NeuroImage 101 (2014) 750–764

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NeuroImage

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Non-Negative Spherical Deconvolution (NNSD) for estimation of fiberOrientation Distribution Function in single-/multi-shell diffusion MRI

Jian Cheng a,⁎, Rachid Deriche b, Tianzi Jiang c, Dinggang Shen a, Pew-Thian Yap a,⁎a Department of Radiology and BRIC, The University of North Carolina at Chapel Hill, USAb Athena Project-Team, INRIA Sophia Antipolis-Méditerranée, Francec Center for Computational Medicine, LIAMA, Institute of Automation, Chinese Academy of Sciences, China

⁎ Corresponding authors.E-mail addresses: jian_cheng@med.unc.edu (J. Cheng),

http://dx.doi.org/10.1016/j.neuroimage.2014.07.0621053-8119/© 2014 Elsevier Inc. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Accepted 28 July 2014Available online 7 August 2014

Keywords:Spherical deconvolutionDiffusion MRIFiber Orientation Distribution FunctionNon-negativity constraintSpherical harmonics

Spherical Deconvolution (SD) is commonly used for estimating fiber Orientation Distribution Functions (fODFs)from diffusion-weighted signals. Existing SD methods can be classified into two categories: 1) ContinuousRepresentation based SD (CR-SD), where typically Spherical Harmonic (SH) representation is used for convenientanalytical solutions, and 2) Discrete Representation based SD (DR-SD), where the signal profile is represented bya discrete set of basis functions uniformly oriented on the unit sphere. A feasible fODF should be non-negativeand should integrate to unity throughout the unit sphere S2. However, to our knowledge, most existing SH-basedSD methods enforce non-negativity only on discretized points and not the whole continuum of S2. MaximumEntropy SD (MESD) andCartesian Tensor FiberOrientationDistributions (CT-FOD) are the only SDmethods that en-sure non-negativity throughout the unit sphere. They are however computational intensive and are susceptible toerrors caused by numerical spherical integration. Existing SDmethods are also known to overestimate the numberof fiber directions, especially in regions with low anisotropy. DR-SD introduces additional error in peak detectionowing to the angular discretization of the unit sphere. This paper proposes a SD framework, called Non-NegativeSD (NNSD), to overcome all the limitations above. NNSD is significantly less susceptible to the false-positivepeaks, uses SH representation for efficient analytical spherical deconvolution, and allows accurate peak detectionthroughout the whole unit sphere. We further show that NNSD and most existing SD methods can be extendedto work on multi-shell data by introducing a three-dimensional fiber response function. We evaluated NNSD incomparison with Constrained SD (CSD), a quadratic programming variant of CSD, MESD, and an L1-norm regular-ized non-negative least-squares DR-SD. Experiments on synthetic and real single-/multi-shell data indicate thatNNSD improves estimation performance in terms of mean difference of angles, peak detection consistency, and an-isotropy contrast between isotropic and anisotropic regions.

© 2014 Elsevier Inc. All rights reserved.

Introduction

Diffusion MRI (dMRI) non-invasively reveals the microstructure ofwhite matter by capturing the diffusion patterns of water molecules.The most widely used dMRI approach, Diffusion Tensor Imaging (DTI),captures one fiber direction per voxel and is incapable of describingcomplex diffusion processes due to its Gaussian diffusion assumption(Johansen-Berg and Behrens, 2009). In view of this, many High AngularResolution Diffusion Imaging (HARDI) (Tuch et al., 2002)methods havebeen developed in recent years to characterize non-Gaussian diffusionand compute quantities such as the Ensemble Average Propagator(EAP) (Wedeen et al., 2005; Descoteaux et al., 2010; Cheng et al.,

ptyap@med.unc.edu (P.-T. Yap).

2010b; Özarslan et al., 2009; Cheng et al., 2012), diffusion OrientationDistribution Function (dODF) (Tuch, 2004; Hess et al., 2006;Descoteaux et al., 2007; Aganj et al., 2010; Cheng et al., 2010a; Chenget al., 2012), and fiber Orientation Distribution Function (fODF)(Tournier et al., 2004; Tournier et al., 2007; Alexander, 2005; Jian andVemuri, 2007; Dell'Acqua et al., 2007; Dell'Acqua et al., 2010;Landman et al., 2012; Weldeselassie et al., 2012).

Spherical deconvolution (SD) has been shown to be effective for es-timating the fODF by assuming that the measured diffusion-weightedsignal can be obtained via spherically convolving a latent fODF with afiber response function estimated from voxels known to be traversedby a single fascicle (Tournier et al., 2004, 2007; Jian and Vemuri, 2007;Johansen-Berg and Behrens, 2009). The fODF can hence be recoveredvia an inverse problem by deconvolving the signal with the estimatedfiber response function. The local peaks (maxima) of the fODF give thecorresponding fiber directions. SD methods can be classified into two

751J. Cheng et al. / NeuroImage 101 (2014) 750–764

categories, 1) Continuous Representation based SD (CR-SD), which isnormally based on the Spherical Harmonic (SH) basis (Tournier et al.,2004, 2007; Anderson, 2005) and 2) Discrete Representation based SD(DR-SD), which is based on a discrete mixture of rotated versions ofthe fiber response function (Jian and Vemuri, 2007; Dell'Acqua et al.,2007, 2010; Landman et al., 2012).

Existing SDmethods in both continuous and discrete representationcategories share some common limitations. First, they often result infalse-positive fiber directions (Tournier et al., 2004, 2007; Alexander,2005; Johansen-Berg and Behrens, 2009; Jian and Vemuri, 2007;Landman et al., 2012; Weldeselassie et al., 2012), especially in low-anisotropy gray matter and cerebrospinal fluid (CSF) regions. Second,they normally fall short in ensuring that the estimated fODF is a properprobability density function, because non-negativity and unit integralthroughout the unit sphere are not explicitly enforced. Most SDmethods, including the popular Constrained SD (CSD) (Tournier et al.,2007) and all DR-SD methods (Jian and Vemuri, 2007; Dell'Acquaet al., 2007; Dell'Acqua et al., 2010; Landman et al., 2012), considernon-negativity only on discretized points but not the whole continuumof the unit sphere S2. To the best of our knowledge, Maximum EntropySD (MESD) (Alexander, 2005) and Cartesian Tensor Fiber OrientationDistributions (CT-FOD) (Weldeselassie et al., 2010; Weldeselassieet al., 2012) are the only existing methods that ensure non-negativitythroughout S2. However, they are computationally inefficient and relyon the error-prone process of numerical spherical integration. Ad-hocnormalization is also employed in these methods to obtain fODFs withunit integral. Some methods estimate continuously non-negativedODFs (Schwab et al., 2012; Cheng et al., 2012; Krajsek and Scharr,2012) and EAPs (Cheng et al., 2012) using eigenvalue distribution ofspherical functions and square root representation. But to our knowl-edge, noneof thesemethods has been proposed to estimate continuous-ly non-negative fODF in a SD framework. Third, for estimation of thefODF with reasonable accuracy, DR-SD methods (Jian and Vemuri,2007; Dell'Acqua et al., 2007, 2010; Landman et al., 2012) require a sig-nificant amount of rotated fiber response functions along directions thatare distributed densely on the unit sphere, significantly increasing thedimensionality and the time cost of the optimization problem. Further-more, since for DR-SD methods (Jian and Vemuri, 2007; Dell'Acquaet al., 2007, 2010; Landman et al., 2012) the local peaks (maxima) ofthe fODF are detected fromdiscretized points on the unit sphere, the an-gular resolution is limited.

To our knowledge, existing SD methods deal only with single-shelldata (i.e., single b-value) and do not consider the radial component ofdiffusion. With advances in dMRI, multi-shell data are increasinglyavailable (e.g., Human Connectome Project (HCP) (Sotiropoulos et al.,2013)). For example, the HCP Q1 data come with three b-values (b =1000/2000/3000 s/mm2). Recent estimation methods such as theball-stick model (Jbabdi et al., 2012), Q-Ball Imaging (Aganj et al.,2010), and other multi-shell HARDI methods (Assemlal et al., 2011;Cheng et al., 2010a, 2010b; Descoteaux et al., 2010; Özarslan et al.,2009) demonstrated that ODF, EAP as well as fiber directions can be es-timated with greater accuracy from multi-shell data compared withsingle-shell data. However, there is currently no existing work on howto perform SD on multi-shell data.

In this paper, we propose a method called Non-Negative SphericalDeconvolution (NNSD) to estimate fODF from both single- and multi-shell data.

The main contributions of this paper are summarized as follows:

• NNSD is the first SH-based SD method to guarantee non-negativitythroughout S2, not only on discretized points on the unit sphere, as inCSD (Tournier et al., 2007). In NNSD, non-negativity is achieved byrepresenting the square root of the fODF as a linear combination of SHbasis functions. Compared with non-SH methods like MESD andCT-FOD, which also guarantee non-negativity on the whole S2, NNSDis significantly faster due to the use of closed-form expression for

spherical convolution. Compared with CSD which suppressesnegative values in discrete samples using iteratively re-weightedregularization, the non-negativity constraint in NNSD is built into itsfODF representation andhenceNNSDworkswell evenwithout any reg-ularization.

• In addition to the non-negativity constraint, NNSD reduces spuriouspeaks by implementing Riemannian gradient descent with an adaptivestopping condition. As a result, the anisotropy values of the fODFsestimated by NNSD in gray and white matter regions exhibit a largecontrast. Existing SD methods result in many false-positive peaks andhence high anisotropy in regions that are less anisotropic.

• Compared with traditional single-shell methods, we show that multi-shell data can be used for fODF estimation with greater robustness.

• Performance evaluation using real data is difficult due to the lack ofground truth. We propose in this paper a measure called peak consis-tency (PC) for quantitative fODF evaluation without exact knowledgeof the ground truth.

Part of thiswork has been reported in our conferencepaper (Cheng et al.,2013a, 2013b). Herein, we provide additional examples, results, deriva-tions, and insights that are not part of this conference publication.

The rest of the paper is organized as follows. SD theory and algorithmsare reviewed in Section Theory. Section Spherical deconvolution revisitedprovides an overview of existing SD methods using single-/multi-shelldata, i.e. CSD (Tournier et al., 2007), a variant of CSD based on quadraticprogramming, MESD (Alexander, 2005), and DR-SD via L1 regularizednon-negative least-squares fitting (L1-NNLS) (Jian and Vemuri, 2007;Landman et al., 2012). Section Non-Negative Spherical Deconvolution(NNSD) describes NNSD and the associated Riemannian gradient descentalgorithm. Two stopping strategies for Riemannian gradient descent arediscussed. Section Methods furnishes evaluation details, including fiberresponse function estimation (Section Estimation of fiber responsefunction), peak detection (Section Peak detection), synthetic data gener-ation (Section Synthetic data simulation and evaluation), and peak con-sistency evaluation (Section Real data evaluation via peak consistency).In Section Experiments, NNSD is evaluated in comparison with themethods discussed in Section Spherical deconvolution revisited.Section Discussion provides additional discussions on various aspects ofNNSD. Section Conclusion concludes this paper.

Theory

Spherical deconvolution revisited

In this section, we describe CSD, MESD, and L1-NNLS, which wereoriginally proposed for single-shell data, and generalize them formultiple-shell data. We also proposed a new implementation of CSDusing quadratic programming.

Constrained SD (CSD)SD (Tournier et al., 2004; Anderson, 2005)methods assume that the

measured signal in each voxel is the product of convolving a latent fODFwith an axisymmetric fiber response function. For u ∈ S2, the fODF isrepresented as

Φ uð Þ ¼XLl¼0

Xlm¼−l

f lmYml uð Þ; ð1Þ

and the axisymmetric 3D fiber response function along the z-axis isrepresented as

H quj 0;0;1ð Þð Þ ¼XLl¼0

hl qð ÞY0l uð Þ; ð2Þ

752 J. Cheng et al. / NeuroImage 101 (2014) 750–764

where Ylm(u) is the l order m degree real SH basis function

(Johansen-Berg and Behrens, 2009; Descoteaux et al., 2007) defined inspherical coordinates as

Yml θ;ϕð Þ ¼

ffiffiffi2

pRe y mj j

l θ;ϕð Þ� �

if −l ≤ m b 0;

yml θ;ϕð Þ if m ¼ 0;ffiffiffi2

pIm yml θ;ϕð Þ� �

if 0 bm ≤ l;

8><>: ð3Þ

yml θ;ϕð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 14π

l−mð Þ!lþmð Þ!

reimϕPm

l cosθð Þ: ð4Þ

Re(⋅) and Im(⋅) denote respectively the real and imaginary parts of acomplex number. Plm(⋅) is the associated Legendre polynomial. Sincethe fODF is antipodal symmetric, only even l's are used. Based on theproperty of spherical convolution and spherical harmonics (Tournieret al., 2004; Anderson, 2005), the diffusion signal can be representedin a closed form as the convolution of the fiber response function withthe fODF (Tournier et al., 2004) using

E quð Þ ¼ZS2H qujrð ÞΦ rð Þdr ¼

XLl¼0

Xlm¼−l

ffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2lþ 1

rf lmhl qð ÞYm

l uð Þ: ð5Þ

Note that in the original formulation (Tournier et al., 2004), signalmeasurements are performed on a single-shell in the q-space,1 fixingq at a certain value. In the current work, we let q vary and define thefiber response function in R3, hence allowing the coefficients {hl(q)} tovary across shells. This generalization allows us to extend CSD to workwith multi-shell data. Based on Eqs. (1) and (2), the SH coefficient vec-tor f = (f00, …, fLL)T of the fODF can be estimated via

minf

M f−Ek k22; ð6Þ

whereM ¼ ffiffiffiffiffiffi4π

2lþ1

phl qið ÞYm

l uið Þ� �is an N � Lþ1ð Þ Lþ2ð Þ

2 basis matrix that is theelement-wise product between the SH basis matrix [Ylm(ui)] and theradial basis matrix

ffiffiffiffiffiffi4π

2lþ1

phl qið Þ� �

,and E = (E1, …, EN)T is the measuredsignal vector, including samples from single shell or multiple shells.The least-squares solution is (MTM)−1MTE (Tournier et al., 2004). Dueto the smoothness of H(qu), hl(q) decreases to zero as l increases, caus-ing instability in estimating flmwhen l is large. Therefore, Tournier et al.(2004) proposed a low-pass Filtered SD (FSD) by down-weighting thehigh-order SH coefficients. However, FSD is known to result in manyfalse-positive peaks because fODF non-negativity is not considered. Inview of this, Constrained SD (CSD) (Tournier et al., 2007) attempts to it-eratively suppress the negative values by a discrete reconstruction ofeach fODF estimate, i.e.

f kþ1ð Þ ¼ argminf

M f−Ek k22 þ λ2CSD L kð Þ f 2

2; ð7Þ

where L i, j(k) = Pi, j if (Pf (k))i b τ, L i, j

(k) = 0 if (Pf (k))i N τ, and P is a matrixfor reconstructing the fODF at discretized points on S2. τ is a thresholdnormally chosen as 0.1 (Tournier et al., 2007). Note that non-negativity is only imposed on a set of discretized points, not throughoutthe whole unit sphere. Although CSD reduces significantly the negativevalues compared with FSD (Tournier et al., 2004), a significant amountof negative values still exist, even at points where non-negativity is im-posed. An implementation of CSD for single shell data is available viaMRtrix2 (Tournier et al., 2012).

1 q = qu is a wave vector in q-space.2 MRtrix: http://www.brain.org.au/software/mrtrix/.

Maximum Entropy SD (MESD)MESD is a CR-SD method that involves solving a non-linear least-

squares problem using the Levenberg–Marquardt algorithm (Alexander,2005). Based on themaximumentropy principle, the fODF is representedas an exponential function of the mixture of fiber response functions, i.e.,

minλ jf gK

i¼0

XNi¼1

ZS2H qijrð ÞΦ rj λ j

n oK

j¼0

�dr−Ei

�2; ð8Þ

Φ rj λ j

n oK

j¼0

�¼ exp λ0 þ

XKj¼1

λ jH q jjr� �0@ 1A; ð9Þ

where N is the number of diffusion-weighted samples. The exponentialrepresentation naturally ensures non-negativity on S2. However, withoutany closed-form expression for spherical convolution as in Eq. (5), thespherical integration in Eq. (8) needs to be approximated numerically.Based on the method of Lagrange multipliers and the maximumentropy principle, K should ideally be equal to N and the sampling points{qj}j = 1

K should correspond to the N sampling points of the diffusion-weighted signal (Alexander, 2005). However, as a trade-off between ac-curacy and speed, only K b N evenly distributed points are used. After es-timating {λj}j = 0

K , ad-hoc normalization of the fODF needs to beperformed for unit integral. CT-FOD (Weldeselassie et al., 2010;Weldeselassie et al., 2012), which achieves non-negativity byrepresenting the fODF using a sum of squares of homogeneous polyno-mials, shares the same limitations as MESD by relying on numericalspherical integral and ad-hoc normalization.

L1-Regularized Non-Negative Least-Squares (L1-NNLS)DR-SD uses a discrete representation of the fODF: {wi = Φ(ui)}

(Jian and Vemuri, 2007; Landman et al., 2012; Yap and Shen, 2012).The SD problem in this case is typically formulated as

minw

Aw−Ek k22 þ λL1 wk k1; s:t: w≥0; ð10Þ

where A is a matrix with columns containing rotated versions of thefiber response function, and λL1 is the sparsity tuning parameter. Werefer to this method as L1-regularized Non-Negative Least Squares(L1-NNLS). The problem in Eq. (10) can be solved using efficient algo-rithms, such as those reported in Lee et al. (2006), and Kim et al.(2007). When λL1 = 0, Eq. (10) becomes the Non-Negative LeastSquares (NNLS) (Landman et al., 2012; Jian and Vemuri, 2007;Dell'Acqua et al., 2007). The Richardson–Lucy SD (Dell'Acqua et al.,2007), another DR-SD method, was shown to converge to the solutionof NNLS in the case of Gaussian noise (De Pierro, 1993). The sparsityof the solution increases as λL1 increases. If λL1 is large enough, all ele-ments in w are zero. Landman et al. (2012) proposed to determine λL1

based on the so-called breakdown parameter λL1∗ = ||2ATE||∞, which isthe minimal λL1 that causes w = 0 (Kim et al., 2007). Experimentally,Landman et al. (2012) set λL1 = 0.1λL1

∗ . We followed this approach forexperiments performed in this paper. Since the fODF in L1-NNLS is rep-resented using discretized points, peak detection is restricted to thesepoints instead of the whole S2, limiting the angular resolution.

Quadratic-Programming CSD (QP-CSD)We propose a quadratic programming implementation of CSD

(QP-CSD) for imposing non-negativity in a manner that is more similarto NNLS, i.e.,

minf

M f−Ek k22; s:t: f 00 ¼ 1ffiffiffiffiffiffi4π

p ; P f ≥ 0; ð11Þ

753J. Cheng et al. / NeuroImage 101 (2014) 750–764

where f 00 ¼ 1ffiffiffiffiffi4π

p ensures unit integral

ZS2Φ uð Þdu ¼

ZS2

XLl¼0

Xlm¼−l

f lmYml uð Þdu ¼ 1: ð12Þ

QP-CSD, unlike CSD, produces the global solution and guarantees non-negativity on fODF points specified by matrix P. Compared with L1-NNLS, QP-CSD works in a low dimension space thanks to the SH repre-sentation and allows peak detection to be performed on the continuoussphere.

Non-Negative Spherical Deconvolution (NNSD)

SH-based SD methods are able to leverage the nice property of hav-ing a closed-form solution to spherical deconvolution. But currentlythey fall short in enforcing non-negativity throughout the unit sphere.To address this issue, we propose a new approach called Non-Negative Spherical Deconvolution (NNSD), which will be describednext.

Square root representation of the fODFThe square root representation has been proposed for the dODF

(Cheng et al., 2009) and the EAP (Cheng et al., 2011), and it has beenused for non-negative dODF and EAP estimation (Cheng et al., 2012).In this work, we propose to utilize the square root representation fornon-negative fODF estimation by letting

Φ ujcð Þ ¼XLl¼0

Xlm¼−l

clmYml uð Þ

!2

¼X2Lα¼0

Xαβ¼−α

XLl;m

XLl0 ;m0

clmcl0m0Qmm0βll0α

0@ 1A|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

f αβ

Yβα uð Þ;

ð13Þ

where

Qmm0βll0α ¼

ZS2Yml uð ÞYm0

l0 uð ÞYβα uð Þdu: ð14Þ

The notation ∑ l,mL is a shorthand for ∑ l = 0

L ∑ m = − ll . Qmm0β

ll0α is aconstant resulting from the integration of three real SH functions(Cheng et al., 2012), which can be calculated from the Wigner 3-jsymbol and Eq. (3). Note that the sum over α is up to 2L, because when

α N 2L, Qmm0βll0α ¼ 0, based on the property of the Wigner 3-j symbol.3

Note that 1) a SH representation offfiffiffiffiffiffiffiffiffiffiffiΦ uð Þp

with maximum order Lcorresponds to a SH representation of Φ(u) with maximum order 2L.Therefore compared with Eq. (1), the square root representation ofEq. (13) is more compact in representing high angular resolutioninformation. This is also contributive to reducing the Gibbs ringing arti-fact associated with the truncation of the SH series (Raffelt et al., 2012).2) The fODF is naturally non-negative throughout S2 due to the squareroot representation.

Based on Eqs. (2), (5) and (13), the diffusion signal can be representedusing convolution as

E quð Þ ¼X2Lα¼0

Xαβ¼−α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rf αβhα qð ÞYβ

α uð Þ

¼X2Lα¼0

Xαβ¼−α

XLl;m

XLl0 ;m0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rclmcl0m0Qmm0β

ll0α hα qð ÞYβα uð Þ

¼ cTK quð Þc;

ð15Þ

3 http://mathworld.wolfram.com/Wigner3j-Symbol.html.

where for any fixed qu, K(qu) is a square matrix of dimension(L + 1)(L + 2)/2 with elements

Kmm0

ll0 quð Þ ¼X2Lα¼0

Xαβ¼−α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rQmm0β

ll0α hα qð ÞYβα uð Þ: ð16Þ

Riemannian gradient descentBased on Eq. (15), we propose to estimate c from the measured

{Ei}i = 1N by minimizing

J cð Þ ¼ 12

XNi¼1

cTK qiuið Þc−Ei� �2 þ 1

2cTΛc; s:t: ck k2 ¼ 1 ð17Þ

where Λ is a diagonal matrix with elements Λlm = λNNSDl2(l + 1)2

for Laplace–Beltrami regularization (Descoteaux et al., 2007), penal-izing the high-order coefficients. The constraint ‖c‖2 = 1 is a result ofthe unit integral, i.e. ∫

S2Φ uð Þdu ¼ 1, and the orthogonality of the SH

basis:

ZS2Φ uð Þdu ¼

XLl;m

XLl0 ;m0

clmcl0m0

ZS2Yml uð ÞYm0

l0 uð Þdu ¼ ck k22 ¼ 1; ð18Þ

which is equivalent to f 00 ¼ 1ffiffiffiffiffi4π

p , as in Eq. (11). The influence of theunit integral constraint on NNSD will be discussed in Section Unitintegral constraint.

We propose a Riemannian gradient descent method on the sphere‖c‖2 = 1 to minimize Eq. (17). The Euclidean gradient of J(c) is

∂ J cð Þ∂c ¼

XNi¼1

2 cTK qiuið Þc−Ei� �

K qiuið Þcþ Λc ð19Þ

and the Riemannian gradient is the projection of Euclidean gradientonto the tangent space of c, i.e.,

∇ J cð Þ ¼ ∂ J cð Þ∂c − cT

∂ J cð Þ∂c

�c: ð20Þ

The Riemannian gradient descent associated with J(c) is

c kþ1ð Þ ¼ Expc kð Þ −dt∇ J cð Þ∇ J cð Þk k2

�; ð21Þ

Expc vð Þ ¼ c cos vk k2 þvvk k2

sin vk k2; ð22Þ

where c(k) is the estimated c in the k-th step, dt is the step size chosenfrom inexact line search in (0, dt0], and Expc(v) is the exponentialmap on the sphere (Cheng et al., 2009). See Algorithm 1 for a summaryof thewhole process.We use the isotropic fODFwith c(0) = (1, 0,…, 0)T

for initialization and choose dt0 = 0.1 based on experimental observa-tions and the fact that ‖c‖2 = 1. We found that NNSD is robust tonoise and λNNSD = 0 works well in the experiments. The stoppingcondition is described next.

Adaptive stopping conditionThe stopping condition for Algorithm 1 is

eJ kð Þ ¼J c k−1ð Þ� �

− J c kð Þ� �J c k−1ð Þ� � b δ: ð23Þ

Parameter δ should be chosen as a small number such that thealgorithm converges to a local minimum of Eq. (17). Experimentallyδ = 10−2 works well in most cases. If δ is further reduced beyond10−2, the fODFs in regions with high anisotropy will increase in

754 J. Cheng et al. / NeuroImage 101 (2014) 750–764

sharpness, but the fODFs in regions with low anisotropy will howeversuffer from spurious peaks due to over-fitting. Therefore, we proposeto adaptively stop the algorithm if GFA(c(k)) b T and eJ kð Þbδ0 or ifGFA(c(k)) ≥ T and eJ kð Þb0:01δ0, where δ0 = 10−2 and

GFA cð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− c200

ck k22

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−c200

qð24Þ

is the Generalized Fractional Anisotropy (GFA) (Tuch, 2004) of thesquare root of the fODF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΦ ujcð Þp

, and T∈ [0, 1] is a threshold dependingon the noise level. We call the NNSD implementationwith this adaptivestopping condition ASC-NNSD. Since gradient descent is initializedusing the isotropic fODF, GFA(c(k)) b T is satisfied for all voxels during

the early stage of optimization. As k increases, eJ kð Þ decreases such that

for a certain step k0 we have eJ k0ð Þ b δ0. If for a voxel eJ k0ð Þ b δ0 and GFA

c k0ð Þ� �bT , we consider the voxel insufficiently anisotropic to proceed

further and hence stop the algorithm early at step k0. If eJ k0ð Þ b δ0 and

GFA c k0ð Þ� �≥T , we consider the voxel anisotropic and further refine

the solution by proceeding with gradient descent. When T = 0, ASC-NNSD becomes NNSD with δ = 0.01δ0 = 10−4; when T = 1, ASC-NNSD becomes NNSD with δ = δ0 = 10−2. Both NNSD and ASC-NNSD converge quickly, normally in a dozen of steps. Based on ourC++ implementation, if L = 6, 1000 fODFs can be estimated within 5s using an ordinary laptop, significantly faster thanMESD as implement-ed in Camino (Alexander, 2005).

Methods

Estimation of fiber response function

SD methods require an estimate of the fiber response function,which is essentially the diffusion signal profile of a single-directionalhighly-coherent fiber bundle. Anderson (2005) computed a SH repre-sentation of thefiber response function using the diffusion signal gener-ated from a tensor model. Tournier et al. (2007) proposed to first fit thetensor model to the data of voxels with high Fractional Anisotropy (FA)and then rotate the diffusion-weighted measurements such that theprincipal directions coincidewith the z-axis. From the rotatedmeasure-ments, the SH coefficients of the fiber response function are computed.Although these twomethods described above were originally proposedfor single-shell data, they can be generalized for multi-shell data usingtwo different approaches:

• First fit diffusion models, such as DTI (Basser et al., 1994) or SphericalPolar Fourier Imaging (Assemlal et al., 2009; Cheng et al., 2010b), tohigh-anisotropy diffusion signal E(q) in R3. Then, reconstruct thesignal in each shell and compute the SH representation (Anderson,2005; Cheng et al., 2010b).

• If a sufficiently large number of measurements are collected for eachshell, the fiber response function can be estimated independently foreach shell using the approach described in Tournier et al. (2007).

Both approacheswere used in the experiments (see Section Parameters).

Peak detection

At each voxel, fiber orientations are given by the peaks or localmaxima of each fODF. For L1-NNLS, the peaks are determined basedon the local maxima of 321 points evenly distributed on a hemisphere,generated by subdividing the faces of an icosahedron. We discard localmaximawith low fODF value and consider only peaks with fODF valuesgreater than themean of theminimal andmaximal fODF values.We as-sume that the fiber crossing angle is no smaller than θΔ=15∘ andwith-in this angle only one peakwith the greatest fODF value is retained. Thechoice θΔ =15∘ is motivated by the fact that the angle is approximatelytwice theminimal angular separation (i.e., 7.9∘) between the 321points.

For all CR-SDmethods, includingNNSD, CSD, QP-CSD, andMESD, wefirst determine candidate local maxima from the same 321 points. Then,using the continuous representation associated with these methods, agradient ascent method, also implemented in MRtrix (Tournier et al.,2012), is performed to give a set of refined peaks.

Synthetic data simulation and evaluation

For evaluation, synthetic data were generated by amixture of tensormodels:

E quð Þ ¼ S uð ÞS 0ð Þ ¼

XKk¼1

pk exp −q2uTDku� �

; b ¼ q2; ð25Þ

where Dk is a 3 × 3 matrix representing a diffusion tensor and K= 1, 2corresponds to the single-direction case and the crossing case, respec-tively. The baseline signal is S(0) = 1 and b is the diffusion weighting(Johansen-Berg and Behrens, 2009). For the crossing case, the tensorsshare the same eigenvalues (λ1, λ2, λ2) and are equally weighted withp1 = p2 = 0.5. We generated Rician noise corrupted signal by

Ei ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE qiuið Þ þ s1ð Þ2 þ s22

qð26Þ

where s1 and s2 are the Gaussian distributed noise with zero meanand standard deviation σ. The signal-to-noise ratio (SNR) is definedas 1

σ (Descoteaux et al., 2007). The methods used for comparisonmight not necessarily detect the correct number of peaks (2 in thiscase). Therefore, we report the success ratio, which tells us the propor-tion of trials that correctly detect the number of peaks. TheMeanDiffer-ence of Angles (MDAs) (Descoteaux et al., 2007) is then computed onlyfor the successful trials.

Real data evaluation via peak consistency

Real data evaluation is challenging due to the lack of ground truth.To make comparison possible, we gauge the correctness of the fiberdirections at each voxel based on the agreement of different datasetsand methods. We assume that if a certain direction is widely agreedupon, then there is a high possibility that the direction reflects thetrue direction. Based on this assumption, we propose here a novel mea-sure, called Peak Consistency (PC), to evaluate peak estimation accura-cy. Assuming we have N datasets (e.g., resulting from differentsampling schemes) pertaining to the same subject, then M methodswill yield MN fODFs at each voxel, resulting in more than MN peaks(at least one peak for each fODF).

Two peaks are said to be consistent if the angle between them is nomore than threshold θC. Our choice of angular threshold θC= 5∘ is basedon the empirical observation that, for moderate Rician noise, the peak

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detection error of the algorithms is around 2∘. This angular thresholdgives a good tolerance to cover the variability caused by noise.

For voxel x, we denote pi,kx as the k-th peak from the i-th fODF

(i ∈ [1 : MN], [a : b] denotes {a, a + 1,…, b}). Since θC b θΔ, for peak

pi,kx , at most one consistent peak can be found from peaks pxj;l

n oKxj

l¼1for a given j ∈ [1 : MN], j ≠ i, where Kj

x denotes the number ofpeaks for the j-th fODF at voxel x. The maximum number of possiblematches for all j ≠ i is hence MN-1. For each peak, we define theSingle-Peak Consistency (SPC)

SPC i; k; xð Þ ¼ Γ i; k; xð Þj jMN−1

; ð27Þ

where Γ(i, k, x)={j | j≠ i, j∈ [1 :NM], l∈ [1 :Kjx],d(pi,kx ,pj,lx)≤ θC}, d(pj,lx ,pi,kx )is the angle between the two peaks, and | ⋅ | denotes the number ofelements in a set. SPC = 0 indicates that no peak is consistent withthe reference peak pi,k

x . SPC = 1 indicates that the reference peak pi,kx is

fully consistent with all other MN-1 peaks. For concurrent consistencyevaluation of multiple peaks, we define the Multi-Peak Consistency(MPC) as

MPC i; xð Þ ¼j∩k∈ 1:Kx

i½ �Γ i; k; xð ÞjMN−1

: ð28Þ

See Fig. 1 for a visual summary of the Single- and Multi-PeakConsistency measures. Given a field of fODFs with voxels {x|x ∈ Ω}, wedefine for a threshold PCT ∈ (0, 1) (0.5 in the current work) the overallSingle- and Multi-Peak Consistency measures as

SPC ið Þ ¼X

x∈ΩkjSPC i; k; xð Þ≥PCT; k∈ 1 : Kx

i

� � ��� ��Xx∈Ω

Kxi

ð29Þ

and

MPC ið Þ ¼X

x∈ΩxjMPC i; xð Þ≥PCTf gj j

Ωj j : ð30Þ

Fig. 1. Peak consistency. The peaks detected from each fODF are indicated by identicallycolored arrows. The cones cover peak orientations subtending an angle θC with thewhite arrows. The white peaks are said to be consistent with the peaks within thesecones. In this case, SPC = 3/5 and SPC = 2/5, respectively for the vertical and horizontalwhite peaks, and MPC = 2/5.

Experiments

Parameters

We compared NNSD and ASC-NNSD with CSD, QP-CSD, MESD,and L1-NNLS, using both synthetic and real data. For CSD, MRtrix(Tournier et al., 2012) was used in all experiments involving single-shell data (Tournier et al., 2007), and an in-house implementation ofCSD was used in experiments involving multi-shell data. The defaultparameters specified in Tournier et al. (2007) and 321 orientationsevenly distributed in a hemisphere were used to generate L(k) inEq. (7) for CSD (Tournier et al., 2007) and P in Eq. (11) for QP-CSD.Accelerated MESD with K = 16 in Eq. (8) was used as suggestedin Camino. Following Landman et al. (2012), we set λL1 = 0.1λL1

∗ forL1-NNLS. To demonstrate the robustness of NNSD, we intentionally setλNNSD = 0 to switch off the Laplace–Beltrami regularization and setT =0.5 for ASC-NNSD. For the synthetic data experiments, the fiber re-sponse function for all methodswas set according to the signal from thetensormodelwith eigenvalues [1.7, 0.2, 0.2] × 10−3 mm2/s. For the realdata experiments, the response function was estimated by MRtrix(Tournier et al., 2012) using voxels with FA greater than 0.7.

Synthetic data

Evaluation with SNR = 1000We would like to investigate how the different methods behave

when the anisotropy of the signal changes and when there is amismatch between the fiber response function and the data. Diffusionsignal was first generated from the mixture of tensor models inEq. (25) with b = 1500 s/mm2, crossing angle 60∘, mean diffusivity0.7 × 10−3 mm2/s, and different FA values (Johansen-Bergand Behrens, 2009) by varying tensor eigenvalues (λ1, λ2, λ3 = λ2).Fig. 2 (SNR = 1000) shows the estimated fODFs when λ1 increasesfrom 0.7 × 10−3 mm2/s (left with FA = 0) to 1.7 × 10−3 mm2/s(right with FA = 0.87). For all methods, the fiber response functionwas the tensor model with eigenvalues [1.7, 0.2, 0.2] × 10−3 mm2/s.Therefore, the response function matches the data on the far right butnot on the left. The results indicate that CSD, QP-CSD and L1-NNLS resultin many spurious peaks for data with low anisotropy, even when theSNR is high. NNSD and ASC-NNSD work well when the signal is aniso-tropic or less so. The fODFs estimated byNNSDwith δ=10−2 are a littlesmoother than the fODFs estimated by NNSD with δ= 10−4. Note thatfor isotropic signal, the fODFs given byNNSD and ASC-NNSD are close tobeing isotropic.

Evaluation with SNR = 30The estimation results when the noise level is increased are shown

in Fig. 2 (SNR = 30). Despite the noise, both NNSD and ASC-NNSD ob-tain significantly smoother results for signal that is more isotropicwhen compared with other methods. ASC-NNSD works best among allmethods in obtaining sharper fODFs for anisotropic signal and smootherfODFs for isotropic signal. The results given by ASC-NNSD can be seen asa combined outcomeofNNSDwith δ=10−2 for the less anisotropic sig-nal and NNSD with δ = 10−4 for the anisotropic signal.

Evaluation of non-negativity and anisotropy1000 Rician-noise realizations of isotropic signal with eigenvalues

[0.7, 0.7, 0.7] × 10−3 mm2/s and anisotropic signal with eigenvalues[1.7, 0.2, 0.2] × 10−3 mm2/s were generated at SNR = 15 and 30 forb = 1500 s/mm2 and 60 evenly distributed directions. From the esti-mated fODFs, we recorded the GFA values (Tuch, 2004) and the propor-tion of directions (among 5121 points uniformly distributed on the unitsphere) with negative fODF values. We ignored negative values close tozero and only took into consideration negative values with absolutevalues larger than 1% of the maximal fODF value. The top subfigure inFig. 3 indicates that CSD is still susceptible to negative values despite

Fig. 2. Synthetic data. fODF estimation results given by different methods under different parameter settings for synthetic data with varying anisotropy.

756 J. Cheng et al. / NeuroImage 101 (2014) 750–764

its non-negativity constraint. QP-CSD results in a significantly lowernumber of negative values than CSD because it enforces a stricter non-negativity requirement than CSD. ASC-NNSD and MESD guaranteenon-negativity throughout S2. Note that the fODFs estimated by L1-NNLS are non-negative on the 321 points used during optimization;however, since fODF values on the other points are unknown,we cannotcompute the proportion of negative values for L1-NNLS. The bottomsubfigure indicates that the GFA values of the fODFs given by CSD, QP-

Fig. 3. Nonnegativity and anisotropy. Proportions of negative values and GFA values of fODFs edeviations.

CSD, MESD and L1-NNLS are high even for isotropic signal, indicatingoverestimation of the number of fiber directions. Only ASC-NNSD yieldssignificant contrast between anisotropic and isotropic signals.

Evaluation of peak accuracy — single shellNoise-corrupted signals were generated from a two-tensor model

with b=1500 s/mm2, eigenvalues [1.7, 0.2, 0.2] × 10−3 mm2/s, differ-ent crossing angles in the range of [30∘, 90∘], and at SNR=10 and20. For

stimated from isotropic (ISO) and anisotropic (ANI) data. The error bars indicate standard

757J. Cheng et al. / NeuroImage 101 (2014) 750–764

peaks detected from the fODFs estimated by eachmethod, we recordedthe success ratio and the MDA as described in Section Synthetic datasimulation and evaluation. The results are shown in Fig. 4 for bothSNR = 10 and SNR = 20. When SNR = 10, MESD, CSD, and NNSDwith L = 10 give low success ratios for small crossing angles, but highsuccess ratios for the large crossing angles. With L=6, when the cross-ing angle is larger than 55∘, ASC-NNSD generally gives a higher successratio than CSD and lowest MDA among all methods. When SNR = 20,MESD, CSD, and NNSD with L = 10 give higher success ratios for bothsmall and large crossing angles. CSD and QP-CSD NNSD with L = 10give better performance than their counterpartswith L=6. This impliesthat, if SNR is high, CSD andNNSDbenefit fromusing a higher order rep-resentation. It is interesting to see that the success ratio of L1-NNLS ishigh for small crossing angle but low for large crossing angle. This indi-cates that L1-NNLS has high sensitivity, but low specificity due to false-positive peaks. Partially due to limited angular resolution, L1-NNLSyields larger MDA values than CSD and NNSD. The MDA performanceof MESD can potentially be improved by increasing parameter K.

Evaluation of peak accuracy — multiple shellsWe performed an evaluation similar to the previous section using

multi-shell data. Since the results reported in the previous sectionindicate that CSD and NNSD generally give better performance thanQP-CSD, MESD, and L1-NNLS, we focus our comparison in this sectionon CSD and NNSD. We chose L = 8, which is a good trade-offbetween angular resolution and robustness. This is also suggested inMRtrix (Tournier et al., 2012). The data were generated by samplingfrom a mixture of tensor models (Caruyer et al., 2011) using b =1500 s/mm2, 3000 s/mm2 in 30 staggered directions per shell. Fig. 5shows the success ratios and MDA values given by CSD and NNSDusing samples respectively from the outer b = 3000 s/mm2 shell(30 samples) and from two shells (60 samples) concurrently.When SNR = 10, ASC-NNSD and CSD give similar success ratios, butASC-NNSD generally gives lower MDA than CSD for both single- and

Fig. 4. Peak accuracy — single shell. Success ratios and MDA valu

two-shell cases. For both ASC-NNSD and CSD, the two-shell datagenerally result in higher success ratios and lower MDA values thansingle-shell data.When SNR= 20, ASC-NNSD gives better performancethan CSD both in terms of success ratio and MDA. For both ASC-NNSDand CSD, the two-shell data result in success ratios similar to thesingle-shell data with b = 3000 s/mm2, but lower MDA values.

Real data

Single-shell dataEvaluation was performed using real human data with b =

2000 s/mm2, 120 gradient directions, 2 mm isotropic resolution,and TR/TE = 12400/116 ms. We set L = 8 for CSD, QP-CSD,NNSD, and ASC-NNSD. MRtrix was used for the estimation of thefiber response function from voxels with FA N 0.7 (Tournier et al.,2004; Tournier et al., 2007). The results are shown in Fig. 6. NNSDwith δ= 10−4 and ASC-NNSD with T= 0.5 give similar results, im-plying that the result is insensitive to T for this data. For each meth-od, the GFA map computed from the estimated fODFs is displayedas the background image. Consistent with the results in Fig. 3, theresults in Fig. 6 for CSD, QP-CSD, MESD, and L1-NNLS show a signif-icant amount of false-positive peaks, which are especially evidentin the regions with low anisotropy, as indicated by the high GFAvalues in gray matter and cerebrospinal fluid regions. NNSD andASC-NNSD dramatically reduced the spurious peaks of the estimat-ed fODFs. See for example the region marked by the yellow circlesin Fig. 6. Compared with CSD, NNSD and ASC-NNSD yield sharperfODFs in anisotropic regions and more isotropic fODFs in isotropicregions. This gives high GFA contrast between these regions. Notethat existing work (Tournier et al., 2004, 2007; Jian and Vemuri,2007; Landman et al., 2012; Weldeselassie et al., 2012) attemptsto avoid such low GFA contrast by computing FA or GFA usingsome other models, such as the tensor model.

es for different crossing angles at SNR = 10 and SNR = 20.

Fig. 5. Peak accuracy— multiple shells. Success ratios and MDA values for different crossing angles at SNR = 10 and SNR = 20.

758 J. Cheng et al. / NeuroImage 101 (2014) 750–764

High spatial resolution multi-shell HCP dataEvaluation was also performed using a high-resolution dataset

(1.25 mm in all dimensions) obtained from the Human ConnectomeProject (HCP).4 fODF estimation from this dataset is challenging becausethe SNR is low due to the small voxel size. This dataset was acquiredusing three shells, with 90 staggered directions per shell, and at b =1000, 2000, 3000 s/mm2.

First, we performed a single-shell evaluation (b = 2000 s/mm2)using NNSD with δ = 10−4, ASC-NNSD with T = 0.5, CSD, QP-CSD,MESD, and L1-NNLS. L = 8 was used in all SH-based methods. Theresults shown in Fig. 7 demonstrate the effectiveness of NNSD andASC-NNSD. In isotropic regions, only NNSD and ASC-NNSD give isotro-pic fODFs, while other methods result in a significant amount of false-positive peaks. See for example the regionmarked by the yellow circles.In anisotropic regions, NNSD and ASC-NNSD yield the sharpest fODFs. Inparticular, the fODFs estimated by CSD are not as sharp as those estimat-ed by NNSD, ASC-NNSD, and MESD. This can be seen in the regionmarked by the yellow squares. Spurious peaks can be observed for L1-NNLS, as indicated by the high anisotropy values, even in regions withisotropic diffusion.

Second, we included the two other shells (b = 1000, 3000 s/mm2)in the estimation, totaling up to90 × 3 samples. The results are shownin Fig. 8. For ASC-NNSD, the b = 1000 s/mm2 data give fODFs that arenot as sharp as the results given by the two other b-values. The b =3000 s/mm2 data give fODFs that are the sharpest but with morefalse-positive peaks. The b = 2000 s/mm2 data seem to yield moremoderate results than among all b-values.

Third, we detected the peaks from the fODFs estimated using the 4sampling schemes (the individual shells and all three shells) and 5

4 http://www.humanconnectome.org/documentation/Q1/.

methods5 (ASC-NNSD, CSD, QP-CSD, MESD, and L1-NNLS) and thencomputed the corresponding peak consistency values defined inSection Real data evaluation via peak consistency. Note that fODFpeaks detected in isotropic regions are not reliable and are hence not in-cluded in the evaluation.Only voxelswithGFA greater than 0.5, estimat-ed based on ASC-NNSD using all three shells, were used for consistencyevaluation. The detected peaks, color-codedwith SPC values, are shownin Fig. 9, indicating that 1) peak consistency is high in anisotropic andsingle directional regions; 2) peak consistency is low in fiber crossingregions; and 3) the lowest peak consistency occurs at boundaries be-tween white matter and gray matter or cerebrospinal fluid regions. Itcan also be observed that for ASC-NNSD, CSD, and QP-CSD, the multi-shell data result in higher SPC than single-shell data. For both single-and multi-shell data, ASC-NNSD yields higher SPC values than othermethods. These observations are confirmed by the overall peak consis-tency values (SPC and MPC) shown in Table 1.

Discussion

Multi-shell SD

Existing fODF estimationmethods such as CSD can only be applied tosingle-shell data.We have generalized SD towork withmulti-shell databy defining the fiber response function in R3. Data with low diffusionweighting have high SNR but low angular resolution, whereas datawith high diffusionweightinghave lowSNRbut high angular resolution.Our method takes advantage of the whole spectrum of the data. Evalu-ation based on both synthetic and real data, as shown in Figs. 5, 6, 8,and 9 and Table 1, indicate that multi-shell data improve peakestimation accuracy compared with single-shell data.

5 Multi-shell versions of MESD and L1-NNLS were not implemented. For peak consis-tency evaluation, we therefore have 4 × 5 − 2 = 18 fODFs per voxel.

Fig. 6. Real data — single shell. Coronal views and close-up views of the fODF fields generated by various methods. The background is the GFA map.

759J. Cheng et al. / NeuroImage 101 (2014) 750–764

It should be noted that for the HCP data, themulti-shell SD methodsuse all 90 × 3 diffusion-weighted images for estimation. This is 3 timesthe number of samples used by single-shell SDmethods. It will be inter-esting to fix the total number of samples and evaluate the advantagesof the two different sampling schemes, e.g., a multi-shell samplingscheme with 90 samples per shell against a single-shell scheme with270 samples. However, choosing a good sampling scheme (Chenget al., 2014) with proper consideration of the tradeoff between angularresolution and SNR is still an open problem beyond the scope of thispaper. The proposed multi-shell SD framework can be used as a toolto evaluate different sampling schemes in the future.

Non-local NNSD (NL-NNSD)

In practice, denoising methods such as non-local mean (Buadeset al., 2005; Descoteaux et al., 2008; Yap et al., 2014) and spatial regular-ization (Goh et al., 2009) can significantly improve SNR and hence fODFestimation accuracy. Our initial implementation of a non-locally regu-larized (Buades et al., 2005) version of NNSD, called non-local NNSD(NL-NNSD) (Cheng et al., 2013a), supports this idea. In the ISBI 2013HARDI Reconstruction Challenge,6 which considered DTI, HARDI, andDSI-like sampling schemes and SNR = 10, 20, and 30, NL-NNSD wasranked the best technique in terms of local fiber orientation estimationfor all sampling categories and all SNRs.7

6 http://hardi.epfl.ch/static/events/2013_ISBI/.7 http://hardi.epfl.ch/static/events/2013_ISBI/_static/talk_Max.pdf.

Unit integral constraint

Unlike NNSD, existing SD methods, such as FSD, CSD, MESD andL1-NNLS, do not explicitly incorporate the unit integral constraint. InA, we show that the solutions given by FSD in Eq. (6) with and withoutthe constraint differ only by the isotropic component, and under someconditions the results given by NNSD with and without the constraintalso differ only by the isotropic component. The constraint is beneficialfor fODF estimation as explained below:

1. As a probability distribution function of orientations, fODF shouldnaturally have a unit integral.

2. The constraint helps mitigate the representation error inher-ent in SD-based fODF estimation methods. Based on Eqs.(5) and (15), the spherical means of E(qu) and H(qu) are re-spectively

E qð Þ ¼ 14π

ZS2E quð Þdu ¼ f 00h0 qð Þ; H qð Þ ¼ 1ffiffiffiffiffiffi

4πp h0 qð Þ: ð31Þ

The value of f00 should ideally be adaptive to the values of thesemeans in order to minimize the representation error. For exam-ple, if the diffusion signal and fiber response function have thesame spherical mean, then one should set f 00 ¼ 1ffiffiffiffi

4πp to avoid rep-

resentation error. This requirement is obvious for q = 0, wherewe expect E 0ð Þ ¼ H 0ð Þ ¼ 1 and h0 0ð Þ ¼

ffiffiffiffiffiffi4π

p, especially when

the fiber response function is based on the tensor model. If thespherical means differ, our analysis in A shows that under

Fig. 7.HCPdata— single shell. Coronal views and close-up views of the fODFfields generated by variousmethods from single shell datawith b=2000 s/mm2. The background is theGFAmap.

760 J. Cheng et al. / NeuroImage 101 (2014) 750–764

some conditions setting f 00 ¼ 1ffiffiffiffi4π

p will result in an fODF with thesame anisotropic component.

3. Due to the constraint, the solution space of SDmethodswith the con-straint is one dimension smaller than the solution space of SDmethods without this constraint. This reduces the feasible solutionsto a smaller space, effectively reducing the complexity of the associ-ated non-convex optimization, and focuses the optimization on theanisotropic components of the fODF.

NNSD without the constraint can be solved by traditionalgradient descent with inexact line search. The stopping conditionEq. (23) and its adaptive version can be used to terminate the algo-rithm. Our experiments indicate that δ=10−4 gives the best results.Fig. 10 shows the fODF results given by NNSD without the constraintusing the synthetic data shown in Fig. 2. The glyphs are normalizedby ∥ c ∥ = 1 for visualization. Compared with the results given byNNSD in Fig. 2 with δ = 10−4, the fODFs given by NNSD without theconstraint are generally similar, but not as sharp. It should be notedthat when δ varies, the fODFs given by NNSD without the constraintvary more dramatically than NNSD with the constraint, indicating thatthe latter is more robust to the choice of δ.

Conclusion

The contribution of this paper is threefold. First, by introducing a R3

fiber response function,we have generalized the existing single-shell SDframework to cater to multi-shell data. The experimental resultsindicate that, compared with single-shell data, multi-shell data givelowerMDA, higher success ratios andmore consistent peaks. Second,

we have proposed a novel SD method, called Non-Negative SD(NNSD), which utilizes a square root representation to imposenon-negativity. Comparisons with existing SD methods, includingCSD, MESD, and L1-NNLS, demonstrate the advantages of NNSD:1) NNSD guarantees non-negativity with a unit integral of the fODFthroughout the whole S 2; 2) NNSD significantly reduces false-positive peaks and yields high GFA contrast between isotropic andanisotropic regions; and 3) due to the SH representation, NNSD is ef-ficient and allows accurate peak detection on S2. Third, we have pro-posed a new metric called peak consistency for evaluation of thedifferent methods in the absence of ground truth. Using this metric,we demonstrated that NNSD gives better estimates of peak orienta-tions in real data. Improvements in fODF estimation obtained withNNSD will be helpful for applications such as fiber tractography(Yap et al., 2011b, 2011c), fiber clustering (Wang et al., 2013), andbrain connectomics (Yap et al., 2010, 2011a; Wee et al., 2010,2012; Shi et al., 2012).

Acknowledgments

This work was supported in part by a UNC BRIC-Radiology start-upfund and NIH grants (EB006733, EB008374, EB009634, AG041721,MH100217, AG042599). Data were provided in part by the HumanConnectome Project, WU-Minn Consortium (Principal Investigators:David Van Essen and Kamil Ugurbil; 1U54MH091657) funded bythe 16 NIH Institutes and Centers that support the NIH Blueprint forNeuroscience Research; and by the McDonnell Center for SystemsNeuroscience at Washington University.

Fig. 8. Single shell vs.multiple shells. Estimation results for differentmethodsusing data fromeach shell (90 samples) and concurrently from the three shells (90× 3 samples). See Fig. 7 forthe b = 2000 s/mm2 results.

761J. Cheng et al. / NeuroImage 101 (2014) 750–764

Appendix A. fODF estimation with and without the unit integralconstraint

Since for each shell the matrix [Ylm(ui)] has orthonormal columns,Eq. (6), in continuous form, is equivalent to

minf

XSs¼1

ZS2

XLl;m

ffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2lþ 1

rf lmhl qsð ÞYm

l uð Þ−E qsuð Þ0@ 1A2

du: ðA:1Þ

Theminimization is quadraticwith respect to flm and the optimal co-efficients can be found at the point where the derivative vanishes as

f �lm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 14π

r XSs¼1

hl qsð Þelm qsð ÞXSs¼1

hl qsð Þ2;ðA:2Þ

where elm(qs) is the SH coefficients of E(qsu). With the unit integralconstraint, we have f �00 ¼ 1ffiffiffiffiffiffi

4πp , but this does not alter flm∗ for l N 0. Thus

Fig. 9. Single peak consistency. Single Peak Consistency (SPC) for different methods and different sampling schemes. Cool and warm colors indicate low and high SPC, respectively.

762 J. Cheng et al. / NeuroImage 101 (2014) 750–764

the difference between the fODFs with and without the constraint isonly the isotropic component.

The NNSD formulation can be seen as an extension of FSD be-cause based on Eq. (15) we can write E(qiui) = cTK(qiui)c =M(qiui)f, where M(qiui) is the row in basis matrix M correspondingto the sample qiui. Without taking unit integral constraint into

Table 1SPC and MPC values for fODF fields estimated by different methods for different sampling sche

b = 1000 s/mm2 b = 2000 s/mm2

SPC MPC SPC MPC

ASC-NNSD 0.2411 0.2503 0.2839 0.29CSD 0.1533 0.1870 0.2245 0.25QP-CSD 0.1811 0.2319 0.2158 0.27MESD 0.1347 0.1752 0.1399 0.18L1-NNLS 0.1080 0.1571 0.1148 0.16

consideration, the unconstrained version of NNSD solves the fol-lowing problem:

minc

XSs¼1

ZS2

X2Lα;β

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rf αβhα qsð ÞYβ

α uð Þ−E qsuð Þ !2

du ðA:3aÞ

mes.

b = 3000 s/mm3 Multi-shell

SPC MPC SPC MPC

31 0.2857 0.2995 0.3845 0.374569 0.2359 0.2793 0.2779 0.323620 0.1938 0.2593 0.2597 0.321815 0.1600 0.210250 0.1030 0.1653

Fig. 10. NNSD without unit integral constraint. Estimation results for NNSD without unit integral constraint for mixture of tensor models with varying anisotropy.

763J. Cheng et al. / NeuroImage 101 (2014) 750–764

s:t: f αβ ¼XLl;m

XLl0 ;m0

clmcl0m0Qmm0βll0α : ðA:3bÞ

Note that by setting α = β = 0 in Eq. (A.3b) we have f 00 ¼ ck k2ffiffiffiffi4π

p .

Without loss of generality, we let, for optimal solution c∗, 1ffiffiffiffiffi4π

p c�k k2 ¼f �00 ¼ 1ffiffiffiffi

4πp þ Δ. The problem given by Eqs. (A.3a) and (A.3b) then becomes

minc

XSs¼1

ZS2

X2Lα;β

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rf αβhα qsð ÞYβ

α uð Þ−E qsuð Þ !2

du ðA:4aÞ

s:t: f αβ ¼XLl;m

XLl0 ;m0

clmcl0m0Qmm0βll0α ; f 00 ¼ 1ffiffiffiffiffiffi

4πp þ Δ: ðA:4bÞ

If Δ = 0, then Eq. (A.4a) is just the NNSD with a unit integralin Eq. (17). If Δ ≠ 0, we let f 100 ¼ 1ffiffiffiffiffi

4πp , and fαβ

1 = fαβ for α N 0. ThenEq. (A.4a) is equivalent to

XSs¼1

ZS2

Δh0 qsð Þ þX2Lα;β

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rf 1αβhα qsð ÞYβ

α uð Þ−E qsuð Þ !2

du

¼XSs¼1

ZS2

X2Lα;β

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rf 1αβhα qsð ÞYβ

α uð Þ−E qsuð Þ !2

du

þ 2XSs¼1

Δh0 qsð Þffiffiffiffiffiffi4π

ph0 qsð Þ−e00 qsð Þð Þ þ

XSs¼1

4πΔ2h0 qsð Þ2:

ðA:5Þ

Since e00(qs), {hα(qs)} andΔ are known constants, the problem givenby Eqs. (A.4a) and (A.4b) is equivalent to

minc

XSs¼1

ZS2

X2Lα;β

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π

2α þ 1

rf 1αβhα qsð ÞYβ

α uð Þ−E qsuð Þ !2

du ðA:6aÞ

s:t: f 1αβ ¼XLl;m

XLl0 ;m0

clmcl0m0Qmm0βll0α ; f 100 ¼ 1ffiffiffiffiffiffi

4πp ðA:6bÞ

which is exactly the NNSD with unit integral constraint.

The conditions for this equivalence are (a) minu∈S2

∑lm f �lmYml uð Þ≥ Δffiffiffiffiffiffi

4πp and (b) L is large enough. Condition (a) is to

ensure that the fODF as represented by {flm1 } is nonnegative; thiscan be satisfied if Δ ≤ 0. Condition (b) is to ensure representationof this nonnegative fODF using {clm∗ } without error.

With these two conditions satisfied, the difference between the fODFsolutions given by NNSD with and without the unit integral constraintis an isotropic component. In practice, since obtaining the solutionsinvolves non-convex optimization, local minima might cause differ-ences even in the anisotropic component.

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