MULTI-TENSOR FIELD SPECTRAL SEGMENTATION FORWHITE MATTER FIBER BUNDLE CLASSIFICATION

Omar Ocegueda Mariano Rivera

Centro de Investigacion en MatematicasGuanajuato, Gto, Mexico, 36240

ABSTRACT

We present an algorithm for segmenting the white-matteraxon fiber bundles from HARDI images. We formulate thesegmentation problem as a Multi-Tensor Field segmentationproblem in which the compartments of each Multi-Tensormay belong to different classes, allowing the algorithm tohandle crossing fiber tracts. Experimental results on twopublicly available synthetic datasets and the fiber-cup phan-tom show that fiber crossings can be effectively separatedby using this approach, and preliminary results on real datashow that the segmentation obtained is consistent with knownanatomical structures.

Index Terms— Brain imaging; Image segmentation;Tensor and vector field analysis

1. INTRODUCTION

Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI) is a technique that allows us to study the anatomicalstructure of living tissues by quantifying the amount of watermolecules that move along a set of M diffusion directionsQ = qiMi=1. When studying DW-MRI signals from thebrain, it is possible to extract useful information related to theorientation of the axon fibers that pass through a given voxel,which can further be used to identify larger anatomical struc-tures in the brain. The identification of anatomical structuresof cerebral white matter from DW-MRI has been extensivelystudied in recent years by the neuroscience community withdifferent objectives in mind [1–5]. Strategies for segmentingthe white matter fiber bundles may be classified into two maincategories: tractography-based clustering methods, and directvolume segmentation methods. Tractography-based cluster-ing methods consist on generating a large set of trajectories(which is called tractography) that traverse areas of the brainwith high diffusivity coherence. The set of trajectories arethen grouped into clusters using a specific similarity met-ric, and the volume segmentation may be finally performedin terms of the number of classified trajectories that passthrough each voxel, see for example Wassermann et al. [11].

Most direct volume segmentation methods formulate theproblem as a binary segmentation problem, in which the bun-dle of interest (foreground) is separated from the rest of thevolume (background), thus requiring a good initialization.Lankton et al. [5] proposed an active-contour evolution algo-rithm that uses local statistics to model the foreground data,the bundle of interest is manually indicated by two endpointswhich are then connected using tractography, the pathwayis then dilated to initialize a level-set binary segmentationmethod. Descoteaux et al. [2] used the coefficients of theSpherical Harmonics representation of the diffusion Orienta-tion Diffusion Functions (ODF) to compute a similarity mea-sure between neighboring ODF’s. Then they used level-setsto segment the volume into foreground-background. Cetingulet al. [1] use manifold learning techniques to learn a sparserepresentation of the ODF at each voxel. They then computea similarity matrix between pairs of voxels using the coeffi-cients of the ODF’s sparse representations and apply spectralclustering. The main drawback of most direct segmentationmethods is that the segmentation task is formulated at thevoxel level: each voxel is classified into one of the classes.However, when two or more fibers cross at a voxel, it is nec-essary to assign such a voxel to different classes. Hagmannet al. [3], and Jonasson et al. [4] proposed to formulate thesegmentation problem on an extended space, called Position-Orientation Space (POS). Under the POS model, the esti-mated ODF’s across the volume may be regarded as a scalarfunction S from a discretization of <3×S2 to <, where S2 isthe unit 2-sphere in <3. In POS, the classification task can besolved by binarizing the 5-dimensional image S (high inten-sity corresponds to the presence of a fiber while low intensitycorresponds to background). The main drawback of POS isthat the discretization of the 5-dimensional space produceshuge hyper-volumes which is highly memory demanding.

Our contribution is the development of an automatic fiberbundle segmentation method with the following character-istics: (i) It does not rely on tractography, (ii) It does notrequire user initialization nor an anatomical atlas to guide thesegmentation (iii) Despite the high dimensionality involved,it is computationally efficient.

2. MULTI-TENSOR FIELD REPRESENTATION OFTHE DW-MRI DATA

The DW-MRI signal Fv : Q → < at each voxel v ∈ Ω maybe represented as a vector Sv ∈ <M , where Ω is the set of allscanned voxels, and the ith element of Sv is Sv,i = Fv(qi),1 ≤ i ≤ M . One of the most widely used tools for model-ing the DW-MRI signals is Diffusion Tensor Imaging (DTI),which models the input signal as a diffusion function:

Sv,i = S0(v)exp(−τqTi Dvqi). (1)

where S0(v) is the standard T2 image at voxel v, Dv is asymmetric positive definite tensor and τ is the effective diffu-sion time. The most important limitation of DTI is that it isunable to resolve axon fiber crossings. One of the most suc-cessful approaches developed to overcome the limitations ofthe DTI model is the Multi-Tensor (MT) Model developed byTuch et al. [10]. Under the MT model, each one of the nvaxon fibers passing through a given voxel v produces a singlediffusion function, as in eq. 1, and the observed signal Sv isrepresented as a positive linear combination of those diffusionfunctions:

Sv,i = S0(v)

nv∑j=1

β(v)j exp(−τqTi D(j)

v qi). (2)

where the positive coefficientsβ(v)j

nv

j=1indicate the volume-

fraction of the voxel v that is occupied by the correspond-ing fiber. If we denote by I = 1, 2, ...,K, where K =max nv : v ∈ Ω, we can think of the set C = Ω × I ofcompartments as an extension of the volume Ω obtained by“splitting” each voxel v inK parts, and each non-empty com-partment represents the presence of a fiber bundle passingthrough the corresponding voxel. In our experiments, weused the Diffusion Basis Functions Model (DBF) proposedby A. Ramırez et al. [8] to fit the multi-tensor model to thedata. Under the DBF, a fixed dictionary of N possible diffu-sion functions is used for all voxels in the volume, so that themodel can be fit to the data by solving a non-negative leastsquares problem:

β∗ = minβ∈<N

||Φβ − Sv||22, s.t. β ≥ 0, (3)

where Φ = [φ1, φ2, ..., φN ] is the dictionary of diffusion func-tions evaluated at the set Q of diffusion directions, and hencemay be regarded as M−dimensional vectors. Each columnφj is then characterized by one tensor Dj . The set of eigen-values of Dj (which is known as the diffusivity profile) is as-sumed to be constant and of the form Λ = (λL, λR, λR),where λR is the radial diffusivity and λL > λR is the longitu-dinal diffusivity. Therefore, each column φj is totally charac-terized by Λ and the principal diffusion direction (PDD) pj ofDj . The DBF model has been shown to be a trusty approachin recent comparative studies 1.

1http://hardi.epfl.ch/static/events/2012 ISBI/

3. MULTI-TENSOR FIELD SPECTRALSEGMENTATION

Since, under the DBF model, the diffusivity profile is con-stant, the DBF representation of the input signal Sv (eq. 2)is totally characterized by the set of positive coefficientsβ(v)j

nv

j=1, and the set of principal diffusion directions of

the tensorsD

(j)v

nv

j=1. After fitting the DBF model to all

voxels of the volume, we obtain a Multi-Tensor Field (MTF),which may be regarded as a function T : C → <3, thatassigns a scaled PDD (a vector in <3) to each compartmentwhere the length ||T (v, j)|| = β

(v)j encodes the positive

coefficient and the unit vector T (v,j)||T (v,j)|| encodes the PDD cor-

responding to the jth tensor in the DBF representation of thesignal Sv at voxel v. This representation is also convenient toencode the empty compartments: T (v, j) = 0 indicates thatthe compartment (v, j) is empty. A segmentation of the MTFcan therefore be defined as a function L : C → L that assignsan element from a set of labels L = 1, 2, ..., ` to each com-partment (v, j) ∈ C in such a way that the MTF T is homoge-neous (in some sense) along regions of C sharing the same la-bel. The requirement that T is homogeneous along regions ofthe same class, needs to be defined more precisely by defininga notion of proximity (or neighborhood) between elements ofC and a notion of similarity (or distance) between elementsof the range of T . In our representation, a fiber bundle isformed by a set of nearby tensors with locally coherent orien-tation, thus a reasonable similarity metric between elementsin the range of T should consider both, spatial proximityand orientation. We used the dissimilarity measure given byD((v, r), (w, s)) = ||v−w||2

d20+ ](r,s)

θ0, and the similarity mea-

sure given by S((v, r), (w, s)) = exp (−D((v, r), (w, s))),where d0, θ0 > 0 are constant parameters and ](·, ·) denotesthe angle between the non-zero vectors given as argument.

Given two neighboring voxels v, w ∈ Ω we first com-pute the best assignment of compartments in v to com-partments in w, by minimizing the sum of the dissimilar-ities between paired tensors. Only paired tensors may beneighbors of each other. We say two paired compartments(v, i), (w, j) ∈ C are neighbors if ](r, s) < θmax, wherer = T ((v, i)), s = T ((w, j)) and θmax ≥ 0 is a constant pa-rameter. The main difficulty in the segmentation of the MTFis that there is no simple way of modeling the variation of T(a vector field) along each class (a fiber bundle). Thus, in-stead of modeling such variations, we define a transformation(“embedding function”) F : C → <d for a fixed d > 0, suchthat tensors that are close to each other according to the abovedistance D are mapped to nearby points in <d. This kind oftransformation is used in several dimensionality reductionand perceptual grouping algorithms. We used the algorithmproposed by Meila and Shi [7], which consists of comput-

ing the d eigenvectorsz(1), z(2), ..., z(d)

of the n × n

similarity matrix corresponding to its d largest eigenvalues.The embedding of the i−th compartment is then computedas (z1i , z

2i , ..., z

di ) ∈ <d. In our case, n is the number of

non-empty compartments, which may be very large, howeversince each compartment is only connected to a subset of itsneighbors in the volume lattice, the resulting similarity matrixis very sparse. Thus, it is possible to compute the requiredeigenvectors using efficient algorithms such as Arpack [6],which we used in our experiments.

After embedding the points in <d it is possible to ap-ply any standard segmentation algorithm, since the mod-els in the embedded space are constant vectors in <d. Weinitialized the segmentation with K-means, and then applyEntropy-Controlled Quadratic Markov Measure Field (EC-QMMF) [9] to obtain the final segmentation. EC-QMMF issuitable for parametric segmentation (i.e., to compute boththe segmentation and the model parameters, which in ourcase are constant vectors in <d), and provides for each datapoint, a discrete distribution indicating the probability of thegiven point to belong to each of the classes.

4. EXPERIMENTS

We performed experiments at three levels of difficulty. Theobjective of the first experiment is to show the segmenta-tion obtained when the true MTF is known (thus, the MTFestimation is not required), this experiment was performedon the publicly available 2012 HARDI Reconstruction Chal-lenge dataset1. The second experiment was performed on thepublicly available fibercup phantom2. In this case the bun-dles can be visually identified, but the MTF must be estimatedfrom the input DW-MRI. The third dataset was obtained froma healthy human brain in realistic clinical conditions. In thiscase, the objective is to show that our proposal can be appliedon realistic HARDI data, and that it is reasonably efficientcomputationally.

4.1. Synthetic data

The dataset consists of two 16 × 16 × 5 synthetic phantoms,one was provided for training, and the other was provided fortesting. The segmentation results are depicted in figures 1 and2, respectively. The angle between crossing fibers in the train-ing set (figure 1) is relatively large compared to those of thetesting set (figure 2). Although the true segmentation is notavailable for either of the datasets, the obtained segmentationis visually correct.

2http://www.lnao.fr/spip.php?rubrique79

(a) (b)

(c) (d) (e)Fig. 1: MTF segmentation obtained on the phantom provided for training atthe 2012 HARDI Challenge. The phantom consists of two straight- and onecircular- fibers crossing at several regions. (a) Input MTF colored accordingto tensor orientation. (b) Segmentation obtained. (c-e) Volume segmentsobtained.

(a) (b)

(c) (d) (e) (f) (g)Fig. 2: MTF segmentation obtained on the phantom provided for testing atthe 2012 HARDI Challenge. The phantom consists of (presumably) five fiberbundles that join in several ways. (a) Input MTF colored according to tensororientation. (b) Segmentation obtained. (c-g) Volume segments obtained.

4.2. Phantom

The data were acquired with a 3T Tim Trio MRI system. Weused the subset consisting of three 64 × 64 slices scannedalong 64 diffusion orientations at a resolution of 3mm, witha b-value of 1500s/mm2. The segmentation is depicted infigure 3. We can observe that the main difficulties occur atkissing and bifurcating bundles (green and blue in figure 3),which will likely be merged as a single bundle. Notice thatthis behavior is also present in Hagmann’s approach [3], sincekissing and bifurcating bundles join even in POS space. How-ever, these segments may be further processed individually bysubdividing them into smaller segments or by running trac-tography constrained to each individual segment.

4.3. Real data

The data were acquired with a Siemens Trio 3T scanner withthe following parameters: single-shot echo-planar imaging,five images with b=1000 s/mm2, 64 unique diffusion orienta-

(a) (b)

(c) (d) (e)Fig. 3: MTF segmentation obtained on the fiber-cup phantom. (a) Esti-mated MTF colored according to tensor orientation. (b) Segmentation ob-tained. (c-e) Volume segments obtained.

tions, TR = 6700ms, TE = 85ms, voxel dimensions equalto 2 × 2 × 2mm3. The SNR is approximately 26. We firstcomputed the MTF from the full dataset and then applied oursegmentation algorithm to a Region of Interest (ROI) consist-ing of 21 central (axial) slices of the brain (approximately30% of the data). The MTF estimation took 12 minutes andthe QMMF-spectral segmentation of the ROI took 2 minutes,for a total of 14 minutes on a 1.7 GHz lap-top using one sin-gle core. In figure 4 we show the segmented MTF in inputspace and embedded space. In this case, we first performed asegmentation in 8 classes and then each class was subdividedin 2 subclasses, illustrating the idea that each segment may befurther processed.

5. CONCLUSION

We presented a method for segmenting the cerebral whitematter fiber bundles from HARDI images. We formulated theproblem as a MTF segmentation problem, which allows usto handle fiber crossings. Experiments conducted on publiclyavailable artificial datasets show that using this approach itis possible to effectively separate crossing fibers, and pre-liminary experiments conducted on a dataset acquired from ahealthy human brain showed that the segmentation obtainedis consistent with known anatomical structures.

Acknowledgments. The authors would like to thank Dr.Alonso Ramırez-Manzanares for providing us with the DW-MRI data. This research was supported in part by CONACyTgrants 131369, 169178 and 131771.

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(a) (b)

(c) (d) (e)Fig. 4: Automatic segmentation obtained on 21 center axial slices from ahealthy human brain. Only 14 out of 16 classes are depicted for the sakeof clarity. Cingulum and corpus callosum were separated. All images arecolored according to the obtained segmentation (a) MTF in input space (b)MTF embedded in <2. (c) Cingulum bundles. (d) Segmentation iso-surfaces.(e) Corpus callosum. The iso-surfaces were smoothed out for visualizationpurposes.

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