3D Stochastic Completion Fields for Mapping Connectivity in Diffusion MRI

3D Stochastic Completion Fields forMapping Connectivity in Diffusion MRI

Parya MomayyezSiahkal and Kaleem Siddiqi, Senior Member, IEEE

Abstract—The 2D stochastic completion field algorithm, introduced by Williams and Jacobs [1], [2], uses a directional random walk to

model the prior probability of completion curves in the plane. This construct has had a powerful impact in computer vision, where it has

been used to compute the shapes of likely completion curves between edge fragments in visual imagery. Motivated by these

developments, we extend the algorithm to 3D, using a spherical harmonics basis to achieve a rotation invariant computational solution

to the Fokker-Planck equation describing the evolution of the probability density function underlying the model. This provides a

principled way to compute 3D completion patterns and to derive connectivity measures for orientation data in 3D, as arises in 3D

tracking, motion capture, and medical imaging. We demonstrate the utility of the approach for the particular case of diffusion magnetic

resonance imaging, where we derive connectivity maps for synthetic data, on a physical phantom and on an in vivo high angular

resolution diffusion image of a human brain.

Index Terms—3D directional random walk, Fokker-Planck equation, completion fields, diffusion MRI, probabilistic connectivity,

spherical harmonics

Ç

1 INTRODUCTION

CONTOUR completion involves “filling in” the missingportions of partly occluded or fragmented outlines in a

visual image. This process is fundamental to computervision, where it constitutes an essential intermediate step inmany applications, including the perceptual grouping ofedge maps, image segmentation, and object recognition. Itsstudy also has a rich history in human vision, includingconsiderations by Kanizsa [3], whose work emphasizes theimportance of both modal and amodal completions in theperception of visual forms. In fact, the notion of illusorycontours, where edges and surfaces that have no physicalreality in the scene are nonetheless perceived, has been andcontinues to be a subject of active research [4]. Mathematicalcontour completion is an ill-posed problem in that aninfinite variety of curves can serve as possible completionswhile being faithful to the edge fragments in an image. As aresult, achieving computational solutions to contour com-pletion is challenging and additional assumptions have tobe imposed, such as the minimization of total squaredcurvature [1], [5], [6] or the minimization of total change incurvature [7].

Among the first attempts to solve the problem of contourcompletion is Ullman’s approach [5], which uses circulararcs to fill the gap between two boundary segments,searching for those which minimize the total curvaturesquared and meet smoothly, while being tangent to the

given edges. Parent and Zucker [8] take a differentialgeometric point of view, using co-circularity in a relaxationlabeling framework to approximate tangent vectors andcurvature fields to be used for curve completion. Similarly,Guy and Medioni [9] implement a tensor voting approachusing a co-circularity kernel. Their kernel generates anextension field starting from a given edge segment byassigning a likelihood of connection to preferred directionsat every point in space. The superposition of extensionfields from two fragments provides a saliency mapdescribing connectivity between the two edges. Shashuaand Ullman propose a saliency network for perceptualgrouping which incorporates the notion of minimizingcurvature squared plus length [10]. The minimization oftotal curvature variation has been explored by Kimia et al.in [7], where Euler spirals are used to bridge the gapbetween edges. The 2D completion algorithm of Williamsand Jacobs [1], [2] models the prior probability distributionof completion curves by the trajectory of particles followinga 2D directional random walk. The maximum likelihoodcurves derived from this model correspond to curves ofleast energy, which, like Shashua and Ullman’s approach,minimize a weighted combination of curvature squaredplus length. The derived probabilistic representation ofpossible connections between two boundary segments isdubbed a stochastic completion field. Williams and Jacobsargue that the shape, salience, and sharpness of thecontours perceived by humans can be represented by themode, magnitude, and variance of this completion field. Inmore recent work, Zweck and Williams [11] achieveinvariance to translation and rotation in 2D by computingcompletion fields using shiftable-twistable functions.

Motivated by Williams and Jacobs’ model, the presentpaper extends this framework to handle curve completionin 3D. While the extension of the directional random walk isstraightforward, the computation of the stochastic comple-tion field in 3D requires special treatment to achieverotation invariance. We accomplish this by developing the

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 35, NO. 4, APRIL 2013 983

. The authors are with the School of Computer Science and Centre forIntelligent Machines, McGill University, McConnell Engineering, 3480University Street, Montreal, QC H3A 2A7, Canada.E-mail: [email protected], [email protected].

Manuscript received 5 Oct. 2011; revised 14 June 2012; accepted 3 Aug. 2012;published online 28 Aug. 2012.Recommended for acceptance by P. Golland.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log NumberTPAMI-2011-10-0716.Digital Object Identifier no. 10.1109/TPAMI.2012.184.

0162-8828/13/$31.00 � 2013 IEEE Published by the IEEE Computer Society

associated Fokker-Planck equation describing the evolutionof the probability density function of particles undergoingthe random walk and by using a spherical harmonics (SPH)basis to achieve a rotation invariant solution on the twospheres. We then tailor the 3D completion field algorithm toinclude additional drift terms to incorporate spatially denseorientation data when available. Whereas the basic methodis applicable to data from motion capture or 3D featuretracking systems, we focus on the particular problem ofderiving completions in diffusion magnetic resonanceimaging (diffusion MRI) data along with probabilisticmeasures of connectivity.

This paper is organized as follows: We start with a briefintroduction to diffusion MRI and review existing modelsfor curve completion in this context. Section 2 develops thetheory of 3D stochastic completion fields. In Section 3, weformulate the Fokker-Planck equation, which describes theevolution in time of the probability density function ofparticles undergoing a 3D directional random walk.Section 4 develops a modification of the model to includeadditional drift terms in order to take into account spatiallydense measurements, should they be available. In Section 5,we extend the model and its underlying Fokker-Planckequation to also include diffusion on the spatial domainvariables. Section 6 develops a rotation invariant numericalmethod for solving the Fokker-Planck equation by usingspherical harmonics to represent functions defined on the2-sphere. In Section 7, we apply the 3D stochastic comple-tion field algorithm to the problem of mapping connectivityin diffusion MRI data. We explain how parameters for thestochastic completion field are selected based on theunderlying diffusion MRI data, and develop a measure toquantify connectivity. Qualitative and quantitative valida-tion of this approach is provided in Sections 7.3.2 and 7.3.3using synthetic multitensor data and the MICCAI 2009Fibre Cup phantom [12]. We also carry out experiments onsome of the major fiber pathways in the human brain. Weconclude with a brief discussion of our results anddirections for future work in Section 8.

1.1 Curve Completion for Mapping BrainConnectivity

The emergence of diffusion MRI [13] has led to noninvasiveways to explore connectivity in fibrous tissue. It has beenused to assist neuroscientists in tasks such as virtual grossanatomical dissections of the brain [14], analysis of thelanguage network [15], [16], human brain parcellation [17],[18], and joint studies of functional/anatomical brainnetworks [19], [20]. Measurements obtained from diffusionMRI are sensitized to the anisotropic Brownian motion ofwater molecules in living tissues, including brain whitematter. This anisotropic direction-dependent motion ofwater molecules is primarily based on the higher diffusivityof molecules along fiber tracts than across them, providinginformation about the local orientation of white matter fiberbundles [21]. A number of fiber tractography algorithmshave been developed to infer the pattern of white matterconnectivity by exploiting the local orientation measure-ments in diffusion MRI [22], [23], [24], [25], [26], [27], [28],[29]. The local fiber orientation is estimated using differentmodels of the diffusion signal, including the diffusion tensor(DT) [30], the orientation distribution function (ODF) [31],

[32], and the fiber orientation density (FOD) [33]. The lasttwo models, which have been developed for high angularresolution diffusion imaging (HARDI) data, are the standardmethods used for tractography in the recent literature.

The first group of tractography algorithms proposed arepurely deterministic schemes which reconstruct the mostlikely fiber pathways by tracing along the fiber ODFmaxima [23], [34], [35]. To improve upon these methodsprobabilistic tractography techniques incorporate measure-ments of noise and model uncertainty [27], [29], [36].Kalman filtering-based methods [37] treat model estimationand tractography simultaneously by recursively updatingthe local model parameters, estimating their mean andvariance, and indicating the most probable direction ofheading. Global tractography methods take a differentapproach by searching for minimal cost pathways betweenregions of interest (ROI) [25], [38], [39]. The major advantageof global methods is their ability to trace through regions oflocal perturbation by optimizing a global criterion. Follow-ing this, a connectivity measure can be assigned to thereconstructed fiber pathways based on the employed energymetric. Approaches which try to find the steady-statesolution of a partial differential equation (PDE) representanother class of probabilistic algorithms. A direct computa-tion of a connectivity map discriminates these PDE-basedalgorithms from other tractography methods.

Among the PDE-based tractography techniques,O’Donnell et al. [42] work directly with the diffusionequation in an anisotropic medium, using 3D tensors asconductivity tensors. The degree of connectivity is approxi-mated by the steady-state flow along the maximum flowpath. Batchelor et al. [41] modify this algorithm by adding aconvection term to the PDE, integrating a measure ofanisotropy into the concentration flow. In [44], Hagemanet al. extend the probabilistic first-order PDE methods bysimulating a fluid flow through a diffusion tensor field. In arelated approach, Fletcher et al. propose a front evolutionscheme to compute the minimal cost of connecting tworegions of interest [45]. Pichon et al. [46] generalized PDE-based methods to HARDI by solving the Hamilton-Jacobi-Bellman equation to extract the minimum cost pathways.

Our 3D stochastic completion field algorithm applied towhite matter connectivity is a PDE-based scheme that hascertain global properties. In particular, the most likelycompletion curves under this model map to a notion of 3Dcurves of least energy. The algorithm is linked to thephysical process to which diffusion MRI measurementshave been sensitized—the anisotropic Brownian motion ofwater molecules. One of the main contributions of theproposed approach is that global tractography is performedwith a true probabilistic nature, where probability valuesare assigned to every state in the volume. Furthermore, asdiscussed in Section 7.2, a measure of connectivity betweentwo regions of interest emerges as an inherent characteristicof the algorithm. This measure has the potential todistinguish real connections from false ones. In thefollowing section, we develop the 3D stochastic completionfield model.

2 3D STOCHASTIC COMPLETION FIELD

The 3D stochastic completion field, first used in [47] for thepurpose of tractography in diffusion MRI, arises as a direct

984 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 35, NO. 4, APRIL 2013

extension of Williams and Jacobs’ 2D model [1]. Thisapproach models the prior probability distribution of 3Dcompletion curves with a 3D directional random walk.

Let a particle’s state be given by its position ðx; y; zÞ andorientation ð�; �Þ in IR3 � S2. At each time step, theparticle’s state is updated according to a 3D directionalrandom walk which is composed of two principal compo-nents: the motion equations which update the position andorientation of the particle and the decay component whichmodels the particle’s exponential decay in time to favorshorter pathways. The motion equations of the directionalrandom walk are given by

dx ¼ sin � cos�dt; dy ¼ sin � sin�dt; dz ¼ cos �dt

d� ¼ ��dWt �ð Þ; d� ¼ ��dWt �ð ÞdWt �ð Þ; dWt �ð Þ � N 0; dtð Þ:

ð1Þ

Here, ðx; y; z; �; �Þ and their corresponding perturbations aregiven with respect to a fixed global coordinate frame. Fig. 1presents a schematic view of the update of a particle’s state ateach time step. Under this model a particle tends to travelwith constant speed in a straight line along its currentdirection of heading, with a slight perturbation in orientationat each step. This perturbation is controlled by two successiveorientation changes which can be interpreted, respectively,as deviations in the local osculating and binormal planes,given by two Brownian motion terms ��dWtð�Þ and��dWtð�Þ. The parameters �� and �� control the magnitudeof the successive orientation changes at each time step.

The second component of the directional random walk isa decay term which depends on the lifetime parameter, �,associated with each particle. This component allows acertain fraction of particles ð1� eð�1=�ÞÞ to decay per unittime. Thus, the longer the trajectory followed by theparticles is, the larger the fraction of particles which dieout is. This results in greater support for shorter pathways.Section 7.1 describes how the parameters of the directionalrandom walk, i.e., ��, ��, and �, are selected at each voxelbased on the available local diffusion MRI data. In Section 5,we develop a more general case with diffusion on the entirestate space, where a perturbation term is also added to thepositional coordinates.

Using the machinery developed by Williams and Jacobsin [1], a stochastic completion field represents the probabilitythat a particle passes through a given location ðx; y; zÞ with

orientation ð�; �Þ in 3D, while completing a path between asource and a sink region in the course of its 3D directionalrandom walk. The source and the sink regions are eachgiven by a fixed collection of starting and ending states,respectively. Using the Markov property underlying therandom walk, this completion field can be computed as theproduct of a stochastic source field and a stochastic sink field.The former is defined as the probability of passing througha particular state ðx; y; z; �; �Þ having started in a particularset of source states (the source region). The latter, in asimilar fashion, represents the probability of passingthrough a particular state ðx; y; z; �; �Þ while ending in asink region. By construction, for chosen source and sinkregions, the 3D stochastic completion field describes thelikely trajectories between them. In Section 7.2, we shallexploit this property to derive probabilistic connectivitymeasures for diffusion MRI data.

2.1 3D Curves of Least Energy

We now show that in analogy to the 2D case [1], themaximum likelihood (discrete) paths generated by the 3Dcompletion field model correspond to a form of elastica [6]or curves of least energy. Using the underlying Markovassumption behind the process, the paths generated by the3D directional random walk can be obtained by a number ofindependent identically distributed (i.i.d.) discrete steps.Consider a given particle which starts its random walk fromthe source voxel p and traces a path �p with n unit lengthsteps ending at sink voxel q. Each step is associated withtwo changes in the orientation: 1) a change in the osculatingplane and 2) a change in the binormal plane, given by�1; . . . ; �n and �1; . . . ; �n, respectively. According to thestochastic differential equations of (1), these orientationchanges are given by two normal distributions with zeromean and variance �2

�dt and �2�dt. In the discrete domain,

we choose the time step to be equal to 1, which simplifiesthe variance terms to �2

� and �2�. Thus, adopting the notation

in [1], the density function for the set of paths leavingsource voxel p under the i.i.d. assumption is given by

P �p� �

¼Yni¼1

e�1�

1

��ffiffiffiffiffiffi2�p e

��2i

2�2�

1


��2i

2�2� : ð2Þ

To get the density function for the set of paths starting atsource p and ending in sink q, i.e., �pq, the above densityfunction is divided by the integral of P ð�pÞ over all pathsending in q, i.e., �q:

P �pq� �

¼ 1R�qP �p� �

d�p

�Yni¼1

e�1�

1


��2i

2�2�

1


��2i

2�2� :

ð3Þ

Taking logarithms of both sides leads to

logP �pq� �

þ C ¼ �Xni¼1

�2i

2�2�

�Xni¼1

�2i

2�2�

� n 1

�þ log ð��

ffiffiffiffiffiffi2�pÞ þ log ð��

ffiffiffiffiffiffi2�pÞ

� �:

ð4Þ

If we consider polygonal arcs with n segments and associatethe orientation changes �i in the osculating plane with

MOMAYYEZSIAHKAL AND SIDDIQI: 3D STOCHASTIC COMPLETION FIELDS FOR MAPPING CONNECTIVITY IN DIFFUSION MRI 985

Fig. 1. The 3D directional random walk. After taking a step in the currentdirection of heading, the particle’s orientation is rotated by �� and �� inthe osculating and binormal planes, respectively.

curvature, and the orientation changes �i in the binormalplane with torsion, the maximum likelihood polygonal pathsin (4) in a discrete sense can be seen to minimize a weightedsum of length, curvature squared and torsion squared. Witha curve of least energy in a continuous sense minimizing

Z�

� tð Þ2dtþ Z

�

� tð Þ2dtþ �Z

�

dt; ð5Þ

which is a natural extension of elastica in 2D (as in [6]) to

3D, the association between the weights in the discrete and

the continuous cases is given by ¼ 12�2

�

, ¼ 12�2

�

, and

� ¼ 1� þ log ð��

ffiffiffiffiffiffi2�pÞ þ log ð��

ffiffiffiffiffiffi2�pÞ.

3 FOKKER-PLANCK EQUATION FOR THE

3D RANDOM WALK

While Monte-Carlo simulation of the stochastic process

formulated in (1) is a straightforward approach for compu-

tation of the stochastic source and sink fields, a parallelizable

and hence more efficient computational model is provided

by exploiting the Fokker-Planck equation of the underlying

stochastic process of the 3D directional random walk, as in

[2]. The Fokker-Planck equation, formulated as a PDE,

describes the time evolution of the probability density

function of particles following a 3D directional random

walk in space. The general form of the Fokker-Planck

equation in N-dimensions is given by Risken [48]:

@P

@tþXNi¼1

@

@Xi

�D

1ð Þið Þ X; tð ÞP

�¼XNi¼1

XNj¼1

@2

@Xi@Xj

�D

2ð Þijð Þ X; tð ÞP

�;

ð6Þ

where Dð1ÞðiÞ for i ¼ 1; . . . ; N are the drift terms and D

ð2ÞðijÞ are

the diffusion coefficients for the i; j ¼ 1; . . . ; N variables on

which the probability density function P depends. As

shown in [47], the drift vector and the diffusion matrix

associated with the 3D directional random walk are given by

Dð1Þ ¼

Dð1ÞðxÞ

Dð1ÞðyÞ

Dð1ÞðzÞ

Dð1Þð�Þ

Dð1Þð�Þ

2666666664

3777777775¼

sin � cos�sin � sin�

cos �00

266664

377775; ð7Þ

Dð2Þ ¼

Dð2ÞðxxÞ D

ð2ÞðxyÞ D

ð2ÞðxzÞ D

ð2Þðx�Þ D

ð2Þðx�Þ

Dð2ÞðyxÞ D

ð2ÞðyyÞ D

ð2ÞðyzÞ D

ð2Þðy�Þ D

ð2Þðy�Þ

Dð2ÞðzxÞ D

ð2ÞðzyÞ D

ð2ÞðzzÞ D

ð2Þðz�Þ D

ð2Þðz�Þ

Dð2Þð�xÞ D

ð2Þð�yÞ D

ð2Þð�zÞ D

ð2Þð��Þ D

ð2Þð��Þ

Dð2Þð�xÞ D

ð2Þð�yÞ D

ð2Þð�zÞ D

ð2Þð��Þ D

ð2Þð��Þ

266666666664

377777777775

¼ diag 0; 0; 0;�2�

2;�2�

2

" #:

ð8Þ

Replacing the above coefficients in (6) leads to the following

partial differential equation:

@P

@t¼ � sin � cos�

@P

@x� sin � sin�

@P

@y� cos �

@P

@z

þ�2�

2

@2P

@�2þ �

2�

2

@2P

@�2� 1

�P :

ð9Þ

Here, P is the probability that a particle passes throughstate ðx; y; z; �; �Þ in the course of its random walk from aseed region. The last term 1=� � P is added to the PDE toincorporate the exponential decay of particles in time. Thestochastic source and sink fields can now be computed bytime integration of the above PDE. We shall achieve this byintroducing a rotation invariant numerical scheme inSection 6.

4 EXTRA DRIFT TERMS IN THE PRESENCE OF

SPATIALLY DENSE MEASUREMENTS

In many pattern analysis problems, e.g., those involving 3Dtracking, motion capture, or diffusion MRI, orientation dataare available at dense spatial locations and this must be takeninto account into the computation of completion shapes. Inthe 3D directional random walk developed in Section 2, theorientation change in a particle’s motion is given bystochastic terms which are entirely Brownian. In such asetting, a larger orientation change has a lower probability ofoccurring. This characteristic leads to lower probabilityvalues being associated with states reached through a curvedpathway, relative to those reached with a straighter one.While this property can be suitable when bridging the gapbetween two edge segments with no other informativefeatures in between, it is desirable to remove the bias whenthe underlying data support larger orientation changes.

To resolve this issue and to take into account spatiallydense orientation information, we propose to add two extraangular advection terms to the stochastic process. Themagnitude of these angular drift terms should incorporatethe available local orientation information. An implementa-tion of these terms is studied to reconstruct diffusion MRIconnectivity in Section 7.1. The introduction of these extraterms modifies the motion component of the 3D directionalrandom walk as follows:

. The particle moves in 3D in the direction of itscurrent heading:

dx ¼ sin � cos�dt; dy ¼ sin � sin�dt;

dz ¼ cos �dt:ð10Þ

. The particle’s orientation is then changed by apply-ing two deviations, one in the osculating plane andone in the binormal plane. Unlike the originaldirectional random walk, a drift term is added tothe orientation changes d� and d� whose magnitudeis proportional to �� and ��. Should a local estimateof orientation be available at a voxel during thepropagation of completion fields, that informationcan be exploited to set �� and ��. In the context ofdiffusion MRI, as described in Section 7.1, thesevalues are chosen for each state based on its angulardifference from the available orientation data, whichprovides a local estimate of the direction of fibers.The angular diffusion terms �2

� and �2�, on the other


hand, govern the amount of random angular changeallowed at each step, contributing to the Browniancomponent of the directional random walk:

d� ¼ ��dtþ ��dWtð�Þ; d� ¼ ��dtþ ��dWtð�Þ;dWtð�Þ; dWtð�Þ � Nð0; dtÞ:

ð11Þ

As before, we also allow a certain fraction of particlesð1� e�1

�Þ to decay per unit time. The net effect of thestochastic differential equations given in (11) is to push theincoming direction toward the most likely orientation ateach voxel (when available) via the associated nonzero driftterms. This encourages the particles to move along thepathways supported by the already available denseorientation data. At the same time, the angular diffusionterms allow the particles to explore other possible trajec-tories by moving along other directions. This can accountfor other contributing random factors (such as Brownianmotion of molecules in diffusion MRI) and can compensatefor noise and other model-dependent inaccuracies.

4.1 Modified Fokker-Planck Equation

The angular drift terms added above modify the originalFokker-Planck equation as follows:

@P


@P


@P

@y� cos �

@P

@z

� ��@P

@��

@P

@�þ�2�

2

@2P

@�2þ �

2�

2

@2P

@�2� 1

�P :

ð12Þ

5 A MODEL WITH DIFFUSION COMPONENTS ON THE

ENTIRE STATE SPACE

It is straightforward to generalize the completion fieldalgorithm to model a 3D directional random walk whichalso includes diffusion components in the spatial domain.To do so, random perturbations are also added to thechange in a particle’s position ðx; y; zÞ. The differentialequations according to which the particle’s state is updatedat each time step are then given by

dx ¼ sin � cos�dtþ �xdWt xð Þ;dy ¼ sin � sin�dtþ �ydWt yð Þ;dz ¼ cos �dtþ �zdWt zð Þ;d� ¼ ��dtþ ��dWt �ð Þ; d� ¼ ��dtþ ��dWt �ð Þ;

dWt xð Þ; dWt yð Þ; dWt zð Þ; dWt �ð Þ; dWt �ð Þ � N 0; dtð Þ:ð13Þ

The corresponding Fokker-Planck equation describing theevolution of the probability density of a particle is thengiven by

@P


@P


@P

@y� cos �

@P

@z

þ �2x

2

@2P

@x2þ�2y

2

@2P

@y2þ �

2z

2

@2P

@z2

� ��@P

@��

@P

@�þ�2�

2

@2P

@�2þ �

2�

2

@2P

@�2� 1

�P:

ð14Þ

Current diffusion MRI modeling techniques lead to a singlediffusion ODF/FOD maximum for a single fiber, a fanningor a branching structure, making it difficult to distinguishbetween these three cases. Whereas the spatial diffusioncoefficients have been set to zero in the experiments in thispaper, the use of nonzero values could be helpful in therecovery of fanning/branching configurations by allowingfor particles to propagate not only in the direction of ODFmaxima but also in a range of possible directions, if there isevidence that such configurations are present in a region.

6 A ROTATIONALLY INVARIANT NUMERICAL

SCHEME

Given a region of interest as the initial source state or thefinal sink state, one needs to integrate the PDE given by(12) over time to compute the stochastic source or sinkfields. We accomplish this using operator splitting, whichdivides (12) into five PDEs, three of which are in theCartesian space, one in the spherical domain, and the finalone incorporates the decay term. Each of these PDEs canbe solved separately using an appropriate stable numericalscheme. Since the advection coefficients for the x, y, and z

coordinates are always smaller than 1ð� sin � cos�;

� sin � sin�;� cos �Þ, a simple explicit numerical method,such as upwind differencing, suffices. An upwind differ-encing method avoids transport errors by approximatingderivatives based on the direction of the underlyingphysical process.

The main challenge in the numerical estimation of (12) isto achieve rotation invariance in the solution of the PDEinvolving the spherical terms. Rotation invariance, in thecontext of the application to diffusion MRI, implies that ameasure of connectivity should not depend on the orienta-tion of the diffusion weighted image (DWI) in a givenglobal coordinate system. Although uniform sampling ofthe � and � parameters (the colatitude and azimuth inspherical coordinates) offers the possibility of solving thespherical PDE using a simple numerical scheme, such as theCrank-Nicholson method, such a sampling pattern suffersfrom the inherent problem of having singularities at thepoles. In particular, the sampling of the spherical surfaceends up being nonuniform since the sampling densityvaries with distance from the poles. A rotation invariantcomputation of (12) requires a uniform sampling of thespherical surface, but this in turn makes the application ofthe finite differencing methods to the spherical domain achallenge since the notion of the nearest neighbor in the �and � coordinates is no longer well defined.

In the section that follows, we introduce a new metho-dology for achieving rotation invariance in the solution ofthe spherical PDE underlying the stochastic completion fieldalgorithm. The key idea is to use a spherical harmonics basisto represent functions defined on a spherical surface [40].While the development in this paper is tailored to thespecific PDE of (12), the basic idea of solving PDEs ondomains with a spherical topology has been of significantinterest in the fields of astronomy, geophysics, and nuclearphysics [50] and can find applications in other computer


vision or medical imaging applications where 3D orientationdata are involved.

Notably, this SPH-based formulation is not linked to anyparticular sampling scheme. Thus, we choose a quasi-uniform point set obtained by an electrostatic repulsionapproach on the sphere, known as the minimum energypoint distribution [26]. It is important to note that there is noknown point set that achieves the analog of uniformsampling on a spherical surface as in euclidean space.However, our goal is to minimize the error associated withthe global properties of the sampling points.

6.1 Spherical Harmonics-Based Numerical Solution

SPHs, normally denoted by Y ml , are the spherical analog of

the Fourier transform basis defined for complex functionson the unit sphere. They form a complete orthonormalsystem of the space of square integrable functions on thesphere L2ðS2Þ. SPHs have been widely used in differentapplication areas dealing with data on a sphere, includinggeophysics, astrophysics, quantum mechanics, and ima-ging. Many algorithms have also been introduced in thediffusion MRI community which have adapted the SPHbasis to model the diffusion signal [32], [33].

Using this basis, any spherical function can be written asP ð�; �Þ ¼

P1l¼0

Plm¼�l c

ml Y

ml ð�; �Þ, where the expansion is

usually truncated at some order L. In this section, wedevelop an algorithm which uses SPH expansion of theprobability function to solve the spherical part of the PDEgiven in (12), i.e.:

@P

@t¼ � ��

@P

@��

@P

@�þ�2�

2

@2P

@�2þ �

2�

2

@2P

@�2: ð15Þ

We use the real form of the spherical harmonics since thefunction to be expanded is a real-valued probability densityfunction. The real SPHs are formulated in terms of thecomplex SPHs as follows:

YYml ¼

ffiffiffi2p

Re Y ml

� ��l � m < 0

Y 0l m ¼ 0ffiffiffi2p

Img Y ml

� �0 < m � l:

8<: ð16Þ

Given N data points on the sphere P ¼ fP1; . . . ; PNg andchoosing an approximation order of L, a linear least-squaresscheme can be used to solve for the unknown SPHcoefficients C ¼ fc0

0; c�11 ; c0

1; c11; . . . ; cLLg [51]. The linear

least-squares approach seeks to find the SPH coefficientswhich satisfy the following overdetermined linear system ofequations while minimizing the 2-norm of the error term atthe sample data points:

YY00 �1; �1ð Þ . . . YYL

L �1; �1ð Þ... . .

. ...

YY00 �N; �Nð Þ . . . YYL

L �N; �Nð Þ

264

375

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Y

�c0

0

..

.

cLL

264

375

|fflffl{zfflffl}C

¼P1

..

.

PN

264

375

|fflfflffl{zfflfflffl}P

: ð17Þ

The mean least-squares error estimation of the SPHcoefficients is computed using the pseudo-inverse of theSPH matrix:

YC ¼ P; C ¼ YþP ¼ YTY� ��1

YTP: ð18Þ

The SPH expansion together with (18) is next used torewrite (15) in terms of SPHs:

@P

@t¼ ��

@P

@��

@P

@�þ�2�

2

@2P

@�2þ �

2�

2

@2P

@�2

@P

@t¼XLl¼0

Xlm¼�l

cml tð Þ �� ; �ð Þ @YYml �; �ð Þ@�

�

� �� ; �ð Þ @YYml �; �ð Þ@�

þ�2� �; �ð Þ

2

@2YYml �; �ð Þ@�2

þ �2� �; �ð Þ

2

@2YYml �; �ð Þ@�2

:

ð19Þ

Note that in (19) the x, y,z-dependence of P is implicit sincethe spherical PDE is solved locally for each voxel,independently of the neighboring voxels. Equation (19)can be written simultaneously for all the sample points inmatrix form, leading to a linear system of ordinarydifferential equations (ODEs):

@P

@t¼ �M�Y� �M�Y� þ��Y�2 þ��Y�2

� �C

¼ �M�Y� �M�Y� þ��Y�2 þ��Y�2

� �Yþ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

D

P;

�� ¼ diag �� 1; �1ð Þ; . . . ; �� N; �Nð Þ �

;

�� ¼ diag �� 1; �1ð Þ; . . . ; �� N; �Nð Þ½ �;M� ¼ diag �� 1; �1ð Þ; . . . ; �� N; �Nð Þ

�;

M� ¼ diag �� 1; �1ð Þ; . . . ; �� N; �Nð Þ½ �;

ð20Þ

where

Y� ¼

@YY0

0 �1;�1ð Þ@� . . .

@YYL

L �1;�1ð Þ@�

..

. . .. ..

.

@YY0

0 �N ;�Nð Þ@� . . .

@YYL

L �N ;�Nð Þ@�

266664

377775;

Y� ¼

@YY0

0 �1;�1ð Þ@� . . .

@YYL

L �1;�1ð Þ@�

..

. . .. ..

.

@YY0

0 �N ;�Nð Þ@� . . .

@YYL

L �N ;�Nð Þ@�

26664

37775;

Y�2 ¼

@2YY0

0 �1;�1ð Þ@�2 . . .

@2YYL

L �1;�1ð Þ@�2

..

. . .. ..

.

@2YY0

0 �N ;�Nð Þ@�2 . . .

@2YYL

L �N ;�Nð Þ@�2

266664

377775;

Y�2 ¼

@2YY0

0 �1;�1ð Þ@�2 . . .

@2YYL

L �1;�1ð Þ@�2

..

. . .. ..

.

@2YY0

0 �N ;�Nð Þ@�2 . . .

@2YYL

L �N ;�Nð Þ@�2

26664

37775:

ð21Þ

In the above equations, the partial derivatives of Y ml with

respect to � and � at a point can be expressed as thecombination of SPHs of different phase factors m and of thesame order l. The obtained linear system of ODEs updatesthe probability values of all the states at a voxel simulta-neously and thus can be applied to any kind of samplingscheme. We use the Runge-Kutta-Fehlberg (RKF45) method,which is an adaptive step size numerical algorithm, to solve(20) in order to avoid too large or too small time steps.


6.2 The Overall Computational Model

The overall computational model is described in (22). Asmentioned before, a time splitting approach is employed tosolve the Fokker-Planck equation given by (12) in five steps.The three first steps use an upwind-differencing algorithmto solve the PDE in x, y, and z coordinates. The sphericalharmonics based formulation of the previous section isimplemented in the fourth step to solve the equation in thespherical domain. And the final step involves the multi-plication by an exponential to take the effect of particle’slifetime into account:

Ptþ1

5

x;y;z;�;� ¼ Ptx;y;z;�;�

� sin � cos�Ptx;y;z;�;� � Pt

x�1;y;z;�;� if sin � cos� > 0

Ptxþ1;y;z;�;� � Pt

x;y;z;�;� if sin � cos� < 0;

(

Ptþ2

5

x;y;z;�;� ¼ Ptþ1

5

x;y;z;�;�

� sin � sin�Ptþ1

5

x;y;z;�;� � Ptþ1

5

x;y�1;z;�;� if sin � sin� > 0

Ptþ1

5

x;yþ1;z;�;� � Ptþ1

5

x;y;z;�;� if sin � sin� < 0;

8<:

Ptþ3

5

x;y;z;�;� ¼ Ptþ2

5

x;y;z;�;�

� cos �Ptþ2

5

x;y;z;�;� � Ptþ2

5

x;y;z�1;�;� if cos � > 0

Ptþ2

5

x;y;zþ1;�;� � Ptþ2

5

x;y;z;�;� if cos � < 0;

8<:

Ptþ4

5x;y;z ¼ DP

tþ35

x;y;z;

P tþ1x;y;z;�;� ¼ e

�1�P

tþ45

x;y;z;�;�:

ð22Þ

Both the source field and the sink field are computed bytime integration of the probability density function,computed by (22), over time starting from the given sourceand sink regions:

P 0 x; y; z; �; �ð Þ ¼Z 1

0

dtP t x; y; z; �; �ð Þ

�Xt0

0

Pt x; y; z; �; �ð Þ:ð23Þ

Examining the set of (22), it is clear that the computation canbe performed locally for each state, which offers thepossibility of parallelizing the algorithm, resulting insignificant computational savings.

7 APPLICATION TO CONNECTIVITY MAPPING IN

DIFFUSION MRI

7.1 Local Selection of Parameters

In the application of the 3D stochastic completion field tofiber tractography, we are interested in those curves of leastenergy between a source and a sink region that are alsosupported by the underlying diffusion MRI data. Onesimple way to incorporate local fiber orientation informa-tion obtained from diffusion MRI acquisitions into thecompletion field framework is to employ a Bayesianapproach where the stochastic completion field is calculatedindependently of the local diffusion MRI FODs/ODFs. Thecompletion field obtained forms the prior probability of

passing through any state in the 3D volume in the processof reconstructing the possible trajectories between tworegions of interest. Subsequently, the probability densityfor the plausible fiber pathways is found by multiplicationof this prior distribution by the FOD/ODF probabilityprofiles, considered as the conditional probability. Themultiplication outputs the posterior probability of the initialcompletion field given the diffusion MRI data. Whereastheoretically simple, this approach requires empiricaladjustments of angular diffusion and advection parameters,and this emerges as a significant challenge without priorknowledge of the fiber pathways being estimated.

The Fokker-Planck equation developed in Section 6provides a more principled way to incorporate the localdiffusion MRI data. As (20) and (22) demonstrate, theproposed numerical scheme affords the possibility of localspecification of the angular advection and diffusion para-meters ��; ��; ��; �� and also the lifetime coefficient �, forevery state. This selection can be done based on thediffusion MRI data available throughout the volume.Empirical tuning of the parameters is no longer required,and this significantly extends the applicability of thestochastic completion field model to fiber tractographyand connectivity measurement.

For the purpose of our experiments, we have used theFODs calculated by the spherical deconvolution method ofAnderson [52]. Nevertheless, the completion field algorithmcan employ any other type of diffusion profile, irrespectiveof the actual method used to estimate it. The basic idea forlocal selection of parameters is to align the headingdirection of the particles with the FOD’s maxima and tothen introduce some perturbation around the headingdirection to account for noise and other sources ofuncertainty. To accomplish this, for all states whoseorientation is within a certain angular difference from theFOD maxima, �� and �� are set equal to the angulardifference between the closest FOD maximum and theirrespective orientation. In the current setting, �� and �� areset to some fixed value to model the effect of the underlyingnoise and inaccuracies. Our future goal is to set theseangular diffusion parameters according to the actual valuesof uncertainty measured at each voxel. Finally, the particle’slifetime � for each state is also chosen based on the angulardifference from the closet fiber maxima. That is, the lifetimecoefficient takes a maximum value, �Max, for those stateswhich are aligned with the available local fiber maxima. Avery small lifetime is assigned to those states far from thefiber maxima which is obtained by scaling �Max with a verysmall value. For other states, the lifetime reduces linearlywith the angular difference, reaching its maximum (�Max)for angular difference of zero. The value of �Max is chosenbased on the number of iterations required to get from thesource to the sink, and vice versa, which means that topropagate along a longer fiber tract, particles are assigned alarger lifetime. Such a selection criterion for �Max reducesthe length dependency of the output connectivity measures.

The strategy described above allows parameters to be setautomatically from the field of FODs. It supports smoothlyvarying fiber tracts while avoiding an excessive penalty forcurved fiber pathways. To compute the value of �Max, the


algorithm is run once using a large enough �Max. Followingthis initial run, the estimated number of required iterationsis used to set the value of �Max in a final second run.

7.2 A Measure of Connectivity

Most of the methods proposed thus far in the literature forquantification of white matter connectivity are based ontractography. Two of the most commonly used connectivitymeasures reported in the literature are: 1) voxel counting,which enumerates the number of times an ROI has beenreached from a seed region employing a Monte-Carlosimulation [27], [29], [53], and 2) the weakest link approach[24], [26], [55], where the lowest confidence value of all tractsegments along a reconstructed streamline specifies theconfidence in it. Confidence in each tract segment iscomputed based on certain criteria defined by the under-lying tracking process or the previously mentioned Monte-Carlo approach. While these connectivity measures havebeen used extensively in the literature, they have limitationsand are heuristic in nature. Furthermore, not all the relevantinformation along the reconstructed fiber tracts is incorpo-rated in a principled fashion. These approaches also sufferfrom the issue of being length dependent, where reducedconnectivity is observed with distance from the seed region.

We suggest a different notion of connectivity, which isgiven by the average of the probability values assigned toall possible states within the source and sink ROIs by thestochastic completion field. Denoting the final probabilitydensity function by P , the connectivity measure ismathematically defined as:

Connectivity ¼P

x;y;z2Src;SnkP

�

P� P x; y; z; �; �ð Þ

#states in all voxels of source & sink ROIs:

ð24Þ

These probability values have been calculated during theevolution of the probability density from the source and thesink regions and thus reflect the information accumulatedalong the connectivity pattern. While the local nature of theFokker-Planck-based algorithm provides the opportunity totune parameters in a data dependent fashion, the length-dependence issue is addressed by setting the maximumparticle lifetime �max based on the minimum number ofiterations required to bridge the gap between the sourceand sink regions. In the present work, the parameterselection process is performed by exploring the use of fiberODF maxima and their consistency with the local directionof heading. However, the local parameter settings can betailored to incorporate other measures such as the under-lying fiber fractional volume, fractional anisotropy (FA), orfiber tract radius if available.

7.3 Experimental Validation

In this section, we evaluate the performance of ouralgorithm by running a series of validation experiments.Qualitative and quantitative validations of our method areprovided using synthetic multitensor data and the MIC-CAI 2009 Fibre Cup phantom, which is made of small-diameter acrylic fibers [12]. We also compute connectivitypatterns on FOD reconstructions of HARDI measurementsacquired in vivo from a human brain. In the series ofvalidation experiments, we show that the computed

connectivity measures are consistent with the knownground truth. We compare the performance of thecompletion field algorithm to that of a standard probabil-istic fiber tractography algorithm based on residual boot-strapping, similar to that of [29]. Comparative results arealso provided between the connectivity measures com-puted by the completion field algorithm and the weakestlink approach. Our results show that the stochasticcompletion field outperforms the other (state of the art)method by providing more consistent measures across thevolume. Throughout our experiments, we use a quasi-uniform minimum energy label set of 100 unit directions,distributed isotropically over a hemisphere, as our set oforientations defining a particle’s state at each voxel.

In all the experiments performed, the probability valuescomputed by the completion field algorithm over the entirestate space are visualized by representing each state, i.e.,every location and orientation ðx; y; z; �; �Þ, by a stick. Tofurther facilitate the visualization of these probabilityvalues, they are first multiplied by a scalar and thentransformed by a sigmoid-like function due to their widerange. The transparency of every stick is then madeinversely proportional to the transformed probability ofthe corresponding state. As mentioned in the previoussection, the connectivity values for different source and sinkpairs are calculated as the average probability of all thestates within them.

7.3.1 Probabilistic Fiber Tractography Using the

Residual Bootstrap

Residual bootstrapping, in the context of high angulardiffusion MRI, is a widely used technique for nonpara-metric characterization of uncertainty in the estimateddiffusion profiles and fiber orientations [29], [54]. Thismethod can approximate probability density functions ofthe true fiber orientations, which can then be used in aprobabilistic framework to perform tractography. Thepotential of such an approach for reconstruction of whitematter fiber tracts has been demonstrated through varioustractography studies [28], [29], [36], [54]. We have thereforechosen to compare our method against a probabilistictractography method similar to the residual bootstrapalgorithm of Berman [29], which is explained in thefollowing paragraphs.

First, a residual bootstrapping scheme is implemented togenerate 1,000 diffusion weighted images. In a residualbootstrap approach, the acquired diffusion weighted signalis fit to an SPH basis, and for each iteration, the residualsfrom this SPH fit are added at random without replacementto the SPH profile to generate a new DWI. The randomplacement of the residuals reflects the noise characteristicsof the original acquisition [29]. For each DWI, the fiberorientation distributions are computed using sphericaldeconvolution and are subsequently used to extract thedirection of maximal diffusion at each voxel. The uncer-tainty profiles are then approximated by a Gaussiandistribution whose parameters are computed by matchingthe FOD maxima across iterations to obtain the mean FODmaxima and their associated variances. We also computethe occurrence rate of each FOD maximum, representingthe number of times a given direction has been observed.Tractography is performed by running standard streamline


tracking in a probabilistic setting. Given a user defined seedregion, tracking is performed iteratively, starting from theseed ROI, where at each step the direction of propagation ischosen randomly from within the uncertainty profile of theclosest FOD maximum.

To measure anatomical connectivity, a confidence valueis associated with each tract segment according to thepreviously computed Gaussian distribution at the corre-sponding voxel. In a final step, the weakest link approach isused to assign a confidence value to each streamline. Thisconfidence value is determined by choosing the support ofthat segment along the tract that is weakest. The con-nectivity index for each voxel to the seed region isdetermined by finding the maximum confidence value overall the streamlines connecting the seed ROI to that voxel. Inour experiments, the connectivity measure between a pairof source and sink ROIs is computed using the aboveapproach by running probabilistic tractography startingform both ROIs. The mutual measure of connectivitybetween the two ROIs is given by taking the average ofconnectivity indices obtained for sink ROI voxels startingfrom the source and vice versa.

7.3.2 Experiment on Synthetic Data

In this first series of experiments, we look at theconnectivity patterns and indices obtained from runningthe completion field algorithm on synthetic data. Theseexperiments are designed to evaluate specific features ofour algorithm, such as its robustness to noise and curvature.

Synthetic data are generated using Alexander et al.’smultitensor model [51]. For diffusion encoding directions, aquasi-uniform minimum energy point set of 100 directions ischosen, similar to the label set used to define the particle’s

state in the directional random walk. A crossing fiberconfiguration is also considered, where a curved pathwaycrosses a straight one. The synthetic data are created byplacing DTs along the fiber pathways where the majoreigenvector of tensors aligns with the tangent vector of thecurves at a given voxel. Partial volume averaging is alsoconsidered for voxels in crossing regions and backgroundvoxels are filled with isotropic tensors with eigenvaluesequal to 600 � 10�6 mm2

s . The eigenvalues for the anisotropicvoxels are chosen based on the desired fractional anisotropy.

Finally, we simulate the effect of noise by adding complexGaussian noise to the raw signal. To investigate the effect ofnoise and change in fractional anisotropy, synthetic datawere generated for two different values of FA (0.84 and 0.6)and three levels of noise (1, 35, and 10). The crossing fiber

configuration and the FODs computed for FA ¼ 0:84 andSNR ¼ 35 are demonstrated in Fig. 2.

To evaluate the performance of our algorithm, fourdifferent ROIs are chosen. The top left image in Fig. 3 showsthe chosen ROIs overlaid on the FA image of the syntheticdata. The completion field algorithm is run for four pairs ofROIs, i.e., AB, CD, BC, and BD, where the first two areassociated with true connections while the latter two are not.As an example, the completion fields obtained for thesepairs are illustrated in Fig. 3 for the same FA and SNR as inFig. 2. One can observe that although our algorithm connectsall four ROI pairs, the associated probability values for theBC and BD pairs are much lower than those for the AB andCD connections. The same behavior is observed as onevaries the level of noise and FA in the synthetic data.

We have investigated the behavior of the completion fieldalgorithm more thoroughly by looking at how the computedconnectivity measures change as the incorporated noise leveland the fractional anisotropy value of the synthetic data aremodified. Table 1 provides a summary of these connectivityindices, obtained for different ROI pairs with differentparameter settings. We observe that the connectivitymeasures associated with the curved and straight pathways


Fig. 2. Left: Crossing fiber configuration used to generate the syntheticdata. Right: FOD reconstruction for FA ¼ 0:84 and SNR ¼ 35.

Fig. 3. Steady-state probability distributions obtained by the completionfield for the synthetically generated DWI data with FA ¼ 0:84 andSNR ¼ 35. Top left: The FA image of the synthetic data together with theseed regions used. For visualization, every state is illustrated by a stick,whose transparency is inversely proportional to a sigmoid transformationof that state probability.

TABLE 1Connectivity Indices of Four ROI Pairs

for Different SNR Levels and FA Values

are of the same order for the whole range of parameter

settings. On the other hand, it can be observed that for highenough SNR (SNR ¼ 1; 35), that latter being a typical SNR

level for in vivo diffusion MRI, the connectivity indicesassociated with the true connections (AB and CD) are at least

an order of magnitude larger than those of the falseconnections. As the SNR level falls, true and false connec-

tions become less distinguishable. Moreover, comparing thevalues in the last three columns of Table 1 with those given in

the first three shows that, for lower values of FA, SNRdecrease affects the connectivity measures more consider-

ably since FOD maxima are modified to a greater extent bythe added noise. We conclude that as long as the SNR level iswithin an acceptable range for real data acquisition, the

completion field algorithm shows promise in differentiatingtrue connections from false ones.

7.3.3 Experiment on a Physical Phantom

We also test our proposed algorithm on the MICCAI 2009Fibre cup phantom by Poupon et al. [12], for which the

ground truth connections are known. The phantom is made

of hydrophobic acrylic fibers whose diameter is of the samescale as that of myelinated axons. Two diffusion MRIdatasets have been provided with different spatial resolu-tions, of which we have selected the higher resolutionimage (3 � 3 � 3 mm3) with an associated b-value of1;500 s=mm2.

The steady-state probability densities output by thecompletion field algorithm for eight different source andsink region pairs are shown in Fig. 4. Based on the groundtruth, it is known that the first five pairs (a)-(e) areassociated with a true fiber tract, while there is no realtract associated with the latter three (f)-(h). The connectiv-ity measures assigned to each ROI pair are summarized inTable 2. For the purpose of comparison, Table 2 alsoprovides the connectivity indices output by the weakestlink approach and the probabilistic tractography algorithmexplained in Section 7.3.1.

The connectivity indices presented in Table 2 demon-strate the failure of the probabilistic tractography method torecover all the existing connections and the inability of theweakest link measure to provide consistent connectivityindices across different ROI pairs. The probabilistic boot-strap algorithm did not find any connection between ROIpairs (f), (g), and (h) and consequently the weakest linkapproach correctly assigns zero connectivity to these cases.However, the tractography fails to recover the nearlystraight connectivity pattern between the source and sinkROIs of (b) and (e), which leads to a zero connectivity indexfor these pairs too. The fact that the connectivity measuresobtained are not consistent across true and false connectionsmakes any kind of inference based on these measuresinaccurate. It is important to note that this failure is due todeficiencies in both the connectivity measure and thetractography algorithm. Nevertheless, both of these meth-ods are among the state-of-the-art algorithms for tracto-graphy and measuring connectivity.

In contrast, the connectivity values computed by the 3Dstochastic completion field algorithm are very promising.As can be observed form the first row in Table 2, thecompletion field algorithm finds a connectivity pattern forall given pairs of ROIs. However, the connectivity valuesassigned to the (f), (g), and (h) ROI pairs are much lower (atleast two orders of magnitude) than those computed for thefirst five pairs, where the latter represent true connections.Additionally, among the five connectivity measures asso-ciated with true connections, those obtained for cases (d)and (e) are an order of magnitude lower that the other three.This is mainly due to the complex configuration of fibertracts at the top part of the phantom. The computation timefor each ROI pair on the phantom, with state spacedimension of 64� 64� 3� 200, was about 15 minutes onaverage, running on four 2.8 GHz Intel Xeon processors.


Fig. 4. Visualization of the completion fields obtained for a set of ROIpairs on the physical phantom. Top-Left: Mean diffusivity image and theROIs used. Every state is illustrated by a stick whose transparency isinversely proportional to a sigmoid transformation of the probability ofthat state.

TABLE 2Connectivity Indices Assigned to Eight Different Seed Region Pairs of the Physical Phantom [12]

7.3.4 Experiment on Human Brain Data

We now present experiments in which the 3D completionfield algorithm is applied to in vivo human brain HARDIdata. The diffusion weighted images were acquired in vivofrom a healthy subject on a Siemens 3T Trio MR scannerusing an 8-channel phased-array head coil. Diffusionencoding was achieved using a single-shot spin-echoplanar sequence with twice-refocused balanced diffusionencoding gradients. A dataset designed for high-angularresolution reconstruction was acquired with 99 diffusionencoding directions, 2 mm isotropic voxel size, 63 slices,b ¼ 3;000 s=mm2, TE ¼ 121 ms, TR ¼ 11:1 s, and GRAPPAparallel reconstruction. The series of experiments wereperformed on FOD reconstructions of the HARDI data ofthe human brain. We also only considered voxels with afractional anisotropy value higher than 0.1.

We have examined the performance of the completionfield algorithm on a selection of ROI pairs chosen fromsome of the major fiber tracts in the brain. This choice ofROIs, for which the associated anatomical connectivity isknown, allows us to assess the validity of the obtainedresults. Out of the four chosen ROI pairs, one is notassociated with any known connection. (The ROIs areactually located in the medial part of the corpus callosum,one anterior and one posterior. The slice shown below theROIs is chosen such that the reconstructed completion fieldcan be observed.) The other three pairs are chosen as part ofthe genu of the corpus callosum, the splenium of the corpuscallosum, and finally as part of the cortico-spinal tract. Onaverage, the computation for each ROI pair took about aday, with a state space dimension of 128 � 128 � 63 � 200,while running on four 2.8 GHz Intel Xeon processors. Fig. 5shows the computed completion fields for the four sets ofsource and sink regions. For reference, the results are

overlaid on the T1-weighted image and the source and sinkregions are shown with red and green blobs. The top rightand the two bottom subfigures in Fig. 5 illustrate trueconnections, while that on the top left shows the con-nectivity pattern of the ROI pair for which a connectionshould not exist. Table 3 also shows the associatedconnectivity measures for the four ROI pairs. As mentionedearlier, the probability value of every state is multiplied by ascalar and then transformed by a sigmoid function to aid invisualization, but this also leads to a reduction in transpar-ency contrast between high and low probability states. Thedifference in the degree of connectivity is reflected in theconnectivity measures which demonstrate that the comple-tion field algorithm is capable of distinguishing trueconnections from false ones. The connectivity measureassociated with the ROI pair located in the cortico-spinaltract is higher than that for the other two connections.Referring to Fig. 5, this higher connectivity strength isconsistent with the presence of more trajectories connectingthese two regions. In all cases the connectivity patternsobtained for the true connections are consistent with theknown anatomy of the underlying regions.

8 CONCLUSION

The 3D stochastic completion field extends Williams andJacobs’ curve completion model in computer vision [1] toprovide a probabilistic representation of the connectivitypattern between two given regions of interest. We haveformulated the computation of the 3D stochastic comple-tion in an efficient local parallel computational framework,exploiting the Fokker-Planck equation of the underlying3D directional random walk. Further, we have introduceda rotation invariant numerical scheme by using a sphericalharmonics basis to represent functions defined on the2-sphere. Adding extra angular drift terms allows forincorporation of dense orientation data into the algorithm.This extension paves the way for applications of thestochastic completion field to computer vision and patternanalysis applications in 3D where dense orientation dataarises. For the particular application we have focused onin this paper, the local nature of the proposed computa-tional model allows diffusion MRI measurements to bedirectly incorporated.

We have demonstrated the potential of this algorithmfor distinguishing real connections from false ones indiffusion MRI through experiments performed on syn-thetic data and on data from a physical phantom. Weobserve that as long as the SNR of the acquired data iswithin an acceptable range, the obtained connectivitymeasures can distinguish true connections from false ones,while the associated completion field renders a reliableprobabilistic representation of the underlying connectivitypatterns. The same observations hold for the physical


Fig. 5. Visualization of the completion fields obtained for four differentsets of ROI pairs. The top left figure shows a pair with no underlyingphysical connection while the other three correspond to regions with realconnections in the brain. Top row, from left to right: False connection,splenium. Bottom row, from left to right: Genu of corpus callosum,cortico-spinal tract.

TABLE 3Connectivity Indices for Four ROI Pairs of Human Data

phantom, where ground-truth knowledge confirms thevalidity of the computed connectivity measures. We havealso demonstrated that the proposed measure qualitativelyoutperforms the weakest link index. Additionally, althoughno ground truth is available for the human brain dataset,the connectivity patterns obtained are consistent with theknown anatomy of the brain.

In future work, we plan to extend this approach tointegrate measurements of the underlying noise anduncertainty in the dataset into the framework. In applicationto diffusion MRI, this can be done through proper selectionof angular diffusion parameters at each voxel, based on theassociated noise and uncertainty levels. The use of comple-tion fields for probabilistic segmentation of the cortical-subcortical regions in the brain and for the classification ofcontrols versus patients suffering from white matter dis-eases are additional fruitful directions to explore.

ACKNOWLEDGMENTS

The authors would like to thank Jennifer Campbell andBruce Pike for many helpful discussions and for providingthe fiber ODF data for our experiments. This work wassupported by grants from NSERC and FQRNT Quebec.

REFERENCES

[1] L. Williams and D. Jacobs, “Stochastic Completion Fields: ANeural Model of Illusory Contour Shape and Salience,” NeuralComputation, vol. 9, pp. 837-858, 1997.

[2] L. Williams and D. Jacobs, “Local Parallel Computation ofStochastic Completion Fields,” Neural Computation, vol. 9,pp. 859-881, 1997.

[3] G. Kanizsa, Organization in Vision: Essays on Gestalt Perception,series Praeger Special Studies. Praeger, 1979.

[4] S. Petry and G. Meyer, The Perception of Illusory Contours. Springer-Verlag, 1987.

[5] S. Ullman, “Filling-In the Gaps: The Shape of Subjective Contoursand a Model for Their Generation,” Biological Cybernetics, vol. 25,no. 1, pp. 1-6, 1976.

[6] D. Mumford, “Elastica and Computer Vision,” Algebraic Geometryand Its Applications. chapter 31, pp. 491-506, Springer-Verlag, 1994.

[7] B. Kimia, I. Frankel, and A. Popescu, “Euler Spiral for ShapeCompletion,” Int’l J. Computer Vision, vol. 54, pp. 159-182, 2003.

[8] P. Parent and S.W. Zucker, “Trace Inference, Curvature Consis-tency and Curve Detection,” IEEE Trans. Pattern Analysis andMachine Intelligence, vol. 11, no. 8, pp. 823-889, Aug. 1989.

[9] G. Guy and G. Medioni, “Inferring Global Perceptual Contoursfrom Local Features,” Int’l J. Computer Vision, vol. 20, pp. 113-133,1993.

[10] A. Shashua and S. Ullman, “Structural Saliency: The Detection ofGlobally Salient Structures Using a Locally Connected Network,”Proc. Second IEEE Int’l Conf. Computer Vision, pp. 321-327. 1988,

[11] J. Zweck and L.R. Williams, “Euclidean Group InvariantComputation of Stochastic Completion Fields Using Shiftable-Twistable Functions,” J. Math. Imaging and Vision, vol. 21, pp. 135-154, 2004.

[12] C. Poupon, B. Rieul, I. Kezele, M. Perrin, F. Poupon, and J.Mangin, “New Diffusion Phantoms Dedicated to the Study andValidation of High-Angular-Resolution Diffusion Imaging(HARDI) Models,” Magnetic Resonance in Medicine, vol. 60,pp. 1276-1283, 2008.

[13] D. LeBihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, andM. Laval-Jeantet, “MR Imaging of Intravoxel Incoherent Motions:Application to Diffusion and Perfusion in Neurologic Disorders,”Radiology, vol. 161, pp. 401-407, Nov. 1986.

[14] M. Catani, R. Howard, S. Pajevic, and D. DK, “Virtual in VivoInteractive Dissection of White Matter Fasciculi in the HumanBrain,” Neuroimage, vol. 17, pp. 77-94, 2002.

[15] S. Frey, J. Campbell, G. Pike, and M. Petrides, “Dissociating theHuman Language Pathways with High Angular ResolutionDiffusion Fiber Tractography,” J. Neuroscience, vol. 28, pp. 11435-11444, 2008.

[16] M. Catani, D. Jones, and D. ffytche, “Perisylvian LanguageNetworks of the Human Brain,” Annals of Neurology, vol. 57,pp. 8-16, 2005.

[17] A. Anwander, M. Tittgemeyer, D. von Cramon, A. Friederici, andT. Knosche, “Connectivity-Based Parcellation of Broca’s Area,”Cerebral Cortex, vol. 17, pp. 816-825, 2007.

[18] M. Rushworth, T. Behrens, and H. Johansen-Berg, “ConnectionPatterns Distinguish 3 Regions of Human Parietal Cortex,”Cerebral Cortex, vol. 16, pp. 1418-1430, 2006.

[19] A. Venkataraman, Y. Rathi, M. Kubicki, C. Westin, and P. Golland,“Joint Generative Model for fMRI/DWI and Its Application toPopulation Studies,” Proc. 13th Int’l Conf. Medical Image Computingand Computer-Assisted Intervention, pp. 191-199, Sept. 2010.

[20] M. Koch, D. Norris, and M. Hund-Georgiadis, “An Investigationof Functional and Anatomical Connectivity Using MagneticResonance Imaging,” Neuroimage, vol. 16, pp. 241-250, 2002.

[21] D. LeBihan, R. Turner, and P. Douek, “Is Water DiffusionRestricted in Human Brain White Matter? An Echo-Planar NMRImaging Study,” NeuroReport, vol. 4, pp. 887-890, July 1993.

[22] S. Mori, B. Crain, V. Chacko, and P. van Zijl, “Three DimensionalTracking of Axonal Projections in the Brain by MagneticResonance Imaging,” Annals of Neurolology, vol. 45, pp. 265-269,1999.

[23] P. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi, “InVitro Fiber Tractography Using DT-MRI Data,” Magnetic Reso-nance in Medicine, vol. 44, pp. 625-632, 2000.

[24] G. Parker, C. Wheeler-Kingshott, and G. Barker, “EstimatingDistributed Anatomical Connectivity Using Fast Marching Meth-ods and Diffusion Tensor Imaging,” Trans. Medical Imaging,vol. 21, pp. 505-512, 2002.

[25] C. Poupon, C. Clark, V. Frouin, J. Regis, I. Bloch, D. LeBihan, andJ. Mangin, “Regularization of Diffusion-Based Direction Maps forthe Tracking of Brain White Matter Fascicles,” NeuroImage, vol. 12,pp. 184-195, 2000.

[26] J. Campbell, K. Siddiqi, V. Rymar, A. Sadikot, and B. Pike, “Flow-Based Fiber Tracking with Diffusion Tensor q-Ball Data: Valida-tion and Comparison to Principal Diffusion Direction Techni-ques,” NeuroImage, vol. 27, pp. 725-736, 2005.

[27] P. Behrens et al., “Characterization and Propagation of Uncer-tainty in Diffusion-Weighted MR Imaging,” Magnetic Resonance inMedicine, vol. 50, pp. 1077-1088, 2003.

[28] D. Jones and C. Pierpaoli, “Confidence Mapping in DiffusionTensor Magnetic Resonance Imaging Tractography Using aBootstrap Approach,” Magnetic Resonance Medicine, vol. 53,pp. 1143-1149, 2005.

[29] J. Berman, S. Chung, P. Mukherjee, C. Hess, E. Han, and R.Henrya, “Probabilistic Streamline q-Ball Tractography Using theResidual Bootstrap,” NeuroImage, vol. 39, pp. 215-222, 2008.

[30] P. Basser, J. Matiello, and D.L. Bihan, “MR Diffusion TensorSpectroscopy and Imaging,” Biophysical J., vol. 66, pp. 259-267, Jan.1994.

[31] D.S. Tuch, “Q-Ball Imaging,” Magnetic Resonance in Medicine,vol. 52, pp. 1358-1372, Dec. 2004.

[32] M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche,“Regularized, Fast, and Robust Analytical q-Ball Imaging,”Magnetic Resonance Medicine, vol. 58, pp. 497-510, 2007.

[33] J. Tournier, F. Calamante, D. Gadian, and A. Connelly, “DirectEstimation of the Fiber Orientation Density Function fromDiffusion-Weighted MRI Data Using Spherical Deconvolution,”NeuroImage, vol. 23, pp. 1176-1185, Sept. 2004.

[34] S. Mori and P.V. Zijl, “Fiber Tracking: Principles and StrategiesaTechnical Review,” NMR in Biomedicine, vol. 15, pp. 468-480, 2002.

[35] D. Jones, A. Simmons, S. Williams, and M. Horsfield, “Non-Invasive Assessment of Axonal Fiber Connectivity in the HumanBrain Via Diffusion Tensor MRI,” Magnetic Resonance in Medicine,vol. 42, pp. 37-41, 1999.

[36] P. MomayyezSiahkal, J. Campbell, P. Savadjiev, G. Pike, and K.Siddiqi, “Beyond Crossing Fibres: Probabilistic Tractography ofComplex Subvoxel Fibre Geometries,” Proc. MICCAI WorkshopDiffusion Modelling and the Fibre Cup, pp. 81-92, Sept. 2009.

[37] J.G. Malcolm, M.E. Shenton, and Y. Rathi, “Filtered Multi-TensorTractography,” IEEE Trans. Medical Imaging, vol. 29, no. 9,pp. 1664-1675, Sept. 2010.


[38] P. Fillard, C. Poupon, and J. Mangin, “A Novel GlobalTractography Algorithm Based on an Adaptive Spin GlassModel,” Proc. 12th Int’l Conf. Medical Image Computing andComputer-Assisted Intervention, pp. 927-934, Sept. 2009.

[39] S. Jbabdi, P. Bellec, R. Toro, J. Daunizeau, M. Pelegrini-Issac, andH. Benali, “Accurate Anisotropic Fast Marching for Diffusion-Based Geodesic Tractography,” Int’l J. Biomedical Imaging,vol. 2008, pp. 1-12, 2008.

[40] P. Momayyez and K. Siddiqi, “Rotation Invariant CompletionFields for Mapping Diffusion-MRI Connectivity,” Proc. 22nd Int’lConf. Information Processing in Medical Imaging, 2011.

[41] P. Batchelor, D. Hill, F. Calamante, and D. Atkinson, “Study ofConnectivity in the Brain Using the Full Diffusion Tensor fromMRI,” Proc. 17th Int’l Conf. Information Processing in MedicalImaging, pp. 121-133, 2001.

[42] L. O’Donnell, S. Haker, and C. Westin, “New Approaches toEstimation of White Matter Connectivity in Diffusion Tensor MRI:Elliptic PDEs and Geodesics in a Tensor-Warped Space,” Proc.Fifth Int’l Conf. Medical Image Computing and Computer-AssistedIntervention, pp. 459-466, Sept. 2002.

[43] M. Pechaud, M. Descoteaux, and R. Keriven, “Brain ConnectivityUsing Geodesics in Hardi,” Proc. 12th Int’l Conf. Medical ImageComputing and Computer-Assisted Intervention, pp. 482-489, Sept.2009.

[44] N. Hageman, A. Toga, K. Narr, and D. Shattuck, “A DiffusionTensor Imaging Tractography Algorithm Based on Navier-StokesFluid Mechanics,” IEEE Trans. Medical Imaging, vol. 28, no. 3,pp. 348-360, Mar. 2009.

[45] P. Fletcher, R. Tao, W. Jeong, and R. Whitaker, “A VolumetricApproach to Quantifying Region-to-Region White Matter Con-nectivity in Diffusion Tensor MRI,” Proc. 20th Int’l Conf. Informa-tion Processing in Medical Imaging, pp. 346-358, 2007.

[46] E. Pichon, C. Westin, and A. Tannenbaum, “A Hamilton-Jacobi-Bellman Approach to High Angular Resolution DiffusionTractography,” Proc. Eighth Int’l Conf. Medical Image Computingand Computer-Assisted Intervention, pp. 180-187, 2005.

[47] P. Momayyez and K. Siddiqi, “3D Stochastic Completion Fieldsfor Fiber Tractography,” Proc. IEEE Workshop Math. Methods inBiomedical Image Analysis, pp. 178-185, June 2009.

[48] H. Risken, The Fokker-Planck Equation: Methods of Solution andApplications. Springer, 1996.

[49] P. Savadjiev, J. Campbell, and B.P.K. Siddiqi, “3D CurveInferencefor Diffusion MRI Regularization and Fibre Tractogra-phy,” Medical Image Analysis, vol. 10, pp. 799-813, Aug. 2006.

[50] B. Fornberg and D. Merrill, “Comparison of Finite Difference andPseudospectral Methods for Convective Flow over a Sphere,”Geophysical Research Letters, vol. 24, pp. 3245-3248, 1997.

[51] D. Alexander, G. Barker, and S. Arridge, “Detection and Modelingof Non-Gaussian Apparent Diffusion Coefficient Profiles inHuman Brain Data,” Magnetic Resonance in Medicine, vol. 48,pp. 331-340, 2002.

[52] A. Anderson, “Measurement of Fiber Orientation DistributionsUsing High Angular Resolution Diffusion Imaging,” MagneticResonance in Medicine, vol. 54, no. 5, pp. 1194-1206, 2005.

[53] T. Yo, A. Anwander, M. Descoteaux, P. Fillard, C. Poupon, and T.Knosch, “Quantifying Brain Connectivity: A Comparative Tracto-graphy Study,” Proc. 12th Int’l Conf. Medical Image Computing andComputer-Assisted Intervention, pp. 886-893, Sept. 2009.

[54] B. Jeurissen, A. Leemans, D.K. Jones, J.D. Tournier, and J. Sijbers,“Probabilistic Fiber Tracking Using the Residual Bootstrap withConstrained Spherical Deconvolution,” Human Brain Mapping,vol. 32, pp. 461-479, 2011.

[55] O. Ciccarelli, A.T. Tossy, G.J. Parker, C.A. Wheeler-Kingshott, G.J.Barker, and P.A. Boulby, “From Diffusion Tractography toQuantitative White Matter Tract Measures: A ReproducibilityStudy,” NeuroImage, vol. 18, pp. 348-359, 2003.

Parya MomayyezSiahkal received the BScdegree in electrical engineering from SharifUniversity of Technology, Iran, in 2004, theMEng degree in signal processing from McGillUniversity in 2007, and the PhD degree incomputer science from the School of ComputerScience, McGill University, Canada, in 2012.Her fields of interest include shape analysis,medical imaging, computer vision, and patternrecognition.

Kaleem Siddiqi received the BS degree fromLafayette College in 1988 and the MS and PhDdegrees from Brown University in 1990 and1995, respectively, all in the field of electricalengineering. He is currently a professor, WilliamDawson scholar, and an associate directorresearch of the School of Computer Science atMcGill University. He is also a member ofMcGill’s Centre for Intelligent Machines. Beforemoving to McGill in 1998, he was a postdoctoral

associate in the Department of Computer Science at Yale University(1996-1998) and held a visiting position in the Department of ElectricalEngineering at McGill University (1995-1996). His research interestsinclude the areas of computer vision, image analysis, and medicalimaging. He is a member of the Phi Beta Kappa, Tau Beta Pi, and EtaKappa Nu. He is a senior member of the IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.


Documents

3D Stochastic Completion Fields for Mapping Connectivity in Diffusion MRI