Estimating diffusion propagator and its moments using directional radial basis functions

Abstract

The ensemble average diffusion propagator (EAP) obtained from diffusion MRI (dMRI) data captures important structural properties of the underlying tissue. As such, it is imperative to derive an accurate estimate of the EAP from the acquired diffusion data. In this work, we propose a novel method for estimating the EAP by representing the diffusion signal as a linear combination of directional radial basis functions scattered in q-space. In particular, we focus on a special case of anisotropic Gaussian basis functions and derive analytical expressions for the diffusion orientation distribution function (ODF), the return-to-origin probability (RTOP), and mean-squared-displacement (MSD). A significant advantage of the proposed method is that the second and the fourth order moment tensors of the EAP can be computed explicitly. This allows for computing several novel scalar indices (from the moment tensors) such as mean-fourth-order-displacement (MFD) and generalized kurtosis (GK) – which is a generalization of the mean kurtosis measure used in diffusion kurtosis imaging. Additionally, we also propose novel scalar indices computed from the signal in q-space, called the q-space mean-squared-displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD), which are sensitive to short diffusion time scales. We validate our method extensively on data obtained from a physical phantom with known crossing angle as well as on in-vivo human brain data. Our experiments demonstrate the robustness of our method for different combinations of b-values and number of gradient directions.

I. Introduction

Diffusion-weighted Magnetic Resonance Imaging (dMRI) is increasingly used to study the neural architecture and connectivity of the brain. It has proven important in clinical settings for investigating many brain disorders such as Alzheimer’s disease, schizophrenia, mild traumatic brain injury etc. [1], [2], [3]. The ensemble average diffusion propagator (EAP), which describes the average displacement of spins within a voxel during the sampling period, provides important structural information of the underlying tissue. Thus, it is important to derive a reliable method to estimate the propagator from noisy measurements.

In the narrow pulse setting (for single pulse field gradient experiment), the EAP, denoted by P(r), is related to the normalized diffusion signal E(q) ≜ S(q)/S(0) measured in the q-space via the Fourier transform:

$$P(\mathit{r})={\int }_{\mathit{q}\in {ℝ}^3}E(\mathit{q})exp(-i2\pi \mathit{q}·\mathit{r})d\mathit{q},$$ (1)

as given in [4], where r is the displacement vector, and S(q) is the diffusion signal measured at q-space point q. The vector q in the q-space is an experimentally controlled parameter with q = (2π)−¹γδG where, γ is the gyromagnetic ratio and δ is the duration of the diffusion sensitizing gradients whose magnitude and orientation are determined by the vector G. Alternatively, E can be written as a function of b-value and a unit vector u ∈ , such that E(b, u) : ℝ⁺ × → [0, 1], where b = γ²δ²(Δ − δ/3)||G||² s/mm² with Δ being the mixing time (i.e., the time between two diffusion-encoding gradients). The fundamental relation given in Eq. (1) implies that P(r) is uniquely determined by E(q) which is measured directly in experiments. Thus, the propagator estimation problem reduces to one of estimating a continuous function E(q) based on scattered measurements acquired in the q-space.

A classical method in dMRI is Diffusion Tensor Imaging (DTI) [5], which assumes that E(q) is a Gaussian function centered at the origin (q = 0). However, this over-simplified assumption has limitations in voxels where there is complex fiber architecture (crossing and kissing fiber bundles). To resolve this issue, High Angular Resolution Diffusion Imaging (HARDI) technique is nowadays becoming standard, which involves acquiring diffusion signal on a single q-shell in several gradient directions [6]. This protocol, in conjunction with various estimation methods [7], [8], [9], [10], is capable of resolving complex angular structures. However, it does not provide information about the signal decay along radial directions (with increasing q-value) which is desirable in studying subtle anomalies of white matter [11], [12].

The q-space imaging theory was first introduced by Paul Callaghan, see [13]. In practice, a imaging technique named Diffusion Spectrum Imaging (DSI) was recently introduced in [14], which involves acquiring measurements over a dense Cartesian grid of points in the q-space, followed by application of discrete Fourier transform to obtain an estimate of the EAP. However, a large number of measurements and a long acquisition time makes it impractical to use DSI in clinical settings. To address this issue, many imaging methods have been proposed, which reduce the number of measurements by using suitable signal models or by representing the signal using a series of functions in q-space [15], [16], [17], [18], [19]. For example, the Bessel Fourier basis [20], the Spherical Polar Fourier (SPF) basis [21], [22] and the spherical ridgelet basis [23] extend the spherical representation of the signal on a single shell to multiple shells with a continuous radial term. On the other hand, MAP-MRI [24] represents the diffusion signal using a combination of Hermite polynomials. Another related work is that of [25], where a linear combination of isotropic Sinc functions scattered in q-space, were used to represent the diffusion signal in q-space. This method is similar in spirit to our work, however the radial basis functions used are substantially different between the two methods. Further, we present several closed form analytical expressions for higher order moments of the EAP using the proposed method, whereas such expressions have not been derived for the Sinc basis functions.

In this work, we use directional radial basis functions for representing the diffusion signal and computing the corresponding EAP. In particular, the diffusion signal is expressed as a linear combination of anisotropic Gaussian basis functions centered at several locations in the q-space. This is in contrast to other mixture models, which typically center the basis functions at the origin in q-space, e.g. [9]. Since the Fourier transform of a Gaussian is another Gaussian, one obtains simple analytical expressions for the EAP, and other quantities derived from the EAP, such as, the orientation distribution function (ODF) and the return-to-the-origin probability (RTOP). Moreover, a significant advantage of the proposed method is that the second order and fourth order moment tensors of the EAP can be computed explicitly, from which we can compute scalar indices such as mean-squared-displacement (MSD), mean-fourth-order-displacement (MFD), generalized kurtosis (GK) and many others. These measures capture higher order statistics of the diffusion propagator (and the q-space signal), which are particularly sensitive to the hindered and restricted components of the diffusion process. We extensively validate our method on a physical phantom data set with known fiber crossing and on an in-vivo human brain data set. Experiments on in-vivo human brain data demonstrate the different properties captured by our novel scalar indices, which could potentially be helpful in investigating subtle abnormalities in the brain tissue. Additionally, we also demonstrate the robustness of our technique to faithfully recover these diffusion measures from a sparse set of measurement, thus allowing for reducing the scan time.

II. Theory

A. Radial basis functions

A radial basis function (RBF) φ(x) with x ∈ ℝ^d is a real-valued function whose value at x depends only on its distance to the origin. In the case of Euclidean distance, φ(x) can be written as ϕ(||x||). RBF’s are typically used for functional approximations of the form

$$s(\mathit{x})=\sum _{n=1}^N{w}_n\varphi (\Vert \mathit{x}-{\mathit{c}}_n\Vert ),$$ (2)

where the function to be approximated, s(x), is represented as a sum of N radial basis functions each associated with a different center c_n and a weighting coefficient w_n. A fundamental theorem derived in [26] shows that, if the function ϕ is continuous and bounded such that ∫_ℝ^d ϕ(x) ≠ 0, then the function s(x) written in the form of (2) is dense in L^p(ℝ^d) (which is the set of p^th power integrable functions). In other words, for a sufficiently large number N and suitably chosen centers c_n’s, the radial basis functions can be used to approximate any well-behaved function up to any degree of precision. Hence, RBF’s are widely used in many modern applications to approximate multivariate functions [27], [28].

A type of commonly used radial basis function is a Gaussian: ϕ(x) = exp(−σ||x||²) where, the normalization term is omitted since it can be absorbed into the weights, and ||x|| denotes the Euclidian distance from x to the origin. In many situations, the so-called Mahalanobis distance is used instead of the Euclidean distance [29]. In general, a Mahalanobis distance between two points x, y ∈ ℝ^d is defined by appositive definite matrix Σ, and is given by: ${\Vert \mathit{x}-\mathit{y}\Vert }_{\mathrm{\sum }}=\sqrt{{(\mathit{x}-\mathit{y})}^T\mathrm{\sum }(\mathit{x}-\mathit{y})}$. Hence, the corresponding radial basis function $\varphi (x)=exp(-{\Vert \mathit{x}\Vert }_{\mathrm{\sum }}^2)$ is a (scaled) multivariate Gaussian.

B. Application in dMRI

We adapt the methodology of RBF’s to represent the dMRI signal continuously in the q-space. Let {q̂₁, …, q̂_N} be N fixed points (chosen a-priori) that are densely scattered in q-space which are also the centers of RBF’s. We represent the normalized diffusion signal using

$$E(\mathit{q})=\sum _{n=0}^N{w}_n{\varphi }_n(\mathit{q}-{\widehat{\mathit{q}}}_n),$$ (3)

with ϕ_n(q − q̂_n) = exp(−(q−q̂_n)^TD_n(q − q̂_n)) and q̂₀ = 0. This representation can also be thought of as a generalization of a scheme that involves placing all the basis functions at the origin. However, using a non-centered set of Gaussians allows for better modeling the signal decay at high q-value. Further, the theory of radial basis functions provides theoretical guarantees on the ability of such a framework to be able to represent any function with high accuracy [28].

The tensor D₀ is computed as in standard DTI [5] and the tensors D_n’s with n ≥1 are assumed to have a cylindrical shape, which have the same set of eigenvectors as D₀ and with eigenvalues σ₀, σ₁, σ₂ such that σ₁ = σ₂. This anisotropic shape of D_n is motivated from the fact that the signal E(q) decays faster along the direction indicated by the principle eigenvector of D₀. While the eigenvectors of D_n could, theoretically, be set to random orientations, however, utilizing the orientation of D₀ could aid in better estimation of the signal decay. We should note that, even though the orientation of D_n is fixed to that of D₀, yet, by an appropriate choice of the weights w_n, one can represent any type of anisotropic function to a very high degree of accuracy [27]. Moreover, using the eigenvectors of D₀ for D_n also leads to simpler expressions for several scalar statistics as shown in Appendix C-A. We note that without the first term w₀ϕ₀(q), Eq. (3) is a standard radial basis function approximation problem. This extra term at the origin provides better fit for measurements from gray matter and CSF areas where signal decays almost isotropically along all directions.

Since the dMRI measurements are assumed to have antipodal symmetry i.e., E(q) = E(−q), we impose equal weights for Gaussian functions placed at antipodal pairs of points:

$$E(\mathit{q})=\sum _{n=0}^N{w}_n[{\varphi }_n(\mathit{q}-{\widehat{\mathit{q}}}_n)+{\varphi }_n(\mathit{q}+{\widehat{\mathit{q}}}_n)].$$ (4)

The weights w_n’s are typically computed such that the mean squared error between E(q) and the measurements is minimized. Given these weights, the estimated signal can be interpolated continuously in the q-space. An advantage of using Gaussian basis functions for representing the measured signal is that its Fourier transform, namely the EAP, can be computed analytically.

C. Estimation of EAP and ODF

Since the reconstructed signal E(q) is a linear combination of Gaussian functions, its Fourier transform is given by a linear combination of the Fourier transforms of the individual basis functions. Hence, the estimated EAP, P(r), is of the form

$$P(\mathit{r})=\mathcal{F}(E(\mathit{q}))=\sum _{n=0}^N{w}_n{\mathrm{\Phi }}_n(\mathit{r}),$$ (5)

with Φ_n = (ϕ_n(q − q_n) + ϕ_n(q + q_n)), where denotes the Fourier transform. In particular, the Fourier transform of the Gaussian function ϕ_n(q) is given by

$${\int }_{{ℝ}^3}exp(-i2\pi \mathit{q}·\mathit{r})exp(-{\mathit{q}}^T{D}_n\mathit{q})d\mathit{q}={\pi }^{\frac{3}{2}}{\mid D_n\mid }^{-\frac{1}{2}}exp(-{\pi }^2{\mathit{r}}^T{D}_n^{-1}\mathit{r}).$$

A translation of a basis function leads to a phase shift of its Fourier transform, i.e. (ϕ_n(q + q̂_n)) = exp(i2πq̂_n · r) (ϕ_n(q)). Hence,

$${\mathrm{\Phi }}_n(\mathit{r})=2{\pi }^{\frac{3}{2}}{\mid D_n\mid }^{-\frac{1}{2}}cos(2\pi {\widehat{\mathit{q}}}_n·\mathit{r})exp(-{\pi }^2{\mathit{r}}^T{D}_n^{-1}\mathit{r}).$$ (6)

Though each individual Φ_n(r) has exponentially decayed oscillation, a combination of many basis functions may provide a positive diffusion propagator (at least within a large range of r) with suitable constraints for the diffusion signal, see Section VI for more detail.

In dMRI, the orientational feature of the underlying tissue can be visualized by using the diffusion orientation distribution function (ODF), which is the marginal of the propagator P(r) on the unit sphere. More specifically, the (solid angle version) ODF, denoted by Ψ(u), is computed from the EAP by evaluating the integral

$$\mathrm{\Psi }(\mathit{u})={\int }_0^{\infty }P(r\mathit{u})r^{2}dr,$$

where u is a unit vector and r is the radial co-ordinate [14]. From the propagator P(r), Ψ(u) can be computed analytically as $\mathrm{\Psi }(\mathit{u})={\sum }_{n=0}^N{w}_n{\mathrm{\Psi }}_n(\mathit{u})$, with ${\mathrm{\Psi }}_n(\mathit{u})={\int }_0^{\infty }{\mathrm{\Phi }}_n(r\mathit{u})r^{2}dr$ which is given by

$${\mathrm{\Psi }}_n(\mathit{u})=\frac{1}{2\pi }{\mid D_n\mid }^{-\frac{1}{2}}{({\mathit{u}}^T{D}_n^{-1}\mathit{u})}^{-\frac{3}{2}}\left(1-\frac{2{(\mathit{u}·{\widehat{\mathit{q}}}_n)}^2}{{\mathit{u}}^T{D}_n^{-1}\mathit{u}}\right)\times exp\left(-\frac{{(\mathit{u}·{\widehat{\mathit{q}}}_n)}^2}{{\mathit{u}}^T{D}_n^{-1}\mathit{u}}\right).$$ (7)

We provide more details for Ψ_n in Appendix A.

One significant advantage of the proposed method is that the moments of the EAP can be computed explicitly. These moments provide statistics that describe important features of the underlying tissue. Thus, they can be potentially used in investigating tissue abnormalities. Let $\mathit{r}={[\begin{array}{ccc}x& y& z\end{array}]}^T$ denote the displacement vector and γ ≜ r⊗r ∈ ℝ⁹, where ⊗ denotes the Kronecker product. Then the covariance matrix of r is defined as

$$R\triangleq {\int }_{{ℝ}^3}{\mathit{rr}}^{T}P(\mathit{r})d\mathit{r},$$ (8)

which is a 3 × 3 positive definite matrix whose eigenvectors determine the principal diffusion directions. The diagonal entries of R are the mean-squared displacements along the x, y and z directions, respectively. Moreover, the fourth order diffusion tensor is given by

$$M\triangleq {\int }_{{ℝ}^3}\mathit{\gamma }{\mathit{\gamma }}^{T}P(\mathit{r})d\mathit{r}.$$

Explicit expressions for R and M are given in Appendix B.

III. Scalar indices derived from the EAP

In this section, we discuss some quantitative scalar indices that could be used to describe various features of the underlying diffusion process. We also provide closed form expressions for these indices.

A. Mean-squared-displacement (MSD) and mean-fourth-order-displacement (MFD)

The mean-squared-displacement (MSD) is proportional to the average amount of diffusion that occurs during the diffusion experiment. It can be computed from the following expression:

$$\text{MSD}={\int }_{{ℝ}^3}{\Vert \mathit{r}\Vert }^{2}P(\mathit{r})d\mathit{r}=\text{trace}(R).$$

The last equality follows from (8) and the fact that ||r||² = trace(rr^T), where trace(·) denotes the trace operator. Similarly, the mean-fourth-order-displacement (MFD), which is defined as

$$\text{MFD}={\int }_{{ℝ}^3}{\Vert \mathit{r}\Vert }^{4}P(\mathit{r})d\mathit{r}=\text{trace}(M),$$

can be computed from the fourth order moment tensor M. This measure is more sensitive to the “tail” of the diffusion propagator, and hence it can capture contributions from the hindered components of the diffusion propagator.

B. Hindered and restricted diffusivity

In standard DTI, the diffusion of water molecules is modeled by a Gaussian propagator G(r) with a covariance matrix $R_g=\frac{1}{2{\pi }^2}{D}_0$. However, the diffusion in human brain tissue is typically restricted or hindered, making the diffusion propagator non-Gaussian [30], [31]. The amount of hindered and restricted diffusivity can be captured by computing the discrepancy between the estimated propagator P(r) and the Gaussian propagator G(r). For this purpose, we compute an angular metric between P(r) and G(r) as in [24].

1) Non-Gaussianity (NG)

Consider two propagators P(r) and Q(r) with their inner product defined as 〈P(r), Q(r)〉 = ∫_ℝ³ P(r)Q(r)dr. Suppose P(r) and Q(r) are expressed in the form of (5) with the weighting coefficients given by w_n and v_n for n = 0, …, N, respectively. Then the inner product can be written as $\langle P(\mathit{r}),Q(\mathit{r})\rangle ={\sum }_{m=0}^N{\sum }_{n=0}^N{w}_n{v}_n{T}_{mn}$ where

$$T_{mn}=\langle {\mathrm{\Phi }}_m(\mathit{r}),{\mathrm{\Phi }}_n(\mathit{r})\rangle =2(\langle {\varphi }_m(\mathit{q}-{\widehat{\mathit{q}}}_m),{\varphi }_n(\mathit{q}-{\widehat{\mathit{q}}}_n)\rangle +\langle {\varphi }_m(\mathit{q}-{\widehat{\mathit{q}}}_m),{\varphi }_n(\mathit{q}+{\widehat{\mathit{q}}}_n)\rangle )$$

with the second equality obtained from Parseval’s theorem and the inner product between two basis functions ϕ_m(q − q̂_m)

$$\langle {\varphi }_m(\mathit{q}-{\widehat{\mathit{q}}}_m),{\varphi }_n(\mathit{q}-{\widehat{\mathit{q}}}_n)\rangle ={\pi }^{\frac{3}{2}}{\mid D_m+D_n\mid }^{-\frac{1}{2}}exp(-{\widehat{\mathit{q}}}_m^T{D}_m{\widehat{\mathit{q}}}_m-{\widehat{\mathit{q}}}_n^T{D}_n{\widehat{\mathit{q}}}_n+{(D_m{\widehat{\mathit{q}}}_m+D_n{\widehat{\mathit{q}}}_n)}^T{(D_m+D_n)}^{-1}(D_m{\widehat{\mathit{q}}}_m+D_n{\widehat{\mathit{q}}}_n)).$$ (9)

Similar to the method used in [24], the angular metric between the two propagators is defined as

$$cos{\theta }_{PQ}=\frac{\langle P(\mathit{r}),Q(\mathit{r})\rangle }{{(\langle P(\mathit{r}),P(\mathit{r})\rangle \langle Q(\mathit{r}),Q(\mathit{r})\rangle )}^{\frac{1}{2}}}.$$

In terms of the representation coefficients in the basis functions, the angular metric is given by the expression

$$cos{\theta }_{PQ}=\frac{{\sum }_{m=0}^N{\sum }_{n=0}^N{w}_n{v}_n{T}_{mn}}{{\left({\sum }_{m=0}^N{\sum }_{n=0}^N{w}_m{w}_n{T}_{mn}{\sum }_{m=0}^N{\sum }_{n=0}^N{v}_m{v}_n{T}_{mn}\right)}^{\frac{1}{2}}}$$ (10)

Consequently, a dissimilarity measure can be defined as $sin{\theta }_{PQ}=\sqrt{1-cos{\theta }_{PQ}^2}$ which takes value between 0 and 1. In practice, one may want to scale the value of the dissimilarity measures in order to obtain better contrast. For this purpose, we used the same family of functions as in [24] which is given by σ(t, ε) = t^3ε/(1−3t^ε + 3t^2ε) where t is the value to be scaled, e.g. t = sin θ_PQ, and the parameter ε > 0. The non-Gaussianity (NG) of the diffusion of water molecules is reflected by the dissimilarity between the estimated EAP P(r) and the Gaussian propagator G(r). To this end, the value of cos θ_PG is given by

$$cos{\theta }_{PG}=\frac{{\sum }_{m=0}^N{w}_m{T}_{m0}}{{\left({\sum }_{m=0}^N{\sum }_{n=0}^N{w}_m{w}_n{T}_{mn}{T}_{00}\right)}^{\frac{1}{2}}}.$$

We define the non-Gaussianity (NG) as

$$\text{NG}\triangleq \sigma (sin{\theta }_{PG},\epsilon )$$

and throughout this paper we use ε = 0.4 as in [24].

2) Difference in covariances

As a distance measure between the diffusion propagators, NG reflects the difference in all the moments of the estimated EAP and a Gaussian propagator. Since the higher order moments of the EAP are determined by the higher order derivatives of the diffusion signal near the origin, they are usually more sensitive to measurement noise. Consequently, a distance measure between only the second order moments R and R_g may be used to measure the hindered or the restricted diffusion though it only partially reflects the difference between the propagators. In the following, we present two methods for comparing the second order moments.

Let r̂ denote the displacement of molecules that is distributed according to a zero-mean Gaussian function G(r̂). If the propagator is non-Gaussian, each displacement vector r̂ is associated with a difference in displacement r̃ such that the true displacement r̂ + r̃ actually has a covariance matrix R (as opposed to R_g). There are many ways to account for the difference in displacement, which could lead to the covariance matrix R. One choice is based on the theory of optimal mass transport, and is also closely related to the Wasserstein metric for comparing probability measures [32]. This measure, which we define as a measure of Difference in Covariances is given by:

$$\text{DC}\triangleq \underset{\stackrel{\sim }{\mathit{r}}}{min}{\int }_{{ℝ}^3}{\Vert \stackrel{\sim }{\mathit{r}}\Vert }^{2}G(\mathit{r})d\mathit{r},$$

such that the covariance matrix of r̂ +r̃ is R. DC has a closed form expression in terms of R and R_g, (see e.g. [33], [34]) and is given by:

$$\text{DC}=\text{trace}\left(R+R_g-2{(R_g^{\frac{1}{2}}R{R}_g^{\frac{1}{2}})}^{\frac{1}{2}}\right).$$

DC measures the difference in the second moment of the diffusion propagator which partially reflects the hindrance or restriction on the diffusing molecules.

Alternatively, a symmetric comparison of the relative difference between R and R_g can be computed using $\text{trace}(R^{-1}{R}_g+R_g^{-1}R)-6$. Based on the inequality $x+\frac{1}{x}\ge 2$ with x > 0, it can be easily shown that this distance measure is nonnegative and it equals to zeros if and only if R = R_g. In practice, we expect that both this distance measure and DC have similar contrast between tissues, however the relative distance may be more sensitive to noise due to the matrix inversion. To this end, we only consider DC in the experiments of the paper.

3) Generalized Kurtosis (GK)

A multivariate generalization of kurtosis was studied in [35] and is given by

$$\text{GK}\triangleq {\langle {({\mathit{r}}^T{R}^{-1}\mathit{r})}^2\rangle }_P={\int }_{{ℝ}^3}{({\mathit{r}}^T{R}^{-1}\mathit{r})}^{2}P(\mathit{r})d\mathit{r},$$

where R is the moment tensor (covariance matrix) of the diffusion propagator P(r). It can be easily shown that, if P(r) is a Gaussian distribution function, then GK = 15. Thus, GK > 15 implies that P(r) has a fatter tail than a Gaussian and is called super-Gaussian. On the other hand, sub-Gaussian diffusion leads to GK < 15. In our framework, GK can be easily computed from: GK = y^TMy where y ∈ ℝ⁹ is obtained by packing all the columns of R⁻¹ into a vector. Thus, this measure has a specific range of scalar values that can be used to interpret the type of the diffusion propagator (Gaussian, sub-Gaussian, super-Gaussian), unlike the “mean kurtosis” measure typically used in the literature [36].

Since MSD and MFD are the second and the fourth order moments of the ℓ₂-norm ||r|| of the displacement, respectively, an alternative way of generalizing the mean kurtosis is to compare MFD and (MSD)². The this end, we define the Generalized Kurtosis of the Norm (GKN) of displacement as

$$\text{GKN}\triangleq \frac{{\langle {\Vert \mathit{r}\Vert }^4\rangle }_P}{{\langle {\Vert \mathit{r}\Vert }^2\rangle }_P^2}=\frac{\text{MFD}}{{(\text{MSD})}^2}.$$

If P(r) is Gaussian, GKN measures the anisotropy of the diffusion process. In particular, it can be shown that GKN = 5/3, if P(r) is isotropic Gaussian, and it equals to 3 for a completely anisotropic Gaussian. If P(r) is not a Gaussian, then the fatter tail of P(r) leads to higher values for GKN.

Since both GK and GKN reflect the kurtosis of the computed propagator, we expect them to have similar contrast between different tissue types. We also note that though the definition of GK is practically less attractive, as it is usually more sensitive to the measurement noise because it needs the computation of a matrix inversion. A quick comparison between these moment based measures and the one obtained by computing the difference between the entire propagators (NG), is that, the measures such as GK and GKN have specific numerical ranges that make it intuitive to understand the physical meaning behind the estimated numbers.

Similar to the scalar statistics derived for the EAP, explicit expressions for the moments of q-space signal also provides several important indices that characterize different properties of the underlying tissue. In Appendix C, we provide explicit expressions for the return-to-origin probability (RTOP), the return-to-the-axis probability (RTAP), the return-to-the-plane probability (RTPP), the q-space mean-squared displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD).

IV. Estimation method

The first step in the estimation procedure is to fix the parameters of the radial basis functions. In this work, the radial basis functions are parameterized by the tensors D_n’s and the centers q̂_n’s. As was explained earlier, the tensors D_n’s (at each voxel) have the same eigenvectors as that of D₀, which in turn is estimated using a standard tensor fitting technique as in DTI. Note that, the eigenvalues for D_n are different from that of D₀ and are fixed a-priori. On the other hand, the centers q̂_n’s, for n=1, …, N, are chosen as a set of dense points in q-space (for all voxels). In the examples of this paper, we chose N = 162 and the centers are distributed on two b-shells along 81 gradients. A detailed description for choosing these parameters is presented in Section V-B. We note that once the basis functions have been defined, the EAP, the ODF and the moments are all a function of the weights w_n. Thus the performance of the proposed method heavily relies on the strategy used in estimating these coefficients. In this section, we discuss different methods for computing the coefficients w_n’s.

Given K measurements of E(q) at {q₁, …, q_K}, we denote e ∈ ℝ^K×1 as the vector of measurements, i.e. e_k = E(q_k) for k = 1, …, K. From the Gaussian basis functions, one constructs a K × (N + 1) dimensional matrix A with

$$A_{k,n+1}={\varphi }_n({\mathit{q}}_k-{\widehat{\mathit{q}}}_n)+{\varphi }_n({\mathit{q}}_k+{\widehat{\mathit{q}}}_n)$$

for n = 0, …, N. We denote by w, a vector of size N +1 whose entries are the coefficients w_n (to be estimated). Typically, the vector w can be estimated such that the mean squared error between Aw and e is minimized.

Since the linear system Aw = e may be underdetermined in cases when only a small number of measurements are available, a simple method is to estimate w as a solution of the Tikhonov regularization problem

$$\underset{\mathit{w}}{min}{\Vert A\mathit{w}-\mathit{e}\Vert }^2+\lambda {\Vert \mathit{w}\Vert }^2$$

with λ ≥ 0 being the weighting coefficient. This optimization problem has a closed form solution given by w_ℓ2 = (A^TA + λI)⁻¹A^Te.

More recently, the ℓ₁ regularization method has been shown to be useful in solving linear inverse problems especially when the vector w is assumed to be sparse. In this case, w is estimated as the solution of the optimization problem:

$$\underset{\mathit{w}}{min}{\Vert A\mathit{w}-\mathit{e}\Vert }^2+\lambda {\Vert \mathit{w}\Vert }_1.$$

This minimization approach, however, does not take into account the decay of the diffusion signal with increasing q-value. Several authors have reported that the observed signal with increasing b-values shows a multi-exponential type of decay (or a power-law decay) [30], [31], [37]. Hence, we assume that the diffusion measurements are monotonically decreasing with increasing b-values. Moreover, the diffusion signal should be positive and the value of the measurement (normalized) at the origin is known to be one. Thus, one can numerically enforce these constraints while estimating the weight vector w by solving the following optimization problem:

$$\begin{array}{l}\underset{\mathit{w}}{min}{\Vert A\mathit{w}-\mathit{e}\Vert }^2+\lambda {\Vert w\Vert }^2\\ \text{s.t.}B\mathit{w}\ge 0,{\mathit{c}}^T\mathit{w}=1.\end{array}$$ (11)

The matrix B is of the form $B={[B_1^T,B_2^T]}^T$ with each column of B₁ being ϕ_n evaluated at a given set of locations for q and each column of B₂ being the difference between the ϕ_n’s along a given set of gradient directions at several b-value shells. Specifically, we choose a set of points on X shells with b-values b = b₁, …, b_X and b₁ ≤ b₂ ≤ … ≤ b_X. On each b-value shell, we use Y points along the directions u₁, …, u_Y. Hence, the total number of points is XY. We denote these points as q_x,y for x = 1, …, X and y = 1, …, Y. The matrix B₁ has size XY × (N + 1) and each column of B₁ is the value of the basis function at the chosen set of XY points. In particular, the (xy, n)th element of B₁ is given by

$$B_1(xy,n)={\varphi }_{n-1}({\mathit{q}}_{x,y}-{\widehat{\mathit{q}}}_{n-1})+{\varphi }_{n-1}({\mathit{q}}_{x,y}+{\widehat{\mathit{q}}}_{n-1}),$$

for x = 1, …, X, y = 1, …, Y and n = 1, …, N + 1. The matrix B₂ has size (X − 1)Y × (N + 1) and the xy × n the entry of B₂ is given by

$$B_2(xy,n)=B_1(xy,n)-B_1((x+1)y,n),$$

for x = 1, …, X−1, y = 1, …, Y and n = 1, …, N + 1. Each entry in the vector c is the value of basis function at the origin, i.e. c_n+1 = 2ϕ_n(q̂_n). We note that these constraints are always feasible and a trivially feasible element is given by w = [1, 0, …, 0]^T.

V. Experiments

We tested our method on a data set acquired from a spherical physical phantom as shown in Figure 1a with a crossing angle of 45 degree. The phantom was developed along the same lines as described in [38]. Grooves of size 1 × 0.7 cm² were filed with polyfil fibers of diameter 15 μm. The fibers were dipped in NaCl solution during the winding process to potentially remove air between fibers. After the winding, the phantom was immediately placed into a bin and its position was fixed by casting it into agarose gel.

Fig. 1.

(a) The spherical phantom used for data acquisition. (b) The colored FA image of the data set used in SPARC. (c) The baseline image of the data set.

The data sets were acquired on a 3T Siemens scanner with voxel size of 2 × 2 × 7 mm³. We acquired 10 separate scans of the phantom with 5 different b-values with b = {1000, 2000, 3000, 4000, 5000} s/mm² and each b-value shell consisted of 81 gradient directions. The 10 scans were averaged to obtain the “gold-standard” data. The experimental parameters for b = 5000 s/mm² were as follows: ||G|| = 20.79 mT/m, TE/TR = 141/3400 ms and δ ≈ Δ = 62 ms. The FOV was 118 × 54 and in-plane GRAPPA acceleration factor of 2 was used during image acquisition. The color coded FA and the B₀ images of the data set are shown in Figure 1b and 1c, respectively.

The test data set was based on one slice of the physical phantom and was acquired as follows: For each of the following number of gradient directions K = {16, 20, 24, 30, 36, 42, 60, 81}, we acquired the diffusion measurement over 2 b-shells corresponding to b = {1000, 3000}s/mm² and b = {1000, 2000}s/mm², respectively. Further, 5 repetitions were acquired for each of these data samples to test the effect of noise on signal reconstruction quality. For data sets with the same number of gradient directions, we compute the signal-to-noise ratio (SNR) for signal S_r(q) at the rth voxel at q as $\frac{\text{Mean}(\{S_r^k(\mathit{q}),k=1,\dots ,5\})}{\text{STD}(\{S_r^k(\mathit{q}),k=1,\dots ,5\})}$ where STD denotes the standard deviation. The average SNR over all gradient directions and over all data sets is considered the SNR in the corresponding voxel. Table I shows the average SNR in different voxels with different b-values.

TABLE I.

The average SNR of the phantom data sets

	b = 0	b = 1000	b = 2000	b = 3000
single-fiber voxels	51	21	13	10
multiple-fiber voxels	26	15	11	9

A. Comparison metrics

We demonstrate the performance of our method on different number of measurements as well as on different b-values using the following comparison metrics.

a) Normalized mean-squared error (NMSE) in signal reconstruction

Let ê_x denote the reconstructed signal at location x that is estimated at the same points in q-space as the “gold-standard” signal e_x,gold. The normalized mean squared error (NMSE) in signal reconstruction was computed using

$$\text{NMSE}=\frac{1}{\mid \mathrm{\Omega }\mid }\sum _{\mathit{x}\in \mathrm{\Omega }}\frac{{\Vert {\widehat{\mathit{e}}}_{\mathit{x}}-{\mathit{e}}_{\mathit{x},\text{gold}}\Vert }^2}{{\Vert {\mathit{e}}_{\mathit{x},\text{gold}}\Vert }^2},$$

where Ω denotes the set of all voxel locations and |Ω| is the total number of voxels. Thus, lower NMSE implies that the reconstructed signal is close to the “gold-standard” signal.

b) Estimated angle (EA)

The orientation of the fibers at each voxel location is indicated by the peaks of the estimated ODF. These peaks are often used in tractography methods to determine the connectivity of the brain. Thus, accurate estimation of the ODF peaks is important in studying white matter fiber bundles. For the case of the phantom data set, the fiber crossing was at a known angle of 45°. The estimated angle (EA) was computed at each of the voxels where a crossing of two fibers was detected.

The estimated angle between the two principal diffusion directions (in case of crossing) was computed as the average estimated angle (EA) given by

$$\text{EA}=\frac{1}{\mid {\mathrm{\Omega }}_2\mid }\sum _{\mathit{x}\in {\mathrm{\Omega }}_2}\mid arccos({\mathit{u}}_{\mathit{x},1}·{\mathit{u}}_{\mathit{x},2})\mid ,$$

where u_x,1 and u_x,2 denote the direction of the two peaks at location x and Ω₂ denotes the set of locations that have two peaks.

c) Percentage of false peaks (PFP)

The estimated angle was computed only at those locations where two peaks were detected. However, in many cases, the recovered signal may miss or overestimate the number of peaks. Thus, the percentage of false peaks (PFP) is used to evaluate the accuracy of peak detection with respect to the gold standard. Based on the results from gold-standard data set, two sets of voxels were selected such that Ω_gold,1 contained only those voxels with one peak, while Ω_gold,2 contained all voxels with two crossing fibers. For each set, the percentage of false peaks (PFP) was computed using:

$$\text{PFP}(\mathrm{i})=\frac{1}{\mid {\mathrm{\Omega }}_{\text{gold},i}\mid }\sum _{\mathit{x}\in {\mathrm{\Omega }}_{\text{gold},i}}\mathcal{B}(n_{\mathit{x}}-i),\text{for}i=1,2,$$

where n_x denotes the number of peaks detected at location x and is a binary function whose value is 1 if n_x = i and 0 otherwise.

d) NMSE in estimation of scalar statistics

Let P_x(0) denote the estimated return-to-origin probability (RTOP) and P_gold,x(0) be the estimated RTOP for the gold-standard data, both computed using the same method (ℓ₁ regularized, 3D-SHORE etc.). The normalized mean-squared error in the estimation of RTOP is computed using

$$\text{NMSE}\text{of}\text{RTOP}=\frac{1}{\mid \mathrm{\Omega }\mid }\sum _{\mathit{x}\in \mathrm{\Omega }}\frac{{\Vert P_{\mathit{x}}(0)-P_{\text{gold},\mathit{x}}(0)\Vert }^2}{{\Vert P_{\text{gold},\mathit{x}}(0)\Vert }^2}.$$

The normalized-mean-squared-errors for RTAP, RTPP, MSD, MFD, MMSDE, NG, DC, GK, GKN, QMSD and QMFD were defined in a similar fashion.

B. Implementation details

We used these metrics to quantify the reconstruction quality of the data using the anisotropic Gaussian basis functions. All the three methods discussed in the previous section, i.e. the ℓ₁ regularized, ℓ₂ regularized and ℓ₂ regularized with constraints, were used to estimate the representation vector w. The centers q̂_n of Gaussian basis function were uniformly distributed on two b-value shells with b = {2000, 4000} along 81 gradient directions on the hemisphere. Thus, N = 162 and the total number of columns of the basis matrix A was 163. We note that the location of center points could vary for different acquisition schemes. Following the theory of radial basis functions [28], as long as the center points are densely placed in q-space and enough measurements are available, the fitted signal will be an accurate estimate of the true signal. In our experiments, we used the same set of q̂_n’s in all the examples (for dense and sparsely sampled data, both in-vivo and phantom) to show the robustness of our choice of the center points. For the ℓ₂ regularized methods, the parameter λ was chosen such that the condition number of A^TA + λI is bounded by 10⁷. This bound is chosen to be large enough so that λ is small and the estimated signal is mainly determined by the least-squares fitting term in the objective function. For the method using ℓ₂ regularization with constraints, the signal was constrained to be non-negative along 81 gradient directions uniformly sampled on the hemisphere. Along each gradient direction, the difference in the signal at the b-value shells {1000, 2000, …, 8000}s/mm² were constrained to be non-negative. The eigenvalues for the Gaussian basis functions ϕ_n were set to σ₀ = 0.0011 mm²/s, σ₁ = 0.0006 mm²/s for the ℓ₂ and ℓ₁ regularization methods, while for the constrained ℓ₂ regularization method, the parameters were set to σ₀ = 0.0015 mm²/s and σ₁ = 0.0008 mm²/s. These parameters were selected using an exhaustive search, so that the estimated angle with 30 gradient directions was as close as possible to the ground truth. We should however emphasize that, we used the same set of parameters for all sampling schemes and data sets, to demonstrate the robustness of the proposed method. Further, the training data set was different than the test data set used to obtain the optimal parameters (σ₀, σ₁).

We compared the proposed methods with the 3D-SHORE method of [19]. In this case, we constructed the basis matrix A according to the 3D-SHORE basis functions and estimated the coefficient vector using ℓ₁ regularization [19], ℓ₂ regularization [39], [19], ℓ₂ regularization with the constraints for monotonicity and positivity of the signal (for a fair comparison with our Gaussian basis function based method). The scale parameter of 3D-SHORE basis and the regularization coefficients were chosen as described in [19].

C. Phantom results

Our main goal in studying phantom data is to quantitatively evaluate the performance of the different estimation methods. Moreover, by comparing the estimation results we can also determine the most economical sampling scheme for the proposed method. In particular, if the measurements are to be sampled on two b-value shells with b = {1000, 3000}, what is the trade off between the performance and the acquisition time? If the scan time allows to acquire only 60 measurements, should we use b-values with b = {1000, 2000} or use b = {1000, 3000} and 30 directions each? The extensive validation we perform in this work is an attempt to answer these questions using the proposed methods.

The estimation results on the “gold standard” data (where the actual angle is 45°) with different methods are summarized in Table II:

TABLE II.

Gold standard error metrics

Method	NMSE	EA	Method	NMSE	EA
ℓ1 Gaussian	0.5%	45.5°	ℓ1 3D-SHORE	0.3%	45.9°
ℓ2 Gaussian	0.5%	44.2°	ℓ2 3D-SHORE	1.0%	48.8°
ℓ2 Gaussian-cons.	1.1%	44.6°	ℓ2 3D-SHORE cons.	1.1%	48.7°

The estimated ODF’s with gold-standard data using the six methods are shown in Figure 7 in Appendix D. All the six methods successfully detected the crossing fibers based on the gold-standard data set. In Figure 7d and Figure 7f, the top-left and bottom-right corners of the glyph’s are not displayed since the estimated ODF’s have negative values in these isotropic regions.

Fig. 7.

Estimated ODF using “gold-standard” data: (a) Gaussian basis with ℓ₂ regularization and constraints,(b) 3D-SHORE basis with ℓ₂ regularization and constraints, (c) Gaussian basis with ℓ₂ regularization, (d) 3D-SHORE basis with ℓ₂ regularization, (e) Gaussian basis with ℓ₁ regularization, (f) 3D-SHORE basis with ℓ₁ regularization.

We first compared our results using the test data set with b = {1000, 3000} and the number of gradient directions K = {16, 20, 24, 30, 36, 42, 60, 81}. For visual inspections, Figure 8 in Appendix D shows the estimated ODF’s using one data set with 30 gradient directions on each shell (60 total measurements). In particular, the figures on the left panel are obtained using the proposed Gaussian basis methods and the results obtained using 3D-SHORE are shown on the right. For the gold-standard data set, as was shown in Figure 7, all the methods successfully identified the crossing fibers. However, with fewer number of samples, the performance of the 3D-SHORE basis methods deteriorated. Many voxels in the crossing region in Figure 8b, 8d and 8f did not detect the fiber crossing.

Fig. 8.

Estimated ODF from measurements on b-value shells with b = {1000, 3000} and with 30 gradient directions each (total of 60 measurements) using: (a) Gaussian basis with ℓ₂ regularization and constraints, (b) 3D-SHORE basis with ℓ₂ regularization and constraints, (c) Gaussian basis with ℓ₂ regularization, (d) 3D-SHORE basis with ℓ₂ regularization, (e) Gaussian basis with ℓ₁ regularization, (f) 3D-SHORE basis with ℓ₁ regularization

1) Estimated angle

The orientation of the fibers is estimated via the peaks of the ODF. A maxima of the ODF is considered as a valid peak if its value is at-least 40% of the highest peak in that ODF. Considering the fact that the estimated ODF’s may have small spurious peaks (see Figure 7f and Figure 8e), this threshold of 40%, was chosen so that these artifacts do not lead to a significant number of incorrect detection in the 3D-SHORE method.

Numerically, the ODF is evaluated on a grid of 2562 points sampled on the sphere. If there are only two peaks detected, the directions corresponding to the peaks are used to compute the estimated angle in that voxel. The average estimated angles for the six methods are shown in Figure 2a. We note that if the number of gradient directions is larger than 30, the estimated angle using all the three Gaussian basis function based methods (red curves) are within 1.5° of the ground truth. Results also show that, the ℓ₂ regularization method with 3D-SHORE basis underestimated the angle with fewer number of samples. The angle obtained using ℓ₁ 3D-SHORE method is within 2° of the ground truth for K ≥ 30.

Fig. 2.

Comparison metrics using measurements on b-value shells with b = {1000, 3000} and gradient directions per shell (x-axis): (a) Estimated angle vs. gradient directions, (b) Percentage of false peaks for two-peak voxels vs. gradient directions, (c) Percentage of false peaks for one-peak voxels vs. gradient directions.

2) Percentage of false peaks

We note that the estimated angles were computed only at voxels in the region of crossing fibers where two peaks were detected in the ODF. To get a complete picture, it is important to compute the percentage of voxels with the correct number of peaks. We computed the percentage of false peaks (PFP) in the region with and without crossing fibers separately.

Figure 2b shows the percentage of false peaks in the region with two fiber crossings. We observe that among the proposed methods, ℓ₂ Gaussian with constraints, has the lowest error rate which is around 4% with 30 gradient directions and is about 11% with 24 gradient directions. The two unconstrained methods using ℓ₁ regularization and ℓ₂ regularization for Gaussian basis functions have similar performances but slightly higher error than the constrained method. On the other hand, all the three methods using 3D-SHORE basis functions have much higher percentage of false peaks. Among those methods, the ℓ₁ regularization based method leads to better results than the other two. Note that, the proposed method performs significantly advantage in detecting the correct number of peaks than the 3D-SHORE method.

Figure 2c shows the percentage of false peaks in the region of one-fiber voxels. In this case, the ℓ₁ and ℓ₂ regularized Gaussian methods have slightly higher error (about 8%). The constrained ℓ₂ Gaussian method however has better performance (about 6% false positives), similar to the 3D-SHORE basis methods.

3) Normalized-mean-squared-errors (NMSE)

Figure 9a in Appendix D shows the NMSE of the reconstructed signal with respect to the gold-standard data. The method using ℓ₂ Gaussian with constraints has the lowest fitting error which is about 1.5% with 30 gradient directions. The ℓ₂ 3D-SHORE with constraints has lower NMSE compared with the other two 3D-SHORE based methods. We note that the performances of both methods, Gaussian and 3D-SHORE have been improved by using the constraints. In order to compare the difference between the estimated signal obtained by Gaussian with constraints and 3D-SHORE with constraints, Figure 3 shows the radial profile (signal as a function of b-value) of the estimated and the gold-standard signal. The first and second row of Figure 3 show the signal in a single-fiber and two-fiber voxel, respectively. Each line in these figures corresponds to the signal along one gradient directions at different b-values. Clearly, all the signals are positive and monotonically decreasing along radial directions, though the signal may seem to be flat along some directions. We also note that the estimated signal obtained by ℓ₂ Gaussian with constraints at high b-values, i.e. at b = 4000, 5000 s/mm², is quite similar to gold standard, which is the main reason that the proposed method (Gaussian with constraints) has lower NMSE and better angular performance.

Fig. 9.

Normalized-mean-squared errors using measurements on b-value shells with b = {1000, 3000} vs. different number of gradient directions.

Fig. 3.

The radial profile of the gold-standard signal, the estimated signals using ℓ₂ Gaussian with constraints and the estimated signal using ℓ₂ 3D-SHORE with constraints, respectively. The estimated signal was based on measurements on b-shells with b = 1000, 3000 s/mm² along 30 gradient directions. Each line in the figures corresponds to the diffusion signal along one gradient direction at different b-values.

The NMSE of the return-to-the-orgin-probability (RTOP) is computed by comparing the estimated RTOP obtained using gold-standard data for each of the methods and the results are shown in Figure 9b. We note that ℓ₂ Gaussian with constraints performs significantly better than other methods whose NMSE in RTOP is about 1.5%. The ℓ₁ Gaussian and ℓ₂ Gaussian methods have similar performance with about 7% error for 30 directions. The NMSE in RTOP for the two ℓ₂ regularized 3D-SHORE basis based methods are all above 50% while for ℓ₁ regularization it is 80%. This further demonstrates that the 3D-SHORE method does not model the high b-value signal correctly and greatly emphasizes the low b-value data, resulting in poor estimate of RTOP. We should emphasize that, the “gold standard” data contains b-values up to 5000, whereas the the test data had b-values of only 1000 and 3000. Thus, higher error in RTOP is due to the fact that the 3D-SHORE method does not extrapolate the data correctly in the high b-value regime.

Since the scalar indices RTAP, RTPP, MSD, MFD, NG, DC, GK, GKN, QMSD and QMFD do not have a closed form expression using the 3D-SHORE methods, we only computed the corresponding normalized-mean-squared-errors using the Gaussian basis function methods as shown in Figure 9c to 9l. The method using ℓ₂ regularization and constraints has the lowest NMSE for RTAP, RTPP, MSD, MFD, NG, DC, QMSD and QMFD among the three methods. These three methods have similar values for NMSE in GK and GKN. We note that the NMSE in RTPP for ℓ₁ Gaussian is larger than 100% while the NMSE of RTPP for ℓ₂ Gaussian with constraints is about 1%.

4) Performance using only lower b-value data

We also tested the performance of all the six methods on data sets with two lower b-value shells with b = {1000, 2000} and with the same number of gradient directions K = {16, 20, 24, 30, 36, 42, 60, 81}. The estimated angle and percentage of false peaks are shown in Figure 4. In particular, Figure 4b shows that the proposed methods using Gaussian basis functions performs better than the methods using 3D-SHORE basis in detecting multiple peaks. Comparing Figure 4b, 4c to Figure 2b and 2c, respectively, we note that the percentage of false peaks using b = {1000, 2000} is higher than using b = {1000, 3000}.

Fig. 4.

Comparison metrics with measurements on b-value shells with b = {1000, 2000}: (a) Estimated angle vs. gradient directions, (b) Percentage of false peaks for two-peak voxels vs. gradient directions, (c) Percentage of false peaks for one-peak voxels vs. gradient directions.

The normalized-mean-squared-error of the reconstructed signal and RTOP for the six methods are shown in Figure 10a and 10b in Appendix D, respectively. The proposed Gaussian basis function based method using ℓ₂ regularization and constraints still has the best performance. The NMSE of RTAP, RTPP, MSD, MFD, NG, DC, GK, GKN, QMSD and QMFD for the three methods using Gaussian basis functions are shown in Figure 10c to 10l, respectively. These plots show similar relations among the three method as was shown in Figure 9. We summarize the normalized-mean-squared-errors using ℓ₂ Gaussian with constraints for 30 gradient directions on two b-value shells with b = {1000, 2000} and b = {1000, 3000} in the following table: We note that all the normalized-mean-squared errors obtained using measurements on b-values shell with b = {1000, 2000} are higher than the corresponding results based on b = {1000, 3000}. Hence, if the scan time only allows to acquire 60 measurements, for the proposed method, we recommend using measurements on b-value shells with b = {1000, 3000} and 30 directions each.

Fig. 10.

Normalized-mean-squared errors using measurements on b-value shells with b = {1000, 2000} vs. different number of gradient directions.

These comparisons using phantom data suggest that the ℓ₂ Gaussian with constraints provides better performance than the other five alternatives. The comparisons also show that if the sampling time allows for 60 samples on b-value shells, a better choice is to sample the signal on the b-value shells of b = {1000, 3000}.

D. In vivo results

We tested our method on in-vivo human brain data with the following scan parameters on a 3T Siemens scanner: spatial resolution of 2.5mm × 2.5mm × 2.5mm, b-values of {900, 2000, 3600, 5600}s/mm² with each b-value shell having 60 gradient directions. This data set was considered as the “gold-standard” data. The test data set was obtained by subsampling the gold-standard data on b-value shells with b = {900, 3600}s/mm² along 30 or 60 gradient directions on each shell. The experimental parameters were as follows: 32-channel head coil, GRAPPA acceleration factor of 2, maximum gradient strength of ||G|| = 31.62 mT/m, TE/TR = 141/3400 ms and δ = 45 ms and Δ = 54 ms. We used all the six methods to estimate the ODF’s for the voxels shown in the rectangular region (white box) in Figure 5g. We used the same parameters for the Gaussian basis functions that were used for the phantom data set.

Fig. 5.

Estimated ODF for the rectangle region in the color FA image (g). Left panel and right panel show respectively the results obtained from gold-standard (GS) data sent and from measurements on two b-value shells with 30 gradient directions per shell (60 total).

The estimated ODF’s with the gold-standard data set and using the ℓ₂ Gaussian constraints, ℓ₂ 3D-SHORE constraints and ℓ₁ 3D-SHORE are shown in the left panel of Figure 5, while the right panel shows the corresponding results using the data set obtained on two b-value shells with b = {900, 3600} and 30 gradient directions (total of 60 measurements). In each of the ODF glyphs, the estimated return-to-the-origin probabilities of each methods are shown in the background. Visually, the estimated ODFs shown in Figure 5a and Figure 5b are very similar with each other. The ODF’s are almost isotropic in voxels where the RTOP is low (dark background). For the ODF’s highlighted in the rectanglular area, where three fiber crossing is detected in gold-standard data, we see that one of the fiber component is missing using the subsampled data; see 5b.

ODF profiles using 3D-SHORE is shown in Figure 5c and Figure 5d. Two of the voxels in the rectangle area of Figure 5d failed to detect some of the peaks. The ODF’s shown in Figure 5f are smoother and sharper than the ODF’s obtained by other methods. However, in the rectangular area of Figure 5c, three voxels failed to detect the correct number of peaks compared to gold standard data. The RTOP shown in Figure 5f are not consistent with the results shown in other figures due to poor extrapolation.

The comparison metrics for the test set with measurements on two b-value shells and 30 gradient directions (per shell) are tabulated in Table IV. The NMSE of the reconstructed signal is slightly higher than the results shown in the phantom data set indicating that the SNR for in vivo data set is possibly even lower than the phantom data (SNR = 9.5). Further, we did not have several acquisitions of the same individual, hence the gold standard data itself was quite noisy. The normalized-mean-squared-errors obtained by the ℓ₂ Gaussian with constraints are consistently lower than the results using other methods. We note that due to low SNR and singularity in the denominator, the NMSE of DC for ℓ₂ and ℓ₁ Gaussian methods are higher than 100%.

TABLE IV.

Error metrics for b = {900, 3600} and 30 directions per shell

	Gaussian			3D-SHORE
	ℓ2-cons.	ℓ2	ℓ1	ℓ2-cons.	ℓ2	ℓ1
NMSE	3.9%	4.0%	3.5%	4.0%	5.9%	6.5%
NMSE of RTOP	0.7%	85.7%	48.8%	46.3%	44.0%	70.4%
NMSE of RTAP	1.4%	65.3%	31.0%	–	–	–
NMSE of RTPP	0.6%	12.6%	52.9%	–	–	–
NMSE of MSD	0.8%	10.7%	8.1%	–	–	–
NMSE of MFD	4.4%	26.0%	19.4%	–	–	–
NMSE of NG	1.1%	9.2%	6.4%	–	–	–
NMSE of DC	11.9%	> 100%	> 100%	–	–	–
NMSE of GK	0.2%	5.4%	3.3%	–	–	–
NMSE of GKN	0.1%	2.9%	1.7%	–	–	–
NMSE of QMSD	1.4%	42.4%	37.7%	–	–	–
NMSE of QMFD	1.8%	46.3%	41.4%	–	–	–

We also computed the comparison metrics for the test set using measurement on two b-values shells and 60 gradient directions (120 total measurements) as shown in Table V. We note that there is no significant improvement in the performance by doubling the number of samples.

TABLE V.

Error metrics for b = {900, 3600} and 60 directions per shell

	Gaussian			3D-SHORE
	ℓ2-cons.	ℓ2	ℓ1	ℓ2-cons.	ℓ2	ℓ1
NMSE	3.6%	3.1%	2.9%	3.6%	5.5%	6.2%
NMSE of RTOP	0.6%	72.5%	40.4%	45.3%	43.9%	71.0%
NMSE of RTAP	1.0%	41.4%	21.7%	–	–	–
NMSE of RTPP	0.4%	8.0%	29.1%	–	–	–
NMSE of MSD	0.8%	8.1%	8.5%	–	–	–
NMSE of MFD	4.7%	18.1%	21.7%	–	–	–
NMSE of NG	0.7%	5.6%	5.6%	–	–	–
NMSE of DC	12.2%	> 100%	> 100%	–	–	–
NMSE of GK	0.2%	4.5%	3.3%	–	–	–
NMSE of GKN	0.1%	2.9%	2.4%	–	–	–
NMSE of QMSD	1.2%	38.6%	30.3%	–	–	–
NMSE of QMFD	1.5%	41.6%	33.2%	–	–	–

We also computed the scalar indices for one coronal slice of the brain as shown in Figure 11 in Appendix E using the gold-standard data set with the proposed Gaussian basis function based method with ℓ₂ regularization and constraints. The return-to-the-origin-probability (RTOP) shown in Figure 11a highlights the white matter areas including the regions with crossing fibers. The return-to-the-axis probability (RTAP) is shown in Figure 11b which shows the contrast between single-fiber white matters and the regions with crossing fibers. The return-to-the-plane probability (RTPP) is shown in Figure 11c which shows lower values in single-fiber white matter region corresponding to high diffusion along the fiber direction. Figure 11d shows the fractional anisotropy (FA) that is obtained from DTI. Figure 11e and Figure 11f are respectively the mean-squared-displacement (MSD) and mean-fourth-order-displacement (MFD), both showing similar contrast between gray/white and CSF areas. The ratio MFD/MSD² corresponds to the images shown in Figure 11j which is the generalized kurtosis of norm (GKN) estimated at each voxel.

Fig. 11.

Estimated scalar indices for one coronal slice using the “gold-standard” data set: (a) return-to-the-origin probability (RTOP), (b) return-to-the-axis probability (RTAP), (c) return-to-the-plane probability (RTPP), (d) Fractional anisotropy (FA), (e) mean-squared-displacement (MSD), (f) mean-fourth-order-displacement (MFD), (g) non-Gaussianity (NG), (h) difference in covariances (DC), (i) general kurtosis (GK), (j) generalized kurtosis of the norm (GKN) of displacement, (k) q-space mean-squared-displacement (QMSD), (l) q-space mean-fourth-order-displacement (QMFD).

The scalar indices that capture hindered diffusivity are shown in the third row of Figure 11. In particular, in Figure 11g, white matter areas which are more densely packed (e.g. corpus callosum, cortico-spinal tract) have higher non-Gaussianity (NG) than areas that have less densely packed fibers. Figure 11h also highlights the white matter areas but the contrast between white matter and gray matter areas is stronger than Figure 11g. Note that, areas of white matter that are affected by partial voluming due to adjoining gray matter or CSF areas also have lower DC.

Figure 11i shows the generalized kurtosis (GK). The values for gray matter and CSF areas are very close to 15 which indicate that the underlying diffusion process is similar to free Gaussian diffusion in these regions. The single-fiber white matter areas show higher GK than the crossing fiber areas. The GKN shown in Figure 11j also shows similar patterns as the generalized kurtosis. One important feature that these measures present is that, both GK and GKN are higher in densely packed white matter areas with predominantly single fiber bundles than in the crossing fiber areas, indicating that these measures provide different contrast than the standard mean kurtosis measure presented in [40]. Note that, the GKN values in the gray matter and CSF areas are very close to 5/3 which is the value corresponding to isotropic Gaussian.

Figure 11k and Figure 11l are respectively the q-space mean-squared-displacement (QMSD) and q-space mean-fourth-order-displacement (QMFD). These two figures have similar contrast as the HD, although the underlying processes could be different (hindered vs restricted diffusion). Unlike MSD and MFD, these two images highlight the white-matter area while MSD and MFD are brighter in the CSF areas.

We also computed the same set of scalar indices using only measurements on two b-value shells with b = {900, 3600} and 30 gradient directions per shell as shown in Figure 12 in Appendix E, respectively. These figures are all very similar to the corresponding results that are obtained for the gold-standard data set, albeit with slight error in the estimation of higher order statistics (GK, GKN). However, the overall contrast between different white and gray matter regions is similar to that of the gold standard data.

Fig. 12.

Estimated scalar indices for one coronal slice using measurements on b-value shells b = {900, 3600} and 30 directions: (a) return-to-the-origin probability (RTOP), (b) return-to-the-axis probability (RTAP), (c) return-to-the-plane probability (RTPP), (d) Fractional anisotropy (FA), (e) mean-squared-displacement (MSD), (f) mean-fourth-order-displacement (MFD), (g) non-Gaussianity (NG), (h) difference in covariances (DC), (i) general kurtosis (GK), (j) general kurtosis of the norm (GKN) of displacement, (k) q-space mean-squared-displacement (QMSD), (l) q-space mean-fourth-order-displacement (QMFD).

VI. Discussion and conclusion

In this work, we presented a novel method to represent and analyze diffusion MRI data from low to high b-value data. A framework using directional radial basis functions was introduced to estimate the three-dimensional q-space signal and the diffusion propagator (EAP) in analytical forms. The properties of the proposed method can be summarized in the following aspects:

• If the goal of a neuroimaging study is to obtain subtle measures of diffusion to obtain accurate micro structural information about the tissue, 60 measurements are needed spread over at least two b-shells (b = 1000 and b = 3000). Note that, a high b-value of 3000 is required for accurate estimation of several diffusion measures. We also note that the proposed method can be applied to diffusion data sets that have different gradient directions on multiple b-shells.

• An important feature of the proposed method using Gaussian basis functions is that many high order statistics, such as the return-to-the-origin probability, mean-squared-displacement, the second and the fourth order moment tensors of the propagator, can be computed in a stable and analytical manner. We also proposed several novel scalar indices that were derived from these higher order moment tensors, such as Generalized Kurtosis (GK) and the Generalized Kurtosis of the Norm (GKN) of the displacement. In particular, the GK considers the diffusion signal from the entire q-space (as opposed to the measure of kurtosis used in diffusion kurtosis imaging, which is only valid up to a maximum b-value of around 2500). Additionally, we proposed two high order statistical quantities computed from the q-space data, which are more sensitive to restricted diffusion. Further, we provided analytical formulations for quantities such as return-to-axis and return-to-plan probabilities.

• The proposed method allows analytical computation of the ODF and hence can be used within any tractography algorithm as well, but requires a slightly sophisticated acquisition scheme with multiple b-values. Thus, the proposed method is quite general and can be used in connectivity analysis as well as to study diffusion properties.

We have showed using experiments on human in-vivo data that the proposed scalar indices reveal different features of the nervous tissue structure. We also have performed extensive validation of our method on a physical phantom data set and demonstrated the robust performance of using the proposed method (Gaussian basis with constraints) on sub-sampled data sets. Hence, the proposed method can be potentially used to analyze dMRI data in neuroscience studies. Next, we discuss the limitations and some possible issues for applying our method in clinical settings.

As was shown in equation (6), each individual basis function in the r-space is an exponentially decaying oscillatory function. Though a linear combination of a large set of basis functions may reduce the oscillations, the computed EAP may not be monotonically decreasing along radial directions. For example, Figure 6 shows the radial profile of the estimated propagator in a single-fiber voxel of the phantom data set using measurements on the b-shells on b = 1000, 2000 s/mm² along 30 directions. We note that the propagator is not monotonically decreasing along some of the fast-decay directions. Since the propagator was not enforced to be positive, we have observed that the propagator may have negative values in some data sets. Though small negative values did not lead to any significant issues in our experiments, we note that it may be useful to enforce positive constraints for the EAP in a similar way as in [24] when applying the proposed method in clinical settings to study brain disorders. In this case, we expect a larger number of basis functions to be used which will certainly increase the computational burden significantly. With our current settings, the computational time for a whole-brain data set is about 6-7 hours using 8 processes.

Fig. 6.

The radial profile of the estimated EAP is a single-fiber voxel of the phantom data set.

Another concern for using the proposed method in clinical settings is the choice of the basis parameters. The set of the parameters used in this paper were learned from a physical phantom data set so that they provide close-to-optimum performances for many data sets. The same set of parameters were also used for in-vivo data set from a healthy adult human brain. We have seen that the estimation result is robust with changes in the parameters, i.e. small difference in the parameters will not lead to significant variations in the estimated results. However, these parameters may not be optimal for neonate or infant brains.

The code base for the proposed method can be downloaded via the link https://github.com/LipengNing/RBF-Propagator. Some of the phantom data sets are also available at the website http://projects.iq.harvard.edu/sparcdmri/Challenge_Data.

TABLE III.

NMSE’s using measurements with 30 gradient directions on two shells

b-value	NMSE of signal	NMSE of RTOP	NMSE of RTAP	NMSE of RTPP
{1000, 2000}	2.0%	5.6%	4.9%	1.1%
{1000, 3000}	1.4%	0.8%	1.9%	0.7%
b-value	NMSE of MSD	NMSE of MFD	NMSE of DC	NMSE of NG
{1000, 2000}	3.7%	36.6%	58.7%	6.3%
{1000, 3000}	1.8%	12.0%	9.9%	1.0%
b-value	NMSE of GK	NMSE of GKN	NMSE of QMSD	NMSE of QMFD
{1000, 2000}	8.8%	2.2%	10.6%	14.1%
{1000, 3000}	4.6%	0.6%	1.7%	2.7%

Appendix A. Derivations for the ODF

We provide the derivations of Ψ_n(u) in (7) that is used for computing ODF. First, we compute the integral ${\int }_0^{\infty }2exp(-a{x}^2)cos(bx)x^{2}dx$ with a ≥ 0, from which we can obtain the expression for Ψ_n for specific values of a and b. Since the integrand exp(−ax²) cos(bx)x² is symmetric with respect to the origin, the above integral equals to the real part of

$${\int }_{-\infty }^{\infty }exp(-a{x}^2-\mathit{ibx})x^{2}dx.$$ (12)

By Cauchy’s Integral Theorem, the integral of exp(−ax² − ibx)x² along a rectangle with vertices at −c, c, $c-i\frac{b}{2a},-c-i\frac{b}{2a}$ equals to zero. Letting c → ∞, we see that (12) equals to

$${\int }_{-\infty }^{\infty }exp\left(-a{(x-i\frac{b}{2a})}^2-ib(x-i\frac{b}{2a})\right){(x-i\frac{b}{2a})}^{2}dx$$

whose real part can be easily computed as

$$\frac{1}{2}{\pi }^{\frac{1}{2}}{a}^{-\frac{3}{2}}\left(1-\frac{b^2}{2a}\right)exp\left(-\frac{b^2}{4a}\right).$$ (13)

Then the expression of Ψ_n(u) in (7) can be obtained by multiplying ${\pi }^{\frac{3}{2}}{\mid D_n\mid }^{-\frac{1}{2}}$ to (13) and substituting a, b with ${\pi }^2{\mathit{u}}^T{D}_n^{-1}\mathit{u}$ and 2π(u · q̂_n), respectively.

Appendix B. Second and fourth order moment tensors of EAP

A. Second order moment

Following (5), R has the expression $R={\sum }_{n=0}^N{w}_n{R}_n$ where R_n is computed from the function Φ_n and is given by

$$R_n=\frac{1}{{\pi }^2}exp(-{\widehat{\mathit{q}}}_n^T{D}_n{\widehat{\mathit{q}}}_n)\left(D_n-2D_n{\widehat{\mathit{q}}}_n{\widehat{\mathit{q}}}_n^T{D}_n\right).$$

B. Fourth order moment

Using the proposed formulation, M can be estimated using $M={\sum }_{n=0}^N{w}_n{M}_n$ with M_n = ∫_ℝ³γγ^TΦ_n(r)dr. Each element of the matrix M_n is of the form ${\int }_{{ℝ}^3}{\mathit{r}}_i^2{\mathit{r}}_j{\mathit{r}}_k{\mathrm{\Phi }}_n(\mathit{r})d\mathit{r}$ for i, j, k ∈ {1, 2, 3} where r_i denotes the ith element of r. Due to symmetry in the indices, the 9 × 9 matrix M_n only has 15 independent elements. A general expression for each element can be derived explicitly and is given by:

$$\begin{array}{l}{\int }_{{ℝ}^3}{\mathit{r}}_i^2{\mathit{r}}_j{\mathit{r}}_k{\mathrm{\Phi }}_n(\mathit{r})d\mathit{r}=\\ 2exp(-{\widehat{\mathit{q}}}_n^T{D}_n{\widehat{\mathit{q}}}_n)({\mathit{\mu }}_i^2{\mathit{\mu }}_j{\mathit{\mu }}_k-\frac{1}{2{\pi }^2}{\mathit{\mu }}_j{\mathit{\mu }}_k{D}_{n,ii}-\frac{1}{2{\pi }^2}{\mathit{\mu }}_i^2{D}_{n,jk}\\ -\frac{1}{{\pi }^2}{\mathit{\mu }}_i{\mu }_k{D}_{n,ij}-\frac{1}{{\pi }^2}{\mathit{\mu }}_i{\mu }_j{D}_{n,ik}+\frac{1}{4{\pi }^4}{D}_{n,ii}{D}_{n,jk}\\ +\frac{1}{2{\pi }^4}{D}_{n,ij}{D}_{n,ik})\end{array}$$

where $\mathit{\mu }=\frac{1}{\pi }{D}_n{\widehat{\mathit{q}}}_n$ and D_n,ij denotes the (i, j)th element of D_n. The matrices R and M provide higher order statistics of the diffusion propagator, from which, several useful scalar indices can be computed as described in the next section.

Appendix C. Statistics from Q-space

A. Return-to-the-origin probability (RTOP)

The net displacement of water molecules between the application of two diffusion sensitizing gradients is inversely proportional to the mean pore volume [24]. An estimate of this net displacement can be computed using the return-to-origin probability (RTOP), given by P(0), which is the integral of the normalized diffusion signal over the entire q-space, i.e. RTOP = ∫_ℝ³ E(q)dq. Following (5) and (6), the closed form expressions of the estimated RTOP is simply given by

$$P(0)=2{\pi }^{\frac{3}{2}}\sum _{n=0}^N{w}_n{\mid D_n\mid }^{-\frac{1}{2}}.$$

In coherently organized white-matter, the diffusion is maximally restricted in the plane defined by the two smaller eigenvalues of the diffusion tensor (plane orthogonal to the main eigenvector). Therefore, [24] suggests to estimate the return-to-the-axis probability given by RTAP = ∫_ℝ² E(q_⊥)dq_⊥, where q_⊥ denotes a (RTAP) q-vector in the plane orthogonal to the principal diffusion direction. RTAP is the reciprocal of the mean cross-sectional area of the pore space. To derive the expression for RTAP using the proposed method, we let u₁, u₂, u₃ be the eigenvectors of the tensor D₀, associated with the three eigenvalues (in descending order). We note that in the proposed method, u₁, u₂, u₃ are also the eigenvectors for D_n for n ≥ 1. Let σ_n1, σ_n2 and σ_n3 denote the three eigenvalues of D_n, such that σ_n1 ≥ σ_n2 ≥ σ_n3. Assuming u₁ is the fiber orientation, the RTAP can computed using

$$\text{RTAP}=2\pi \sum _{n=0}^N{w}_n{({\sigma }_{n2}{\sigma }_{n3})}^{-\frac{1}{2}}exp(-{\sigma }_{n1}{({\mathit{u}}_1·{\widehat{\mathit{q}}}_n)}^2).$$

Similarly, the return-to-the-plane probability (RTPP) can be computed through a one-dimensional integral along the fiber orientation, i.e., RTPP = ∫_ℝE(q_//)dq_//, where q_// denotes the direction along the fiber axis. This measure is the inverse of the mean length of the cylinders. The expression for RTPP is given by

$$\text{RTPP}=2{\pi }^{\frac{1}{2}}\sum _{n=0}^N{w}_n{\sigma }_{n1}^{-\frac{1}{2}}exp(-{\sigma }_{n2}{({\mathit{u}}_2·{\widehat{\mathit{q}}}_n)}^2-{\sigma }_{n3}{({\mathit{u}}_3·{\widehat{\mathit{q}}}_n)}^2).$$

It has been observed that the diffusion signal decay is biexponential in tissues with restricted compartments and small pore sizes [12]. Hence, the measurements in the high q-value regime are more sensitive to the restricted diffusion of water molecules. To this end, it is natural to derive the higher-order statistics of E(q), i.e ∫_ℝ³||q||^kE(q)dq for k > 0, in order to study the restricted diffusivity of the diffusion process. For k = 0, we obtain the return-to-the-origin probability P(0). In the following, we introduce the second-order moment and the fourth-order moment tensors of the three-dimensional signal E(q) using the proposed method.

B. Q-space mean-squared-displacement (QMSD)

Since the propagator is the Fourier transform of the diffusion signal, the higher order moments of the signal are equal to the higher order derivatives of the propagator at the origin scaled by certain constants. Thus, it would be informative to compute the higher order moments of the signal in q-space. The second-order moment tensor of E(q) is defined as R_q ≜ ∫_ℛ² qq^T E(q)dq which is a 3 × 3 positive-semidefinite matrix. In the proposed method, R_q is estimated as

$$R_q=2{\pi }^{\frac{3}{2}}\sum _{n=0}^N{w}_n{\mid D_n\mid }^{-\frac{1}{2}}\left(\frac{1}{2}{D}_n^{-1}+{\widehat{\mathit{q}}}_n{\widehat{\mathit{q}}}_n^T\right).$$

As an analogy to the mean-squared-displacement (MSD), we define the q-space mean-squared-displacement (QMSD) as

$$\text{QMSD}\triangleq {\int }_{{ℝ}^2}{\Vert \mathit{q}\Vert }^{2}E(\mathit{q})d\mathit{q}=\text{trace}(R_q).$$

The reciprocal of QMSD was referred to as the q-space inverse variance (QIV) in [41], [42]. For Gaussian propagators, R_g equals to the inverse of the covariance R of P(r). Hence, the inverse of QMSD has similar contrast as MSD in different tissue types. For non-Gaussian propagators, QMSD may have different contrast than MSD. Note that, other measures, such as anisotropy and norm of the matrix R_q can also be computed, to obtain more information about the covariance function of the signal in q-space.

C. Q-space mean-fourth-order-displacement (QMFD)

To introduce the fourth-order moment tensor, we define p = q ⊗ q ∈ ℝ⁹. Then the fourth-order moment tensor of E(q) is defined as M_q ≜ ∫ _ℝ³ pp^T E(q)dq. In the proposed method, M_q is estimated as $M_q={\sum }_{n=0}^N{w}_n{M}_{q,n}$ with M_q,n = ∫_ℝ³pp^T[ϕ_n(q − q̂_n) + ϕ_n(q + q̂_n)]dq being a symmetric matrix of size 9 × 9. Each element of the matrix M_q,n is of the form $M_{q,n}={\int }_{{ℝ}^3}{\mathit{pp}}^T[{\varphi }_n(\mathit{q}-{\widehat{\mathit{q}}}_n)+{\varphi }_n(\mathit{q}+{\widehat{\mathit{q}}}_n)]d\mathit{q}$ for i, j, k ∈ {1, 2, 3} where q_i denotes the ith element of q. Due to symmetry in the indices, the 9 × 9 matrix M_q,n only has 15 independent elements. The expression for ${\int }_{{ℝ}^3}{\mathit{q}}_i^2{\mathit{q}}_j{\mathit{q}}_k[{\varphi }_n(\mathit{q}-{\widehat{\mathit{q}}}_n)+{\varphi }_n(\mathit{q}+{\widehat{\mathit{q}}}_n)]d\mathit{q}$ is given as

$$\begin{array}{l}2{\pi }^{\frac{3}{2}}{\mid D_n\mid }^{-\frac{1}{2}}({\widehat{\mathit{q}}}_{n,i}^2{\widehat{\mathit{q}}}_{n,j}{\widehat{\mathit{q}}}_{n,k}+\frac{1}{2}{\widehat{\mathit{q}}}_{n,j}{\widehat{\mathit{q}}}_{n,k}{(D_n^{-1})}_{ii}\\ +\frac{1}{2}{\widehat{\mathit{q}}}_{n,i}^2{(D_n^{-1})}_{jk}+{\widehat{\mathit{q}}}_{n,i}{\widehat{\mathit{q}}}_{n,k}{(D_n^{-1})}_{ij}+{\widehat{\mathit{q}}}_{n,i}{\widehat{\mathit{q}}}_{n,j}{(D_n^{-1})}_{ik}\\ +\frac{1}{4}{(D_n^{-1})}_{ii}{(D_n^{-1})}_{jk}+\frac{1}{2}{(D_n^{-1})}_{ij}{(D_n^{-1})}_{ik})\end{array}$$

where q̂_n,i denotes the ith entry of q̂_n and ${(D_n^{-1})}_{ij}$ denotes the (i, j)th element of $D_n^{-1}$. Similar to mean-fourth-order-displacement (MFD), we define the q-space mean-fourth-order-displacement (QMFD) as

$$\text{QMFD}\triangleq {\int }_{{ℝ}^2}{\Vert \mathit{q}\Vert }^{4}E(\mathit{q})d\mathit{q}=\text{trace}(M_q).$$

We note that E(q) is not a probability density function since its integral is not equal to one. Hence, the range for scalar indices introduced in Section III-B are not applicable to E(q). Though we can still compute the scalar statistics in q-space using R_q and M_q, their values lie on the positive real line (without any bounds).

Both the measures introduced above, QMSD and QMFD, are more sensitive to slow diffusion compartments or restricted diffusion. This is in contrast to the measures derived from the diffusion propagator, which are more sensitive to hindered diffusion.

Appendix D. Experimental results using the phantom data sets

Figures 7 and 8 show the estimated ODF’s for the phantom data sets. Figures 9 to 10 show the normalized-mean-squared errors.

Appendix E. Scalar statistics for the in vivo data sets

Figures 11 to 12 show the estimated scalar statistics for the in vivo data sets.