THE EQUIVALENCE OF LINEAR SPHERICAL DECONVOLUTION AND MODEL-FREELINEAR TRANSFORM METHODS FOR DIFFUSION MRI

Justin P. Haldar and Richard M. Leahy

Signal and Image Processing Institute, University of Southern California, Los Angeles, CA 90089

ABSTRACT

This work provides a theoretical analysis of linear spheri-cal deconvolution methods in diffusion MRI, building offof a theoretical framework that was previously developedfor model-free linear transforms of the Fourier 2-sphere. Itis demonstrated that linear spherical deconvolution meth-ods have an equivalent representation as model-free lineartransform methods. This perspective is used to study thecharacteristics of linear spherical deconvolution from thepoint of view of the diffusion propagator. Practical results areshown with experimental brain MRI data.

Index Terms— Diffusion Magnetic Resonance Imaging,Spherical Deconvolution, Linear Transform, Orientation Dis-tribution Function, q-Space

1. INTRODUCTION

Diffusion MRI is a powerful tool for the study of white mat-ter fiber pathways in the brain [1]. An important objectivein high angular resolution diffusion MRI (HARDI) is to es-timate white matter fiber orientations from data sampled ona sphere in q-space, and a wide variety of different meth-ods have been proposed to address this problem [2]. Whilemodel-free linear transform methods [3, 4] have strong theo-retical characterizations, model-based methods like sphericaldeconvolution (SD) [5–12] are observed to also perform wellin practice, but depend on specific modeling assumptions andlack the same level of theoretical characterization. This pa-per introduces a new framework for understanding the familyof linear SD methods [5–7] at the same level of theoreticaldepth that is available for linear transform methods. This en-ables a new model-free understanding of the performance ofSD methods, helps to explain some of the current problemsfacing SD techniques [12], and sheds new light on the distinc-tion between estimated fiber orientation distributions (FODs)and orientation distribution functions (ODFs) [2].

This paper is organized as follows. In Section 2, webriefly review SD. In Section 3, we demonstrate that many

This work was supported in part by the following researchgrants: NIH-P41-RR013642, NIH-R01-EB000473, NIH-R01-NS074980,and NIH-P50-NS019632.

linear SD methods can be viewed within the linear trans-formation framework that we introduced in our previouswork [4]. In Section 4, we use these results to theoreti-cally compare SD-based linear transforms to the Funk-RadonTransform (FRT) [3] and the Funk-Radon and Cosine Trans-form (FRACT) [4]. Finally, discussion and conclusions arepresented in Section 5.

2. REVIEW OF SPHERICAL DECONVOLUTION

2.1. Diffusion MRI and HARDI

Diffusion MRI is based on the fact that water molecules inthe body are constantly undergoing random walks (Brown-ian motion) through their environment as a result of ther-mal fluctuations. We assume that each imaging voxel can beassociated with a three-dimensional probability distributionfunction f∆ (x) known as the Ensemble Average Propagator(EAP), which describes the probability (averaged over the en-semble of water molecules in the voxel) that a diffusing watermolecule will be found at a spatial displacement of x ∈ R3

relative to its starting position after a time period of length ∆.Under the q-space formalism, we can model the acquired dif-fusion data in each imaging voxel via the Fourier transform:

E (q) = C

∫R3

f∆ (x) exp(−i2πqTx

)dx, (1)

where q ∈ R3 is the q-space sampling location defined bythe diffusion encoding gradients, and C is an unknown scalefactor. In a HARDI experiment, measurements are confinedto the surface of a sphere of a given radius. Without loss ofgenerality, this work will assume the sphere has radius 1, i.e.,q ∈ S2 where S2 =

{q ∈ R3 : ‖q‖`2 = 1

}.

One approach to identifying white matter fiber orienta-tions is to estimate the ODF [2–4], which can be defined aseither

ODFFRT (u) =

∫ ∞0

f∆ (αu) dα (2)

or

ODF (u) =

∫ ∞0

f∆ (αu)α2dα (3)

for u ∈ S2. These definitions leverage the fact that wa-ter molecules tend to diffuse preferentially along directions

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aligned with white matter fibers, such that peaks in the ODFfrequently coincide with white matter orientations. The ODFdefinition in (2) is estimated by the FRT [3], while the ODFdefinition in (3) is estimated by the FRACT and a variety ofother methods [4].

2.2. Spherical Deconvolution

One of the limitations of the ODF definitions in (2) and (3)is that the ODF does not directly correspond with an FOD,whose value along any given direction should be proportionalto the number of white matter fibers oriented along that di-rection. Instead, the ODF suffers from blurring due to thefact that diffusion in a single fiber can still occur along direc-tions orthogonal to that fiber. SD attempts to overcome thislimitation by deconvolving the “fiber function” that is respon-sible for this blurring [2]. The basic assumption of SD is thatwhite matter fibers in the brain are homogeneous, such thatthe ideal noiseless diffusion signal on the sphereE (q) can berepresented as

E (q) =

∫S2

K (q,u)F (u) du, (4)

where F (q) is the FOD (defined for q ∈ S2), andK (q,u) isa fiber function that describes the measurements that would beobserved at positions q ∈ S2 for a white matter fiber popula-tion oriented along direction u ∈ S2. It is common to assumethat K (q,u) is orientationally invariant and possesses axialsymmetry with respect to the u-axis, such that the value ofK (q,u) depends only on the value of uTq.

Let the diffusion signal be represented in the sphericalharmonic (SH) basis as

E (q) =

∞∑`=0

∑m=−`

c`mYm` (q) (5)

with the SH basis functions defined as

Y m` (q) =

√2`+ 1

4π

(`−m)!

(`+m)!Pm` (cos θ) exp (imφ) , (6)

where θ ∈ [0, π] is the polar angle and φ ∈ [0, 2π] is theazimuthal angle for the vector q, and Pm` (·) is the associatedLegendre function of degree ` and orderm. Similarly, let κ`mand f`m be the (`,m)th SH coefficients of the kernelK (q,k)(where k is the unit vector along the z-axis) and the FODF (q), respectively. It has been shown [13] that

c`m = 2π

√4π

2`+ 1κ`0f`m (7)

for each m = −`, . . . , ` and ` = 0, 1, . . . ,∞. As a result ofthis relationship, some previous SD work [5–7]1 noted that if

1It should be noted that [7] made slightly different assumptions than [5,6],and treated a slightly different SD problem than the one presented in thiswork. However, the mathematics of using the SH decomposition to computelinear SD were largely the same.

E (q) and K (q,k) are provided, then the SH coefficients ofthe FOD can be obtained as a simple linear rescaling of theSH coefficients of the diffusion measurements:

f`m = w`c`m (8)

for each m = −`, . . . , ` and ` = 0, 1, . . . ,∞, where we haveintroduced the SH rescaling weights

w` =

√2`+ 1

4π√πκ`0

. (9)

3. LINEAR TRANSFORMS ASSOCIATED WITHSPHERICAL DECONVOLUTION

In our previous work [4], we showed that for any G (t) ∈L1 ([−1, 1]), we can express the action of the linear transfor-mation

O (u) =

∫S2

G(uTq

)E (q) dq (10)

in the SH domain as

o`m = λ`c`m, (11)

for each m = −`, . . . , ` and ` = 0, 1, . . . ,∞, where o`m andc`m are the SH coefficients of O (q) and E (q), respectively,and

λ` = 2π

∫ 1

−1

G (t)P` (t) dt. (12)

Note that, similar to the computation used for SD in (8),the transform in (10) represents a simple rescaling of the SHcoefficients of the measured data by a factor that depends on` but not on m. In this work, we observe that any methodthat rescales SH coefficients in this way can be interpreted interms of a linear transform of the form (10). In particular,we note that the λ` coefficients in (12) are simply the scaledinner products of G (t) with the associated Legendre func-tions. Since the associated Legendre functions are orthogo-nal, an algorithm that filters SH coefficients according to (8)is equivalent to a transformation of the form (10) with

G (t) =1

4π

∞∑`=0

w` (2`+ 1)P` (t) . (13)

The advantage of having (13) is that our previous work [4]provided a theoretical characterization of transforms of theform (10) for arbitrary kernel functions, allowing us to di-rectly adapt the tools developed in [4] for the analysis of lin-ear SD.

In particular, for kernels of the form (13), we can derivefrom [4] that the estimated FOD is related to the EAP via

F (u) = C

∫R3

f∆ (x) g(x‖, x⊥

)dx, (14)

505

where x‖ = xTu is the component of x along the axis defined

by u, x⊥ =√‖x‖2`2 − |x

Tu|2 is the component of x orthog-

onal to u, and g(x‖, x⊥

)is a response function for the linear

transform (similar to a point-spread function), which equals:

g(x‖, x⊥

)=

2π∫ 1

−1G (τ) J0

(2πx⊥

√1− τ2

)e(−i2πτx‖)dτ.

(15)

This characterization was important for deriving both theFRT and the FRACT, since the kernel functions for thesetransforms were chosen specifically so that g

(x‖, x⊥

)≈

δ (x⊥) (for the FRT) and g(x‖, x⊥

)≈ x2

‖δ (x⊥) (for theFRACT), which enables approximation of (2) and (3), re-spectively [3, 4]. In this work, this characterization alsoallows us to compare the characteristics of the FODs esti-mated with linear SD to the ODFs estimated by the FRT andFRACT. Such analysis is particularly enlightening because itenables a model-free theoretical analysis of the characteris-tics of linear SD that does not depend on the accuracy of theSD model in (4). One of the main limitations of (4) is thatit assumes that all white matter fibers have similar diffusioncharacteristics and that these characteristics can be accuratelyestimated in advance. In reality, white matter fiber bundlesare heterogeneous (e.g., different fibers can have axons withdifferent radii, and the diffusion characteristics of specificfiber bundles can change in the presence of disease), making(4) inaccurate for any choice of K (u,k). This mismatch canseverely impact the quality of white matter fiber orientationestimates [12].

4. CHARACTERIZATION OF SPHERICALDECONVOLUTION WITH TENSOR FIBER

FUNCTIONS

To explore the model-free characteristics of linear SD, weconsider the transform kernels G (t) and response func-tions g

(x‖, x⊥

)that would be obtained from fiber functions

K (u,k) associated with the standard diffusion tensor imag-ing (DTI) model. In particular, we consider the case whereK (u,k) is an axially-symmetric diffusion tensor:

K (u,k) = e−buTDu = e

−b(λ⊥+|kTu|2(λ‖−λ⊥)

), (16)

where b denotes the diffusion encoding b-value, and D is anaxially-symmetric diffusion tensor oriented along the z-axis,with axial and radial eigenvalues of λ‖ and λ⊥, respectively.DTI-based fiber functions have been used previously for lin-ear SD [5–7], and have the characteristic that, dropping con-stant scale factors, the shape of K (u,k) depends only on thesingle parameter γ = b ·

(λ‖ − λ⊥

).

We have computed transform kernels for linear SD (usingthe same filtered version of linear SD as described in [5] witha maximum SH degree of 12) with DTI-based fiber functionsspanning a range of different γ. These kernels are are shown

(a) SD (γ=1) (b) SD (γ=5) (c) SD (γ=10) (d) SD (γ=50)

(e) FRT (f) FRACT (ξ=0.05) (g) FRACT (ξ=0.34) (h) FRACT (ξ=0.5)

-MAX MAX0

Fig. 1. (a-d) Transform kernels for linear SD for fiber func-tions with different γ: (a) γ = 1, (b) γ = 5, (c) γ = 10,and (d) γ = 50. (e) Transform kernel for the truncated FRT.(f-h) Transform kernels for the truncated FRACT for differentvalues of ξ: (f) ξ = 0.05, (g) ξ = 0.34, and (h) ξ = 0.5.

(a) SD (γ=1) (b) SD (γ=5) (c) SD (γ=10) (d) SD (γ=50)

(e) FRT (f) FRACT (ξ=0.05) (g) FRACT (ξ=0.34) (h) FRACT (ξ=0.5)

Fig. 2. Response functions g(x‖, x⊥

)corresponding to the

different transform kernels shown in Fig. 1.

in Fig. 1. In addition, the figure also shows transform ker-nels associated with the FRT and the FRACT, truncated to amaximum SH degree of 12. The FRACT has a single parame-ter ξ that controls its characteristics [4], and we show imagesfor several values of ξ. The corresponding response func-tions g

(x‖, x⊥

)for all of these transform kernels are shown

in Fig. 2.Notably, these figures illustrate that in certain cases, linear

SD has striking similarities to the FRT and the FRACT, par-ticularly as γ grows larger. It should also be noted that the re-sponse functions g

(x‖, x⊥

)are such that the FODs obtained

with linear SD can be viewed as approximations of the ODFsdefined in (2) and (3). Thus, while SD methods aim to achievebetter orientation resolution than techniques based on ODFestimation, the FOD estimated with linear SD is strongly re-lated to ODFs obtained with linear transform methods, and

506

(a) SD (γ=1) (b) SD (γ=5) (c) SD (γ=10) (d) SD (γ=50)

(e) FRT (f) FRACT (ξ=0.05) (g) FRACT (ξ=0.34) (h) FRACT (ξ=0.5)

Fig. 3. Real brain FOD and ODF estimates from a crossingfiber region in the human pons, where the FODs/ODFs corre-spond to the different transform kernels shown in Fig. 1.

thus would be expected to have similar resolution limitations.To examine this, we also applied linear SD, the FRT, and

the FRACT to diffusion data from a crossing fiber region inthe human pons. Data acquisition parameters and preprocess-ing steps were the same as described in [4] and [14], respec-tively. Results are shown in Fig. 3. It should be noted thatfor this dataset, a data-driven estimate of the fiber function(from high anisotropy brain voxels that are expected to rep-resent single fibers [5]) for SD would lead to a γ value ofapproximately 2. As can be seen, linear SD performs rela-tively well as long as the γ used to define the kernel func-tion is large enough, and that the failure of linear SD occursin a situation where the kernel function has lower anisotropythan what would be expected from real fibers, in which casethe response function g

(x‖, x⊥

)also yields an integral that

bears little resemblance to the ODF computations in (2) or (3).For this data, the FRACT demonstrates higher angular resolu-tion characteristics than SD or the FRT. While other linear SDmethods exist [6, 7] that would have slightly different resolu-tion characteristics, they also did not outperform the FRACTin our experiments.

5. DISCUSSION AND CONCLUSION

This work established a model-free theoretical framework forunderstanding linear SD methods, and provided new links be-tween linear SD and linear transform methods like the FRTand the FRACT. Our results demonstrated that there is anequivalence between linear SD and linear transform methods,and that linear SD methods appear to estimate FODs usingcomputations that would also yield approximate ODFs. Thishelps to reestablish the idea that an estimated ODF can be asuseful as a direct estimate of the FOD in the determination ofwhite matter fiber orientations.

While this paper considered the case of linear SD, thereis an emerging class of nonlinear SD methods [8–12] that

will have different characteristics than the linear case exam-ined here. Though our theoretical analysis suggests that linearSD is relatively robust to the choice of fiber function, resultspresented in [4, 12] suggest that this robustness does not ex-tend directly to nonlinear SD, and that care must be taken toachieve good results.

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