Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: Fiber tract mapping from diffusion tensor MRI
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Title: Fiber tract mapping from diffusion tensor MRI
Series Title: Department of Computer and Information Science and Engineering Technical Report ; 01-004
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Language: English
Creator: Vemuri, B.C.
Chen, Y.
Rao, M.
McGraw, T.
Wang, Z.
Mareci, T.
Affiliation: University of Florida -- Department of Computer and Information Science and Engineering
University of Florida -- Department of Mathematics
University of Florida -- Department of Mathematics
University of Florida -- Department of Computer and Information Science and Engineering
University of Florida -- Department of Computer and Information Science and Engineering
University of Florida -- Department of Biochemistry
Publisher: Department of Computer and Information Sciences, University of Florida
Place of Publication: Gainesville, Fla.
Copyright Date: 2001
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Fiber Tract Mapping from Diffusion Tensor MRI *


B. C. Vemuri'


Y Chen2
1Dept. of CISE


M. Rao2
2Dept. of Mathematics
University of Florida
Gainesville, Fl. 32611

UF-CISE TR01-004


T. McGraw', Z. Wang'
3Dept. of Biochemistry


Abstract

To understand evolving 1,ari. '1i.- in the central nervous
system (CNS) and develop cet rti've treatments, it is essen-
tial to correlate the nerve fiber connectivity with the vi-
sualization of function. Such information is fundamental
in CNS processes since anatomical connections determine
where information is passed and processed. Dirtu.\ion ten-
sor imaging (DTI) can provide the fundamental informa-
tion required for viewing structural connectivity. However,
robust and accurate acquisition and processing (l1i .0. di1 ,
are needed to accurately map the nerve connectivity. In
this paper we present a novel, ul- ... aihl for automatic
fiber tract mapping in the CNS specifically, the spinal cord.
The automatic fiber tract mapping problem will be solved
in two phases, namely a data ',' .. 'rii,, phase and a fiber
tract mapping phase. In the former ,a.'. 'hri,,g; is achieved
via a new weighted TV-norm minimization which strives to
smooth while retaining all relevant detail. Existence and
uniqueness results for this minimization are presented in
brief For the fiber tract mapping, a smooth 3D vector field
.,1. i ... 1ri the dominant anisotropic direction at each spa-
tial location is computed from the smoothed data. Fiber
tracts are then determined as the smooth integral curves of
this vector field in a variational framework. To facilitate vi-
sualization of the computedfiber tracts, we overlay the 3D
fibers on a volume rendering of the DT-MR scan. Exam-
ples are presented for DT-MR data sets from a normal and
injured rat spinal cords respectively.


1 Introduction

Fundamental advances in understanding living biological
systems require detailed knowledge of structural and func-
tional organization. This is particularly important in the
nervous system where anatomical connections determine

*This research was in part funded by the NSF grant IIS-9811042 and
NIH RO1-RR13197


the information pathways and how this information is pro-
cessed. Our current understanding of the nervous system
is incomplete because of a lack of fundamental structural
information [13] necessary to understand function. For the
entire central nervous system, understanding and treating
evolving pathology, such as spinal cord injury, depends
on a detailed understanding of the anatomical connectiv-
ity changes and how they relate to function. For example,
an evolving spinal cord lesion undergoes an initial response
to an insult which is followed by a sequence of secondary
events leading to further tissue degradation [37]. A method
for defining the structure-function relationship is needed
that can be used in the whole living organism that facilitates
the study of such an evolving dynamics process.
Recently MR imaging has been used to study the struc-
tural connectivity within whole living organisms. The MR
measurement of water translational self-diffusion provides
a method that can be used to study structural connectivity
with ubiquitous indigenous material, water. In highly orga-
nized nervous tissue, like white matter, diffusion anisotropy
can be used to visualize fiber tracts. Douek, et al. [14] have
used diffusion measurements along three orthogonal axes to
estimate diffusion anisotropy in human brain white matter.
From this data, they produced a color map of the fiber ori-
entations reflective of white matter organization. Recently
MR measurements have been developed to measure the ten-
sor of diffusion. This provides a complete characteriza-
tion of the restricted motion of water through the tissue that
can be used to infer tissue structure and hence fiber tracts.
The development of diffusion tensor acquisition, process-
ing, and analysis methods provides the framework for creat-
ing fiber tract maps based on this complete diffusion tensor
analysis [12, 17, 19, 21].
For automated fiber tract mapping, prior to estimating
the diffusion tensor, the raw data must be smoothed while
preserving relevant detail. The raw data in this context con-
sists of seven directional images acquired for varying mag-
netic field strengths. Note that atleast seven values at each
3D grid point in the data domain are required to estimate


T. Mareci3










the six unknowns in the symmetric 2-tensor and one scale
parameter. The data smoothing or de-noising can be formu-
lated using variational principles which in turn require solu-
tions to PDEs. Recently, there has been a flurry of activity
on the PDE-based smoothing schemes. In [25], Perona and
Malik developed an anisotropic di/ttu.\in scheme for im-
age smoothing. The basic idea of this nonlinear smoothing
scheme was to smooth the image while preserving the edges
in it. This was done by using the following equation Is =
div(c(VI)VI), where I is the image to be smoothed and
It describes its evolution over time, and c(VI) is a decreas-
ing function of VI. Catte et al., [5], Nitzberg and Mumford
[22] and Alvarez et al. [1] recognized the ill-posedness of
the Perona-Malik diffusion and proposed modifications to
overcome the same. Since then, several nonlinear diffusion
methods have been developed and a good account of these
can be found in [5, 1, 8, 23, 31, 33]. All of these are non-
linear models and differ in the diffusivity coefficient and/or
the diffusion term. Some of them are also supplemented
with a reactive term. Another popular framework for im-
age smoothing is the total variation or TV norm framework
pioneered by Rudin et.al., [28] and further developed by
Chan et.al., [6] and Strong and Chan [32]. The total vari-
ation methods yield nonlinear diffusion equations that are
always derived from variational principles using the TV
norm. In [20], Malladi and Sethian propose a unified ap-
proach to noise removal and image segmentation using the
concept of min-max curvature flow. Based on the image
data, a min/max switch was designed to select min(n, 0.0)
or max(n, 0.0) so that the curvature based curve evolution
smoothes out small oscillations, but maintains the essen-
tial properties of the shape. Results of implementation were
shown on a variety of images yielding quality noise removal
and image segmentation. In [31], Shah developed a com-
mon framework for curve evolution, image de-noising and
segmentation, and anisotropic diffusion. In this work, a
new segmentation functional was developed which lead to
a coupled system of PDEs, one of them performed nonlin-
ear smoothing of the input image and the other smoothed
an "edge strength" function. Shah [31] demonstrated that
all the existing curve evolution and anisotropic diffusion
schemes reported in literature can be viewed as special
cases of his method. In [9], Chen et. al., a nonlinear diffu-
sion equation supplemented with reactive terms for achiev-
ing edge preserving smoothing was presented. All of the
methods discussed thus far are primarily applicable to the
selective 'i, ,. hri,,i ofscalar valued images.

Smoothing vector valued images has been less popular
than the scalar valued image data sets. In this context,
Whitaker and Gerig introduced anisotropic vector-valued
diffusion which was a direct extension of the work by Per-
ona and Malik [25] to vector-valued images. The selec-
tive term in their work was based on the the gradient of


the vector valued image, which is the Jacobian matrix. In
[30] Sapiro et.al., introduced a selective smoothing tech-
nique where the selection term is not simply based on the
gradient of the vector valued image. Instead, he showed
that the stopping term should be a quantity related to the
eigen values of the Riemanian metric tensor computed from
the underlying surface defined by the vector valued image.
They applied their selective smoothing technique to smooth
noisy color images leading to impressive results. A very
general flow called the Beltrami flow as a general frame-
work for scalar and vector valued image smoothing was in-
troduced in Kimmel et. al., [18] and it was shown that most
flow-based smoothing schemes may be viewed as special
cases in their framework. A generalization of the TV norm
to handle vector-valued image smoothing was presented in
Blomgren and Chan [3]. They showed that their general-
ization was natural and had desirable properties such as the
rotational invariance in the image space etc. Existence and
uniqueness of a solution to their evolution equation is yet to
be explored but is not difficult to establish. There are many
other PDE-based image smoothing techniques that we have
not covered here but will refer the interested reader to a re-
cent survey by Weickert [35] and also the special issue of
the IEEE Transactions on Image Processing on PDE-based
image processing [4].

Very briefly, we propose a novel and efficient weighted
total variation (TV) norm based image smoothing scheme
where in the raw image data (one image for each of the 7
directions) S is smoothed using a PDE which is obtained
as a consequence of a weighted TV norm minimization de-
fined for vector valued functions. The selective term in
our work, is based on the eigen values of a diffusion ten-
sor D that can be computed initially from the raw im-
age data using the relationship S = S,,. ifl- Eij bijDij,
where, S is the vector of signal/image measurements taken
along seven directions X, Y, Z, XY, YZ, XZ, XYZ, So
is a constant, bij is the magnetic field strength (which is
a constant for a given direction) and Dij are the entries
of the (3, 3) matrix representing the diffusion tensor mea-
suring the diffusion of water inside the body being im-
aged. The selective term in this case g(s) = 1/(1 + s)
where s = FA is the fractional anisotropy defined as [2]
FA = V+(A ) A) ) where, A1, A3 and
are the largest, smallest and average eigen values of the
diffusion tensor D respectively. This selection criteria pre-
serves the dominant anisotropic direction while smoothing
the rest of the data. Another measure that works quite well
is s = (A1 A3)/A3. This selection criteria preserves the
dominant anisotropic direction while smoothing the rest of
the data. Note that since we are only interested in the fiber
tracts which correspond to the streamlines of the dominant
anisotropic direction, it is apt to choose such a selective










term as opposed to one that preserves edges in signal in-
tensity as was done in [24].



1.1 Finding Stream Lines

Water in the brain preferentially diffuses along white mat-
ter fibers. By tracking the direction of fastest diffusion, as
measured by MRI, non-invasive fiber tracking of the brain
can be accomplished. Fibers tracks maybe constructed by
repeatedly stepping in the direction of fastest diffusion. The
direction along which the diffusion is dominant corresponds
to the direction of eigen vector corresponding to the largest
eigen value. In Conturo et. al., [11], fiber tracks were
constructed by following the dominant eigenvector in 0.5
mm steps until a predefined measure of anisotropy fell be-
low some threshold. This usually occurred in grey matter.
The tensor, D, was calculated at each step from interpolated
DT-MRI data. This tracking scheme is primarily based on
heuristics and is not grounded in well founded mathemati-
cal principles. Using methods well grounded in mathemati-
cal principles will allow us to better understand/quantify the
strengths and weakness of the method/algorithm leading to
a better overall performance.
In Mori et.al., [21] fiber tracking was achieved using sev-
eral heuristics. The tracking algorithm starts from a voxel
center and proceeds in the direction of the major axis of the
diffusion ellipsoid. When the edge of the voxel is reached,
the direction is changed to that of the neighboring voxel.
Tracking stops when a measure of adjacent fiber alignment
crosses a given threshold. One possible measure is the
sum of inner products of nearby data points. This method
was able to reconstruct well-known pathways through a rat
brain. This method is also based on several data dependent
heuristics for achieving the fiber tract mapping.
Given the dominant eigen vector field of the diffusion
tensor in 3D, tracking the fibers (space curves) along this
dominant eigen vector field is basically equivalent to find-
ing the stream lines/integral curves in 3D of this vector field.
Finding integral curves of vector fields is a well researched
problem in the field of Fluid Mechanics [10]. The sim-
plest solution would be to numerically integrate the given
vector field using a stable numerical integration scheme
such as a fourth order Runge-Kutta integrator [27]. How-
ever, this may not yield a regularized integral curve. In
this paper, we pose the problem of finding stream lines
of the dominant eigen vector field of the diffusion tensor
in a variational framework incorporating smoothness con-
straints which regularize the integral curve. The variational
principle formulation leads to a PDE which can be solved
using efficient numerical techniques. We present the com-
puted 3D fiber tracts produced for normal and injured spinal
cords of rats using volume rendering techniques.


2 Image De-noising and Diffusion
Tensor Computation

In this paper, we propose a novel technique for smoothing
vector valued data that will be used in computing the dif-
fusion tensor field and mapping out the fiber tracts. The
novelty lies in the formulation that leads to a PDE which is
different from the traditionally used PDEs in literature for
vector valued image selective smoothing. The difference
lies in both the selective term used as well as the fact that
the PDE is derived from a minimization principle which
does not involve arc length minimization as is used tradi-
tionally in most selective image smoothing schemes that are
based on minimization principles. Note that there are sev-
eral PDE-based schemes in literature that are not based on
minimization principles [31, 34, 29, 20, 36].
Smoothing the raw vector valued image data is posed
as a variational principle involving a first order smoothness
constraint on the solution to the smoothing problem. Let
S(X) be the vector valued image that we want to smooth
where, X = (x, y, z) and let S(X) be the unknown smooth
approximation of the data that we want to estimate. We
propose a weighted TV-norm minimization for smoothing
the vector valued image S. The variational principle for
estimating a smooth S(X) is given by


min (X) = gj(AX+, A) IVS(X)I +
s .,i


p/2 |5 'dX
i=1
where, Q is the image domain and p is a regularization
factor. The first term here is the regularization constraint
on the solution to have a certain degree of smoothness
and selective smoothing is achieved by the term g(A) =
1/[1 + {(A+ A_)/A+}2], where A is the eigenvalue of the
diffusion tensor computed from the initial data. This func-
tion has very small value (approaching zero) as the relative
difference in the largest and smallest eigen values becomes
large stopping the smoothing at such locations and vice
versa. Since, the anisotropy in the image is well captured by
the direction of the dominant eigen value, it is apt for us to
preserve any discontinuities in the anisotropy while smooth-
ing the data. Note that it is not the edges (local maxima in
the gradient) in the DT image that we are interested in but
its the anisotropy or the lack thereof that is crucial for the
fiber tract mapping. The aforementioned selective smooth-
ing criteria is therefore well justified and is also supported
by the superior quality of the preliminary results (in com-
parison to the competing method described in [24]). The
second term in the variational principle makes the solution
faithful to the data to a certain degree. The parameter p con-
trols how close the smooth approximation should be to the










given data. The gradient descent of the above minimization
is given by

as., /g(A+, A_)VSj
O -=div ft) -p(Si -S) i = 1,...,7


O anx+ = 0 and S(X,t = 0)= S(X)
(2)
Note that this nonlinear PDE is ditirent from the tradi-
tional nonlinear dtirti,.\ ion equations that are found in litera-
ture [31, 29, 20] which are curve evolution based schemes.
The main difference is that there is NO V S I multiplica-
tive factor in the first term on the right hand side. What dif-
ference does this make? Firstly, its asymptotic solution con-
verges to the correct minimizer of 1 and the proof of con-
vergence does not require the use of the viscosity methods
as in [1]. Moreover, as in the traditional curve/surface evo-
lution based nonlinear diffusion equations, if we include the
aforementioned multiplicative factor I| S I it is not clear
if the asymptotic solution of the gradient descent equation
2 converges to the true minimizer of 1. The above vari-
ational principle is a generalization of the traditional TV-
norm for the scalar valued functions but differs in obvious
ways from the generalization presented in Blomgren and
Chan [3]. The main difference being that we use a weighted
TV-norm and our generalization does not have a square root
and the square under the summation as in [3]. The exis-
tence and uniqueness questions for such a weighted TV-
norm minimization have not been answered in literature to
date. We present a sketch of this proof in the next section.
The above nonlinear PDE can be solved using efficient
and stable numerical schemes. In this paper, we used an im-
plicit method namely the Crank-Nicholson scheme [26]. It
can also be solved using the lagged-diffusivity scheme dis-
cussed in [7] but we found the former to be more effective
and stable.

2.1 Estimating the Stream Lines/Integral
Curves

Once the diffusion tensor has been robustly estimated, the
fiber tracts may be mapped by choosing seed points in the
image lattice and using numerical integration techniques to
determine the integral curves of the eigen vector field cor-
responding to the dominant eigen values. Several numer-
ical integration schemes exist in literature [27]. The most
widely used and stable numerical integration scheme for or-
dinary differential equations is the Runge-Kutta scheme of
order four (RK4) [27]. The solution obtained by directly
using the RK4 may not be at times desirable since there
are no regularization constraints on the resulting integral
curves/fibers which are space curves in this case. In order
to have these space curves not exhibit very sharp twists and


turns (e.g., those with covers), we can formulate the in-
tegral curve estimation problem as a variational principle.
Thus, given the eigen vector field v = (vl, v2, v3)T corre-
sponding to the dominant eigen values, our formulation of
the variational principle involves minimizing the following
functional:


minE(p) = rmin cic'/) + '(p) v(c(p)) 2dp

(3)

where, c(p) = (x(p), ,/(/'i z(p))T, is the integral curve we
want to estimate and p E [0, 1] is the parameterization of
the curve, Q is the domain over which the curves are to be
determined and v(c(p)) is the vector field v restricted to the
curve c(p). The first term in this functional E(p) is seeking
to minimize the Li norm of the first derivative of the curve
i.e., seeking smooth curves and the second term requires
that the tangent to the smooth curve that we seek be close
to the the given dominant eigen vectors in an L2 sense. The
gradient descent i.e., the Euler-Lagrange expressed as an
initial boundary value problem, of the variational principle
in equation 3 is given by


ct = c'(p) kn + 3[c"(p) V(c(p))c'(/') +
3V (c(p))(c'(p) v(c(p))


where k is the curvature of the space curve, 3 is a regular-
ization parameter and

(Vix Vly Vlz
V = Dv = v2x V 2z
03x V 3z/

VT = the transpose of V. The curvature k is given by

/ c'(p) N
ip \c'Qu)J,,

The above initial boundary value problem can be solved
numerically using a variety of methods. We propose to use
the Crank-Nicholson implicit method which is a very stable
scheme (see [26]). Each iteration of this numerical iterative
scheme requires the solution a sparse banded (tridiagonal
and positive definite) linear system which can be solved in
O(n) time, where n is the size of the linear system equal
to the number of discrete points on the space curve. As
an initial condition for solving this PDE, we use the results
obtained by simply integrating the given vector field numer-
ically using a fourth order Runge-Kutta method [27]. The
computed integral curves are superimposed on the original
DTI data and visualized using a volume renderer. Results
of this volume visualization are presented in section 4.










3 Existence and Uniqueness Results

In this section, we will briefly outline the approach for es-
tablishing the existence and uniqueness of a solution to the
weighted TV-norm minimization equation 1. We will actu-
ally present the well-posedness result for a general weighted
TV-norm minimization and our minimization problem (1),
is included in this framework.
Consider the problem
k
min E(u) = min / l{g Vui +
uEBV(n,R"n)nL2 uEBV(n,Rk)nL2 i =1

IUi I,2}
2
(5)
To study the well-posedness of this problem, it is necessary
to introduce the concept of weighted TV norms for func-
tions of bounded variation. Recalling definition of bounded
variation (BV) spaces [15, 16].
Definition 1: Let Q C R" be an open set and let f =
(fi,... ,fk) E L(Q,Rk). Define


with ao < g < ai for x E ,, where ao and aI are positive
constants. Define the weighted total variation norm of f(x)
with the weight function g, fo g VfI by


SglV I = glVfi

k
=: sup fi(x)div((x))dx ,

where
) =:{> = 3D( < ,... ,,) e l (,nR")
l0(x))l < g, for all x E a}.


(10)


Theorem 1 Suppose that g satisfies all the assumptions in
Definition 4 and I E L2(, Rk). Then, the minimization
problem (5) has a unique solution u E BV(Q, R") n L2.

Proof Outline 1 The uniqueness of the minimum in (5)fol-
lows by the strict convexity of the functional


, |+(1/:'l -1 I-}.


(11)


jVf


k
Ej^h


k
: Esup
@i= e' l.In


fi(x)div(O(x


where
S=: 0{ = (1, ... d) E Co (, Rn)


1(x) < 1,
for all x E t


Definition 2: A function f E L' (, Rk) is said
have bounded variation in Q, if JI VfI < oo. We def
BV( Rk) as the space of all functions in L1 (Q, Rk) w
bounded variation.
Iff E BV(Q,Rk), Vfi (i = 1,... ,k) then Vfi is
R" valued Radon vector measure. This means


Sfi(x)div(J(x))dx


)) dx (The second term is strictly convex). The existence is proved
) as follows: Let ua, be a minimizing sequence. Then ua,
(6) is bounded in L2 because of the second term in (5). Us-
ing the convexity of the functional, we may assume that un
converges in L2 to say u. Then, we use the lower semi-
continuity of the norm is BV spaces to conclude that u is a
minimizer

(7) In order to use the gradient descent method for solving
(5), we consider the following evolution problem:
to
n OAu, = div(g(Vu/jV,, ')) (ui Ii), x E Q, t > 0


vith

an


for all E Co-(, R").
Definition 3: fm E BV(Q, R) converges weakly in
BV(Q, Rk) to a function f E BV(Q, Rk), if for each i =
1,... ,k


Ui(x,0) = Ii(x), x E O~n
On


(12)


0, x E OQ, t > 0,
(13)


where i = 1,... k, and n is the outward unit norm to 0.
Definition 4: A function u = (U1,... ,U ) E
L2(0, T; BV(Q, Rk)nL2) is called a weak solutionof (12)-
(13), if for each i = 1,... ,k Otui E L2(0,T; L2()),
ui = Ii, a.e. in Q, and ui satisfies


lim J Vf mi


) 01Vfi,
./n


for all 0 E C o(Q, R"), where f,m is the ith element of
fm.
Definition 4: Let Q C R" be an open set and f =
(fi,... ,fk) E L k(Q,Rk) and let g be continuous in Q,


\! Ju. r-ui)+ g vi1

- (ui Ii)(vi ui) f v.,
Q 0 Q


(14)


for a.e. s E [0,T], all v = (v,... ,Vk) E
L2(0, T; BV(Q, Rk) L2). We can now prove the follow-
ing theorem.


- f VA(x) ().
Qa










Theorem 2 The problem (12)-(13) has an unique solution
u E L2(0, T; BV(Q, Rk)) nL in the sense of(14). More-
over, as t -+ oo u(., t) converges weakly in BV(Q, Rk)nL2
to a function u,, which solves (5).

Proof Outline 2

Otui = divg( Vui)(p-v (u j,)

The existence here is well known. We show that the solu-
tion Up is indeed in L". We obtain uniform estimates of the
norms to show that the limit as p -+ 1 of the asymptotic lim-
its up -+ oo exists and is indeed a minimizer i.e., a solution
in BV of(5).


4 Experimental Results

In this section, we present two sets of experiments on, ap-
plication of the proposed selective smoothing to the raw im-
age data yielding smoothed tensor fields, and the computed
fiber tract maps from these smoothed data for the case of a
normal and an injured rat spinal cord respectively.
In both the experiments, we first smooth the seven 3D di-
rectional images using the novel selective smoothing tech-
nique outlined in section 2. Following this, the diffusion
tensor is estimated from the smoothed data using a stan-
dard least squares technique. The fractional anisotropy, the
color trace of the diffusion tensor (sum of the diagonal terms
of the diffusion tensor in an RGB color space), the domi-
nant eigen value as well as the color map of the direction
cosines of the eigen vector corresponding to the dominant
eigen value are computed. The latter color map depicts the
standard axis (X, Y, Z) toward which direction of diffusion
in the data is dominant and the dominant eigen value cor-
responds to the magnitude of this dominant diffusion di-
rection. Red corresponding to the X component, Green for
the Y component and Blue for the Z component. This color
coding will indicate the standard direction (X, Y or Z) along
which the dominant eigen vector has the strongest compo-
nent. Images obtained as a result of these computations
from raw data, smoothed data using a competing smoothing
method outlined in Parker et.al., [24] and smoothed data us-
ing our proposed method are depicted for the two data sets.
In addition, estimated fiber tracts from the smoothed data
using the proposed fiber tract/integral curves computation
scheme are also depicted for the data sets. The results of
smoothing for the two examples are shown in Figures 1 and
3 which are organized as follows: first row contains im-
ages computed from raw (noisy) data, second row contains
images computed using methods in [24] and third row con-
tains computed images using the proposed image smooth-
ing technique. Note the superior performance of the pro-
posed smoothing scheme in comparison to the method in
Parker et .al., [24].


Figure 2 depicts the computed 3D fiber tracts for a nor-
mal rat spine and is organized as follows: On the top left,
computed fiber tracts are shown in green. These fiber tracts
are supposed to run along the length of the spinal cord in the
white matter which is exactly what our computations reveal.
The top right image depicts a volume rendered DT-MR data
set with the fiber tracts overlayed in green. The bottom row
shows two different cut away views of the overlayed fiber
tracts on the volume rendered DT-MR scan of this normal
rat spinal cord.
Figure 3, depicts the results obtained from an injured rat
spine by application of the proposed smoothing in compar-
ison to results obtained from raw data and the competing
method in [24]. Fiber tract mapping is also shown for this
injured spine in figure 4. These results are organized as in
the figures 1 and 2. As evident, the injury has caused a large
cavity down the length of the spine and there are no fibers in
this region. Also evident is the fact that our data smoothing
results are superior to smoothing performed by the Perona
and Malik scheme [25] which was used in [24]. Also, the
visual quality of the fiber tracts is satisfactory.
In the above presented results, what is to be noted is that
we have demonstrated a proof of concept for the proposed
data smoothing and fiber tract mapping algorithms in the
case of the normal and injured rat spinal cords respectively.
The quality of results obtained is reasonably satisfactory for
visual inspection purposes but quantitative validation needs
to be performed and will be the focus of our future efforts.






5 Conclusions


In this paper, we presented a new weighted TV-norm min-
imization formulation for smoothing vector-valued data
specifically tuned to computation of smooth diffusion tensor
MR images. Existence and uniqueness of a solution for the
weighted TV-norm minimization is outlined. The smoothed
vector valued data was then used to compute a diffusion
tensor image using standard least squares technique. Fiber
tracts in 3D were computed as the integral curves of the
dominant eigen vector field obtained from the diffusion ten-
sor image. The integral curve computation was formulated
in a variational framework as well and solved using efficient
numerical schemes. Finally, results of fiber tract mapping of
a normal and an injured rat spinal cord were depicted using
standard volume rendering techniques. The computed fiber
tracts are quite accurate when inspected visually. However,
quantitative validation of the computed fiber tracts is essen-
tial and will be the focus of our future efforts.





































Figure 1: Normal Cord, left to right: FA, Color Trace, di-
rection cosines of the dominant eigen vector, and the dom-
inant eigen value. First row: results computed from raw
data, Row-2: results computed using Perona-Malik diffu-
sion, Row-3: results from the proposed smoothing.


Figure 2: Normal rat spinal cord: computed fibers over-
layed on volume rendered DT-MR data and its cut away
views.


Figure 3: Injured Cord, left to right: FA, Color Trace, di-
rection cosines of the dominant eigen vector and the dom-
inant eigen value. First row: results computed from raw
data, Row-2: results computed using Perona-Malik Diffu-
sion, Row-3: results from the proposed smoothing.










(a) (b)












(c) (d)

Figure 4: Injured rat spinal cord: computed fibers overlayed
on volume rendered DT-MR data and its cut away views.










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