Fiber Tract Clustering on Manifolds With Dual Rooted-Graphs

Andy Tsai1,2 Carl-Fredrik Westin1 Alfred O. Hero III3 Alan S. Willsky2

1 Dept. of Radiology 2 Dept. of Electrical Eng. 3 Dept. of Electrical Eng.Brigham & Women’s Hospital & Computer Science & Computer Science

Harvard Medical School Mass. Institute of Technology University of MichiganBoston, MA 02115 Cambridge, MA 02139 Ann Arbor, MI 48109

Abstract

We propose a manifold learning approach to fiber tractclustering using a novel similarity measure between fibertracts constructed from dual-rooted graphs. In particular, togenerate this similarity measure, the chamfer or Hausdorffdistance is initially employed as a local distance metric toconstruct minimum spanning trees between pairwise fibertracts. These minimum spanning trees are effective in cap-turing the intrinsic geometry of the fiber tracts. Hence, theyare used to capture the neighborhood structures of the fibertract data set. We next assume the high-dimensional inputfiber tracts to lie on low-dimensional non-linear manifolds.We apply Locally Linear Embedding, a popular manifoldlearning technique, to define a low-dimensional embeddingof the fiber tracts that preserves the neighborhood struc-tures of the high-dimensional data structure as captured bythe method of dual-rooted graphs. Clustering is then per-formed on this low-dimensional data structure using thek-means algorithm. We illustrate our resulting clusteringtechnique on both synthetic data and on real fiber tract dataobtained from diffusion tensor imaging.

1. Introduction

Diffusion tensor imaging is an emerging MRI-basedtechnology designed to measure the diffusivity of the wa-ter molecules in local tissue beds. Of particular interest isthe application of this technique to the brain parenchyma.Specifically, by taking advantage of the property that watermolecules diffuse preferentially along the length of the ax-onal tracts and less so in the direction perpendicular to thetracts, the white matter fiber structures can be characterizedand their connectivity mapped. This information can thenbe organized for use in surgical planning and in studyinga variety of disorders including neurodegenerative diseases,addiction, epilepsy, and mental disorders. A good review ofdiffusion magnetic resonance imaging can be found in [11].

Accurate and efficient visualization of this large andcomplicated 3D fiber tract data set to gain clinical insightis extremely difficult. Motivated by this shortcoming, thispaper focuses on the problem of clustering the fiber tractsinto natural anatomical bundles for ease of display and tofacilitate information exchange and interpretation.

1.1. Previous Works

Over the past few years during the time when diffusiontensor imaging modality has gained immense popularity,fiber tract clustering has likewise gained significant atten-tion with the development of various clustering techniques.In general, these algorithms all share the common themeof first defining a similarity metric between the fiber tracts,and then employing an algorithm for clustering based onthe established similarity measure. For example, a clus-tering algorithm similar to k nearest-neighbors approach isproposed in [6] with the similarity metric between pairedfiber tracts defined in terms of the length ratio and the Eu-clidean distance between the corresponding segments of thepaired fiber tracts. A fuzzy c-means clustering algorithm ispresented in [14] incorporating various distance measuresbetween fiber tracts including the dot product of the cor-responding tangents of the tracts and the average distancebetween points along the tracts. In [16], an agglomera-tive hierarchical clustering method is used in conjunctionwith a distance metric based on shortest distances betweenpoints on the tracts as defined in [15]. In [5], various pair-wise distances between tracts (including closest point dis-tance, symmetric chamfer distance, and symmetric Haus-dorff distance) and geometric characteristics of fibers (in-cluding length, center of mass, and second order moment)are utilized for threshold-based clustering. In [10], B-splinerepresentations of fiber tracts are used for comparison be-tween those fiber tracts extracted from the subject to thosefrom an atlas, and then based on the labeled atlas of the fibertracts, the subject’s fiber tracts are clustered.

Of particular interest to this paper is the work described

11-4244-1180-7/07/$25.00 ©2007 IEEE

in [3] which is the first to utilize manifold learning as animage processing tool for visualizing fiber tracts. Inspiredby this work, three additional techniques have followed suit[2, 9, 12], and they deserve special mention as they sharestrong ties with the algorithm proposed in this paper. Thesetechniques, very much like ours, employ spectral methodsof various flavors for clustering with each method utiliz-ing an affinity matrix constructed from a different fiber tractsimilarity measure. In [2], an Euclidean feature space (com-posed of the means and covariances of the points buildingup the fiber tracts) is used as a similarity measure for pair-wise fiber tracts. Radial basis functions are employed tomap this feature space to weights of an undirected graphwhich are then partitioned into coherent sets using the nor-malized cut criterion for clustering. In [9], a co-occurrencematrix containing the number of times two fibers share thesame voxel is used as the affinity matrix, and eigenvalue de-composition is performed on this affinity matrix to obtain aset of eigenvectors for clustering by a k-means algorithm.In [12], a k-way normalized cut procedure is proposed forclustering with an affinity matrix composed of symmetrizedHausdorff distances between pairwise tracts.

1.2. Contributions of Our Work

In the literature, there appears to be disproportionatelymore effort and emphasis placed on the development of abetter clustering algorithm and less so on the design of a bet-ter similarity measure between the fiber tracts, even thoughthe latter is more important in determining the success of afiber tract clustering algorithm. The challenge in the clus-tering arena is to find the most appropriate distance measurethat will farthest separate the fiber tracts belonging to dif-ferent clusters while keeping fiber tracts of the same clusterclose by. Popular distance measures including the chamferand the Hausdorff distances have been proposed that ade-quately capture the local relationship of the fiber tracts buttend to lack the ability to capture the global structure ofthe input data set. In this paper, we propose a similaritymeasure based on dual rooted diffusion [7] which providesa more geometrically descriptive measure of the similaritybetween fiber tracts. Importantly, it captures both the localand the global intrinsic geometry of the data set in a princi-pled and effective manner.

Similar to the approaches taken by others [2, 9, 12], ourproposed distance measure is then incorporated into a man-ifold clustering algorithm, which in our case is Locally Lin-ear Embedding, for data partitioning. Manifold learning ap-proaches seek to define a low-dimensional embedding ofthe input data points that preserves the neighborhood struc-ture of the high dimensional point set. We believe that thismethodology will be an effective mechanism to reveal theunderlying meaningful low dimensional information hiddenwithin high dimensional observations for successful cluster-

ing of the input fiber tract data.

1.3. Paper Organization

The rest of this paper is organized as follows. In Sec-tion 2, we introduce the similarity measure between pair-wise fiber tracts that we use in our technique. Section 3describes how we utilize a variant of the Locally Linear Em-bedding method for manifold clustering. Section 4 presentspreliminary results of our algorithm using both syntheticand actual diffusion tensor imaging data. We offer our con-cluding remarks and future research directions in Section 5.

2. Fiber Tract Similarity Measure

In this section, we introduce two well studied distancemeasures for 3D space curves, the chamfer [1] and theHausdorff [8] distances. Both of these distance measuresprovide some form of local similarity measure betweenfiber tracts. As a more effective means of capturing thelocal and global relationships between the fiber tracts, wedescribe how either one of these distance measures can beincorporated into a framework of dual rooted graph diffu-sion to obtain a novel fiber tract similarity metric which iscapable of capturing the intrinsic geometry of the data set.

2.1. Chamfer and Hausdorff Distances

Let XM = {x1, ...,xM} be the set of M fiber tracts in ahigh dimensional vector space R

d. A reasonable and popu-lar distance measure between fiber tract xi = {xip}P

p=1 and

fiber tract xj = {xjq}Qq=1 is the chamfer distance which is

given by the average of the distances between each pointxip ∈ xi and its closest point in xj :

dchamfer(xi,xj) =1n

∑xip∈xi

minxjq∈xj

‖xip − xjq‖

where ‖ · ‖ denotes the Euclidean norm. A different butequally popular distance measure between fiber tract xi andfiber tract xj is the Hausdorff distance [8] which is given bythe maximum1 of the distances between each point xip ∈ xi

and its closest point in xj :

dHausdorff (xi,xj) = maxxip∈xi

{min

xjq∈xj

‖xip − xjq‖}.

Both of these distance measures can easily be madesymmetric by taking the average between d(xi,xj) andd(xj ,xi) so as to obey the metric properties. To a certaindegree, both these measures are effective in capturing thelocal neighborhood structures of the input fiber tract data setbut inadequate in capturing distant relationships. Of note,

1Others have also proposed the use of the median distance instead ofthe average (chamfer) or the maximum (Hausdorff) distance.

an inherent problem with Hausdorff distance is that a pointin xi that is farthest from any point in xj dominates andmay inappropriately skew this distance measure.

2.2. Dual Rooted-Graphs

Motivated by the notion of a diffusion distance built uponrandom walks on graphs [4], a novel and more robust simi-larity criterion between high dimensional data points (suchas fiber tracts) is introduced in [7]. This measure is suitedfor clustering on smooth manifolds, and is effective in cap-turing the intrinsic geometry of the input data.

The specifics of this algorithm based on dual rootedgraphs is described here. For each fiber tract x ∈ XM , re-cursively grow a minimum spanning tree (MST) rooted inx in the following manner. Start at the root node of thetree x at time k = 0 with the set MST0(x,XM ) = x. LetMSTk(x,XM ) denote the set of fiber tracts in the tree attime k rooted at x. With each successive discrete time pointk, add a fiber tract closest to the root node x that is in XM

but not already in MST(k−1)(x,XM ). Chamfer or Haus-dorff distance is used here to measure the distance betweentwo fiber tracts in R

d.2 At the end of time k = M − 1, allthe fiber tracts in XM will be included in MSTk(x,XM )arranged in an ascending order of distance to the root nodex. Time stamps of when each fiber tract is added to thetree is also recorded. This process is repeated M times toproduce M fully grown trees with each fiber tract x ∈ XM

serving as a root node in this set of M MSTs.Next, define the hitting time τ(xa,xb) between the fiber

tracts xa and xb in XM as the time k when the two MSTsrooted in xa and xb intersect, i.e.

τ(xa,xb)=min{k :MSTk(xa,XM )∩MSTk(xb,XM )�=0}.

From an implementational stand point, the hit timeτ(xa,xb) is determined by parsing MSTk(xa,XM ) andMSTk(xb,XM ) sequentially until a common point isfound between the two MSTs. Once τ is found, the totalpath length between xa and xb is calculated by summing upthe pairwise distances between sequential fiber tracts withineach of the two MSTs up to the hit time τ . A M × M sym-metric square matrix T containing the distances betweenevery pairwise fiber tracts in Xm can be generated in thisfashion. We believe that this proposed methodology cap-tures the local structure of the input data via the chamfer orthe Hausdorf distance while the global structure is capturedthrough the complexity of the paths taken between any pairof fiber tracts in the data set via the MSTs.

3. Manifold Clustering With LLE

Various methods can be employed to analyze matricesof pairwise distances for spectral clustering, and we chose

2In fact, any reasonable distance metric between curves can be used.

a variant of the Locally Linear Embedding (LLE) as de-scribed in [13]. The goal of LLE is to map high dimensionalinputs XM to low dimensional outputs YM using local lin-ear reconstruction weights W. To accomplish that, LLEfirst attempts to represent the input data manifold locally byreconstructing each data point xi as weighted combinationof its neighbors through the weights W. Specifically, weseek W as below:

W =argminW

M∑i=1

‖xi −∑

j∈N (i)

Wijxj‖2 s.t. ∀i∑

j

Wij=1 .

However, instead of calculating the reconstructionweights based on the above equation using fiber tract dataxi, we opt to use the method described in [13] and calcu-late them based on the pairwise distances of the fiber tracts.In particular, given the distance matrix T computed as de-scribed in Section 2, the nearest K neighborsNK(i) of fibertract xi is identified by parsing T to find the K smallestnon-zero elements corresponding to each fiber tract. Know-ing the neighborhood structure of each fiber tract, the localcovariance matrix Cij of fiber tract xi and its K neighbor-ing fiber tracts xj with j ∈ NK(i) can be derived by com-puting the following:

Cij =12(Di + Dj − Dij − D0),

where Dij is the square of the distance between the ith andthe jth neighbors as provided by T , Dl =

∑z Dlz , and

D0 =∑

ij Dij [13]. In terms of Cij , the optimal recon-

struction weights W to best reconstruct each fiber tract xi

from its neighbors are given by:

wj =

∑k C−1

jk∑lm C−1

lm

.

The optimal weights W is an M × M sparse matrix cal-culated to capture the neighborhood structure of the fibertracts. Based on these weights, the next step is to mapthe high dimensional observation data XM to a low dimen-sional vectors YM by minimizing an embedding quadraticcost functional:

Y=argminY

M∑i=1

‖yi −∑

j∈N (i)

Wijyj‖2 s.t.Y1 = 0YY′ = IM

.

Importantly, only the geometric information encoded by theweights W is used to construct the embedding and not theinput data XM . Since this embedding cost functional isquadratic in Y, it can be estimated by solving a sparseM × M eigenvector problem. The eigenvectors associatedwith the smallest d positive eigenvalues define the best ddimensional fit. Finally, as is common practice, a k-meansmethod is applied to partition the resulting d eigenvectorsfor clustering. The number of clusters d is defined a priori.

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Figure 1. Trivial synthetic fiber tract clustering example. (a) Input data consisting of 121 synthetic fiber tracts each described by 50 datapoints. (b) Correct partitioning of fiber tracts into 11 bundles with K = 4 and d = 10.

4. Preliminary Results

Results based on both synthetic and real fiber tract dataare presented in this section to illustrate the performance ofour clustering algorithm. All algorithms are implemented inMatlab. In section 4.1, we show simulation results specif-ically designed to illustrate some of the features and capa-bilities of our algorithm as described earlier in the paper.Section 4.2 demonstrates the performance of our approachby applying it to the clustering of a real fiber tract data setobtained from diffusion tensor imaging.

4.1. Synthetic Dataset

The purpose of the first synthetic example as shown inFigure 1 is to demonstrate some of the basic features ofour manifold clustering algorithm, specifically, its ability tocluster a 3D data set (121 fiber tracts each consisting of 50data points) into multiple bundles (11) accurately and in anefficient manner (28 seconds on a 3.1GHz Intel Xeon pro-cessor). The second synthetic example shown in Figure 2is intended to demonstrate the robustness of our clusteringalgorithm in a more hostile environment–one corrupted byadditive noise, with complicated fiber structures, and havingvarying fiber tract lengths. The original synthetic data withcomplicated fiber tract structures is displayed in Figure 2(a).Segments of varying size fiber tracts are removed to arriveat the data set shown in Figure 2(b). Gaussian noise withzero mean and standard deviation of 1 is added to generatethe input noisy test data set shown in Figure 2(c). Based onthis test data set, our algorithm generated the color labeledclusters as shown in Figure 2(d).

The intent of the third synthetic example shown in Fig-ure 3 is to explicitly demonstrate the added capabilities ofour proposed distance measure over the more conventionalchamfer and Hausdorff distance measures. The input datashown in Figure 3(a) contains two interlacing “U” shapeddata sets. Specifically, the data sets are facing each otherwith each data set consisting of 3D parallel fiber tracts. Bydesign, a tail of each data set is sandwiched between thewings of the other data set. As a result, the parallel fibertracts near these tail regions are in close proximity to theparallel fiber tracts from the other data set. When only theHausdorff (Figure 3(b)) or the chamfer (Figure 3(c)) dis-tance is used as the similarity measure within a LLE man-ifold clustering algorithm, erroneous clustering results oc-cur. However, our proposed similarity measure successfullyclustered the input data as shown in Figure 3(d). This ex-ample illustrates the effectiveness of our proposed distancemeasure in capturing not only the local but also the globalstructure of the data set.

4.2. Fiber Traces from DT-MRI

The corpus callosum is a white matter structure locatedjust ventral to the cortex that connects the left and rightcerebral hemispheres to allow communication between thetwo halves of the brain. Subdividing the corpus callosuminto anatomically defined portions is not well defined butof much importance, especially in study normal develop-ment and in understanding mental and neurodegenerativedisorders. We apply our method in partitioning the fibertracts of the corpus callosum into anatomical bundles. Fig-ure 4 demonstrates the results of our fiber tract clustering

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(c) (d)Figure 2. Synthetic fiber tract clustering example in a hostile environment. (a) Original data consisting of 126 synthetic fiber tracts eachdescribed by 50 data points. (b) Random removal of varying size segments to generate a data set with fiber tracts lengths varying between4–50 data points. (c) Corruption of the data by additive Gaussian noise (μ = 0 and σ = 1) to generate the input data set. (d) Correctpartitioning of fiber tracts into 6 bundles with K = 8 and d = 5 in less than 23 seconds on a 3.1 GHz Intel Xenon processor.

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Figure 3. Synthetic example consisting of two interweaving “U” shaped data sets. (a) Input data consisting of 246 parallel fiber tracts. (b)Clustering result based on the Hausdorff distance. (c) Clustering result based on the chamfer distance. (d) Clustering result based on theproposed distance measure. (K = 4 and d = 1 for all these simulations.)

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Figure 4. Clustering of 1665 corpus callosum fiber tracts obtained from diffusion tensor imaging with K = 20 and d = 11. (a) Sagittalview. (b) Coronal view.

algorithm in dividing a patient’s 1665 corpus callosum fibertracts into 12 clusters.

5. Conclusions and Future Work

We have outlined a manifold clustering algorithm uti-lizing Local Linear Embedding and a new similarity mea-sure between fiber tracts built on the concept of dual rooted-graphs to yield a more robust and principled clustering algo-rithm that accounts for both the local and global geometryof the input data for effective partitioning. The preliminaryresults demonstrate improved visualization of the connec-tivity of the fiber tracts for clinical use. A natural extensionof this work is to incorporate our proposed distance metricwithin other spectral clustering techniques. Much neededimprovement is needed in speeding up the construction ofthe MSTs to reduce the calculation of the distances betweenthe fiber tracts. We are actively investigating the clinicalutility of our algorithm for disease understanding and treat-ment.

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