produce the surface data used in this study, AIRI scans were first acquired with

a 1.5 T MRI scanner (Siemens Magnetom 42SPA, Siemens Medical Systems, Iselin, NJ)

using :3D magnetization-prepared rapid gradient echo (!P'R AGE) sequences. The gradient









echo images were transferred for post-processing. The sequence parameters included

the following: obtained in sagittal plane, 250 mm field of view, repetition/echo times of

10/4 ms, T1 weighted sequence of 300 ms, 100 flip angle, 130x256 matrix, and 180 mm

slab with 128 partitions producing 1.25 mm gapless sections. The segmentation of the

hippocampal surfaces from the MRI data was done manually by a trained neuroanatomist,

and a smooth surface was fit to the marked points by using a deformable pedal surface

as described by Vemuri and Guo in [28]. Each segmented hippocampal surface obtained

from the application of this technique was represented by a 40 x 21 mesh of points on

that surface, periodic in one direction. (This mesh is the instantiation of the surface

parametric domain as discussed in Sec. 2.4.) Homologous hippocampi were brought

into rigid alignment through application of the iterative closest points (ICP) algorithm

followingf the segmentation. The rigid alignment step included extraction of volume data

and normalization of the shapes with respect to this data.

2.6.2 Discrete Representation of the Deformation IVap and the Segmentation
Curve

An intrinsic map between two structures that are topologically equivalent to cylinders

can be stored (in discretized form) as a 2-vector function over an m x n grid, where

m and n are the grid point counts in the u and v parameter directions. The 2-vector

at each grid point on the domain grid represents the displacement vector between the

domain grid point and the corresponding target grid point. Since we are representing a

cylinder, which is periodic in one direction, with a grid, we must enforce this periodicity

through a module arithmetic implemented with respect to one of the grid coordinate

directions ('u'). The grid must then be bounded in the remaining ('v') parametric

direction, with the two bounded ends corresponding to the "top" and "bottom" of

the cylinder being parameterized. The v-coordinates at the poles (columns 1 and a

of the grid) are excluded from the map evolution process, and so we conduct energy

minimization with respect to mn displacement u-coordinates and m(n 2) displacement









mn m(n-2)


Figure 2-4. Hessian sparsity structure.


v-coordinates, which can be stored as a single variable vector of size 2m(n 1). Note

that this optimization is constrained in that the v component must everywhere respect

the boundaries corresponding to the top and bottom of the cylinder. We choose also to

restrict the modular family of solutions for the a components (corresponding to 3600

rotations around the cylindrical axis) to falling within a modulus above or below the

initial conditions. We evolve the system under these constraints using the LargeScale

version of Matlab's 'fmincon' function, a subspace trust region method based on the

interior-reflective Newton method described in [6]. We exploit the inherent sparsity

structure of the Hessian (as described in Sec. 2.6.3). The Riemannian characteristics

themselves, whose values of course drive the energy minimization process, are calculated

through analytical differentiation of a bicubic surface fit to the given points.

The segmenting curve is simply stored and evolved according to well established level

set curve evolution techniques (see [20] for extensive discussion). The one modification

present in our case is that length and area computations respect the distances between

grid points by using the appropriate Riemannian metric at each point (as opposed to the

uniform Euclidean metric used for flat domains and targets).

2.6.3 Sparsity of the Hessian

The trust region method which conducts the energy minimization with respect to

the map (for a fixed segmentation) can exploit the fact that local changes in the map

can only effect local changes in the induced metric structure. To make this point clear,

let us consider the fact that the FFF is defined as in Eqn. 2-1, where X is the surface










spatial location 3-vector and u and v are the 2D parametric coordinates. The FFF at

a point depends only on the derivatives of X with respect to u and v. Furthermore,

for a fixed segmentation, the energy itself is a function only of the FFF structure on

the target manifold imposed by the evolving map. But consider each component of the

2m(n 1)-vector which we are evolving: infinitesimal variation in one of these components

can only perturb the metric structure on the target manifold locally, as the FFF definition

shows that it depends only on X, and X, at the given (u, v), and these partial derivatives

themselves are inherently local. It should then be immediately clear that mixed second

partial can only be nonzero when both variables being differentiated against lie within

the same neighborhood (as defined by the differentiation scheme): this leads to the

Hessian structure depicted in Fig. 2-4, wherein the solid diagonals represent the nonzero

matrix elements (and the map variable vector structure has been explained in Sec. 2.6.2).

Exploitation of this sparsity structure is necessary for a second-order optimization scheme

to achieve computational feasibility.

2.7 Results

To illustrate the method's performance, we first present a visual display of its output

and follow by demonstrating the classification power of the extracted features. The input

dataset consisted of 60 L/R hippocampus pairs, collected and formatted as described in

Sec. 2.6.1. Clinicians provided a trinary classification of this dataset: LATL (left atrophied

temporal lobe) epileptic, RATL (right atrophied temporal lobe) epileptic, and Control. In

this step, 6 samples were discarded due to ambiguity of their clinical class membership,

leaving the final set at 54 patient samples (15 LATL, 16 RATL, 23 Control).

2.7.1 Output Visualization

Our result visualization format can best be understood through reference to

Fig. 2-1, in which we present a holistic schematic. For each sample, we illustrate (1)

the deformation map f as the warp necessary to carry points in P1 to their correspondents

in P2, (2) the segmentation yl and the deformation energy function e as they exist in P1,










and (3) the segmentation ]*1 as it exists on S1 through the paranletrizing function front

P1 (there is a corresponding region on S2 as well, not depicted here but fully specified for

given f and ]*1 through the paranletrizing functions: y2 is specified by f and ]*,). As noted

in Fig. 2-1, the segmentation is depicted on S1 through the shading of its interior and

exterior with red and blue respectively: the curve(s) itself is then of course the boundary

between the shaded regions.

We first consider the case of a pair of cylinders, where one nienter of the pair

has had its surface distorted according to an outward normal vector field of magnitude

dictated by a Gaussian function confined to a known support. Since the support is

known, we can use its segmentation as one quantification of accuracy. Fig. 2-5 illustrates

the obtained results. Here, N.~' of the distortion area is included in the segmentation,

and (I' of the undistorted area is included in the segmentation. This means that the

segnienting curves lie slightly within the distorted regions, as can he seen in the figure.

Even the 11' I!- lin !--!1h. 11 .! !" cannot necessarily be seen completely as error in this

context: since the protrusions are far less severe towards the boundaries of their support

(the feet of the hills), they are accordingly far more similar in deformation level to the

undistorted regions than to the heavily distorted regions towards the support centers. As

such, the segmentation arrived at is in compliance with the two-nican framework (which

produces a high/low segmentation). While we do later note that it is possible to refine this

framework (see conclusion), it is important to understand that this manner of grouping is

not inherently a limitation, particularly considering the intended application. We follow

by demonstrating robustness of the segmentation in the face of reparanletrization of

the surface of comparison and random noise in the surface point cloud (see caption for

parameter levels). When repeated under these conditions, the segmentation remains HCIII'.

consistent at the pixel resolution with that obtained prior.

For the real cases described previously, we present the results obtained on one

niember of each class (LATL, R ATL, and control), as Figs. 2-6, 2-7, and 2-8, respectively.











Figure 2-5. Synthetic results: case of cylinders with surface distortion.


(a) Segmentation of prominent distortions from cylindrical surface (left: init.; middle: 5
its.; right: 2() its.).









(b) Performance under surface noise (a = ).()1 for unit cylinder) and reparanietrization
(left: warp field, a = ().2 in each component; middle: segmented energy field in
parametric domain; right: converged result, at 2() its.).


Each figures depicts a mirrored left hippocampus on the left side of each panel and the

corresponding right hippocampus to which it is compared on the right side. This aspect

of the study (inherently) lacks ground truth, and a "correct" result is thus one in which

the segmented regions indeed appear to be the portions of one hippocampus which are

shaped least like the corresponding portions of the hippocampus to which they are being

compared. This is readily evident in the presented figures. The provision of an input

device to allow for expert manual segmentation (closed curves confined to the surface)

for validation is itself a research problem. This does not mean, however, that we cannot

evaluate the quality of the data extracted from the real set. Immediately following, we

present a classification analysis as evidence.

2.7.2 Classification Analysis

As it is clear that volume does indeed contain information relevant in the shape

comparison of homologfues, it is unsurprising that it is a significant feature in distinguishing

epileptics from controls. However, it is equally evident that the complex process of






Figure 2-6. Segfmentation of distortion between hippocampi of LATL set member.


i"a


(a) Top left: naive initialization; Top right: 3 iterations; Middle left:
10 iterations; Middle right: 20 iterations; Bottom: final (50
iterations)


rl


(b) Warp of parametric domain
induced by evolution process.


(c) Final segmented distortion
energy e graph as function
over parametric domain
(arbitrary units,
intercomparable between
patients) .


~41


F










Figure 2-7. Segmentation of distortion between hippocampi of RATL set member.


4


O a


4


(a) Top left: naive initialization; Top right: 3 iterations; Middle left:
10 iterations; Middle right: 20 iterations; Bottom: final (50
iterations)


(b) Warp of parametric domain
induced by evolution process.


(c) Final segmented distortion
energy e graph as function
over parametric domain
(arbitrary units,
intercomparable between
patients) .












Figure 2-8. Segmentation of distortion between hippocampi of control set member. (The
smaller of the two segmented regions has been chosen for enlargement.)


(a) Top left: naive initialization; Top right: 3 iterations; Middle left:
10 iterations; Middle right: 20 iterations; Bottom: final (50
iterations)


(b) Warp of parametric domain
induced by evolution process,
in parametric pixels.


(c) Final segmented distortion
energy e graph as function
over parametric domain
(arbitrary units,
intercomparable between
patients).


g: g;










neurodegeneration associated with epilepsy cannot he captured in full so simply. The

utility of volume as a sole basis for classification is thus inherently limited. We have

provided a fine-grained shape difference characterization with a mind to overcoming

precisely this limitation. The test performance of a classifier should thus increase

when our extracted features are included as input. Note that we have worked with

volume-normalized data to deliberately orthogonalize the extracted features with respect

to volume. Since volume itself is assumed to be a relevant feature, it is possible that

the best test classification accuracy will be obtained by using volume and local shape

difference information in conjunction.

Since we produce a segmented energy function over the entire surface parametricc)

domain, we have options as to how to present features to the classifier algorithm. We

can collect summary statistics that profit from both segmentation and deformation

quantification, such as mean deformation energy (e as defined in Eqn. 2-4) inside and

outside of the segmenting curves. Alternately, once the shapes have been volume-normalized

and mutually registered, we can compare the (quasi-)continuous e functions across patients

by forming feature vectors from the e values at corresponding sample locations. There is

also an array of supervised learning methods from which to select a preferred classification

algorithm: we choose the ones observed to give best test performance on all feature sets

examined, including volume alone (as further detailed below).

Fig. 2-10a demonstrates the success rates in classifying controls vs. epileptics (LATL

and R ATL groups combined into a single set) and Fig. 2-10b demonstrates analogous

results for the problem of separating R ATL and LATL members. The training and test

statistics were collected through a standard leave-one-out cross validation procedure: test

results should be regarded as the reliable performance indicators (and high discrepancy

between training and test percentages as evidence of overfitting). The feature vectors

being compared are (1) volume alone, (2) volume with a set of 6 summary statistics

derived from the algorithm output (total area within the curve(s), total area outside of










Figure 2-9. Epilepsy classification results: optimal test accuracies over all classifiers shown
in bold.

SVM w/ PB SVM w/ RB K(FD w/ PB
Training Test Training Test Training Test
VOL 79.59 77.78 79.63 79.56 77.81 77. 78
VOL+6SUM 92.31 81.48 88.638 81.48 80.43 77. 78
E 100.00 85.19 94.44 81.48 94.51 87.04
E+VOL+6SUM 100.00 88.89 96.26 81.48 99.97 85.19
(a) Control vs. epilepsy

SVM w/ PB SVM w/ RB K(FD w/ PB
Training Test Training Test Training Test
VOL 83.25 80.65 80.65 80.65 79.78 77.42
VOL+6SUM 93.55 90.32 93.76 87.10 93.87 80.65
E 100.00 70.97 100.00 67.74 99.89 67.74
E+VOL+6SUM 100.00 74.19 100.00 70.97 93.23 67.74
(b) LATL vs. RATL


the curve(s), total energy e within the curve(s), total energy e outside of the curve(s),

mean energy e within the curve, mean energy e outside of the curve), (3) the function

e at 600 sampled locations, and (4) a concatenation of feature vectors (1), (2), and (3).

The chosen abbreviations for these feature vectors are 'VOL', 'VOL+6SUM', 'E', and

'E+VOL+6SUM', respectively. The classifiers used are (1) support vector machine with

polynomial basis (SVM w/ PB), (2) support vector machine with radial basis (SVM w/

RB), and (3) kernel Fisher discriminant with polynomial basis (K(FD w/ PB).

The results of both studies make the method's success clear. In the case of distinguishing

epileptics from controls, all experimental feature vectors demonstrated superior test

classification accuracy, with a maximum performance of 88 I' on 'E+VOL+6SUM'

as classified by SVM w/ PB, as compared to 79. J.' .~ for 'VOL' as classified by SVM w/

RB. Note in particular that feature vector 'E' outperforms 'VOL' as well, despite the

fact that 'E' contains no volume information. This portion of the experiment confirms

our notion that local distortion information is an important complement to volume

data, and so__~-r-;- that it can in some cases outweigh volume in independent relevance.










While we do not observe this latter phenomenon in the LATL vs. RATL case, we do

still observe substantial outperformance of 'VOL' by 'VOL+6SUM' (90.t:-' vs. 80.I. .' .),

again demonstrating the algorithm's extraction of clinically relevant shape information

not represented by volume. The large difference between training and test accuracies

for the other two feature vectors indicates overfitting, which is enabled by their higher

dimensionalities. This indicates that an overly fine-grained approach can confound

classification on this set, but that subregion identification and characterization remains

nonetheless crucial. We have not come across clinical test classification accuracies of this

level in any of the literature we have sure- i-n I









CHAPTER 3
USSR: UNIFIED SMOOTHING, SEGMENTATION, AND REGISTRATION (FOR
MULTIPLE IMAGES)

3.1 Motivation for Unification of Image Processing Operations

We have previously discussed the history of the simultaneous segmentation and

registration subfield, and noted the nature of the interdependence which motivates

unifying the two image processing operations (IPO) in the first place. But given that

smoothing (a.k.a. reconstruction) is often performed as a preprocessing step for the benefit

of other operations, we might wish to consider whether it too has a rightful place in a

unified processing framework. In fact, the natural interdependence of segmentation and

smoothing was elucidated in the seminal paper by Mumford and Shah[15], in which the

optimal image reconstruction is defined in terms of a segmenting curve or curves specifying

the boundaries of the piecewise smooth regions. In the curve evolution implementation

of Mumford-Shah minimization by Tsai et al. [25] the mutual assistance of segmentation

and smoothing is made explicit: a preliminary solution to each alternately serves as

input to the other. As such, tentative smooth reconstructions enable the curve to find

object boundaries, and the identification of object boundaries by the curve enables the

smooth reconstruction to remain maximally faithful to the initial data. But as already

pointed out, registration (indeed, any data-driven IPO) depends on smoothing results, as

a smoothing process by definition reconstructs the data upon which it operates. Based on

the observations made prior, we can observe the system relationship depicted in Fig. 3-1.

All interdependences are direct with the exception of the dependence of smoothing on

registration. However, considered in the context of simultaneous segmentation, this

relationship clearly emerges.

Despite inherent connection between the three IPOs, and the evident option of

unifying them under a single variational principle, such proposals are recent and limited in

number (some references which could be considered to fall into this category are Vemuri

et al.[27], Pohl et al. [18], and Unal et al.[26]). Moreover, each of the works just cited









Smoothing









feature identification
Registration : Segmentation
data redundancy

Figure 3-1. Nature of interdependence between segmentation, smoothing, and
registration.

contains at least one of the following restrictions, simplifications, or limitations: (1)
reliance on explicit ,,il 1-;0. (2) low-dimensional registration parametrization (i.e. rigid
or affine), (3) lack of guaranteed symmetry/unhiasedness with respect to input arguments
or consistency (through composition) of registration maps, (4) reliance on piecewise
constant reconstruction model, and/or (5) restriction to image pairs (rather than larger

sets). Through an extension to the Mumford-Shah functional and the formulation of
the problem on a single canonical domain, we have devised a variational framework for
simultaneous segmentation, smoothing, and (nonrigid) registration of image suites which
eliminates all of the aforementioned limitations. The energy functional can accurately
be summarized as a linear combination of an intensity-based matching penalty on the
evolving registration maps with Mumford-Shah functionals of each image through the
correspondences defined by those maps. An important aspect is that instead of working

with K different segmentation/smoothing problems, the registration allows us to solve a

single joint smoothing and segmentation problem on a shared domain (~D), as dictated by
the evolving correspondences.









3.2 Derivation of Framework for Flat Image Sets

In this section, we present our unified variational framework. Let {I1, IK} denote

the set of K input images, and Ii a smooth reconstruction of the corresponding Ii. The

outputs of the USSR algorithm are

1. A collection # of registration maps (diffeomorphisms) #sy between ly and Ii,

2. A collection of smoothed images Ii, 1 < i < K.
3. A collection of segmenting contours C1, CK On I1, ,IK, TOSpectively.

The outputs are subject to the following compatibility constraints:



3. #ii = id is the identity map for all i.

The compatibility constraints ensure the consistency among all K images of the

results of all three operations given the registration maps #sy and the segmenting contour

Ci. A general form of the energy functional can be written down immediately as





+ Length(Cs) MatchingCost(#, I)

+ egularization(i#).

The first three (summation) terms represent the combination of Mumford-Shah functionals

of each of the input images. As discussed previously, these terms represent an integrated

approach to assessing the quality of smoothing and segmentation on each image. The last

two terms measure the quality of registration and its smoothness. The interaction between

the registration and segmentation is explicit in the compatibility constraint above, while

the interaction between the registration and smoothing is given by Matching Cost(#, I),

which measures the quality of the registration using the smoothed versions of the input

images. The interdependence of the three operations thus appears naturally.









A direct approach to solving this constrained variational problem is difficult.

However, a slight adjustment in perspective allows for a much more efficient computational

strategy (Figure 3-2). Instead of computing the K2 TeglStration maps #sy and K contours

Ci, we will compute K diffeomorphisms Gi between Ii and a canonical domain D1

and a segmenting contour C on D. This simplifies the computation enormously while

simultaneously satisfying all of the constraints mentioned above: the registration #sy is

defined as eg a 4,-1 and the segmentingr contour on I, is defined as C, = #(C).


Figure 3-2. Unbiasedness and compatibility can be guaranteed by performing all
computations on a canonical domain D.


Mathematically, the canonical domain D and the K registration maps provide

parameterizations for the K input images with a common domain. Computations on each

individual image can now be covariantly formulated on D using the concept of pullbacks.



1 In practice, D is taken to be a rectangular domain with the same size as the images.









For instance, the integral on the left


can be calculated on D as on the right hand side. There are two additional factors: the

pull-back metric gi2 and Je, the determinant of the Jacobian of As.
Through the K registration maps Qi,- #K, we can "pull all of the integrals .1I :

to the canonical domain, giving an energy function that can be evaluated on D:

K:
8(#, C,1) = i ( ())-i( ( )| A





+ Length(Q,(C))+ MC(m,1)+ Reg(m)

where Jo, is the determinant of the Jacobian of es. We choose the thin-plate spline (TPS)

to model the nonrigid deformation. The I control points p {cl, cz} are fixed in D,

and each registration map #4 is uniquely specified by the images of these control points in

Ii. One way to regularize the deformation is to use the bending energy


)2 +2( )22 xy(3)
8 2 iixiyd2

For the sake of computational efficiency, we use a sum of square intensity differences

from the mean smoothed image for the matching cost term. (A preferable but far more

computationally costly alternative would be to insert an information theoretic registration

penalty based on the joint probability density function defined by the registration maps

# and the smoothed images I. However, devising a sufficiently accurate approximation



2 We refer the reader to [16] for the definition of pull-back metric gi Briefly, the metric
gi is the distorted version of the standard Euclidean metric on Ci under the nonlinear map










which scales well with K is itself an ongoing research problem. As such we for now confine

ourselves to a single modality.) Finally, the length of the contour #4(C) can be computed

directly on D as well. Let y(t) denote a parametrization of the contour C: the length of

the contour Ci = #4(C) is then given by the integral





where des is the Jacobian of es. The full energy function is thus given as



i=1D


i= 1
KC K

i= 1


i= 1


3.3 Derivation of System Euler-Lagrange Equations

Let I IK denote the K input images. Also we let D denote the common

domain and # the registration map. A = (01, UK), Where Os is the registration map

#4 : D Ii. The notation here is slightly confusing: we are using Ii to denote both

the image (which is a function) and the image domain (which is the function domain).

However, the confusion should be easily resolved when the context is clear.

The energy functional is given by



i= 1 i= 1 i= 1



i= 1~ C4e(> (

The last two terms do not explicitly depend on Ii and therefore, they do not contribute

to the Euler-Lagrange equations. In the equation above, we have abused the notation









slightly using Ii (and likewise for Ii) to denote Ii(@s(x)). Js denotes the determinant of the

Jacobian of As, and its appearance in the equation above is to ensure that each integral

is equal to the corresponding (push-forward) integral on the images Ii. gi denotes the

pull-back of the usual Euclidean metric under As, and |VE|l~ is the squared norm of the

gradient of Ii using this pull-back metric.

The Euler-Lagrange equation will be a system of K equations (treating each Ii,

1 < i < K as a variable). We will use the formula that the EL equations for an energy
functional of the form


Fp(f1 ,--- ,: fK 1 K 1

are the K equations (for 1 < i < K)




Back to Equation 1, the contributions from the first and third terms are easy to see

since these terms do not have derivatives of Ii. We compute the EL equation for la with

1 < & < K. The contribution from the first term





is simply -2(la Ih) Ja. The contribution from the third term is (let mi=)


202( i (i mI i (1 0 &
i=1,i &

Finally, we consider the contribution from the second term. We note that the volume

form for the metric gi is T Julti1 The second term can be rewritten as


T(Is = p |V~| Jedxdy pl |~a|, dw.

We have

=6 -201 6IAhade 201l = 6S laaIABh










where ah is the Laplace-Beltrami operator


ay,Ii = g'"( ,


anld r0 are th~e C.!-lon!!. symbolhOs. TheC equa~tion A, I = 0 is a2n elliptic partial

differential equation. The first term contains second partial derivatives of Ii, and the

nonlinear second term contains the first partial derivatives of Ii and the metric tensor g .

Putting everything together, the Euler-Lagrange Equation for la is

1 Ji 1
PlA,,,Is + ( -Is) Ja + a2 K (Ii ml)- (1 K )(la ml)) = 0.
i= 1,i &

We note that for every i, Js > 0 everywhere.

3.4 Computational Framework Details

Working in the canonical domain D ensures the satisfaction of all compatibility

constraints, and the absence of bias towards any image or images. To solve the variational

problem, we follow the standard approach of using appropriate gradient flows. In the

following, gi will denote the Riemannian metric on D given as the pullback of the

Euclidean metric on Ii by the diffeomorphism Gi. The gradient flow with respect to

the contour is given by

dC 1 out s
2 f (c I Is )2 _I 2 iiL
i= 1
1 ^ out sin
+1C 2rI II Is | | | V, Ii |, | JoN
i= 1




3 iiiN,(3-5)

in ^ out
where N is the unit normal field of the contour, and Ii Ii are the smoothed images

inside and outside of the contour C, respectively. In the above, as is the curvature of

C with respect to the Riemannian metric gi, and rli is a function that captures (details










omitted) the difference between the metric gi and the standard Euclidean metric on D,

with respect to which the contour will be evolving (i.e., the normals of the contour C are

defined by the Euclidean metric). Both quantities me, rl can be easily incorporated into a

level set framework. In particular, the curvature as with respect to the metric g is given as

(4 is the level-set function)



with the gradient and divergence for the Riemannian metric g given by,


(V,)' gi V,V" = (c ),


where g"" are the components of the local metric tensor.

Next, we derive the gradient for the deformation parameter p of the thin-plate spline

basis function

88~ ~ K 8i
i=1 1 1i-C1i "r

+ 034 (3-6)


The last summand can be computed in closed form using Equation 3-2, and the derivative

inside of the first integral can be evaluated using the chain rule, resulting in a formula

containing image gradients of Ii. The derivative with respect to the curve length can be

derived using Equation 3-3:





The latter integral can be evaluated on the contour C for each #4. Finally, the Euler-Lagrange

Equation for Ii, 1 < i < K is


P14, Ii + (Ii Ii) + P2 4 I) = 0, (3-7)










where o(Jo, I) is a term involving the smoothed images Ii and the determinants of the

Jacobians Je and Asi is the Beltrami-Laplacian operator for the metric gi. The above

equations give a system of nonlinear second-order elliptic equations, which can be solved

using an iterative method, such as quI I- -N. li.-on [14]. The entire algorithm is summarized

below.

Algorithm Summary

For an input set of K intensity images {I1, IK}, the output of the algorithm is a set

of K piecewise-smooth images {I1, IK a Set of K registration maps {@1, ,UK)

between the corresponding Ii/Ii and the shared domain D, and a closed contour C in D.

Those output parameters are evolved as follows:

1. Optimize motion parameters p using gradient descent with gradient given by
Equation 3-6. Update the deformation fields es.
2. Evolve the level set function 4 using Equation 3-5. The contour is updated as the
zero level set of ~.
3. Solve the nonlinear elliptic PDEs in Equation 3-7. This updates the smoothed
images li.
4. If the difference between consecutive iterates is below a pre-chosen tolerance, stop,
else go to Step 1.

3.5 Results on Image Sets

We here present visual examples of simultaneous smoothing, segmentation, and

registration, obtained through a level set contour evolution and thin-plate spline

implementation. In Figures 3-3 and 3-4, the inputs are trios of hands making similar

gestures, but in visibly different places and mannerS3 The segmentation, smoothing, and

registration results are all contained directly in the di;1 i-. II outputs: recall as well that

the segmentation curves correspond across all images in a set through the registration

maps. The examples shown here therefore demonstrate the success of the registration in

matching even such fine features as single fingers, when those fingers are undergoing the




3 Thanks to M.R.C. Tarrosa for acquiring the input images.










fairly complex motion of spreading apart from one another. The (correct) identification of

the fingers in the first column of Figure :3-4 is possible only because of the coupling of the

segmentation of that image to the segmentation of the other two (in which the gaps are

more readily distinguished).

Of course, dense correspondence is also implicitly being obtained between the hand

features (e.g. the crease of the palm) as part of this process. The third sample case

makes this correspondence explicit: in Fig. :3-5, we see the isolation of brain ventricles

in human 1\RI cross sections. Note that ventricle identification in the second (middle)

image is far more difficult than in the first or third, and yet an excellent result is achieved,

thanks to the process coupling. In Fig. :3-6, two alternate methods of visualizing the dense

registration alignment are provided. On the left, we have the effect of each Ai on a set

of concentric circles resident in shared domain D. Since Agi =L 4, 0 O for atll i and ji,

this display provides a visual intuition of how structures in one image are carried onto

another. On the right, we have the initial and final average image as seen from domain

D (i.e. the average pullback). As D is the domain of correspondence, note that the

sharpness of the pullback is a visual method of assessing the success of the registration in

matching intensity values. As we are here segmenting the ventricles, it is the improvement

in clarity and definition of the ventricles with which we are concerned. This effect is

clearly evidenced. (Note that the unusual warping of the ventricles as seen from this

perspective is of no significance. The !....ple !1 >.11. which resides in the domain is

merely a correspondent intermediary between input images, and need not represent a

realistic input, merely a smooth deformation of a realistic input.)





















































Figure 3-3. Processingf of variations on a hand position. Each column represents the
evolution for a single input image in the trio, with the original input on top,
and the final smooth reconstruction at bottom. The evolving segmentation
and registration are depicted through the red curve and blue grid, respectively.
Note the success in corresponding the fingers between all images.























































Figure :3-4. Processing of various openings of the hand, organized as in Figure :3-:3. Note
the identification of the fingers in the first column despite their near closure in
the input image, under heavy noise. This owes to the "implicit atlasingt
inherent in the method.








































Figure 3-5.







Figure 3-6.


Identification of the ventricles of the brains of three different patients (in
respective rows) in cross-sectional MRI scans. Note the successful
identification in the case of the second input, despite the relative fineness of
the structure. The final smoothed images are di;1l li-. I in the last column.



Dj~plw-~ and analysis of inter-image registrations for brain MRI dataset
di;1l li-. I in Fiff. 3-5.


(a) Concentric circles as warped by the final As for
each i (row) in Fig. 3-5, i increasing from left to
right. This is one method of visualizingf the
consistent correspondences between all three pairs.


(b) An alternate visualization of
registration success, as the
sharpness of the average intensity
pullback of the structure being
segmented. On the left is the
initial average (under identity
mappings), on the right the
converged result. The sharpening
of the ventricle is dramatic.










CHAPTER 4
APPLICATION OF USSR FR A1\EWORK( TO SHAPE ANALYSIS

4.1 Motivation for Extending USSR to Shape Analysis Problem

In Ch. 2, we have presented a novel approach to shape comparison which benefits

from being conceptualized around local notions of shape characterization. As such we

are able to provide a fineness of granularity beneficial in many applications, including the

chosen example of hippocampal .I- immetry measurement. However, one could note some

aspects of the proposed framework which invite either extension or improvement:

Lack of theoretical symmetry/unbiasedness. A glance at the variational
principle used in the previously proposed method, Eqn. 2-3, will reveal the fact that
of the two surfaces accepted as inputs to the algorithm, one is arbitrarily chosen as
the domain of energy evaluation and integration. A measure of .l-i lln. ( ry between
two surfaces should itself he theoretically symmetric with respect to permutation of
those surfaces, yet the principle cannot guarantee this as stated.
Restriction to two-mean framework. As an extension to the C'I .I.-Vese
image segmentation scheme, the proposed method can parcellate surfaces only into
regions of either high or low .I-i-mmetry, modelling each only on the basis of its
mean .I-i-int...~ r -y level. While there have been proposals for the extension of the
mean-value framework for image segmentation to a fixed number of regions greater
than two (Chlung5 and Vese, [5]) and a varying number of regions which is itself
optimized over the course of the minimization (Brox and Weickert, [2]), and while
it is true that such implementations could be applied directly to the scheme under
discussion, they would still not eliminate the simplification of mean-based modelling.
Restriction to surface pairs. There is no reason to presume that a user of a
surface shape comparison method will be interested only in comparing pairs of
objects. One might instead wish to identify regions of high shape variability across
subpopulations. Yet the algorithm as described does not admit more than two
inputs, and there is no trivial extension to the multiple input case.

Fortunately, the above list is a subset of the limitations of the prior art in unified

smoothing, segmentation, and registration as enumerated in Sec. :3.1. That set of

limitations is precisely what the USSR algorithm of Ch. :3 was designed to address, in

the domain of image processing. Since our original proposal for local shape .I- i-mmetry

analysis was based on moving standard image processing techniques to the manifold and

having them operate on the metric structures of those manifolds, we might look to adapt










USSR similarly. In so doing, we would produce a technique similar in functionality to the

proposal of Ch. 2, but with certain practical and theoretical superiorities.

4.2 Modification of Variational Principle

Since the proposals of Chs. 2 (local.l-i-inin.! Ir y analysis) and 3 (USSR) are both

variational in nature, their synthesis will be as well. The bulk of the task of defining the

extended shape analysis method is the specification of the variational principle to be

minimized. The first shape analysis proposal was defined through Eqns. 2-3 and 2-4,

which set the form of the functional in terms of the deformation energy, and defined that

deformation energy, respectively. The modified proposal will be defined through two

analogous equations, as follows.

4.2.1 Movement of Computation to Shared Canonical Domain

Glancing at the variational principle driving the original .I-i-n.l!!!! I1 y analysis,

Eqn. 2-3, one sees that of the two surfaces input to the method, S1 and S2, S1 is

arbitrarily chosen as a reference for computation and integration of the .I-i-all!. i 1-y

function (e, Eqn. 2-4), and its parametric domain P1 used as the domain of computation.

Symmetry with respect to the order of the surfaces therefore cannot be guaranteed. One

might note that there is at least one trivial solution to this issue, which is to change the

functional to the following:








However, repeating computation and integration of all quantities on the other domain is

needlessly expensive and lacks theoretical elegance. Another price paid for such a heuristic

solution would be the lack of a clear extension to the case of multiple (more than two)

input surfaces. Something more general must be sought.

Facing the same issue regarding multiple image registration, USSR took the approach

of moving all function computations and integration to a shared canonical domain. But








thanks to the deliberate casting of the shape analysis problem into a framework essentially
equivalent to that of image segmentation and registration, as discussed in Sec. 2.4, USSR's
approach can be applied without much difficulty. Pictorially, the framework will change
from the depiction in Fig. 2-1 to that of Fig. 4-1.


Figure 4-1.


The reformulated version of the local .I-i-all...~ i ly analysis proposal: All K
input surfaces are corresponded through the K maps fi between their
parametric domains Pi and shared canonical domain D. Using the metric
information of the gi, we can define an energy function over D which can be
simultaneously segmented via y, which in turn appears on each Si as yi
through fi, segmenting the surfaces themselves based on group .I-i-mmetry
structure.


137









At its most general level, the system energy functional will now take the form


S(fl, fK ) = ed,4


+ J0 SegmentationEnergydAI

+~ 7 urveRegularizationEnerg.ydn

which by design accepts input of an arbitrary number of surfaces K. To complete the

specification of the modified variational principle, we will need to redefine .I-i-n.l!!!! i t-y

measure e so as to be computable from domain D and unbiased with respect to the

multiple surfaces parametrized by D, and specify the scheme by which the e field will be

segmented over D. Sees. 4.2.2 and 4.2.3 discuss the manner in which this is done.
4.2.2 Functional Form

The form of the original shape analysis functional (omitting explicit domain metric

notation, weighting constants, etc.) is




for domain 12, curve C, and .l--~!~! in t..Iy function e. The second and third term represent

the data terms from ('I! Iil-Vese segmentation as applied to the surface .I-i-all!. i 1-v

parcellation problem: as such they are the source of the two-mean restriction. In

the USSR framework, this restriction was avoided by recognizing the commonly-used

('I! .Il-Vese ("cartoon") model as a limiting case of the general 1\umford-Shah functional,

and implementing the minimization of the latter instead. Using the above terminology, the
form of 1\umford-Shah is


ni Pc d JoC'l JCd

with the first term representing reconstruction smoothness, the second data fidelity, and

the third curve length. The data terms in ('I! .Il-Vese correspond directly to the data










and smoothness terms in the above (there is no separate smoothness term in C'I .Il-Vese

because the limiting assumption of piecewise constancy has already been incorporated

through the region means). Therefore the basic form of our revised functional will be

produced by substituting the latter pair for the former:


/ e dA+ |V|A+ |e-|2A+ d


4.2.3 Asymmetry Measure

In Sec. 4.2.2, we have specified the form of the extended shape analysis functional in

terms of the function representing local deviation from isometry, e. However, the definition

of e given by Eqn. 2-4 is binary in the sense of input arguments, implicitly accepting

exactly two locations, one from each parametric domain of the surfaces under comparison.

This leads to the restriction of the entire algorithm to input pairs, as indicated in Sec. 4.1.

Since we seek to enable the processing of arbitrarily sized "populations" of surfaces,

e must be redefined in terms of a variable number of arguments. But USSR has already

addressed a similar issue, in defining a single registration data term for arbitrarily many

images (a term in Eqn. 3-4). For (smoothed) images lI1 IK, this data penalty function

was given as the sum of differences of individual images from the mean intensity image:



i= 1

Using USSR's registration data term as inspiration, we can replace Eqn. 2-4 with



i= 1 j= 1

Just as, in the case of images, Eqn. 2-4 is designed to assign high registration energy to

pixel locations (as corresponded through the maps) with high variation in intensity across

images, Eqn. 4-1 will, in the shape comparison context, tend to penalize regions of large

FFF difference across the K (arbitrarily many) input shapes.









Related to Eqn. 4-1, there exists an important question regarding the weighting

(area) of the pixel elements in the canonical domain. In the version of the algorithm in

Ch. 2 (as specified by Eqn. 2-3) the issue is rendered trivial by the fact that integration is

arbitrarily carried out on S1, which, under a piecewise constant quadrature scheme, simply

equates to summing the per-pixel deformation energies on parametric domain P1 under

weighting by the areas of the corresponding surface elements. Since those areas are simply

given by the determinants of the FFFs stored at the pixels, in discrete form the integral

fzI fdA becomes the summation term C,1 f|G|, where the summation is of course over
domain pixel elements, with implicit correspondence between the pixels and FFF matrices

G. Under that scheme, the presence of a bias in the energy calculation/integration was

allowed for. However, this is precisely one of the listed limitations to be eliminated in the

modification we are proposing (and is partly the motivation of the use of the canonical

domain in Sec. 4.2.1). The new e function and the integration thereof must therefore

remain unbiased with respect to the input surfaces while capturing in an intuitive manner

the fact that canonical pixels represent comparisons between surface patches of variable

size (which should be accounted for in their integration). Weighting the domain pixels

by the mean element area is such a solution, and is the one we adopt. As such, the

quadrature scheme will involve the factor CE | |j~.
4.2.4 Final System Variational Principle

Using the developments of Sec. 4.2.1, 4.2.2, and 4.2.3, and in particular the

definition of e given by Eqn. 4-1, we can state the modified system variation principle









as follows:


S(fl fK ) =1 e | |dA






+J 02 6-6 dA
j= 1

+ 03 ds (4-2)

where dA is the trivial uniform metric on D. To represent the areas of the surface patches

under comparison in an unbiased manner, this metric is weighted by C3K= -1 'I

4.3 Extension of Mumford-Shah to Case of Weighted Integral

4.3.1 Necessity

In Sec. 4.2.2, we have noted the fact that a Mumford-Shah-based segmentation

scheme forms a component of the modified .I-i-int...~ I1y analysis proposal as codified in

the variational principle of Eqn. 4-2. Theory and practice for the minimization of the

classic Mumford-Shah functional are well-established (as in [15] and [25]), but the terms

appearing in the variational principle proposed here carry an important alteration which

demands a corresponding alteration in the theory. As discussed in Sec. 4.2.3, the domain

of computation D cannot be justifiably considered "f! II in that each pixel corresponds

to K (number of input surfaces) different surface patches, each in general of different

area. As such, the integrals (approximated by summations in the discrete case) are

weighted by a function defined over D chosen to reflect the sizes of the area elements

under correspondence (as detailed, again, in Secs. 4.2.3 and 4.2.4). The price of being

able to perform Mumford-Shah minimization over the familiar two-dimensional pixel grid

is that the Euler-Lagrange equation used in the curve evolution of [25] must be rederived

for the case of weighted integration. This is done below, in Sec. 4.3.2. An interesting

consequence of the extension is revealed, in that an additional term (henceforth referred to









as the "metric correction operator") arises in the differential equation. Additionally, it is

easily noted why this operator reduces to zero in the special case of unity weighting.

4.3.2 Derivation

We will here derive the Euler-Lagrange equation for the general case of a functional

consisting of the Mumford-Shah data and smoothness terms with weighted integration:




where g represents the arbitrary weighting function defined over the domain of integration

D, within which exists closed partitioning curve(s) C. The derivation is a simple matter of

using standard variational calculus in conjunction with a simple application of the formula

for integration by parts in W2:


/i Vu Vvdx uVv vdo- u,vdx,:

where u and v are two functions defined on domain R, and v is the normal vector of the

boundary 80 of R. This equation is also referred to as the first Green's identity.

Now, we seek a critical point of 8 with respect to I, in a functional sense. That is, for

scalar t and any function h defined on D,

I+t h) 8 (I)
him = .
tto t

We can easily write a first-order approximation to S(I + th) in t around t = 0:


S(I + th) a S(I + th)|It=o + t ((a h ~




= (I) -2it g(I -I I) -AdA+ 2at gVI VhdA









Substitutingf into the critical point equation, we have

(I+ th) (I) (I) -2Pt SD g(I -I) AdAl + 2at SD:c gVI VhdA (I)
him h lm
tto t tto t


Now, we make use of the first Green's theorem, through appropriate identification of
factors in the above expressions with those in the theorem's general form as given:

Vv = gVI


Then Green's theorem tells us that


/ AgVI vda= (h~V (gVI) + 9 -gVI)dA

But if we impose a Neumann boundary condition at the region boundaries, then by
definition we have
VI vda = 0


and so


(hV (gVI) + V6 gVI)dA = 0


96 -i glVIdA


AV (gVI)dA


This allows a substitution into the expression derived from the critical point equation:


D /(algVI -i 96 p(I -) I) ldA


(-aV (VI -pg(I -Ijih)dA =


Using the general fact that for any functions u and v,

V (uVv) = Vu Vv + uAv









(a two-dimensional version of the product rule), we can restate this as


/ (f"ghl ghl -ahI -i Vyi -ghailld 4 S hif5yl i3yl -oVI -g Vy -agI)d, = 0I

Since this equation must hold for arbitrary h, the other term in the integrand must he

zero identically:

/3yl 73yl cOVI Vy ~gaI = 0

This is the Euler-Lagrange equation for the general case of the 1\umford-Shah

terms under integration weighted by function g. For the case in which y represents an

area element size (positive definite), division by g is well-defined, and we can write the

expression in the form
Vy
/SI /SI c0 I ca I = 0

ca Vy a ~
I- VII-aI= I

(1 a Vy ~ ni

Note that in the special case in which y is a constant function (e.g. unity), the above

reduces to



which is the Euler-Lagrange eqyuation use~d in Ithe desc~ent in [25]. The operator 'j V,

which acts upon I, can he thought of as the metric correction operator for the more

general case.

4.4 Changes in Implementation

Though the proposal under discussion involves some important conceptual distinctions

from the original shape analysis method of Ch. 2, the conversion of the implementation

of the latter into that of the former can he structured so as to involve a minimal number

of relatively confined changes. One obvious (and trivial, in the sense of implementation)

point of alteration is that the definitions used in the computation of quantities 8 and e,










given by Eqns. 2-3 and 2-4, must be replaced by Eqns. 4-2 and 4-1 respectively. Beyond

this, the only noteworthy structural changes to the program involve the allowance of a

variable number K of input surfaces (as point clouds, all preprocessed identically with

resultant parameterizations as described in Sec. 2.6.1), the maintenance and calculation of

the K map functions from shared domain D to each of these surfaces, the change in the

optimizer's Hessian sparsity structure to account for those changes in the structure of the

map vector, and the replacement of the C'I .Il-Viese-based curve evolution module with the

Mumford-Shah-based approach appropriate for the new system energy.

4.4.1 Alteration of Map Representation and Hessian Structure

The essential nature of the map representation remains the same as was initially

presented in Sec. 2.6.2: we are still working with maps between intrinsically two-dimensional

surfaces by representing them as discrete vector fields over the parameterizing domains.

In the framework of Ch. 2, there existed a single map from S1 to S2 represented as a

2-vector function over the parametric domain of S1, P1, representing the corresponding

locations of each pixel on P2. With a discrete sampling grid of size m x n, two components

in the function, and a zero-order boundary condition on the first and last columns, the

map variables could be represented as a single 2m(n 1)-vector (the concatenation of

the columnwise-vectorized row (u) components followed by likewise vectorization of the

variable column (v) components). Of course, with known sampling size and map variable

vector structure, it is trivial to convert back and forth between the vector (for input to the

optimization method) and grid (for interpolation) representations as necessary.

To represent the map variables in the new version of the algorithm, little need change:

instead of representing a single map from P1 to P2, We must now represent K maps,

each from canonical domain D to one of the K Pi. Each of these maps can be structured

in a manner identical to the above, producing a total of 2Km(n 1) components.

For convenient reshaping between the vector and grid representations, we order these

components via successive concatenations of columnwise-vectorized u-component grids 1









through K, followed likewise by concatenations of the vectorized v-component grids, as in

Fig. 4-2.

um m(n-2)






Figure 4-2. The structure of the argument map vector, used as input to the optimization
method and with respect to which the system Hessian is defined (shown
transposed into a row vector). The a coordinates of the K maps (K segments
with mn elements each) are concatenated first, followed by the variable v
coordinates likewise (K segments of m(n 2) elements), for a vector of total
length 2Km(n 1).


However, alteration of the map vector structure also necessitates alteration of the

Hessian sparsity structure input to the optimizer for efficiency purposes (as explained

and diagrammed in Sec. 2.6.3). Following the earlier logic, however, the correction is

not difficult. We can first partition the Hessian into (2K)2 Submatrices, where each

submatrix is defined by having its rows correspond to one component (either a or v) of a

single map fi, and likewise its columns to one component of fj (j possibly equal to i). Of

these submatrices, we can then identify four types: (1) the row coordinates of the entries

correspond to a coordinates of fi and the column coordinates of the entries correspond to

a coordinates of fj, (2) the row coordinates of the entries correspond to a coordinates of

fi and the column coordinates of the entries correspond to v coordinates of fj, (3) the row

coordinates of the entries correspond to v coordinates of fi and the column coordinates of

the entries correspond to a coordinates of fj, and (4) the row coordinates of the entries

correspond to v coordinates of fi and the column coordinates of the entries correspond to

v coordinates of fj. Let us call each of these submatrices H,,, H,,, H,,, H,,, respectively:

their structures are the top left, top right, bottom left, and bottom right submatrices of

the matrix depicted in Fig. 2-4. The new system Hessian is obtained from the old through

replacement of each submatrix with a K x K block of replications of that matrix.










4.4.2 Switch from Chan-Vese to Mumford-Shah Segmentation

The most involved implementation alteration necessitated by the modified proposal

is the replacement of the C'I .Il-Viese-based curve evolution method with one based

on Mumford-Shah energy minimization. Tsai and Yezzi have demonstrated in [25]

how a level set curve evolution implementation of Mumford-Shah minimization can

be formulated for image functions over bounded rectangular domains (with naturally

uniform associated metrics). In Sec. 4.3.2, we have demonstrated that the difference

between the descent equation for the image reconstruction (for fixed curve position)

in the general case of integral weighting by a general C' function and the special (and

common) case of uniform unit weighting is the appearance of a single additional linear

operator defined in terms of the weighting function and its first derivative. This means

that we can easily base the implementation of the curve evolution (and now, simultaneous

energy reconstruction) module of the modified .I-i-mmetry analysis proposal on the

Tsai-Yezzi implementation through appropriate inclusion of the metric correction operator

derived in the referenced section, and implementation of the u-periodicity demanded

by the cylindrical topology of the parametric domain. Since [25] omits most of the

implementation details of functional minimization, and since this method forms the core

of the curve evolution and piecewise-smooth energy reconstruction module, we will first

discuss our chosen implementation approach in Sec. 4.4.2.1, and the inclusion of the metric

correction operator in 4.4.2.2.

4.4.2.1 Implementation Details for Mumford-Shah Minimization

For an image function I, we have already seen that for fixed curve C, the equation for

the optimal piecewise smooth reconstruction I is





where I and I are in columnwise-vectorized form, 1 is the appropriately sized identity

matrix, and adi, is a modified version of the usual discrete Laplacian operator. It is easy










to see that once that modified Laplacian has been specified, I can be solved for directly

(or, for the sake of computational efficiency, approximated by an iterative method to a

prespecified error tolerance: the method of conjugate gradients serves this purpose well).

The formation of Ags,, then, is the only nontrivial portion of optimal image reconstruction,

for a fixed curve position.

adi, differs from the usual Laplacian operator a in that it must respect the Neumann

boundary condition across the curve C. In a discrete implementation, over a regular grid,

a sensible way of enforcing this condition is to nullify any finite difference derivatives

which involve quantities from opposite sides of the segmenting curve. This discrete notion

of the Neumann condition is what produces the difference between a and Adis. However,

it is easy to see that the ordinary Laplacian can be formed through the linear combination

of standard finite difference operators:


a = Dye Day + Dyv Day


where the four terms on the right side are the forward difference in n, backward difference

in n, forward difference in v, and backward difference in v, respectively. But these four

matrices can easily be modified to respect the discrete Neumann condition through

left multiplication by a diagonal boolean matrix indicating pixel locations at which the

difference operator crosses the curve boundary: in the level set representation, a curve

crossing simply means a change of sign of the level set function at the two pixels used in

the difference equation (the nonzero entries in the corresponding difference operator row).

These .. Ininglt; matrices are of course a function of the current position of the curve,

and must thus be recalculated at each step. For each curve position, there are four such

matrices, one to act on each of the four derivative matrices used in forming the Laplacian,

and we can then simply recalculate the disconnected Laplacian for the current curve

position as

adis = ZFUDFU ZBUDBU + ZFVDFV ZBVDBy









Solving for I then yields the optimal piecewise smooth function reconstruction, in

accordance with weighting constants a~ and P as set in the variational principle.

For any curve position, then, we are able to calculate I, which allows evaluation of the

system energy. This in turn enables us to implement the descent with respect to the curve

position, which is



-~~~~ I-lst2_ _in 2)





in continuous form. The computation of curvature a can be done in the level set

framework as discussed in detail in [20]. The computation of the other terms is straightforward

once a method for calculating lost and Ii" has been determined. These terms refer to the

reconstructed function values at the curve boundary if they are considered to lie outside

or inside of the curve, respectively. In the discrete implementation, of course, function

values are only provided at pixel locations, and these will in general not lie "on" the curve:

our concern is whether at any step, a pixel .ll11 Il:ent to the curve should be moved to the

other side of it. A sufficiently accurate and fast approach to computing (defining, really)

the terms lost and Ii", then, is to advance the curve outward everywhere by one pixel

(bringing all border- Il11 ..:ent outside pixels to the interior), recompute I to serve as Ii",

and to repeat the process using inward curve motion for computation of lowt. With these

quantities, the energies of inward and outward motion can be compared for all bordering

pixels, and the curve updated accordingly. Again, numerical methods for level set curve

maintenance and updating are found in the provided reference.

4.4.2.2 Inclusion of Periodicity and Metric Correction Operator

Relative to the implementation described in Sec. 4.4.2.1, there is nothing to do to

produce the curve evolution and piecewise smooth function reconstruction module of the

modified proposal but to implement the a periodicity necessitated by the domain topology










and the metric correction operator derived in Sec. 4.4.2.2. The former is simply a matter

of appropriately altering the finite difference operators throughout the program (including

in the construction of Vass): call the modified difference operator matrices DFUper and

Dsuver. For the latter task, the continuous operator Vandi is given in discrete form as

ZF Uper DF Uper +ZB Uper DB Uper
2 g ZFUperDFUper + ZBUperDBUper
g 2
2 DvZs~v g ZFVDFV + ZBVDBy
g 2

We then need only include this term in the discrete implementation of solution of

Eqn. 4-3, and otherwise proceed as directed in Sec. 4.4.2.1.










CHAPTER 5,
CONCLUDING RE1\ARK(S

In this dissertation, we have presented a method for local shape analysis of homologous

structures, a general image processing framework for groups of images in which suboperations

provide mutual assistance, and a proposal for improving the former using the insights

of the latter. Through visual observation and statistical validation, we were able to

demonstrate the practical benefits of the novel algorithms. If there is an overarching

theme in their formulation, it is generality: while the implementation details given

herein (certainly in the case of Ch. 2) are specific to the preprocessing steps and

f~~lia I(f ir_ /representation of the data, the formulations themselves remain quite broadly

applicable. Far from being specific to the hippocampi of epileptics, the shape analysis

proposal is conceived around general genus zero surfaces for which symmetry is expected.

This of course covers a wide variety of practical applications, in addition to the theoretical

contribution. As we have noted in Sec. 2.4.1, the particular application and representative

dataset chosen did influence formulation in that the higher-frequency SFF information

was completely discarded, as both numerically problematic and somewhat conceptually

irrelevant in the case of that application. However, if desired for either theoretical or

practical reasons, one could reincorporate this aspect of the shape characterization into

the formulation in a straightforward manner. (Again, the caveat is that to do so implicity

places substantial trust both in the accuracy of the curvature data represented by the

input shape and in the importance of that data in characterizing the shape.) Likewise,

the USSR proposal is arguably the most general image processing framework of its sort

heretofore -II_a---- -1. While certain aspects of the "theoretical implementation" we have

used, such as the use of thin-plate spline regfulariz ation and sum-of-s quare- differences

intensity data penalty on the registration component, do influence/limit the method's

performance and scope of applicability, they can easily be modified as seen fit. For

instance, if one wishes to pl li- the higher computational cost of an information-theoretic









match measure in exchange for the multi-modality it would afford, the substitution is

trivial. It is perhaps due to the general applicability of these methods in their respective

(and distinct) domains that their synthesis can be formulated so naturally. But in

emphasizing generality, we should not neglect the role of recognition of interdependence.

As the field of computer vision has matured, its individual subareas have become

increasingly (and at times exhaustingly) well studied and understood. But there remains a

good deal of unexplored space between these topics, and with it a chance to yield a whole

greater than the sum of its parts.










REFERENCES


[1] S. Bouix, J. C. Pruessner, D. L. Collins, and K(. Siddigi, "Hippocampal shape analysis
using medial surfaces," N~ ~. ur...nny vol. 25, no. 4, pp. 1077-1089, 2005.

[2] T. Brox and J. Weickert, "Level set segmentation with multiple regions," IEEE
Treen~srctions on Image Processing, vol. 15, no. 10, pp. :321:3-3218, 2006.

[:3] T. F. C'I I1. and L. A. Vese, "Active contours without edges," IEEE Transactions on
Image Processing, vol. 10, no. 2, pp. 266-277, 2001.

[4] L.-T. C'I. in_ P. Burchard, B. Merriman, and S. Osher, "Motion of curves constrained
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[5] G. Changl5 and L. A. Vese, "Energy minimization based segmentation and denoising
using a muiltil .w;r level set approach," in Energy Afinimization M~ethods in C'omputer
Vision and Pattern Recognition, 2005, pp. 4:39-455.

[6] T. Coleman and Y. Li, "An interior, trust region approach for nonlinear minimization
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[7] J. G. Csernansky, L. W.'11_ D. Jones, D. Rastogi-Cruz, J. A. Posener,
G. Heydebrand, J. P. Miller, and 31. I. Miller, "Hippocampal deformities in
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[8] J. Csernansky, L. W .11. J. Swank, J. Miller, 31. Gado, D. McE~eel, 31. Miller,
and J. Morris, "Preclinical detection of alzheimer's disease: hippocampal shape
and volume predict dementia onset in the elderly," N. ;,, e.:l.Un.;p vol. 25, no. :3, pp.
78:3792, 2005.

[9] R. H. Davies, C. J. Twining, P. D. Allen, T. F. Cootes, and C. J. Taylor, "Shape
discrimination in the hippocampus using an mdl model," in IPM~I, 200:3, pp. :38-50.

[10] 31. P. do Carmo, Di~ferential Geometry of C'urve~s and S;, il r., Prentice Hall, 1976.

[11] ----, Rievermnian Geometry. Birkhauser, 1992.

[12] G. Gerig, K(. E. Muller, E. O. K~istner, Y.-Y. Chi, 31. C'I I1:.~-~ 3. Styner, and J. A.
Lieberman, "Age and treatment related local hippocampal changes in schizophrenia
explained by a novel shape analysis method," in flC'C'AI, 200:3, pp. 65:3660.

[1:3] H. Jin, A. J. Yezzi, and S. Soatto, "Region-based segmentation on evolving surfaces
with application to :3d reconstruction of shape and piecewise constant radiance," in
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[ 14] P. K~nabner and L. Angermann, Numerical M~ethods for Elliptic and Petrabolic Partial
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[15] D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions
and associated variational problems," Commun. Pure Appl. M~ath, vol. 42, no. 4, pp.
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[16] B. O'Neill, El~ I,to ,lr.J; t Differential Geometry. Academic Press, 1997.

[17] K(. Pohl, J. Fisher, W. Crimson, R. K~ikinis, and W. Wells, "A' l.i-, I1, model for
joint segmentation and registration," N. or;,,n.:ly.l vol. 31, no. 1, pp. 228-239, 2006.

[18] K(. Pohl, J. Fisher, J. Levitt, M. Shenton, R. K~ikinis, W. Crimson, and W. Wells,
"A unifying approach to registration, segmentation, and intensity correction," in
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[19] F. J. P. Richard and L. D. Cohen, "A new image registration technique with free
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[20] J. A. Sethian, Level Set M~ethods and Fast M~arching M~ethods. Cambridge Press,
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[21] L. Shen, J. Ford, F. Makedon, and A. Saykin, "Hippocampal shape analysis:
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[23] M. Styner, G. Gerig, J. Lieberman, D. Jones, and D. Weinberger, "Statistical shape
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[24] M. Styner, J. A. Lieberman, and G. Gerig, "Boundary and medial shape analysis of
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[25] A. Tsai, A. J. Y. Jr., and A. S. Willsky, "Curve evolution implementation of
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[26] G. B. Unal and G. G. Slabaugh, "Coupled pdes for non-rigid registration and
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[27] B. Vemuri, Y. C!. li, and Z. Wang, "Registration-assisted image smoothing and
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[28] B. Vemuri and Y. Guo, "Snake pedals: Compact and versatile geometric models with
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[29] F. Wang and B. C. Vemuri, "Simultaneous registration and segmentation of
anatomical structures from brain mri," in M~ICCAI, 2005, pp. 17-25.

[30] L. Wa y1 S. C. Joshi, M. I. Miller, and J. G. Csernansky, "Statistical analysis of
hippocampal. .-i-mmetry in schizophrenia," N~ ~.:lr..l:ay vol. 14, no. 3, pp. 531-545,
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[31] Y. Wang, M.-C. Chiang, and P. M. Thompson, jl!utual information-based 3d surface
matching with applications to face recognition and brain m1 .pi l~rs; in Proc. Int'l.
Conf. on Comp~uter V/ision, 2005, pp. 527-534.

[32] P. P. Wyatt and J. A. Noble, "Map mrf joint segmentation and registration of
medical images," M~edical Image Aithe;i vol. 7, no. 4, pp. 539-552, 2003.

[33] C. Xiaohua, J. Brady, and D. Rueckert, "Simultaneous segmentation and registration
of medical images," in M~ICCAI, 2004, pp. 663-670.

[34] A. Yezzi, L. Ziillei, and T. K~apur, "A variational framework for joint segmentation
and registration," in IEEE-MM~BIA, 2001, pp. 44-51.

[35] Y.-N. Young and D. Levy, "Registration-based morphing of active contours for
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79-96, 2005.










BIOGRAPHICAL SKETCH

Nicholas Lord has been, above all things, poor at waking up to alarm clocks. He did

not grow up with any particular academic direction, and to the best of his knowledge still

does not possess one. He likes the fact that 'Caucasian' rhymes with 'Jamaican'. He can

do the Frug, Robocop, and Freddie, but cannot do the Smurf. He is seldom accused of

excessive reverence. He is unsure whether there exists a finer food than fried red snapper

with chili garlic sauce. He interprets praise of Richard Linklater movies as a deliberate

attempt to get on his bad side, and takes David Cronenberg more seriously than most.

He has performed tasks for the purpose of securing sustenance, and will no doubt do so

again at some point in the future. Finally, he is a bit old-fashioned in that he draws a

distinction between close friends and total strangers, and as such believes that .Inyione

wishing to hear meaningful details of his life's history can either offer him a job or buy

him a drink.


Figure 5-1. Your author: a wascawwy wabbit.





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formorethanIaskedfor, formorethanIdeserved, formorethanIcanrepay 3

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There'snothingoriginalaboutstrikingamodesttoneinanacknowledgmentsection,but,seriously,Ididnearlynothing.Iwasthefortunatebeneciaryofaconstantstreamofsupport,guidance,andinspiration,withoutwhichtheprobabilityofmygettingthroughthepastfourandahalfyearswouldhavebeenzero.Thanksareinorderto,atbareminimum,thefollowing: Denise,whotooktheleadinseducingmeintochangingmymaster'sdegreeprogramapplicationintotheonethatresultedinallofthis.Whatwastobeahandfulofadditionalclassesforasubstantialincreaseinsalaryinsteadbecameaminiaturelifetimeofselfdoubtandsporadicinsanity.Giventhechancetodoitagain,I'dgladlyletherleadmeastrayoncemore. TheUniversityofFlorida,theDepartmentofComputerandInformationSciencesandEngineering,andtheAlumniFellowshipFund,forgivingmetheopportunitytocompletethisdegree,andwithnetreatmentthroughout.Itwasgoodtobehomeforawhile(andIstandbymyanti-northeasternrantsnowmorethanever). MyadvisorDr.Vemuri,whogavemeaspotinoneofthebestcomputervisionlabsintheworldandheldmetoitsstandards.Or,moreaccurately,heldmetoitsstandardsafterwaitingthroughtwoyearsofmygettingmystutogether.Theexperiencewasmostlybrutal,aswithalmostanythingworthdoing.Icansafelyrecommendhimtoallmasochistsinterestedintheeld:ifmyexperienceisindicative,theworldwillbeyouroysteronceit'sover.IsupposeIcannallyretirethatvoodoodoll. Therestofthemembersofmydissertationcommittee,fortheirvariouscontributions:myexternalmember,Dr.Hager,fordonatinghistime;Je,forworkingcloselywithmeonallofmypapers,forbeingthesourceofmuchofwhatwasinspiredaboutthem,andforalwaysndingawaytomakerigorousexplanationsofmathematicalarcanitiesdigestible;AnandandArunava,forbeingthemostintelligent,engaging,andinspirationallecturersI'veeverhadthepleasuretositwith,andformakingIndianslessconfusingtome. MomandDad,whowereputonthisearthtodeectthebulletwhenIattempttoshootmyselfinthefoot.Everytimemyneglectgetsmeintotrouble,Icallthesamepeopletobailmeoutofit.I'msureI'llhavetokickthehabiteventually,butIwouldn'thavebeenabletowalkthisfarwithoutthatcrutch.Beyondthelast-minuterides,calls,andshipments,whateverIneededtomakemylifemorecomfortable,alaptop,acar,whatever,iteitherliterallyorgurativelyshoweduponmydoorstep.Fortellingmethateverythingwasgoingtobealright,andmakingsurethateverythingreallywas.Loveyouboth,pardontheslipintosecondperson. Nate,forbeingthecoolestbrotherpossible,andKristan,forsharingthewealth. 4

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GrandmaPaulineandLarry,forensuringthatasometimes-destitutegraduatestudentwasabletodothetravellingthathebothwantedandneededtodo.It'salotmorefunwithoutthescraping,andfewhavetheluxuryofavoidingit. Myentireextendedfamily(intheThirdWorldsense:thosewhoknow,know),includingtheabove,forbeingthere,andneverdoubting. ThemanypeoplewhomI'vecalledlabmatesovertheseyears,forbeinganabsurdlybrilliantcrop.WhenIjoined,itwaswiththehopethatIwouldnallybeabletoconsidermyselfasbeingsurroundedbypeers:Iovershotandwoundupwithmysuperiorsinstead.Fewthingsaremorehumblingthanconsideringwhatsomeofthemhavealreadymanagedtodowiththeiryounglives.SpecialthanksisagainduetoSanthoshforhisassistanceinthestatisticalanalysisportionofmyrstjournalpaper,withoutwhichImightstillbeworkingonmyrstjournalpaper. TheentireDeVicenteclan,fortakingmeininthemidstofthatlittletransitionalperiodofmine,andresistingthetemptationtoinvoketheBakerAct.StarcraftIIwillbeoutbeforelong,ifanyone'sdown. Raneeandhisfamily,foradesperatelyneededtripoutofDodge,andMichele,foradesperatelyneededfemaleear.ButIdohopethatbitaboutthehyphenatedsurnamewasajoke. Ko,foranintroductiontotheSanFranciscoBayAreawhichborderedonprofessional.Ifthatengineeringthingofhisdoesn'tpanout,hehasabrightcareerasatourguideaheadofhim.Also,forthethirteen-oddyearsprior,overwhichIwasexposedtosomuchhealthycompetitionthatIamnowunabletomistakethepathologicalvarietyforanythingotherthanwhatitis. JoshandSakar,forbeingquitehairy,andforcontinuingtheonelongconversationthattheyandIhavebeenhavingforsometimenow. Barth,forwastingnearlyeveryworkdayofeveryworkweektalkingtomeonAIM.(Bysymmetry,Imustnothavegottentoomuchdoneeither.)Also,forfallingasleepatstoplights,screeningSecretaryforhismother,andingeneralservingasareminderofhowgooditisthatGermanylosttheSecondWorldWar. DanandTiany,forthatspotontheairmattressinthelivingroom. Anna,foralovelypackingjob. Lisa,whotookpainfullyawfulsingingandgently,patientlysculpteditintomildlylesspainfullyawfulsinging.IenviedherjobuntilIrealizedthatitmustrequiretheconstitutionofasaint.(BonuspointsforlikingthetimbreofSerj'svoice.) SiGungChanPoiforbringingWahLumtoFlorida,SifuDaleforbringingittoGainesville,SiHingAustinforrunningit,andSiHingMattforkeepingitgoingas 5

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Allofthemusicians,writers,andlmmakerswho'vemademylifeworthliving,andallofthewomenwho'vedonethesame.IbelieveVonnegutsaidafewthingsaboutthistopicbeforeheleftus. Maggie,foreverything.There'stoomuchformetoeventry.Tobediscussedoverscones. 6

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1PROPOSALSUMMARY .............................. 12 1.1LocalSurfaceShapeAsymmetryAnalysis .................. 12 1.2USSR:UniedSegmentation,Smoothing,andRegistration ......... 12 1.3ApplicationofUSSRFrameworktoShapeAnalysis ............. 13 2LOCALSURFACESHAPEASYMMETRYANALYSIS ............. 15 2.1Background:HippocampalShapeAnalysis .................. 15 2.2Background:SimultaneousRegistrationandSegmentation ......... 16 2.3ApplicationofRegistrationandSegmentationtoShapeAnalysis ...... 17 2.4DerivationofFrameworkforClosed2DSurfaces ............... 18 2.4.1PracticalDierencesBetweenFFFandSFFasMatchingCriterion 23 2.5SystemEnergyFunctional ........................... 25 2.6ImplementationDetails ............................. 27 2.6.1HippocampalSurfaceDataAcquisitionandFormatting ....... 27 2.6.2DiscreteRepresentationoftheDeformationMapandtheSegmentationCurve ................................... 28 2.6.3SparsityoftheHessian ......................... 29 2.7Results ...................................... 30 2.7.1OutputVisualization .......................... 30 2.7.2ClassicationAnalysis ......................... 32 3USSR:UNIFIEDSMOOTHING,SEGMENTATION,ANDREGISTRATION(FORMULTIPLEIMAGES) ............................ 39 3.1MotivationforUnicationofImageProcessingOperations ......... 39 3.2DerivationofFrameworkforFlatImageSets ................. 41 3.3DerivationofSystemEuler-LagrangeEquations ............... 44 3.4ComputationalFrameworkDetails ...................... 46 3.5ResultsonImageSets ............................. 48 4APPLICATIONOFUSSRFRAMEWORKTOSHAPEANALYSIS ...... 53 4.1MotivationforExtendingUSSRtoShapeAnalysisProblem ........ 53 4.2ModicationofVariationalPrinciple ..................... 54 4.2.1MovementofComputationtoSharedCanonicalDomain ...... 54 7

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............................. 56 4.2.3AsymmetryMeasure .......................... 57 4.2.4FinalSystemVariationalPrinciple ................... 58 4.3ExtensionofMumford-ShahtoCaseofWeightedIntegral .......... 59 4.3.1Necessity ................................. 59 4.3.2Derivation ................................ 60 4.4ChangesinImplementation .......................... 62 4.4.1AlterationofMapRepresentationandHessianStructure ...... 63 4.4.2SwitchfromChan-VesetoMumford-ShahSegmentation ....... 65 4.4.2.1ImplementationDetailsforMumford-ShahMinimization 65 4.4.2.2InclusionofPeriodicityandMetricCorrectionOperator 67 5CONCLUDINGREMARKS ............................. 69 REFERENCES ....................................... 71 BIOGRAPHICALSKETCH ................................ 74 8

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Figure page 2-1Illustrationofshapeanalysisframework. ...................... 19 2-2Apairofcylindricalsurfaceswithidentical\Gaussianbumps"ofoppositeorientation.TheextrinsicorientationinformationisdiscardedinusingtheFFFasthematchingcriterion:inthisframework,thetwosurfacesareconsideredidentical. ...... 24 2-3Apairofcylindricalsurfaceswhicharedierentiatedonlybysurfacenoise.Underthe(identity)mapwhichwouldcorrespondthenoise-freeversions,theSFFscorrespondpoorly. .................................. 25 2-4Hessiansparsitystructure. .............................. 29 2-5Syntheticresults:caseofcylinderswithsurfacedistortion. ............ 32 2-6SegmentationofdistortionbetweenhippocampiofLATLsetmember. ...... 33 2-7SegmentationofdistortionbetweenhippocampiofRATLsetmember. ..... 34 2-8Segmentationofdistortionbetweenhippocampiofcontrolsetmember.(Thesmallerofthetwosegmentedregionshasbeenchosenforenlargement.) ..... 35 2-9Epilepsyclassicationresults:optimaltestaccuraciesoverallclassiersshowninbold. ........................................ 37 3-1Natureofinterdependencebetweensegmentation,smoothing,andregistration. 40 3-2UnbiasednessandcompatibilitycanbeguaranteedbyperformingallcomputationsonacanonicaldomainD. .............................. 42 3-3Processingofvariationsonahandposition.Eachcolumnrepresentstheevolutionforasingleinputimageinthetrio,withtheoriginalinputontop,andthenalsmoothreconstructionatbottom.Theevolvingsegmentationandregistrationaredepictedthroughtheredcurveandbluegrid,respectively.Notethesuccessincorrespondingthengersbetweenallimages. .................. 50 3-4Processingofvariousopeningsofthehand,organizedasinFigure 3-3 .Notetheidenticationofthengersintherstcolumndespitetheirnearclosureintheinputimage,underheavynoise.Thisowestothe\implicitatlasing"inherentinthemethod. .................................... 51 3-5Identicationoftheventriclesofthebrainsofthreedierentpatients(inrespectiverows)incross-sectionalMRIscans.Notethesuccessfulidenticationinthecaseofthesecondinput,despitetherelativenenessofthestructure.Thenalsmoothedimagesaredisplayedinthelastcolumn. ...................... 52 9

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3-5 ...................................... 52 4-1Thereformulatedversionofthelocalasymmetryanalysisproposal:AllKinputsurfacesarecorrespondedthroughtheKmapsfibetweentheirparametricdomainsPiandsharedcanonicaldomainD.Usingthemetricinformationofthegi,wecandeneanenergyfunctionoverDwhichcanbesimultaneouslysegmentedvia,whichinturnappearsoneachSiasithroughfi,segmentingthesurfacesthemselvesbasedongroupasymmetrystructure. ................. 55 4-2Thestructureoftheargumentmapvector,usedasinputtotheoptimizationmethodandwithrespecttowhichthesystemHessianisdened(showntransposedintoarowvector).TheucoordinatesoftheKmaps(Ksegmentswithmnelementseach)areconcatenatedrst,followedbythevariablevcoordinateslikewise(Ksegmentsofm(n2)elements),foravectoroftotallength2Km(n1). .... 64 5-1Yourauthor ...................................... 74 10

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Localityisanimportantbutoft-ignoredaspectofshapeasymmetryquantication,andsegmentationisonemethodbywhichtomakethelocalityofananalysisexplicit.Wehavethusformulatedanapproachtosurfaceregistrationandshapecomparisonwhichfeaturesanintegratedsegmentationcomponentforthepurposeofsimultaneouslyidentifyingandseparatingregionsbytheirevolvingdeformationcharacteristics.Intherstinstantiation,weachievethiseectthroughanadaptationoftheChan-VeseapproachforimagesegmentationtotheproblemofsegmentingtheRiemannianstructuresoftheverysurfacescomprisingthedomainofthesegmentation.Wehavesuccessfullyusedthemethod'soutputonhippocampalpairsinanepilepsyclassicationproblem,demonstratingimprovementoverglobalmeasures.NotingthataChan-Vese-basedapproachtosimultaneoussegmentationandregistrationisinherentlylimited,wehavealsodevelopedauniedapproachtosegmentation,smoothing,andnonrigidregistrationofimagesviaextensionoftheMumford-Shahfunctional,devisedinsuchawayastobeapplicablesymmetricallyandconsistentlytomultiple(twoormore)inputs.Toconclude,weproposeanextensionofthisuniedframework(dubbed\USSR")tothepreviouslyconsideredproblemof2Dsurfaceshapeanalysis(asymmetryquanticationandlocalization),conferringthebenetsofunbiasedness,consistency,multipleinputprocessing,andnontrivialdataeldreconstruction. 11

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30 ]thatanalysisofregionalasymmetriescouldimprovediseaseclassicationcapability.Severalmethodsforne-grainedregionalhippocampalshapeanalysishavesincebeensuggested.Gerigetal.[ 12 ]includedamedialshaperepresentationwithageanddrugtreatmentdatainanexploratorystatisticalanalysisofthehippocampus'slinktoschizophrenia.Shenetal.[ 21 ]conductedastatisticalanalysisbasedonthesphericalharmonic(`SPHARM')representationmethod.Styneretal.[ 23 ]testedthepowerofaSPHARM-basedmedialrepresentationtoseparatemonozygoticfromdizygotictwinsthroughlateralventricularstructure,andschizophrenicsfromnormalsthroughhippocampalandhippocampus-amygdalanstructures.Daviesetal.[ 9 ]devisedaminimumdescriptionlengthframeworkforstatisticalshapemodelingandextractedmodesofvariationbetweennormalandschizophrenicpopulations.Bouixetal.[ 1 ]employedmedialsurfacesinalocalwidthanalysis.Usingaviscousuidowmodel,Csernanskyetal.[ 8 ]computeddieomorphicmapsofpatienthippocampiontoareference,producingadenseinward/outwarddeformationeldovereachhippocampalsurface.ThesurfaceitselfwasadditionallymanuallysegmentedtoallowfortheregionalcomparisonofthedeformationeldsaspartofanattempttoseparatehealthyindividualsfromthoseexhibitingdementiaoftheAlzheimer'stype.Allreportedresultshaveindicatedthebenetofincorporatingregionalinformationintotheanalysis.Thisisunsurprising,as 15

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24 ]thecontrastbetweentheprimaryabnormalitylocalizationinthehippocampaltailfoundinthatworkandthelocalizationintheheadreportedin[ 7 ]).Furthermore,whilestatisticallysignicantdierencesofhippocampalshapehavebeenidentiedbetweendiseasedandnormalsamplepopulations,reliableclassicationofasizeablenumberofpatientswithrespecttothosecategorieshasnotoccurredpreviously. 34 ],whointroducedtheideaofexplicitinterdependenceofsegmentationandregistration:registrationmethodscanmakeuseofthefeaturedetectioninherentinsegmentation,whilesegmentation(particularlyinthecaseofnoisyorincomplete/occludeddata)canutilizetheredundancyprovidedbycorrectlyregisteredimagesofthesamestructurestosegmentsaidstructuresmorerobustlyjointlythanindividually.Theirproposalinvolvedtheunicationofmutualinformation(MI)-basedat2Drigidimageregistrationwithalevelsetimplementationofapiecewiseconstant(Chan-Vese)segmentationschemethroughavariationalprinciple.Wyattetal.[ 32 ]accomplishedmuchthesamething,solvinginsteadamaximumaposteriori(MAP)probleminaMarkovrandomeld(MRF)framework.Xiaohuaetal.[ 33 ]extendedthistothenonrigidregistrationcase,andUnaletal.[ 26 ]addressedthesameproblemusingcoupledpartialdierentialequations(PDEs).In[ 29 ],F.Wangetal.presentedasimultaneousnonrigidregistrationandsegmentationtechniqueformulti-modaldatasets.RichardandCohen[ 19 ]proposedavariationalframeworkforcombiningregionsegmentationandregistration(matching)usingfreeboundaryconditions.Jinetal.[ 13 ]simultaneouslyevolvedasurfacesegmentationandaradiance-discontinuity-detecting 16

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35 ]combinedjointsegmentationand(scaledrigid)registrationwithmorphingactivecontoursforsegmentationofgroupsofCTimages.Pohletal.[ 17 ]producedanexpectationmaximization(EM)-basedalgorithmforsimultaneousaneregistrationandsubcorticalsegmentationofMRIsusingalabeledcorticalatlas. 31 ]),inwhichtheauthorspresentedatechniquetononrigidlyregistersurfacesusinga2D(parametric)dieomorphicmapconstructedfromRiemanniansurfacestructureinformation.Theintegratedmanifoldsegmentationisenoughtodistinguishourapproachfromthiswork,butadditionally,ourdenitionof 17

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22 ]forseparatingphysicalmotionfromsurfacedeformationinregistrationofcorrespondentregions.) 3 ],forinstance,thestatisticisthemeanofeachcurve-denedregion).Further,dierentialqualities(e.g.gradient,Laplacian)oftheintensityprolescanproveusefulinidentifyingedges,corners,andotherusefulfeatures.Whenconductingpairwisesegmentationandregistrationofshapesratherthanimages,however,theappropriatemeasuresareperhapslessobvious.Itremainstoustodenethem. 18

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Illustrationofshapeanalysisframework:TheinputtothealgorithmisapairofhomologousshapesS1andS2.UseofappropriateboundaryconditionsallowsustoassociatetheshapeswithatrectangularpatchesP1andP2respectively.Thesepatches,nowconsideredasparametricdomains,storeallRiemannianmetricinformationg1andg2oftheirrespectivesurfacesintermsoftherstfundamentalformmatricesG1andG2.ThemetricinformationisthenmatchedbetweenP1andP2throughavariationalprinciplewhichdrivesbothahomeomorphicmapfofP1ontoP2(dieomorphicexceptpossiblyonthecurves)andasegmentingcurve1inP1whichiscarriedbyfintoP2as2.ThisprocessregistersandsegmentsthesurfacesS1andS2viatheparametriccorrespondence.NotethatthemapfbetweentheparametricdomainscanbevisualizedasthedeformationofaregulargridrepresentingtheleftparametricdomainP1,andthatthesegmentationinteriors(red)andexteriors(blue)canberepresentedonS1throughshading. Beforedeningamatchingcriterion,werstshowhowitispossibletocasttheshaperegistrationandsegmentationproblemintoacomputationalframeworksimilartotheimageframeworkdescribedabove(saveforthematchingcriterionitself)throughappropriateparametrization.Toaccomplishthis,oneneednoteonlythefollowingsimpletopologicalfacts:(1)thesurfacesofhippocampicanbethoughtofas2DRiemannianmanifoldsembeddedinIR3,(2)thesemanifoldsaretopologicallyequivalenttothesphere(closedandgenuszero),(3)theexclusionofapairofpolesfromthesphereproduces 19

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4 ]).TheapproachcanbevisualizedinthemannerdepictedinFig. 2-1 Clearly,thematchingcriterionwhichwedeneinplaceofimageintensitymustcommunicateshapeinformationaboutthesurfacesbeingregisteredandsegmented.Wenowtaketheopportunitytodenetherstandsecondfundamentalformsofaparametrizedsurface,andsubsequentlymotivatetheirconsideration: whereXisthe3Dsurfacemap,uandvarethemapparameters(equivalently,thecoordinatesystemoftheparametricdomain),Nisthesurfacenormal(denedasXuXv Butawell-knowntheoremduetoBonnet(asstatedin[ 10 ])readsasfollows: 20

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2{1 and 2{2 asfollows:)FFF=0B@EFFG1CASFF=0B@effg1CA 1 mayappeararcaneinitsgivenform,abriefexplanationshouldilluminateitsapplicationtotheprobleminquestion.ThesetVinthetheoremrepresentsapatchofatwo-dimensionalparametricdomain(suchasP1andP2inFig. 2-1 ).E>0andG>0arenecessaryconditionsforthosequantitiestobevalidcoecientsintheFFFofanysurfaceparametrizedonV,whichisobviousfromthedenitioninEqn. 2{1 .EGF2>0,therequirementthattheFFFmatrixbepositivedenite,isnecessaryformapregularity,whichisinturnanecessaryconditionfordieomorphism,theonlyclassofmapswithwhichwearehereconcerned.(Sincep 21

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Whatthetheoremthentellsusisthatiftheseconditionsaresatised,thenViseverywherelocallyparametrizingsomeregularsurfacethroughadieomorphicmapxwithFFFandSFFspeciedbyE,F,G,e,f,andg.Infact,notonlydoessuchasurfaceexist,itisdetermineduptorigidmotion,andassuchitisuniqueinthesenseofshape.Therefore,ifweconsideronlymapsxwhicharedieomorphicovertheentiretyoftheir(connected)domains,thenforgiventopologywecanstatethattheFFFandSFFcontainallpossibleshapeinformation.Thesetensoreldsarethusanaturalchoiceforthematchingcriterionsought. 11 ],Chs.2-5and2-8,foradetaileddiscussionofwhythisisso.2 22

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31 ]).Becauseoftheirlowerdierentialorder(andthusgreaterrelativestability)andintrinsicnature,wechoosetobaseourapproachoncomparisonbetweentheFFFtensorsofthetwosurfaceparametricdomains.Giventopologicalequivalence,FFFequivalenceeverywherethroughabijectivedeformationmapdenesthatmapasaglobalisometry:thisindicatesthattheextentoffailuretomatchthischaracteristicserveswellasameasureofshapedissimilarity. ThesortofinformationcontainedintheSFF(andthusdiscardedinomittingitfromthematchingcriterion)canbevisualizedthroughpairsofsurfaceswithSFFeldswhichdierunderparametrizationswhichmatchtheirFFFeldsexactly.Fig.~refg:inandoutdisplayssuchacase.Inthisexample,itiseasytoseethatthereisanisometricmappingbetweenthetwosurfaces(andthuszeroenergyinthesenseofdeviationfromisometry),andwhatthatisometricmappingis.However,undersaidmapping,theSFFeldsdierovertheportionofthedomainparametrizingthetwoprotrusions(invertedrelativetooneanother):allcurvaturesareequalbutoppositeinsign.Viewedsittinginitsembeddingspace(R3),thisfactisobvious,buttotheproverbial\bugonthesurface", 23

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Figure2-2. Apairofcylindricalsurfaceswithidentical\Gaussianbumps"ofoppositeorientation.TheextrinsicorientationinformationisdiscardedinusingtheFFFasthematchingcriterion:inthisframework,thetwosurfacesareconsideredidentical. However,thequalityofthematchcriterionisapplication-dependent.Forinstance,whenconsideringthecaseofhippocampuspairs,weseektoidentifyregionsoflocalizedgrowthorshrinkage(inlinewiththeclinicalmotivation).ThesortofshapedierencerepresentedinFig. 2-2 doesnotrepresentsuchacase.Further,anexampleofthedrawbackofSFF(essentially,curvature)matchingisillustratedinFig. 2-3 Asregardsthisapplicationandmanyothers,anydierencebetweenthetwosurfacesisattributabletohigh-frequencynoiseinthepointclouddata:thatis,thereisnomeaningfuldierencebetweenthesurfaces.However,thetrivialisometricmap(identity)betweenthenoise-freeversionsofthesurfaceswillbynomeansprovidealow-magnitudedierenceintheirsecondfundamentalformsunderthatmapping.Infact,innitesimalperturbationsinsurfacedataaretheoreticallycapableofproducingsurfacecurvaturesofarbitrarilylargemagnitudesattheperturbationlocations.Theproblemwithusingthecurvaturestructureofasurfaceasanidentifyingcharacteristicisthefactthatitisinherentlyofhighorder:matchingmapsthroughtheirsecondderivativescannotbe 24

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Apairofcylindricalsurfaceswhicharedierentiatedonlybysurfacenoise.Underthe(identity)mapwhichwouldcorrespondthenoise-freeversions,theSFFscorrespondpoorly. expectedtobestable,and,worsestill,inmanycasescannotbeexpectedtobephysicallymeaningfulevenwithnumericalconsiderationscastaside.Thisfactisespeciallytruefortheinputsconsideredherein,whichexplainsouromissionofthisaspectofthesurfacecharacterization. Wesolvethesimultaneoussegmentationandregistrationproblemoutlinedaboveusingavariationalframework.TheenergyfunctionalEisdenedasafunctionalofapair 25

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where ein=RS1inedA ThequantityedenedinEqn. 2{4 provides,ateachpointonS1,ameasureofsimilaritybetweentheRiemannianstructuresg1andg2astheycorrespondunderthecurrentestimateoff.Assuchitrepresentsthelocaldeviationfromisometry,anditsintegraloverthedomain(wehaveadoptedtheconventionofusingP1,theparametricdomainofS1)providesaglobalmeasureofhowfarthegivendeformationfisfromanisometry.ThisistherstterminthesystemenergyrepresentedinEqn. 2{3 .Inlocal(parametric)coordinates,itisgivenbythedistancebetweenthetwomatricesG1andJtG2J: whereJistheJacobianoffwhenfisexpressedfromthelocalcoordinatesystemofP1ontothatofP2,andG1andG2are22positivedenitematricesexpressingthemetricsg1andg2inthecoordinatesystemsofP1andP2respectively,asinEqn. 2{1 .ThesecondterminErestrictsthelengthofthesegmentingcurve,soastoachievethe 26

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2{3 isachievedbyalternatingbetweenthedeformationmapandsegmentationestimationprocesses.Whenoptimizationstepsareconstrainedtobesucientlysmall,thisalternationapproximatesminimizationofEsimultaneouslywithrespecttobothfand1.TheEuler-LagrangeequationforEqn. 2{3 canbederivedthroughstandardcalculusofvariations.Theequationforfcontainsafourth-orderexpressioninvolvingfanditsderivatives.Becauseofthehighdierentialorderofthisanalyticexpression,weinsteadchoosetominimizethefunctionalEwithrespecttofdirectlythroughaconstrainedoptimizationprocess(moredetailinSecs. 2.6.2 and 2.6.3 ).Minimizationwithrespectto1,ontheotherhand,isimplementedviaalevelsetsegmentationasdescribedbyChanandVesein[ 3 ].Theonlydierencebetweenthismoduleofourmethodandthatdescribedthereinisourgeneralizationoftheprocesstogenuszero2-manifolds,whichentailsrespectingnonuniformsurfacelengthandareaelementsinaccordancewiththemetricinformation. 27

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28 ].Eachsegmentedhippocampalsurfaceobtainedfromtheapplicationofthistechniquewasrepresentedbya4021meshofpointsonthatsurface,periodicinonedirection.(ThismeshistheinstantiationofthesurfaceparametricdomainasdiscussedinSec. 2.4 .)Homologoushippocampiwerebroughtintorigidalignmentthroughapplicationoftheiterativeclosestpoints(ICP)algorithmfollowingthesegmentation.Therigidalignmentstepincludedextractionofvolumedataandnormalizationoftheshapeswithrespecttothisdata. 28

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Hessiansparsitystructure. 6 ].WeexploittheinherentsparsitystructureoftheHessian(asdescribedinSec. 2.6.3 ).TheRiemanniancharacteristicsthemselves,whosevaluesofcoursedrivetheenergyminimizationprocess,arecalculatedthroughanalyticaldierentiationofabicubicsurfacettothegivenpoints. Thesegmentingcurveissimplystoredandevolvedaccordingtowellestablishedlevelsetcurveevolutiontechniques(see[ 20 ]forextensivediscussion).TheonemodicationpresentinourcaseisthatlengthandareacomputationsrespectthedistancesbetweengridpointsbyusingtheappropriateRiemannianmetricateachpoint(asopposedtotheuniformEuclideanmetricusedforatdomainsandtargets). 2{1 ,whereXisthesurface 29

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2-4 ,whereinthesoliddiagonalsrepresentthenonzeromatrixelements(andthemapvariablevectorstructurehasbeenexplainedinSec. 2.6.2 ).Exploitationofthissparsitystructureisnecessaryforasecond-orderoptimizationschemetoachievecomputationalfeasibility. 2.6.1 .Cliniciansprovidedatrinaryclassicationofthisdataset:LATL(leftatrophiedtemporallobe)epileptic,RATL(rightatrophiedtemporallobe)epileptic,andControl.Inthisstep,6sampleswerediscardedduetoambiguityoftheirclinicalclassmembership,leavingthenalsetat54patientsamples(15LATL,16RATL,23Control). 2-1 ,inwhichwepresentaholisticschematic.Foreachsample,weillustrate(1)thedeformationmapfasthewarpnecessarytocarrypointsinP1totheircorrespondentsinP2,(2)thesegmentation1andthedeformationenergyfunctioneastheyexistinP1, 30

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2-1 ,thesegmentationisdepictedonS1throughtheshadingofitsinteriorandexteriorwithredandbluerespectively:thecurve(s)itselfisthenofcoursetheboundarybetweentheshadedregions. Werstconsiderthecaseofapairofcylinders,whereonememberofthepairhashaditssurfacedistortedaccordingtoanoutwardnormalvectoreldofmagnitudedictatedbyaGaussianfunctionconnedtoaknownsupport.Sincethesupportisknown,wecanuseitssegmentationasonequanticationofaccuracy.Fig. 2-5 illustratestheobtainedresults.Here,86%ofthedistortionareaisincludedinthesegmentation,and0%oftheundistortedareaisincludedinthesegmentation.Thismeansthatthesegmentingcurveslieslightlywithinthedistortedregions,ascanbeseeninthegure.Eventhe14%\misclassication"cannotnecessarilybeseencompletelyaserrorinthiscontext:sincetheprotrusionsarefarlessseveretowardstheboundariesoftheirsupport(thefeetofthehills),theyareaccordinglyfarmoresimilarindeformationleveltotheundistortedregionsthantotheheavilydistortedregionstowardsthesupportcenters.Assuch,thesegmentationarrivedatisincompliancewiththetwo-meanframework(whichproducesahigh/lowsegmentation).Whilewedolaternotethatitispossibletorenethisframework(seeconclusion),itisimportanttounderstandthatthismannerofgroupingisnotinherentlyalimitation,particularlyconsideringtheintendedapplication.Wefollowbydemonstratingrobustnessofthesegmentationinthefaceofreparametrizationofthesurfaceofcomparisonandrandomnoiseinthesurfacepointcloud(seecaptionforparameterlevels).Whenrepeatedundertheseconditions,thesegmentationremains100%consistentatthepixelresolutionwiththatobtainedprior. Fortherealcasesdescribedpreviously,wepresenttheresultsobtainedononememberofeachclass(LATL,RATL,andcontrol),asFigs. 2-6 2-7 ,and 2-8 ,respectively. 31

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Syntheticresults:caseofcylinderswithsurfacedistortion. (b)Performanceundersurfacenoise(=0:01forunitcylinder)andreparametrization(left:warpeld,=0:2ineachcomponent;middle:segmentedenergyeldinparametricdomain;right:convergedresult,at20its.). 32

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SegmentationofdistortionbetweenhippocampiofLATLsetmember. (b)Warpofparametricdomaininducedbyevolutionprocess. (c)Finalsegmenteddistortionenergyegraphasfunctionoverparametricdomain(arbitraryunits,intercomparablebetweenpatients).

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SegmentationofdistortionbetweenhippocampiofRATLsetmember. (b)Warpofparametricdomaininducedbyevolutionprocess. (c)Finalsegmenteddistortionenergyegraphasfunctionoverparametricdomain(arbitraryunits,intercomparablebetweenpatients).

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Segmentationofdistortionbetweenhippocampiofcontrolsetmember.(Thesmallerofthetwosegmentedregionshasbeenchosenforenlargement.) (b)Warpofparametricdomaininducedbyevolutionprocess,inparametricpixels. (c)Finalsegmenteddistortionenergyegraphasfunctionoverparametricdomain(arbitraryunits,intercomparablebetweenpatients).

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Sinceweproduceasegmentedenergyfunctionovertheentiresurface(parametric)domain,wehaveoptionsastohowtopresentfeaturestotheclassieralgorithm.Wecancollectsummarystatisticsthatprotfrombothsegmentationanddeformationquantication,suchasmeandeformationenergy(easdenedinEqn. 2{4 )insideandoutsideofthesegmentingcurves.Alternately,oncetheshapeshavebeenvolume-normalizedandmutuallyregistered,wecancomparethe(quasi-)continuousefunctionsacrosspatientsbyformingfeaturevectorsfromtheevaluesatcorrespondingsamplelocations.Thereisalsoanarrayofsupervisedlearningmethodsfromwhichtoselectapreferredclassicationalgorithm:wechoosetheonesobservedtogivebesttestperformanceonallfeaturesetsexamined,includingvolumealone(asfurtherdetailedbelow). Fig. 2-10a demonstratesthesuccessratesinclassifyingcontrolsvs.epileptics(LATLandRATLgroupscombinedintoasingleset)andFig. 2-10b demonstratesanalogousresultsfortheproblemofseparatingRATLandLATLmembers.Thetrainingandteststatisticswerecollectedthroughastandardleave-one-outcrossvalidationprocedure:testresultsshouldberegardedasthereliableperformanceindicators(andhighdiscrepancybetweentrainingandtestpercentagesasevidenceofovertting).Thefeaturevectorsbeingcomparedare(1)volumealone,(2)volumewithasetof6summarystatisticsderivedfromthealgorithmoutput(totalareawithinthecurve(s),totalareaoutsideof 36

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Epilepsyclassicationresults:optimaltestaccuraciesoverallclassiersshowninbold. SVMw/PB SVMw/RB KFDw/PB Training Test Training Test Training Test VOL 79.59 77.78 79.63 77.78 VOL+6SUM 92.31 77.78 E 100.00 85.19 94.44 81.48 94.51 100.00 81.48 99.97 85.19 SVMw/RB KFDw/PB Training Test Training Test Training Test VOL 83.25 77.42 VOL+6SUM 93.55 87.10 93.87 80.65 E 100.00 67.74 99.89 67.74 E+VOL+6SUM 100.00 70.97 93.23 67.74 Theresultsofbothstudiesmakethemethod'ssuccessclear.Inthecaseofdistinguishingepilepticsfromcontrols,allexperimentalfeaturevectorsdemonstratedsuperiortestclassicationaccuracy,withamaximumperformanceof88.89%on`E+VOL+6SUM'asclassiedbySVMw/PB,ascomparedto79.56%for`VOL'asclassiedbySVMw/RB.Noteinparticularthatfeaturevector`E'outperforms`VOL'aswell,despitethefactthat`E'containsnovolumeinformation.Thisportionoftheexperimentconrmsournotionthatlocaldistortioninformationisanimportantcomplementtovolumedata,andsuggeststhatitcaninsomecasesoutweighvolumeinindependentrelevance. 37

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38

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15 ],inwhichtheoptimalimagereconstructionisdenedintermsofasegmentingcurveorcurvesspecifyingtheboundariesofthepiecewisesmoothregions.InthecurveevolutionimplementationofMumford-ShahminimizationbyTsaietal.[ 25 ],themutualassistanceofsegmentationandsmoothingismadeexplicit:apreliminarysolutiontoeachalternatelyservesasinputtotheother.Assuch,tentativesmoothreconstructionsenablethecurvetondobjectboundaries,andtheidenticationofobjectboundariesbythecurveenablesthesmoothreconstructiontoremainmaximallyfaithfultotheinitialdata.Butasalreadypointedout,registration(indeed,anydata-drivenIPO)dependsonsmoothingresults,asasmoothingprocessbydenitionreconstructsthedatauponwhichitoperates.Basedontheobservationsmadeprior,wecanobservethesystemrelationshipdepictedinFig. 3-1 .Allinterdependencesaredirectwiththeexceptionofthedependenceofsmoothingonregistration.However,consideredinthecontextofsimultaneoussegmentation,thisrelationshipclearlyemerges. DespiteinherentconnectionbetweenthethreeIPOs,andtheevidentoptionofunifyingthemunderasinglevariationalprinciple,suchproposalsarerecentandlimitedinnumber(somereferenceswhichcouldbeconsideredtofallintothiscategoryareVemurietal.[ 27 ],Pohletal.[ 18 ],andUnaletal.[ 26 ]).Moreover,eachoftheworksjustcited 39

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Natureofinterdependencebetweensegmentation,smoothing,andregistration. containsatleastoneofthefollowingrestrictions,simplications,orlimitations:(1)relianceonexplicitatlasing,(2)low-dimensionalregistrationparametrization(i.e.rigidorane),(3)lackofguaranteedsymmetry/unbiasednesswithrespecttoinputargumentsorconsistency(throughcomposition)ofregistrationmaps,(4)relianceonpiecewiseconstantreconstructionmodel,and/or(5)restrictiontoimagepairs(ratherthanlargersets).ThroughanextensiontotheMumford-Shahfunctionalandtheformulationoftheproblemonasinglecanonicaldomain,wehavedevisedavariationalframeworkforsimultaneoussegmentation,smoothing,and(nonrigid)registrationofimagesuiteswhicheliminatesalloftheaforementionedlimitations.Theenergyfunctionalcanaccuratelybesummarizedasalinearcombinationofanintensity-basedmatchingpenaltyontheevolvingregistrationmapswithMumford-Shahfunctionalsofeachimagethroughthecorrespondencesdenedbythosemaps.AnimportantaspectisthatinsteadofworkingwithKdierentsegmentation/smoothingproblems,theregistrationallowsustosolveasinglejointsmoothingandsegmentationproblemonashareddomain(D),asdictatedbytheevolvingcorrespondences. 40

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1. Acollectionofregistrationmaps(dieomorphisms)ijbetweenIjandIi,1i;j
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3-2 ).InsteadofcomputingtheK2registrationmapsijandKcontoursCi,wewillcomputeKdieomorphismsibetweenIiandacanonicaldomainD Figure3-2. UnbiasednessandcompatibilitycanbeguaranteedbyperformingallcomputationsonacanonicaldomainD. Mathematically,thecanonicaldomainDandtheKregistrationmapsprovideparameterizationsfortheKinputimageswithacommondomain.ComputationsoneachindividualimagecannowbecovariantlyformulatedonDusingtheconceptofpullbacks. 42

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canbecalculatedonDasontherighthandside.Therearetwoadditionalfactors:thepull-backmetricgi ThroughtheKregistrationmaps1;;K,wecan\pullalloftheintegralsback"tothecanonicaldomain,givinganenergyfunctionthatcanbeevaluatedonD:E(;C;^I)=KXi=1ZDjIi((x))^Ii((x))j2JidA+KXi=1ZDnCjr^Ii((x))j2Jidx+KXiLength(i(C))+MC(;^I)+Reg(): Forthesakeofcomputationaleciency,weuseasumofsquareintensitydierencesfromthemeansmoothedimageforthematchingcostterm.(Apreferablebutfarmorecomputationallycostlyalternativewouldbetoinsertaninformationtheoreticregistrationpenaltybasedonthejointprobabilitydensityfunctiondenedbytheregistrationmapsandthesmoothedimages^I.However,devisingasucientlyaccurateapproximation 16 ]forthedenitionofpull-backmetricgiBriey,themetricgiisthedistortedversionofthestandardEuclideanmetriconCiunderthenonlinearmapi. 43

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wherediistheJacobianofi.ThefullenergyfunctionisthusgivenasE(;C;^I)=KXi=1ZDjIi^Iij2Jdx+1KXi=1ZDjr^Ii(i(x))j2giJdx+2KXi=1ZDj^IiPKj=1^Ij TheenergyfunctionalisgivenbyE(;C;^I)=KXi=1ZDjIi^Iij2Jidxdy+1KXi=1ZDjr^Iij2giJidxdy+2KXi=1ZDj^IiPKj=1^Ij Thelasttwotermsdonotexplicitlydependon^Iiandtherefore,theydonotcontributetotheEuler-Lagrangeequations.Intheequationabove,wehaveabusedthenotation 44

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TheEuler-LagrangeequationwillbeasystemofKequations(treatingeach^Ii,1iKasavariable).WewillusetheformulathattheELequationsforanenergyfunctionaloftheformZF(f1;;fK;f1x;;fKxf1y;fKy)dxdy @x@F @y@F BacktoEquation1,thecontributionsfromtherstandthirdtermsareeasytoseesincethesetermsdonothavederivativesof^Ii.WecomputetheELequationforIhwith1hK.ThecontributionfromthersttermZDjIh^Ihj2Jhdxdy

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Puttingeverythingtogether,theEuler-LagrangeEquationfor^Ihis1gi^Ih+(Ih^Ih)Jh+2(KXi=1;i6=h1 2KXi=1(Ii^Iiout)2(Ii^Iiin)2JiN+1 whereNistheunitnormaleldofthecontour,and^Iiin;^IioutarethesmoothedimagesinsideandoutsideofthecontourC,respectively.Intheabove,iisthecurvatureofCwithrespecttotheRiemannianmetricgi,andiisafunctionthatcaptures(details 46

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@xk;rgVi=1 Next,wederivethegradientforthedeformationparameterofthethin-platesplinebasisfunction ThelastsummandcanbecomputedinclosedformusingEquation 3{2 ,andthederivativeinsideoftherstintegralcanbeevaluatedusingthechainrule,resultinginaformulacontainingimagegradientsof^Ii.ThederivativewithrespecttothecurvelengthcanbederivedusingEquation 3{3 :@RCds 47

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14 ].Theentirealgorithmissummarizedbelow.AlgorithmSummary 1. OptimizemotionparametersusinggradientdescentwithgradientgivenbyEquation 3{6 .Updatethedeformationeldsi. 2. EvolvethelevelsetfunctionusingEquation 3{5 .Thecontourisupdatedasthezerolevelsetof. 3. SolvethenonlinearellipticPDEsinEquation 3{7 .Thisupdatesthesmoothedimages^Ii. 4. Ifthedierencebetweenconsecutiveiteratesisbelowapre-chosentolerance,stop,elsegotoStep1. 3-3 and 3-4 ,theinputsaretriosofhandsmakingsimilargestures,butinvisiblydierentplacesandmanners 48

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3-4 ispossibleonlybecauseofthecouplingofthesegmentationofthatimagetothesegmentationoftheothertwo(inwhichthegapsaremorereadilydistinguished). Ofcourse,densecorrespondenceisalsoimplicitlybeingobtainedbetweenthehandfeatures(e.g.thecreaseofthepalm)aspartofthisprocess.Thethirdsamplecasemakesthiscorrespondenceexplicit:inFig. 3-5 ,weseetheisolationofbrainventriclesinhumanMRIcrosssections.Notethatventricleidenticationinthesecond(middle)imageisfarmoredicultthanintherstorthird,andyetanexcellentresultisachieved,thankstotheprocesscoupling.InFig. 3-6 ,twoalternatemethodsofvisualizingthedenseregistrationalignmentareprovided.Ontheleft,wehavetheeectofeachionasetofconcentriccirclesresidentinshareddomainD.Sinceij=i1jforalliandj,thisdisplayprovidesavisualintuitionofhowstructuresinoneimagearecarriedontoanother.Ontheright,wehavetheinitialandnalaverageimageasseenfromdomainD(i.e.theaveragepullback).AsDisthedomainofcorrespondence,notethatthesharpnessofthepullbackisavisualmethodofassessingthesuccessoftheregistrationinmatchingintensityvalues.Asweareheresegmentingtheventricles,itistheimprovementinclarityanddenitionoftheventricleswithwhichweareconcerned.Thiseectisclearlyevidenced.(Notethattheunusualwarpingoftheventriclesasseenfromthisperspectiveisofnosignicance.The\implicitatlas"whichresidesinthedomainismerelyacorrespondentintermediarybetweeninputimages,andneednotrepresentarealisticinput,merelyasmoothdeformationofarealisticinput.) 49

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Processingofvariationsonahandposition.Eachcolumnrepresentstheevolutionforasingleinputimageinthetrio,withtheoriginalinputontop,andthenalsmoothreconstructionatbottom.Theevolvingsegmentationandregistrationaredepictedthroughtheredcurveandbluegrid,respectively.Notethesuccessincorrespondingthengersbetweenallimages. 50

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Processingofvariousopeningsofthehand,organizedasinFigure 3-3 .Notetheidenticationofthengersintherstcolumndespitetheirnearclosureintheinputimage,underheavynoise.Thisowestothe\implicitatlasing"inherentinthemethod. 51

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Identicationoftheventriclesofthebrainsofthreedierentpatients(inrespectiverows)incross-sectionalMRIscans.Notethesuccessfulidenticationinthecaseofthesecondinput,despitetherelativenenessofthestructure.Thenalsmoothedimagesaredisplayedinthelastcolumn. Figure3-6. Displayandanalysisofinter-imageregistrationsforbrainMRIdatasetdisplayedinFig. 3-5 3-5 ,iincreasingfromlefttoright.Thisisonemethodofvisualizingtheconsistentcorrespondencesbetweenallthreepairs. (b)Analternatevisualizationofregistrationsuccess,asthesharpnessoftheaverageintensitypullbackofthestructurebeingsegmented.Ontheleftistheinitialaverage(underidentitymappings),ontherighttheconvergedresult.Thesharpeningoftheventricleisdramatic.

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2 ,wehavepresentedanovelapproachtoshapecomparisonwhichbenetsfrombeingconceptualizedaroundlocalnotionsofshapecharacterization.Assuchweareabletoprovideanenessofgranularitybenecialinmanyapplications,includingthechosenexampleofhippocampalasymmetrymeasurement.However,onecouldnotesomeaspectsoftheproposedframeworkwhichinviteeitherextensionorimprovement: 2{3 ,willrevealthefactthatofthetwosurfacesacceptedasinputstothealgorithm,oneisarbitrarilychosenasthedomainofenergyevaluationandintegration.Ameasureofasymmetrybetweentwosurfacesshoulditselfbetheoreticallysymmetricwithrespecttopermutationofthosesurfaces,yettheprinciplecannotguaranteethisasstated. 5 ])andavaryingnumberofregionswhichisitselfoptimizedoverthecourseoftheminimization(BroxandWeickert,[ 2 ]),andwhileitistruethatsuchimplementationscouldbeapplieddirectlytotheschemeunderdiscussion,theywouldstillnoteliminatethesimplicationofmean-basedmodelling. Fortunately,theabovelistisasubsetofthelimitationsofthepriorartinuniedsmoothing,segmentation,andregistrationasenumeratedinSec. 3.1 .ThatsetoflimitationsispreciselywhattheUSSRalgorithmofCh. 3 wasdesignedtoaddress,inthedomainofimageprocessing.Sinceouroriginalproposalforlocalshapeasymmetryanalysiswasbasedonmovingstandardimageprocessingtechniquestothemanifoldandhavingthemoperateonthemetricstructuresofthosemanifolds,wemightlooktoadapt 53

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2 ,butwithcertainpracticalandtheoreticalsuperiorities. 2 (localasymmetryanalysis)and 3 (USSR)arebothvariationalinnature,theirsynthesiswillbeaswell.Thebulkofthetaskofdeningtheextendedshapeanalysismethodisthespecicationofthevariationalprincipletobeminimized.TherstshapeanalysisproposalwasdenedthroughEqns. 2{3 and 2{4 ,whichsettheformofthefunctionalintermsofthedeformationenergy,anddenedthatdeformationenergy,respectively.Themodiedproposalwillbedenedthroughtwoanalogousequations,asfollows. 2{3 ,oneseesthatofthetwosurfacesinputtothemethod,S1andS2,S1isarbitrarilychosenasareferenceforcomputationandintegrationoftheasymmetryfunction(e,Eqn. 2{4 ),anditsparametricdomainP1usedasthedomainofcomputation.Symmetrywithrespecttotheorderofthesurfacesthereforecannotbeguaranteed.Onemightnotethatthereisatleastonetrivialsolutiontothisissue,whichistochangethefunctionaltothefollowing:ZS1n1edA+Zjd1(t)=dtjg1dt+ZS1injeeinj2g1dA+ZS1outjeeoutj2g1dA+ZS2n2edA+Zjd2(t)=dtjg2dt+ZS2injeeinj2g2dA+ZS2outjeeoutj2g2dA Facingthesameissueregardingmultipleimageregistration,USSRtooktheapproachofmovingallfunctioncomputationsandintegrationtoasharedcanonicaldomain.But 54

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2.4 ,USSR'sapproachcanbeappliedwithoutmuchdiculty.Pictorially,theframeworkwillchangefromthedepictioninFig. 2-1 tothatofFig. 4-1 Figure4-1. Thereformulatedversionofthelocalasymmetryanalysisproposal:AllKinputsurfacesarecorrespondedthroughtheKmapsfibetweentheirparametricdomainsPiandsharedcanonicaldomainD.Usingthemetricinformationofthegi,wecandeneanenergyfunctionoverDwhichcanbesimultaneouslysegmentedvia,whichinturnappearsoneachSiasithroughfi,segmentingthesurfacesthemselvesbasedongroupasymmetrystructure. 55

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4.2.2 and 4.2.3 discussthemannerinwhichthisisdone. 56

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4.2.2 ,wehavespeciedtheformoftheextendedshapeanalysisfunctionalintermsofthefunctionrepresentinglocaldeviationfromisometry,e.However,thedenitionofegivenbyEqn. 2{4 isbinaryinthesenseofinputarguments,implicitlyacceptingexactlytwolocations,onefromeachparametricdomainofthesurfacesundercomparison.Thisleadstotherestrictionoftheentirealgorithmtoinputpairs,asindicatedinSec. 4.1 Sinceweseektoenabletheprocessingofarbitrarilysized\populations"ofsurfaces,emustberedenedintermsofavariablenumberofarguments.ButUSSRhasalreadyaddressedasimilarissue,indeningasingleregistrationdatatermforarbitrarilymanyimages(aterminEqn. 3{4 ).For(smoothed)images^I1^IK,thisdatapenaltyfunctionwasgivenasthesumofdierencesofindividualimagesfromthemeanintensityimage:KXi=1j^IiPKj=1^Ij 2{4 with Kj2(4{1) Justas,inthecaseofimages,Eqn. 2{4 isdesignedtoassignhighregistrationenergytopixellocations(ascorrespondedthroughthemaps)withhighvariationinintensityacrossimages,Eqn. 4{1 will,intheshapecomparisoncontext,tendtopenalizeregionsoflargeFFFdierenceacrosstheK(arbitrarilymany)inputshapes. 57

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4{1 ,thereexistsanimportantquestionregardingtheweighting(area)ofthepixelelementsinthecanonicaldomain.IntheversionofthealgorithminCh. 2 (asspeciedbyEqn. 2{3 )theissueisrenderedtrivialbythefactthatintegrationisarbitrarilycarriedoutonS1,which,underapiecewiseconstantquadraturescheme,simplyequatestosummingtheper-pixeldeformationenergiesonparametricdomainP1underweightingbytheareasofthecorrespondingsurfaceelements.SincethoseareasaresimplygivenbythedeterminantsoftheFFFsstoredatthepixels,indiscreteformtheintegralRS1fdAbecomesthesummationtermPP1fjGj,wherethesummationisofcourseoverdomainpixelelements,withimplicitcorrespondencebetweenthepixelsandFFFmatricesG.Underthatscheme,thepresenceofabiasintheenergycalculation/integrationwasallowedfor.However,thisispreciselyoneofthelistedlimitationstobeeliminatedinthemodicationweareproposing(andispartlythemotivationoftheuseofthecanonicaldomaininSec. 4.2.1 ).Thenewefunctionandtheintegrationthereofmustthereforeremainunbiasedwithrespecttotheinputsurfaceswhilecapturinginanintuitivemannerthefactthatcanonicalpixelsrepresentcomparisonsbetweensurfacepatchesofvariablesize(whichshouldbeaccountedforintheirintegration).Weightingthedomainpixelsbythemeanelementareaissuchasolution,andistheoneweadopt.Assuch,thequadratureschemewillinvolvethefactorPKj=1jJtGjJ Kj. 4.2.1 4.2.2 ,and 4.2.3 ,andinparticularthedenitionofegivenbyEqn. 4{1 ,wecanstatethemodiedsystemvariationprinciple 58

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KjdA+1ZDnjr^ej2KXj=1jJtGjJ KjdA+2ZDj^eej2KXj=1jJtGjJ KjdA+3ZCds wheredAisthetrivialuniformmetriconD.Torepresenttheareasofthesurfacepatchesundercomparisoninanunbiasedmanner,thismetricisweightedbyPKj=1jJtGjJ Kj. 4.3.1Necessity 4.2.2 ,wehavenotedthefactthataMumford-Shah-basedsegmentationschemeformsacomponentofthemodiedasymmetryanalysisproposalascodiedinthevariationalprincipleofEqn. 4{2 .TheoryandpracticefortheminimizationoftheclassicMumford-Shahfunctionalarewell-established(asin[ 15 ]and[ 25 ]),butthetermsappearinginthevariationalprincipleproposedherecarryanimportantalterationwhichdemandsacorrespondingalterationinthetheory.AsdiscussedinSec. 4.2.3 ,thedomainofcomputationDcannotbejustiablyconsidered\at"inthateachpixelcorrespondstoK(numberofinputsurfaces)dierentsurfacepatches,eachingeneralofdierentarea.Assuch,theintegrals(approximatedbysummationsinthediscretecase)areweightedbyafunctiondenedoverDchosentoreectthesizesoftheareaelementsundercorrespondence(asdetailed,again,inSecs. 4.2.3 and 4.2.4 ).ThepriceofbeingabletoperformMumford-Shahminimizationoverthefamiliartwo-dimensionalpixelgridisthattheEuler-Lagrangeequationusedinthecurveevolutionof[ 25 ]mustberederivedforthecaseofweightedintegration.Thisisdonebelow,inSec. 4.3.2 .Aninterestingconsequenceoftheextensionisrevealed,inthatanadditionalterm(henceforthreferredto 59

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Now,weseekacriticalpointofEwithrespectto^I,inafunctionalsense.Thatis,forscalartandanyfunctionhdenedonD,limt!0E(^I+th)E(^I)

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Now,wemakeuseoftherstGreen'stheorem,throughappropriateidenticationoffactorsintheaboveexpressionswiththoseinthetheorem'sgeneralformasgiven:rv=gr^Iu=h andsoZD(hr(gr^I)+rhgr^I)dA=0ZDhr(gr^I)dA=ZDrhgr^IdA Usingthegeneralfactthatforanyfunctionsuandv,r(urv)=rurv+uv

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Sincethisequationmustholdforarbitraryh,theothertermintheintegrandmustbezeroidentically:g^IgIr^Irgg^I=0 ThisistheEuler-LagrangeequationforthegeneralcaseoftheMumford-Shahtermsunderintegrationweightedbyfunctiong.Forthecaseinwhichgrepresentsanareaelementsize(positivedenite),divisionbygiswell-dened,andwecanwritetheexpressionintheform^IIrg g^I^I=0^I rg gr^I ^I=I rg gr )^I=I(4{3) Notethatinthespecialcaseinwhichgisaconstantfunction(e.g.unity),theabovereducesto(1 )^I=I; 25 ].Theoperator rg gr,whichactsupon^I,canbethoughtofasthemetriccorrectionoperatorforthemoregeneralcase. 2 ,theconversionoftheimplementationofthelatterintothatoftheformercanbestructuredsoastoinvolveaminimalnumberofrelativelyconnedchanges.Oneobvious(andtrivial,inthesenseofimplementation)pointofalterationisthatthedenitionsusedinthecomputationofquantitiesEande, 62

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2{3 and 2{4 ,mustbereplacedbyEqns. 4{2 and 4{1 respectively.Beyondthis,theonlynoteworthystructuralchangestotheprograminvolvetheallowanceofavariablenumberKofinputsurfaces(aspointclouds,allpreprocessedidenticallywithresultantparameterizationsasdescribedinSec. 2.6.1 ),themaintenanceandcalculationoftheKmapfunctionsfromshareddomainDtoeachofthesesurfaces,thechangeintheoptimizer'sHessiansparsitystructuretoaccountforthosechangesinthestructureofthemapvector,andthereplacementoftheChan-Vese-basedcurveevolutionmodulewiththeMumford-Shah-basedapproachappropriateforthenewsystemenergy. 2.6.2 :wearestillworkingwithmapsbetweenintrinsicallytwo-dimensionalsurfacesbyrepresentingthemasdiscretevectoreldsovertheparameterizingdomains.IntheframeworkofCh. 2 ,thereexistedasinglemapfromS1toS2representedasa2-vectorfunctionovertheparametricdomainofS1,P1,representingthecorrespondinglocationsofeachpixelonP2.Withadiscretesamplinggridofsizemn,twocomponentsinthefunction,andazero-orderboundaryconditionontherstandlastcolumns,themapvariablescouldberepresentedasasingle2m(n1)-vector(theconcatenationofthecolumnwise-vectorizedrow(u)componentsfollowedbylikewisevectorizationofthevariablecolumn(v)components).Ofcourse,withknownsamplingsizeandmapvariablevectorstructure,itistrivialtoconvertbackandforthbetweenthevector(forinputtotheoptimizationmethod)andgrid(forinterpolation)representationsasnecessary. Torepresentthemapvariablesinthenewversionofthealgorithm,littleneedchange:insteadofrepresentingasinglemapfromP1toP2,wemustnowrepresentKmaps,eachfromcanonicaldomainDtooneoftheKPi.Eachofthesemapscanbestructuredinamanneridenticaltotheabove,producingatotalof2Km(n1)components.Forconvenientreshapingbetweenthevectorandgridrepresentations,weorderthesecomponentsviasuccessiveconcatenationsofcolumnwise-vectorizedu-componentgrids1 63

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4-2 Figure4-2. Thestructureoftheargumentmapvector,usedasinputtotheoptimizationmethodandwithrespecttowhichthesystemHessianisdened(showntransposedintoarowvector).TheucoordinatesoftheKmaps(Ksegmentswithmnelementseach)areconcatenatedrst,followedbythevariablevcoordinateslikewise(Ksegmentsofm(n2)elements),foravectoroftotallength2Km(n1). However,alterationofthemapvectorstructurealsonecessitatesalterationoftheHessiansparsitystructureinputtotheoptimizerforeciencypurposes(asexplainedanddiagrammedinSec. 2.6.3 ).Followingtheearlierlogic,however,thecorrectionisnotdicult.WecanrstpartitiontheHessianinto(2K)2submatrices,whereeachsubmatrixisdenedbyhavingitsrowscorrespondtoonecomponent(eitheruorv)ofasinglemapfi,andlikewiseitscolumnstoonecomponentoffj(jpossiblyequaltoi).Ofthesesubmatrices,wecanthenidentifyfourtypes:(1)therowcoordinatesoftheentriescorrespondtoucoordinatesoffiandthecolumncoordinatesoftheentriescorrespondtoucoordinatesoffj,(2)therowcoordinatesoftheentriescorrespondtoucoordinatesoffiandthecolumncoordinatesoftheentriescorrespondtovcoordinatesoffj,(3)therowcoordinatesoftheentriescorrespondtovcoordinatesoffiandthecolumncoordinatesoftheentriescorrespondtoucoordinatesoffj,and(4)therowcoordinatesoftheentriescorrespondtovcoordinatesoffiandthecolumncoordinatesoftheentriescorrespondtovcoordinatesoffj.LetuscalleachofthesesubmatricesHuu,Huv,Hvu,Hvv,respectively:theirstructuresarethetopleft,topright,bottomleft,andbottomrightsubmatricesofthematrixdepictedinFig. 2-4 .ThenewsystemHessianisobtainedfromtheoldthroughreplacementofeachsubmatrixwithaKKblockofreplicationsofthatmatrix. 64

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25 ]howalevelsetcurveevolutionimplementationofMumford-Shahminimizationcanbeformulatedforimagefunctionsoverboundedrectangulardomains(withnaturallyuniformassociatedmetrics).InSec. 4.3.2 ,wehavedemonstratedthatthedierencebetweenthedescentequationfortheimagereconstruction(forxedcurveposition)inthegeneralcaseofintegralweightingbyageneralC1functionandthespecial(andcommon)caseofuniformunitweightingistheappearanceofasingleadditionallinearoperatordenedintermsoftheweightingfunctionanditsrstderivative.Thismeansthatwecaneasilybasetheimplementationofthecurveevolution(andnow,simultaneousenergyreconstruction)moduleofthemodiedasymmetryanalysisproposalontheTsai-Yezziimplementationthroughappropriateinclusionofthemetriccorrectionoperatorderivedinthereferencedsection,andimplementationoftheu-periodicitydemandedbythecylindricaltopologyoftheparametricdomain.Since[ 25 ]omitsmostoftheimplementationdetailsoffunctionalminimization,andsincethismethodformsthecoreofthecurveevolutionandpiecewise-smoothenergyreconstructionmodule,wewillrstdiscussourchosenimplementationapproachinSec. 4.4.2.1 ,andtheinclusionofthemetriccorrectionoperatorin 4.4.2.2 dis)^I=I; 65

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disdiersfromtheusualLaplacianoperatorinthatitmustrespecttheNeumannboundaryconditionacrossthecurveC.Inadiscreteimplementation,overaregulargrid,asensiblewayofenforcingthisconditionistonullifyanynitedierencederivativeswhichinvolvequantitiesfromoppositesidesofthesegmentingcurve.ThisdiscretenotionoftheNeumannconditioniswhatproducesthedierencebetweenanddis.However,itiseasytoseethattheordinaryLaplaciancanbeformedthroughthelinearcombinationofstandardnitedierenceoperators:=DFUDBU+DFVDBV

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Foranycurveposition,then,weareabletocalculate^I,whichallowsevaluationofthesystemenergy.Thisinturnenablesustoimplementthedescentwithrespecttothecurveposition,whichis@C 20 ].Thecomputationoftheothertermsisstraightforwardonceamethodforcalculating^Ioutand^Iinhasbeendetermined.Thesetermsrefertothereconstructedfunctionvaluesatthecurveboundaryiftheyareconsideredtolieoutsideorinsideofthecurve,respectively.Inthediscreteimplementation,ofcourse,functionvaluesareonlyprovidedatpixellocations,andthesewillingeneralnotlie\on"thecurve:ourconcerniswhetheratanystep,apixeladjacenttothecurveshouldbemovedtotheothersideofit.Asucientlyaccurateandfastapproachtocomputing(dening,really)theterms^Ioutand^Iin,then,istoadvancethecurveoutwardeverywherebyonepixel(bringingallborder-adjacentoutsidepixelstotheinterior),recompute^Itoserveas^Iin,andtorepeattheprocessusinginwardcurvemotionforcomputationof^Iout.Withthesequantities,theenergiesofinwardandoutwardmotioncanbecomparedforallborderingpixels,andthecurveupdatedaccordingly.Again,numericalmethodsforlevelsetcurvemaintenanceandupdatingarefoundintheprovidedreference. 4.4.2.1 ,thereisnothingtodotoproducethecurveevolutionandpiecewisesmoothfunctionreconstructionmoduleofthemodiedproposalbuttoimplementtheuperiodicitynecessitatedbythedomaintopology 67

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4.4.2.2 .Theformerissimplyamatterofappropriatelyalteringthenitedierenceoperatorsthroughouttheprogram(includingintheconstructionofrdis):callthemodieddierenceoperatormatricesDFUperandDBUper.Forthelattertask,thecontinuousoperatorrdisg grdisisgivenindiscreteformasZFUperDFUper+ZBUperDBUper gZFUperDFUper+ZBUperDBUper gZFVDFV+ZBVDBV WethenneedonlyincludethisterminthediscreteimplementationofsolutionofEqn. 4{3 ,andotherwiseproceedasdirectedinSec. 4.4.2.1 68

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Inthisdissertation,wehavepresentedamethodforlocalshapeanalysisofhomologousstructures,ageneralimageprocessingframeworkforgroupsofimagesinwhichsuboperationsprovidemutualassistance,andaproposalforimprovingtheformerusingtheinsightsofthelatter.Throughvisualobservationandstatisticalvalidation,wewereabletodemonstratethepracticalbenetsofthenovelalgorithms.Ifthereisanoverarchingthemeintheirformulation,itisgenerality:whiletheimplementationdetailsgivenherein(certainlyinthecaseofCh. 2 )arespecictothepreprocessingstepsandformatting/representationofthedata,theformulationsthemselvesremainquitebroadlyapplicable.Farfrombeingspecictothehippocampiofepileptics,theshapeanalysisproposalisconceivedaroundgeneralgenuszerosurfacesforwhichsymmetryisexpected.Thisofcoursecoversawidevarietyofpracticalapplications,inadditiontothetheoreticalcontribution.AswehavenotedinSec. 2.4.1 ,theparticularapplicationandrepresentativedatasetchosendidinuenceformulationinthatthehigher-frequencySFFinformationwascompletelydiscarded,asbothnumericallyproblematicandsomewhatconceptuallyirrelevantinthecaseofthatapplication.However,ifdesiredforeithertheoreticalorpracticalreasons,onecouldreincorporatethisaspectoftheshapecharacterizationintotheformulationinastraightforwardmanner.(Again,thecaveatisthattodosoimplicityplacessubstantialtrustbothintheaccuracyofthecurvaturedatarepresentedbytheinputshapeandintheimportanceofthatdataincharacterizingtheshape.)Likewise,theUSSRproposalisarguablythemostgeneralimageprocessingframeworkofitssortheretoforesuggested.Whilecertainaspectsofthe\theoreticalimplementation"wehaveused,suchastheuseofthin-platesplineregularizationandsum-of-square-dierencesintensitydatapenaltyontheregistrationcomponent,doinuence/limitthemethod'sperformanceandscopeofapplicability,theycaneasilybemodiedasseent.Forinstance,ifonewishestopaythehighercomputationalcostofaninformation-theoretic 69

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[1] S.Bouix,J.C.Pruessner,D.L.Collins,andK.Siddiqi,\Hippocampalshapeanalysisusingmedialsurfaces,"Neuroimage,vol.25,no.4,pp.1077{1089,2005. [2] T.BroxandJ.Weickert,\Levelsetsegmentationwithmultipleregions,"IEEETransactionsonImageProcessing,vol.15,no.10,pp.3213{3218,2006. [3] T.F.ChanandL.A.Vese,\Activecontourswithoutedges,"IEEETransactionsonImageProcessing,vol.10,no.2,pp.266{277,2001. [4] L.-T.Cheng,P.Burchard,B.Merriman,andS.Osher,\Motionofcurvesconstrainedonsurfacesusingalevel-setapproach,"JournalofComputationalPhysics,vol.175,no.2,pp.604{644,2002. [5] G.ChungandL.A.Vese,\Energyminimizationbasedsegmentationanddenoisingusingamultilayerlevelsetapproach,"inEnergyMinimizationMethodsinComputerVisionandPatternRecognition,2005,pp.439{455. [6] T.ColemanandY.Li,\Aninterior,trustregionapproachfornonlinearminimizationsubjecttobounds,"SIAMJournalonOptimization,vol.6,pp.418{445,1996. [7] J.G.Csernansky,L.Wang,D.Jones,D.Rastogi-Cruz,J.A.Posener,G.Heydebrand,J.P.Miller,andM.I.Miller,\Hippocampaldeformitiesinschizophreniacharacterizedbyhighdimensionalbrainmapping,"TheAmericanJournalofPsychiatry,vol.159,no.12,pp.2000{2006,2002. [8] J.Csernansky,L.Wang,J.Swank,J.Miller,M.Gado,D.McKeel,M.Miller,andJ.Morris,\Preclinicaldetectionofalzheimer'sdisease:hippocampalshapeandvolumepredictdementiaonsetintheelderly,"Neuroimage,vol.25,no.3,pp.783{792,2005. [9] R.H.Davies,C.J.Twining,P.D.Allen,T.F.Cootes,andC.J.Taylor,\Shapediscriminationinthehippocampususinganmdlmodel,"inIPMI,2003,pp.38{50. [10] M.P.doCarmo,DierentialGeometryofCurvesandSurfaces.PrenticeHall,1976. [11] ||,RiemannianGeometry.Birkhauser,1992. [12] G.Gerig,K.E.Muller,E.O.Kistner,Y.-Y.Chi,M.Chakos,M.Styner,andJ.A.Lieberman,\Ageandtreatmentrelatedlocalhippocampalchangesinschizophreniaexplainedbyanovelshapeanalysismethod,"inMICCAI,2003,pp.653{660. [13] H.Jin,A.J.Yezzi,andS.Soatto,\Region-basedsegmentationonevolvingsurfaceswithapplicationto3dreconstructionofshapeandpiecewiseconstantradiance,"inProc.EuropeanConf.onComputerVision,no.2,2004,pp.114{125. [14] P.KnabnerandL.Angermann,NumericalMethodsforEllipticandParabolicPartialDierentialEquations.Springer,2003. 71

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[15] D.MumfordandJ.Shah,\Optimalapproximationsbypiecewisesmoothfunctionsandassociatedvariationalproblems,"Commun.PureAppl.Math,vol.42,no.4,pp.577{685. [16] B.O'Neill,ElementaryDierentialGeometry.AcademicPress,1997. [17] K.Pohl,J.Fisher,W.Grimson,R.Kikinis,andW.Wells,\Abayesianmodelforjointsegmentationandregistration,"Neuroimage,vol.31,no.1,pp.228{239,2006. [18] K.Pohl,J.Fisher,J.Levitt,M.Shenton,R.Kikinis,W.Grimson,andW.Wells,\Aunifyingapproachtoregistration,segmentation,andintensitycorrection,"inMICCAI,2005,pp.310{318. [19] F.J.P.RichardandL.D.Cohen,\Anewimageregistrationtechniquewithfreeboundaryconstraints:Applicationtomammography,"inProc.EuropeanConf.onComputerVision,2002,pp.531{545. [20] J.A.Sethian,LevelSetMethodsandFastMarchingMethods.CambridgePress,1996. [21] L.Shen,J.Ford,F.Makedon,andA.Saykin,\Hippocampalshapeanalysis:Surface-basedrepresentationandclassication,"inMedicalImaging2003:ImageProcessing,SPIEProceedings5032,2003,pp.253{264. [22] S.SoattoandA.J.Yezzi,\Deformotion:Deformingmotion,shapeaverageandthejointregistrationandsegmentationofimages,"inProc.EuropeanConf.onComputerVision,2002,pp.32{57. [23] M.Styner,G.Gerig,J.Lieberman,D.Jones,andD.Weinberger,\Statisticalshapeanalysisofneuroanatomicalstructuresbasedonmedialmodels,"MedicalImageAnalysis,vol.7,no.3,pp.207{220,2003. [24] M.Styner,J.A.Lieberman,andG.Gerig,\Boundaryandmedialshapeanalysisofthehippocampusinschizophrenia,"inMICCAI,2003,pp.464{471. [25] A.Tsai,A.J.Y.Jr.,andA.S.Willsky,\Curveevolutionimplementationofthemumford-shahfunctionalforimagesegmentation,denoising,interpolation,andmagnication,"IEEETransactionsonImageProcessing,vol.10,no.8,pp.1169{1186,2001. [26] G.B.UnalandG.G.Slabaugh,\Coupledpdesfornon-rigidregistrationandsegmentation,"inProc.IEEEConf.onComputerVisionandPatternRecognition,vol.1,2005,pp.168{175. [27] B.Vemuri,Y.Chen,andZ.Wang,\Registration-assistedimagesmoothingandsegmentation,"inProc.EuropeanConf.onComputerVision,vol.4,2002,pp.546{559.

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[28] B.VemuriandY.Guo,\Snakepedals:Compactandversatilegeometricmodelswithphysics-basedcontrol,"IEEETrans.PatternAnal.Mach.Intell.,vol.22,pp.445{459,2000. [29] F.WangandB.C.Vemuri,\Simultaneousregistrationandsegmentationofanatomicalstructuresfrombrainmri,"inMICCAI,2005,pp.17{25. [30] L.Wang,S.C.Joshi,M.I.Miller,andJ.G.Csernansky,\Statisticalanalysisofhippocampalasymmetryinschizophrenia,"Neuroimage,vol.14,no.3,pp.531{545,2001. [31] Y.Wang,M.-C.Chiang,andP.M.Thompson,\Mutualinformation-based3dsurfacematchingwithapplicationstofacerecognitionandbrainmapping,"inProc.Int'l.Conf.onComputerVision,2005,pp.527{534. [32] P.P.WyattandJ.A.Noble,\Mapmrfjointsegmentationandregistrationofmedicalimages,"MedicalImageAnalysis,vol.7,no.4,pp.539{552,2003. [33] C.Xiaohua,J.Brady,andD.Rueckert,\Simultaneoussegmentationandregistrationofmedicalimages,"inMICCAI,2004,pp.663{670. [34] A.Yezzi,L.Zollei,andT.Kapur,\Avariationalframeworkforjointsegmentationandregistration,"inIEEE{MMBIA,2001,pp.44{51. [35] Y.-N.YoungandD.Levy,\Registration-basedmorphingofactivecontoursforsegmentationofctscans,"MathematicalBiosciencesandEngineering,vol.2,pp.79{96,2005.

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NicholasLordhasbeen,aboveallthings,pooratwakinguptoalarmclocks.Hedidnotgrowupwithanyparticularacademicdirection,andtothebestofhisknowledgestilldoesnotpossessone.Helikesthefactthat`Caucasian'rhymeswith`Jamaican'.HecandotheFrug,Robocop,andFreddie,butcannotdotheSmurf.Heisseldomaccusedofexcessivereverence.Heisunsurewhetherthereexistsanerfoodthanfriedredsnapperwithchiligarlicsauce.HeinterpretspraiseofRichardLinklatermoviesasadeliberateattempttogetonhisbadside,andtakesDavidCronenbergmoreseriouslythanmost.Hehasperformedtasksforthepurposeofsecuringsustenance,andwillnodoubtdosoagainatsomepointinthefuture.Finally,heisabitold-fashionedinthathedrawsadistinctionbetweenclosefriendsandtotalstrangers,andassuchbelievesthatanyonewishingtohearmeaningfuldetailsofhislife'shistorycaneitheroerhimajoborbuyhimadrink. Figure5-1. Yourauthor:awascawwywabbit. 74