A Theoretical Signal Processing Framework for Linear Diffusion MRI: Implications for Parameter Estimation and Experiment Design

Abstract

The data measured in diffusion MRI can be modeled as the Fourier transform of the Ensemble Average Propagator (EAP), a probability distribution that summarizes the molecular diffusion behavior of the spins within each voxel. This Fourier relationship is potentially advantageous because of the extensive theory that has been developed to characterize the sampling requirements, accuracy, and stability of linear Fourier reconstruction methods. However, existing diffusion MRI data sampling and signal estimation methods have largely been developed and tuned without the benefit of such theory, instead relying on approximations, intuition, and extensive empirical evaluation. This paper aims to address this discrepancy by introducing a novel theoretical signal processing framework for diffusion MRI. The new framework can be used to characterize arbitrary linear diffusion estimation methods with arbitrary q-space sampling, and can be used to theoretically evaluate and compare the accuracy, resolution, and noise-resilience of different data acquisition and parameter estimation techniques. The framework is based on the EAP, and makes very limited modeling assumptions. As a result, the approach can even provide new insight into the behavior of model-based linear diffusion estimation methods in contexts where the modeling assumptions are inaccurate. The practical usefulness of the proposed framework is illustrated using both simulated and real diffusion MRI data in applications such as choosing between different parameter estimation methods and choosing between different q-space sampling schemes.

1. Introduction

Diffusion MRI has the unique ability to noninvasively quantify the Brownian motion characteristics of water molecules as they take random walks through their local microenvironment. Within biological tissues, these random walks are strongly constrained by the local tissue microarchitecture, which means that the diffusion MRI signal can be used to infer structural tissue features at microscopic spatial scales that are otherwise inaccessible through conventional millimeter-scale MRI. When applied to the brain, diffusion MRI has emerged as an especially important tool for quantifying tissue microstructural characteristics that change as a result of brain development, plasticity, and pathology, as well as for reconstructing the white matter pathways that connect different brain regions (Le Bihan and Johansen-Berg, 2012; Tournier et al., 2011).

In the diffusion MRI literature, the process of molecular diffusion is often summarized by a probability distribution P_Δ(r) known as the Ensemble Average Propagator (EAP). The EAP describes the ensemble-average probability density that a randomly selected water molecule from a given voxel will be found at a spatial displacement of r ∈ ℝ³ from its original position after undergoing random Brownian motion for a time interval of duration Δ.

The EAP is a popular summary of the diffusion process because, under the narrow-pulse approximation and assuming a standard Stejskal-Tanner diffusion encoding scheme (Stejskal and Tanner, 1965), the EAP is related to the measured diffusion MRI data E(q) at q-space location q ∈ ℝ³ through a Fourier transform relationship (Callaghan, 1991):

$$E(\mathbf{q})=\int P_{\mathrm{\Delta }}(\mathbf{r})e^{-i2\pi {\mathbf{q}}^T\mathbf{r}}d\mathbf{r}+n(\mathbf{q}),$$ (1)

where n(q) represents measurement noise and it is assumed that the data E(q) has been normalized such that E(0) = 1 in the absence of noise.

Almost all of the quantitative diffusion MRI literature¹ can be viewed as estimating the EAP (or quantitative measures derived from the EAP) from a finite set of M q-space measurements E(q_m) for m = 1, 2, …, M. As a simple well-known example, the popular diffusion tensor imaging (DTI) model of Basser et al. (1994) is equivalent to assuming that the EAP has the form of a zero-mean multivariate Gaussian distribution, with the covariance matrix of the Gaussian corresponding to the conventional diffusion tensor. As a result, DTI model fitting is equivalent to EAP estimation under a specific parametric model of the EAP.

The linear Fourier relationship of Eq. (1) may be viewed as especially advantageous because concepts from standard signal processing and linear systems theory (e.g., point-spread functions, the sampling theorem, the Nyquist rate, etc.) could potentially be applied to characterize and design both q-space sampling strategies and EAP estimation strategies. However, to the best of our knowledge, existing literature has largely ignored these possibilities for theoretical analysis. Instead, most methods for diffusion MRI signal modeling and experiment design have been developed heuristically based on some combination of intuition, model-based approximation of the EAP, and empirical validation.

There are two exceptions to our previous generalization. Specifically, in seminal work, Tuch (Tuch, 2002; Tuch et al., 2003; Tuch, 2004) relied on signal processing/linear systems theory to characterize an approach he proposed called the Funk-Radon Transform (FRT), and was able to show theoretically that the FRT estimated a “blurred” version of a specific EAP-based integral of interest. More recently, we generalized the FRT by introducing the Funk-Radon and Cosine Transform (FRACT) (Haldar and Leahy, 2013b) and several variations (Haldar and Leahy, 2013a; Varadarajan and Haldar, 2015b). These FRACT-based approaches were built off of Tuch’s original theory and showed that the FRT estimation approach could be modified to yield improved diffusion estimation with much better theoretical characteristics. However, a limitation of the existing theoretical analyses is that they focus narrowly on the estimation of a specific EAP-derived quantity (obtained by radial integration of the EAP) known as the Orientation Distribution Function (ODF), rather than considering more general features that can potentially be extracted from the EAP. In addition, the existing approaches focused narrowly on diffusion encoding schemes that sample data on a small number of densely-sampled shells in q-space, rather than allowing arbitrary q-space sampling patterns.

This work introduces a novel and general theoretical framework that precisely describes the relationship between estimated EAP measures and the true original EAP. The framework is applicable to arbitrary linear EAP estimation methods and is compatible with arbitrary q-space sampling patterns. As will be illustrated, this has applications ranging from the design of q-space sampling patterns, to the selection of EAP estimation methods, and to a more detailed and more precise theoretical understanding of recently-reported empirically-observed diffusion MRI aliasing phenomena (Lacerda et al., 2016; Tian et al., 2016). A preliminary account of portions of this work was previously presented in (Varadarajan and Haldar, 2016).

The fact that our framework is only valid for linear estimation methods may seem like a limitation of the proposed approach, due to the modern popularity of nonlinear diffusion estimation methods that are used with DTI (Basser et al., 1994; Koay et al., 2006), diffusion kurtosis imaging (Jensen and Helpern, 2010), non-negativity constrained spherical deconvolution (Tournier et al., 2007), diffusion basis spectrum imaging (Wang et al., 2011), NODDI (Zhang et al., 2012), and various other ODF/EAP estimation methods (Aganj et al., 2010; Menzel et al., 2011; Bilgic et al., 2012; Merlet and Deriche, 2013; Rathi et al., 2014). However, it is important to note that there are a large number of powerful existing linear diffusion estimation methods, and the development of new linear diffusion estimation methods is still an active area of research with demonstrated potential for success (Ning et al., 2015a). A few examples of linear methods include q-ball imaging (Tuch, 2002; Tuch et al., 2003; Tuch, 2004; Hess et al., 2006; Descoteaux et al., 2007), FRACT (Haldar and Leahy, 2013b; Varadarajan and Haldar, 2015b), linear spherical deconvolution (Tournier et al., 2004; Anderson, 2005; Descoteaux et al., 2009; Haldar and Leahy, 2013a), inverse Fourier transform (Callaghan, 1991; Cory and Garroway, 1990; Cohen and Assaf, 2002; Wedeen et al., 2005; Wu and Alexander, 2007; Wu et al., 2008; Paquette et al., 2016), generalized q-sampling imaging (GQI) (Yeh et al., 2010), and linear methods for estimating basis function expansions of the EAP (Assemlal et al., 2009; Cheng et al., 2010; Descoteaux et al., 2011; Hosseinbor et al., 2013; Özarslan et al., 2013; Ning et al., 2015b; Zucchelli et al., 2016; Fick et al., 2016; Avram et al., 2016). In addition, it is also worth noting that nonlinear methods often rely on parametric diffusion models. These parametric diffusion models are often idealized and approximate in nature, and their performance can deteriorate unexpectedly in the presence of modeling errors (Jbabdi et al., 2012; Sotiropoulos et al., 2013; Haldar and Leahy, 2013b). In contrast, the theoretical framework we develop for linear methods will still provide insight into performance even when the modeling assumptions of the linear estimation methods are violated, and will be accurate as long as the approximations underlying the q-space model of Eq. (1) are valid.

2. Theory

2.1. Background on Linear Estimation Methods

In its most general form, our proposed theoretical framework is applicable to linear diffusion estimation methods that generate an estimate of an EAP parameter τ(θ) of interest according to

$$\widehat{\tau }(\mathit{\theta })=\sum _{m=1}^M{g}_m(\mathit{\theta })E({\mathbf{q}}_m)$$ (2)

for some linear coefficients g_m(θ), m = 1, …, M, that may depend on additional variables θ but do not depend on the measured data values. The variables θ can be arbitrary and chosen in a context-dependent way, e.g., we will show examples where θ represents a spatial displacement vector and other examples where θ represents an orientation vector.

In practice, common EAP parameters that are estimated in this way include the EAP P_Δ(r) itself; the ODF (Tuch, 2002, 2004)

$$\mathit{ODF}(\mathbf{u})\propto {\int }_0^{\infty }{P}_{\mathrm{\Delta }}(\alpha \mathbf{u})d\alpha$$ (3)

with u ∈ 𝒮² where 𝒮² is the 2-sphere; the constant solid angle (CSA) ODF (Wedeen et al., 2005; Tristán-Vega et al., 2009; Barnett, 2009; Aganj et al., 2010)

$${\mathit{ODF}}_{\mathit{CSA}}(\mathbf{u})={\int }_0^{\infty }{P}_{\mathrm{\Delta }}(\alpha \mathbf{u}){\alpha }^{2}d\alpha$$ (4)

with u ∈ 𝒮²; the return-to-origin probability (RTOP) (Hürlimann et al., 1995; Mitra et al., 1995; Özarslan et al., 2013)

$$\mathit{RTOP}=P_{\mathrm{\Delta }}(\mathbf{0});$$ (5)

the return-to-axis probability (Özarslan et al., 2013); the return-to-plane probability (Özarslan et al., 2013); the mean-squared displacement (Cohen and Assaf, 2002; Wu and Alexander, 2007; Wu et al., 2008; Ning et al., 2015b); the mean fourth-order displacement (Ning et al., 2015b); and the generalized kurtosis (Ning et al., 2015b). All of these EAP-derived parameters can be interesting in practical applications because they are often sensitive to different aspects of the tissue microarchitecture.

While there are many linear diffusion estimation methods that have been proposed, it is worth noting that only a few of them were written in the form of Eq. (2) in the original references. Methods that directly use expressions of this form were usually derived using numerical approximations (to account for finite/non-uniform data sampling) of analytic transform integrals, and include the inverse Fourier transform (Callaghan, 1991; Cory and Garroway, 1990; Cohen and Assaf, 2002; Wedeen et al., 2005; Wu and Alexander, 2007; Wu et al., 2008), the Generalized q-sampling imaging (GQI) approach (Yeh et al., 2010), and the Funk-Radon transform (FRT) approach (Tuch, 2002, 2004). For example, the inverse Fourier transform approaches often estimate the EAP according to

$${\widehat{f}}_{\mathrm{\Delta }}(\mathbf{r})=\sum _{m=1}^M{w}_m{e}^{i2\pi {\mathbf{q}}_m^T\mathbf{r}}E({\mathbf{q}}_m),$$ (6)

where the w_m coefficients are appropriate numerical quadrature weights that can be used to account for non-uniform q-space sampling density. In this case, $g_m(\mathit{\theta })=w_m{e}^{i2\pi {\mathbf{q}}_m^T\mathit{\theta }}$.

On the other hand, many recent linear approaches are based on the use of basis function expansions (Tournier et al., 2004; Anderson, 2005; Hess et al., 2006; Descoteaux et al., 2007, 2009; Assemlal et al., 2009; Cheng et al., 2010; Descoteaux et al., 2011; Hosseinbor et al., 2013; Özarslan et al., 2013; Haldar and Leahy, 2013b,a; Varadarajan and Haldar, 2015b; Ning et al., 2015b; Zucchelli et al., 2016; Fick et al., 2016), which require some manipulation in order to be expressed in the form of Eq. (2). For completeness, we describe the mapping of such methods into Eq. (2) in Appendix A.

2.2. Theoretical Characterization of Linear Estimation

We can gain theoretical insight into linear diffusion estimation methods by inserting the data acquisition model from Eq. (1) into the linear EAP parameter estimation equation from Eq. (2), which leads to

$$\begin{gathered} \widehat{\tau }(\mathit{\theta })=\sum _{m=1}^M{g}_m(\mathit{\theta })\left(\int P_{\mathrm{\Delta }}(\mathbf{r})e^{-i2\pi {\mathbf{q}}_m^T\mathbf{r}}d\mathbf{r}+n({\mathbf{q}}_m)\right) \\ =\int P_{\mathrm{\Delta }}(\mathbf{r})\left(\sum _{m=1}^M{g}_m(\mathit{\theta })e^{-i2\pi {\mathbf{q}}_m^T\mathbf{r}}\right)d\mathbf{r}+\sum _{m=1}^M{g}_m(\mathit{\theta })n({\mathbf{q}}_m). \end{gathered}$$ (7)

While we could analyze Eq. (7) directly, further simplifications are possible if we assume a pure diffusion process in which the EAP is zero-mean, real, and symmetric with P_Δ(r̃) = P_Δ(− r̃) for all possible r̃. It is well known that in this case and in the absence of noise, E(q̃) = E(−q̃) for all possible q̃ (Tuch, 2002; Bracewell, 2000). This symmetry leads to the simplification

$$\widehat{\tau }(\mathit{\theta })=\underset{\text{Noise-Free}\text{Contribution}}{\underbrace{\int P_{\mathrm{\Delta }}(\mathbf{r})g(\mathit{\theta },\mathbf{r})d\mathbf{r}}}+\underset{\text{Noise-Only}\text{Contribution}}{\underbrace{\sum _{m=1}^M{g}_m(\mathit{\theta })n({\mathbf{q}}_m)}}.$$ (8)

with

$$g(\mathit{\theta },\mathbf{r})\triangleq \sum _{m=1}^M{g}_m(\mathit{\theta })cos(2\pi {\mathbf{q}}_m^T\mathbf{r}).$$ (9)

Compared to Eq. (7), Eq. (8) has the advantages of not involving any complex-valued numbers and implicitly incorporating common q-space and EAP symmetry assumptions.

Equation (8) provides the foundation for our theoretical analysis of linear diffusion estimation methods. We believe that this result is powerful because it provides a direct relationship between the linearly-estimated EAP parameter τ̂(θ) with the true EAP P_Δ(r) and the noise n(q), and can be used to assess the accuracy and noise-sensitivity of arbitrary linear estimators.

As indicated in Eq. (8), there are two distinct terms that contribute to the estimated EAP parameter: a noiseless contribution from the true original EAP and a noise-only contribution term. We will discuss these two contributions in more detail in the following subsections.

2.2.1. Contributions to the Linear Estimate from the True EAP

In the absence of noise, Eq. (8) simplifies to

$${\widehat{\tau }}_{\text{noise-free}}(\mathit{\theta })=\int P_{\mathrm{\Delta }}(\mathbf{r})g(\mathit{\theta },\mathbf{r})d\mathbf{r}.$$ (10)

This equation provides a direct relationship between the true original EAP P_Δ(r) and the deterministic noise-free component of the estimated EAP parameter τ̂(θ). Interestingly, this relationship is very similar to the standard point-spread function relationship that is commonly used to characterize the quality of MRI image reconstruction from sampled k-space data (Liang and Lauterbur, 2000). Analysis of the function g(θ, r) in this relationship can be very insightful, because the shape of this function will determine the resolution/accuracy/bias of the estimator. To be consistent with the spatial response function terminology used for similar theoretical analyses in MRI image reconstruction (Mareci and Brooker, 1984, 1991; Dydak et al., 2001; Greiser and von Kienlin, 2003), we refer to g(θ, r) as the EAP response function (ERF).

As an illustrative example, consider the case where we are interested in using a linear estimation method to estimate the EAP at a point θ = r′ from sampled data. In this case, the ideal true value of P_Δ(r′) can be expressed as

$$P_{\mathrm{\Delta }}({\mathbf{r}}^{\prime })=\int P_{\mathrm{\Delta }}(\mathbf{r})\delta (\mathbf{r}-{\mathbf{r}}^{\prime })d\mathbf{r},$$ (11)

or equivalently (due to symmetry) as

$$P_{\mathrm{\Delta }}({\mathbf{r}}^{\prime })=\int P_{\mathrm{\Delta }}(\mathbf{r})\left(\frac{\delta (\mathbf{r}-{\mathbf{r}}^{\prime })+\delta (\mathbf{r}+{\mathbf{r}}^{\prime })}{2}\right)d\mathbf{r},$$ (12)

where δ(·) denotes the Dirac delta function.

In addition, we can also use Eq. (10) to find that any linear estimate of P̂_Δ(r′) from noiseless sampled data will be related to the true EAP through the relationship

$${\widehat{P}}_{\mathrm{\Delta }}({\mathbf{r}}^{\prime })=\int P_{\mathrm{\Delta }}(\mathbf{r})g({\mathbf{r}}^{\prime },\mathbf{r})d\mathbf{r}.$$ (13)

for some ERF g(r′, r) defined through Eq. (9).

However, while Eq. (13) has similar integral form to Eqs. (11) and (12), the estimated EAP value is unlikely to perfectly match the true value unless the ERF satisfies

$$g({\mathbf{r}}^{\prime },\mathbf{r})=\frac{\delta (\mathbf{r}-{\mathbf{r}}^{\prime })+\delta (\mathbf{r}+{\mathbf{r}}^{\prime })}{2}.$$ (14)

Since it is impossible for a finite cosine series in the form of Eq. (9) to perfectly reproduce a delta function, we can infer that our linear estimate of P̂_Δ(r′) is, at best, related to a “blurred” version of the EAP. Importantly, our theoretical framework now provides a mechanism for constructing the ERF and understanding this blurring behavior, allowing us to predict the resolution characteristics of different linear estimation schemes and providing opportunities for optimizing the design of q-space sampling schemes and associated EAP estimation methods.

As another illustrative example, consider the case where we are interested in estimating the CSA ODF from Eq. (4) along the direction specified by θ = u′ = [0, 0, 1]^T. In this case, the ideal CSA ODF from Eq. (4) can be rewritten as

$$\begin{gathered} {\mathit{ODF}}_{\mathit{CSA}}({\mathbf{u}}^{\prime })={\int }_0^{\infty }{P}_{\mathrm{\Delta }}({[0,0,z]}^T)z^{2}dz \\ =\frac{1}{2}\int P_{\mathrm{\Delta }}(\mathbf{r})z^2\delta (x)\delta (y)d\mathbf{r}, \end{gathered}$$ (15)

while a linearly estimated CSA ODF from finitely-sampled noiseless data could be expressed using Eq. (10) as

$$O\widehat{D}{F}_{\mathit{CSA}}({\mathbf{u}}^{\prime })=\int P_{\mathrm{\Delta }}(\mathbf{r})g({\mathbf{u}}^{\prime },\mathbf{r})d\mathbf{r}.$$ (16)

for some ERF g(u′, r) defined through Eq. (9).

Similar to our above analysis of EAP estimation and our previous analysis of ODF estimation (Haldar and Leahy, 2013b), we should not expect to always get perfect estimation of the CSA ODF unless the ERF g(u′, r) = z²δ(x)δ(y)/2. And because we cannot achieve Dirac delta functions using the finite cosine series from Eq. (9) that defines the ERF, the CSA ODF is also expected to be a “blurred” version of the desired CSA ODF. As for the previous case, our theoretical framework again provides a mechanism for understanding this blurring through the ERF, and provides opportunities for optimizing experiment design and parameter estimation methods to achieve an ERF with desirable characteristics.

These two illustrative examples are representative of our general approach to theoretical characterization of linear estimation methods, which can also be used to analyze linear estimation for many other EAP parameters such as those described by Eqs. (3)–(5). Concrete numerical demonstrations of the ERF for different q-space sampling and linear estimation schemes will be presented later in Section 3.

2.2.2. Contributions to the Linear Estimate from Noise

In addition to the noise-free contribution from Eq. (10) which determines the resolution/accuracy/bias of the linear estimator, it is also important to understand the noise-only contribution which determines the precision/variance of the estimator. In the absence of signal, Eq. (8) simplifies to

$${\widehat{\tau }}_{\text{noise-only}}(\mathit{\theta })=\sum _{m=1}^M{g}_m(\mathit{\theta })n({\mathbf{q}}_m).$$ (17)

Analysis of this expression requires additional information about the characteristics of the noise samples n(q_m).

The statistical modeling of diffusion MRI images can be quite complicated (Dietrich et al., 2008; Aja-Fernández and Tristán-Vega, 2013; Varadarajan and Haldar, 2015a), and our description will be presented as generically as possible to accommodate all of the different possible signal distributions that may be encountered in practice (ranging from Rician or non-central chi noise for magnitude images, to complex Gaussian noise for complex images). The main assumption we will make is that each noise sample n(q_m) is a random variable that is statistically independent from the other samples. For convenience, we will use μ_m to denote the expected value < n(q_m) > (where we have used <·> to denote statistical expectation of a random variable) and ${\sigma }_m^2$ to denote the variance < |n(q_m)|² − |μ_m|² > of each noise perturbation n(q_m).

Given this notation, standard results from probability theory indicate that the mean of τ̂_noise-only(θ) is given by

$$<{\widehat{\tau }}_{\text{noise-only}}(\mathit{\theta })>=\sum _{m=1}^M{g}_m(\mathit{\theta }){\mu }_m,$$ (18)

and that the second-moment of τ̂_noise-only(θ) is given by

$$<{\mid {\widehat{\tau }}_{\text{noise-only}}(\mathit{\theta })\mid }^2>={\left(\sum _{m=1}^M{g}_m(\mathit{\theta }){\mu }_m\right)}^2+\sum _{m=1}^M{\mid g_m(\mathit{\theta })\mid }^2{\sigma }_m^2.$$ (19)

As a result, if we know the mean and variance of the noise perturbations in our data, it is also straightforward to use the g_m(θ) values to compute the mean and variance associated with any linear EAP parameter estimate.

In many cases, the means of each noise sample are either approximately zero (e.g., in the case of zero-mean Gaussian noise or high-SNR magnitude images) or can be estimated (e.g., in the case of low-SNR magnitude images with a rectified noise floor (Aja-Fernández and Tristán-Vega, 2013; Varadarajan and Haldar, 2015a)) to demean the noise. In these cases, the noise-only component of the estimate will have approximately zero mean and an easily-calculated variance.

3. Results

In this section, we show several illustrative examples of calculating ERFs for commonly used linear estimation methods, and demonstrate that the theoretically-derived ERF characteristics are predictive of the practical performance characteristics observed through substantially more complicated empirical evaluation. Moreover, we demonstrate that the insight provided by ERFs can help to simplify a number of practical diffusion acquisition and parameter estimation decisions that are often encountered in practical experiments. While there are many interesting practical questions raised by our theoretical analysis, our main objective for this paper is not to provide comprehensive answers to these questions or to make any specific recommendations for how researchers should acquire or process their data – instead, our objective is simply to demonstrate that our proposed theoretical analysis tools have the potential to help answer a range of interesting questions across a range of different acquisition schemes, estimation schemes, and diffusion imaging contexts. As a result, the examples we have chosen to show are intentionally diverse, and we intentionally move rapidly from one context to the next without digging too deeply into any specific application.

The estimation implementations we have used in this section are available as part of the open-source BrainSuite (http://brainsuite.org/) software (Shattuck and Leahy, 2002).

3.1. Theoretical Insights into Estimation Parameter Selection

Estimation methods typically have estimation parameters that need to be chosen by the user prior to data analysis (e.g., the number of basis functions to use or the regularization parameter for a linear basis function expansion method, as discussed in Appendix A). In existing implementations, these parameters are usually either chosen heuristically based on visual assessment, or are chosen based on extensive empirical evaluation of different possibilities. In this section, we investigate the usefulness of ERFs for simplifying these kinds of decisions.

For the sake of concreteness, we consider the case of EAP estimation using the 3D Simple Harmonic Oscillator based Reconstruction and Estimation (3D-SHORE) model (Özarslan et al., 2013), and will assess the usefulness of ERFs for assisting with the choice of model order parameters. The 3D-SHORE model uses a linear basis function expansion as explained in Appendix A. Following Merlet and Deriche (2013), we consider the orthonormal 3D-SHORE basis functions given by:

$${\mathrm{\Phi }}_{n\ell m}(\mathbf{q})=\sqrt{\frac{2(n-\ell )!{\Vert \mathbf{q}\Vert }_2^{2\ell }}{{\zeta }^{(\ell +\frac{3}{2})}\mathrm{\Gamma }(n+\frac{3}{2})}}{e}^{-\frac{{\Vert \mathbf{q}\Vert }_2^2}{2\zeta }}{L}_{n-\ell }^{\ell +\frac{1}{2}}\left(\frac{{\Vert \mathbf{q}\Vert }_2^2}{\zeta }\right)Y_{\ell }^m\left(\frac{\mathbf{q}}{{\Vert \mathbf{q}\Vert }_2}\right),$$ (20)

for ℓ = 0, 1, …, L, n = ℓ, ℓ + 1, …, N, and, m = −ℓ, −ℓ + 1, …, ℓ − 1, ℓ, where N is the maximum radial order parameter and L is the maximum angular order parameter of the 3D-SHORE basis. In this expression, $L_n^{\alpha }(·)$ is the generalized Laguerre polynomial of order n with parameter α, $Y_{\ell }^m(·)$ is the real and symmetric version of the spherical harmonic basis function of order ℓ and degree m (Descoteaux et al., 2007), and Γ(·) is the Gamma function. Assuming a nominal diffusion coefficient of 1.25×10⁻³ mm²/s and effective diffusion time of 39.6 ms and using formulas from Cheng et al. (2010) and Merlet and Deriche (2013), we set the 3D-SHORE scale parameter ζ =256 mm⁻². Given this choice of basis functions, the 3D-SHORE reconstruction procedure was implemented using regularization as described in Eqs. (A.2)–(A.5) from Appendix A, using Laplace-Beltrami regularization to encourage smooth angular and radial variations of the EAP. The Laplace-Beltrami operator was implemented as described by Cheng et al. (2010) and Merlet and Deriche (2013), with the regularization parameter set to λ = 10⁻⁸ (Cheng et al., 2010).

For the results shown in this subsection, we considered a multi-shell q-space sampling scheme that we generated using the generalized multi-shell electrostatic repulsion technique described by Caruyer et al. (2013). Our design uses 3 shells (b-values of 1000, 2000, and 3000 s/mm² with an effective diffusion time of 39.6 ms), with 64 samples per shell.

Based on this reconstruction procedure and q-space sampling scheme, we inserted the results of Eq. (A.5) into Eq. (9) to analytically compute ERFs for EAP estimation for different choices of the radial model order parameter N (i.e., N = 3, 6, 10, and 12), with the angular order parameter bounded by N. Figure 1 shows illustrative results for this case. Note that the ERF is a 3D function of r (or a 6D function of r and r′), but 3D/6D images are hard to visualize. We have chosen to only show a 2D image of the x–z plane, noting that the features of the ERF do not change substantially with rotation about the x-axis (i.e., the ERFs in this case are approximately axially symmetric). Additionally, due to the fairly uniform q-space sampling pattern, the ERFs in this case are also approximately rotation invariant with respect to r′, so we have chosen to only show r′ = [15, 0, 0]^T μm. We have also chosen to only show the region in which x and z are smaller than 50 μm, because in practical in vivo brain imaging scenarios at room temperature, a negligible fraction of spins would diffuse farther than 50 μm over a 39.6 ms time interval.

Figure 1.

Comparison of the ERFs g(r′, r) for 3D-SHORE EAP estimation for different maximum radial orders N. Specifically, the ERF is shown as a function of x and z (i.e., r = [x, 0, z]^T) for r′ = [15, 0, 0]^T μm. The green and pink lines superimposed on these images define 1D axes that pass through the peaks of the ERFs, with reference to Fig. 2.

If EAP estimation were perfect, we would see two delta functions in the ERF at ±r′. However, instead of observing delta functions, we instead see broad peaks in the ERF that are centered in approximately the correct positions, indicating that our estimate of the EAP at r′ will be a blurred version of the true EAP. In order to obtain the highest possible resolution for our EAP estimate, it would desirable for the widths of these dominant peaks (i.e., the mainlobe width using signal processing terminology (Harris, 1978)) to be as small as possible. In addition, in order to avoid leakage of information from one part of the EAP into a distant region of the EAP, it is important for the magnitude of the ERF in regions outside of the dominant peaks (i.e., the sidelobe level using signal processing terminology (Harris, 1978)) to be as small as possible.² As can be seen from Fig. 1, the mainlobe and sidelobe characteristics have clear differences for different values of N. These mainlobe and sidelobe characteristics are potentially even easier to visualize in 1D plots passing through the ERF peaks, as shown in Fig. 2.

Figure 2.

A more detailed look at the 3D-SHORE ERFs shown in Fig. 1 for EAP estimation with different maximum radial orders N. The 1D plots of the ERF values correspond to the (top) pink lines along the x-axis from Fig. 1 and (bottom) green lines parallel to the z-axis from Fig. 1 that pass through the dominant ERF peaks. The plots also show the positions of the ideal ERF peaks at r′= ±[15, 0, 0]^T μm.

As can be seen from these figures, all four different choices of N lead to peaks that are centered in approximately the correct location. However, the ERF behavior is different in each case. Specifically, it can be observed that the mainlobe sizes and shapes are fairly similar for N = 6, 10, and 12, while the mainlobe for N = 3 is much wider. Based on the structure of the ERFs, we would expect the N = 3 case to yield very different EAP estimation performance compared to the other cases, which are expected to be much more similar to one another.

Simulations were performed to examine how ERF characteristics correlate with practical EAP estimation quality. Data was synthesized by modeling the EAP as the linear combination of two diffusion tensor models. Each diffusion tensor component was given the same relative volume fraction. The principal diffusion tensor eigenvalue for both tensors was systematically varied between 0.5×10⁻³ mm²/s and 3×10⁻³ mm²/s, while the relative orientations of the principal eigenvectors were systematically varied between 0 and 90 degrees, leading to a total of 400 different synthetic EAPs. The two smallest eigenvalues of the diffusion tensors were always set to 0.2×10⁻³ mm²/s.

For qualitative illustration, Figure 3 shows estimated EAPs resulting from applying 3D-SHORE to simulated noiseless data with the tensors crossing at 90 degrees and the largest diffusion tensor eigenvalue set to 1.4 × 10⁻³ mm²/s. As would be expected based on the ERFs, the N = 3 case has much different behavior than the other cases, and leads to estimation performance that is much worse. Specifically, the N = 3 case has failed to resolve the orientations of the two tensors, while the orientations of the two tensors are clearly resolved in all the remaining cases. The loss of angular resolution in the N=3 case is not surprising based on the much larger ERF mainlobe width observed in Figs. 1 and 2 for this case.

Figure 3.

EAPs estimated using 3D-SHORE with different maximum radial orders N from noiseless two-tensor simulated data. For simplicity, we only show the 2D plane containing the principal directions of both tensors instead of showing the full 3D estimated EAPs.

Quantitative results are shown in Fig. 4. Specifically, we quantified the 3D-SHORE EAP estimation error at r′ = [15, 0, 0]^T μm for each of the 400 simulated noiseless datasets. Error was computed using the absolute difference between the estimated EAP P̂_Δ(r′) and the ground truth EAP P_Δ(r′), i.e., Error = |P_Δ(r′)−P̂_Δ(r′)|. For comparison, we also computed the equivalent full-width at half-maximum (FWHM) of the ERF for the same EAP location r′ = [15, 0, 0]^T μm. The FWHM was computed by computing the maximum value of the mainlobe peak, and then computing the volume of the mainlobe where the ERF magnitude was larger than half the maximum value. Finally, the FWHM was computed as the diameter of an equivalent sphere with the same volume. Note that the FWHM is a common metric used to assess the resolution of a point-spread function in a conventional imaging experiment, with smaller FWHM values corresponding to higher resolution (Liang and Lauterbur, 2000). As can be seen in Fig. 4, the FWHM of the ERF follows the same trend as the EAP estimation error, and both curves have a minimum at N = 6. Quantitatively, the correlation coefficient between the FWHM of the ERF and the median of the EAP estimation error is R = 0.9650, which is highly significant (p < 10⁻⁵). These results provide a direct indication of the potential usefulness of the ERF, since the theoretically-derived FWHM curve could be used to choose the optimal radial order without requiring extensive simulations to come to the same conclusion.

Figure 4.

(Left) Empirical EAP estimation error at r′ = [15, 0, 0]^T μm for each of the 400 simulated noiseless multi-tensor datasets, plotted as a function of the maximal radial order N of the 3D-SHORE basis. The data is shown using a box and whisker plot, in which the median estimation error is shown using a horizontal black line, while the green box shows the interquartile range between the 25th and 75th percentiles of the empirical distribution. The dashed red “whiskers” are lines spanning the range of the empirical distribution above and below the interquartile range, while observations beyond the whisker length are marked as outliers and denoted with red plus symbols. Samples are marked as outliers if their distance away from the interquartile range is more than 1.5 times the length of the interquartile range. (Center) The corresponding FWHM of the ERF as a function of N. (Right) The theoretical noise variance (normalized to be invariant to σ by dividing by σ²) as a function of N, calculated using Eq. (19) assuming σ_m = σ = 1 for all m.

However, it should also be noted that the estimation error and ERF characteristics are both functions of r′, and that the behavior observed in Fig. 4 is specific to r′ = [15, 0, 0]^T μm. We do not show extensive results for other values of r′ due to space constraints, but generally speaking, we have observed that in regions where the true EAP has large intensity, the empirically observed estimation error follows the same basic trend as the FWHM. On the other hand, in regions where the true EAP intensity is small (e.g., at r′ = [35, 0, 0]^T μm in Fig. 3, at which the true EAP value is approximately zero), we observe that the EAP estimation error instead follows the same trend as the maximum sidelobe level of the ERF. Both of these observations are consistent with intuition – when the EAP value is large, poor resolution (as measured using the FWHM) is likely to be one of the primary sources of estimation error. On the other hand, in regions where the EAP is very small, the errors are much more likely to be dominated by leakage of information from distant regions where the EAP has higher intensity. It is also worth mentioning that, perhaps surprisingly, the optimal choice of N, as identified using either the minimum value of the empirical estimation error or the theoretical characteristics of the ERF, has a strong dependence on r′. While N = 6 was optimal for r′ = [15, 0, 0]^T, we observe that larger values of N should be preferred (both empirically and theoretically) for larger r′.

While the previous results considered EAP estimation quality in the noiseless case, our theoretical framework can also provide a theoretical measure of the noise variance using Eq. (19). We have shown a plot of the value of Eq. (19) as a function of N in Fig. 4, assuming μ_m = 0 and σ_m = 1 for all m. Interestingly (and perhaps coincidentally), we observe that, in this case, the noise variance has its largest value at N = 6, which is exactly the same radial order at which the FWHM has its minimum value. As a result, our theoretical framework suggests that the optimal choice of N will represent a trade-off between bias and variance in this case. Empirical results using noisy simulated data confirm this conclusion, though are not shown here due to space constraints and because we illustrate this trade-off in a different context in the sequel.

It is also worth mentioning that while the ERF is easily calculated to provide insight into the theoretical characteristics of a given estimator, it does not necessarily illuminate the reasons why a given estimator will have a given theoretical characteristics. For instance, the ERF alone does not provide direct insight into what causes the case with N = 6 to have the smallest FWHM in this example. To answer questions like this, it would be necessary to perform a more detailed investigation into the specific numerical characteristics of 3D-SHORE with this specific sampling pattern.

3.2. Theoretical Insights into Resolution/Noise Trade-offs

In this subsection, we use our theoretical framework to explore the trade-offs between resolution and noise variance in the context of RTOP estimation from multi-shell diffusion data. For the purpose of illustration, we estimated RTOP using a different estimation model from that of our previous example: a linear basis function expansion method using the Spherical Polar Fourier (SPF) basis described by Assemlal et al. (2009). The estimation procedure was implemented by simply replacing the 3D-SHORE basis with the SPF basis in the 3D-SHORE implementation from the previous subsection, leaving all other aspects the same. We used a maximum radial order of 6 and a maximum angular order of 8 for the SPF basis, and the SPF scale parameter was set to ζ =457 mm⁻² (based on a nominal diffusion coefficient of 0.7×10⁻³ mm²/s).

For further illustration, we considered three different multi-shell q-space sampling schemes. The first sampling scheme was identical to that described in the previous subsection (3 shells, with 64 orientations per shell, and b-values of 1000, 2000, and 3000 s/mm²). The remaining two sampling schemes were identical to the first, except that we modified the b-values of each shell: one scheme used shells with b-values of 1500, 3000, and 4500 s/mm², while the other scheme used shells with b-values of 2000, 4000, and 6000 s/mm².

The ERFs for SPF-based RTOP estimation are shown in Fig. 5. We observe from this figure that the mainlobe width of the ERF becomes smaller as we move from smaller to larger b-values. This observation is quantified using the FWHM (calculated using the same method from the previous subsection) of the ERF, which is plotted in Fig. 6. This theoretical characterization suggests that RTOP estimation is expected to be more accurate using the higher b-value sampling scheme, at least in situations when the SNR is high. On the other hand, we also used Eq. (19) to calculate the expected noise variance in each case. This is also plotted in Fig. 6, and we observe that the noise variance increases as we move from smaller to larger b-values. This suggests that, holding all other aspects of parameter estimation fixed, the choice of sampling scheme will represent a trade-off between bias and variance for RTOP.

Figure 5.

The ERFs for SPF-based RTOP estimation for different multi-shell sampling schemes. The three b-values used for each multi-shell scheme are shown in units of s/mm².

Figure 6.

Plots of the theoretically-derived ERF FWHM and noise variance (normalized to be invariant to σ by dividing by σ²) values for SPF-based RTOP estimation.

To test these theoretical predictions, we considered the same set of noiseless two-tensor EAP simulation models described in the previous subsection, and have added simulated zero-mean Gaussian noise with noise variance σ². The Gaussian noise variance was designed to achieve signal-to-noise ratios (SNRs) of 500, 150, 80, and 40, with SNR defined as E(0)= σ. Subsequently, we calculated the magnitude of this noisy simulated data, resulting in Rician-distributed measurements. Subsequently, SPF-based RTOP estimation was applied to this data, and the resulting RTOP estimation error distributions are plotted in Fig. 7.

Figure 7.

Plots of the empirically-observed RTOP estimation error for SPF-based estimation with different multi-shell sampling schemes. The data is shown using box and whisker plots, using the same formatting conventions as in Fig. 4.

The results from Fig. 7 demonstrate that at high SNR, the empirical RTOP estimation error follows the same trend as the FWHM of the ERF, while at low SNR, the empirical RTOP estimation error follows the same trend as the theoretical noise variance. Quantitatively, the correlation coefficient between the FWHM of the ERF and the median of the RTOP error is R = 0.9995 for the highest SNR case (significant with p < 0.05), while the the correlation coefficient between the theoretical noise variance and the median of the RTOP error is R = 0.9595 for the lowest SNR case (while the correlation is very high, this result is not statistically significant, though we suspect we would observe significance in this case if we had analyzed more than three experiment designs). At intermediate SNR values, the empirical RTOP error can be viewed as a linear combination of the theoretically-derived FWHM and noise variance trends. These empirical results are consistent with our theoretical predicitions, and support the potential usefulness of our theoretical framework for practical decisions associated with noise/resolution trade-offs and q-space sampling scheme selection.

3.3. Theoretical Comparisons of Different Estimation Methods

In many situations, it is common for diffusion MRI users to be unsure about which estimation procedure will give the best results with their data. In this subsection, we demonstrate that our theoretical framework can provide valuable information to assist users in making this kind of decision.

For concreteness, we consider the problem of ODF estimation from single-shell diffusion data using the FRT (Tuch, 2004; Descoteaux et al., 2007) and FRACT (Haldar and Leahy, 2013b), and ODF estimation from multi-shell diffusion data using GQI (Yeh et al., 2010) and 3D-SHORE (Özarslan et al., 2013; Merlet and Deriche, 2013). In our implementations of FRT and FRACT, we used a spherical harmonic basis expansion with maximum angular order L = 8 and Laplace-Beltrami regularization with λ = 0.006 (Descoteaux et al., 2007). Our implementation of 3D-SHORE used the same settings as in Section 3.1, with maximum radial order N = 8 and maximum angular order L = 8. The FRT and GQI were designed to estimate the ODF from Eq. (3), while FRACT and 3D-SHORE were designed to estimate the CSA ODF from Eq. (4).

For multi-shell ODF and CSA ODF estimation, the acquisition was chosen identical to the Human Connectome Project (HCP) protocol (Sotiropoulos et al., 2013), which has three shells at b-values of 1000, 2000, and 3000 s/mm², with 90 samples per shell and diffusion timing parameters of Δ = 43.1 and δ = 10.6 ms. For single-shell ODF and CSA ODF estimation, we simply used the outermost (b = 3000 s/mm²) shell from the HCP protocol.

The ERFs for these different cases are shown for orientation u′ = [0, 0, 1]^T (see Eq. (16) for notation conventions) in Fig. 8. With this choice of u′, the ideal ERF is g(u′, r) = δ(x)δ(y)=2 for ODF estimation using Eq. (3), while the ideal ERF is g(u′, r) = z²δ(x)δ(y)=2 for CSA ODF estimation using Eq. (4). As can be seen in Fig. 8, the different sampling and reconstruction approaches yield ERFs that approximate these ideal ERFs with different levels of accuracy.

Figure 8.

Comparison of the ERFs g(u′, r) for different ODF (FRT and GQI) and CSA ODF (FRACT and 3D-SHORE) estimation methods for single-shell and multi-shell data. Specifically, the ERFs are shown as a function of x and z (i.e., r = [x, 0, z]^T) for the orientation u′ = [0, 0, 1]^T. The green and pink lines superimposed on the top row of images define 1D axes that pass through the peaks of the ERFs, with reference to the 1D plots shown in the bottom two rows.

We observe that along the z direction (parallel to the ODF orientation being estimated), the ERFs for the ODF-estimation methods FRT and GQI are approximately constant (as desired in the ideal ODF case) for a range of z values near the origin, although the ERFs undesirably decay away for larger values of z. Specifically, the ERF value is within 20% of the ideal ERF value for z values up to |z| = 15 μm for FRT and |z| = 18 μm for GQI. Similarly, the ERFs for CSA ODF-estimation methods FRACT and 3D-SHORE approximately follow a z² trend (as desired in the ideal CSA ODF case) for a range of z values near the origin, although the ERFs also undesirably decay away for larger values of z. The ERF value is within 20% of the ideal ERF value for z values up to |z| = 21 μm for FRACT and |z| = 31 μm for 3D-SHORE.

We expect that the width of the mainlobe in the x – y plane (orthogonal to the ODF orientation being estimated) will be a strong predictor of the angular resolution of the ODF estimate. Interestingly, we observe that the two estimation methods used with single-shell data (FRT and FRACT) have smaller mainlobe widths along the x-dimension than the corresponding ODF or CSA ODF estimation methods used with multi-shell data (GQI and 3D-SHORE). Specifically, the mainlobe FWHMs along the x dimension are calculated as 11.08 μm for FRT versus 13.96 μm for GQI, and 12.07 μm for FRACT versus 16.61 μm for 3D-SHORE. This observation is perhaps surprising, because the single-shell data is a subset of the multi-shell data, and it would generally be expected that the use of more data would lead to improvements in angular resolution. Nevertheless, empirical tests using an EAP model with two crossing diffusion tensors (similar to those described in previous subsections) validate that FRT demonstrates quantitatively better angular resolution than GQI and that FRACT demonstrates quantitatively better angular resolution than 3D-SHORE for the q-space sampling schemes and estimation parameters we assumed for this analysis. It should be noted, however, that the single-shell methods have produced ERFs with larger sidelobes than the multi-shell methods in this case (the ratio between the highest sidelobe value and the value of the ERF peak was 0.42 for FRT versus 0.28 for GQI, and 0.38 for FRACT versus 0.20 for 3D-SHORE), such that the use of more data in the multi-shell case is apparently helping to reduce the amount of potential signal leakage from distant regions of the true EAP.

Results obtained by applying these four ODF estimation methods to real human brain data distributed by the HCP are shown in Fig. 9. Qualitatively, this figure demonstrates the expected behavior, with the FRT having apparently higher angular resolution (more pronounced separation between ODF peaks in a region of known crossing fibers) than GQI, and the FRACT having apparently higher angular resolution than the 3D-SHORE estimate. Quantitative empirical simulation results are shown in supplementary Fig. S1, which similarly suggest that the FWHM of the ERF follows the same trend as empirically-observed angular resolution.

Figure 9.

ODFs (FRT and GQI) and CSA ODFs (FRACT and 3D-SHORE) estimated using from real in vivo HCP brain data. ODFs and CSA ODFs are shown from a region of crossing white matter fibers, as marked in the color-coded fractional anisotropy (FA) image.

Overall, this investigation has shown that ERFs can also provide valuable insight that can inform practitioners about the expected characteristics of different diffusion estimation schemes. However, we would like to caution readers to not draw the conclusion that single-shell sampling is always better than multi-shell sampling, or that FRT and FRACT are always better than GQI and 3D-SHORE. These findings were true for the specific data sampling schemes and parameter estimation settings that we considered in this investigation, although results may differ substantially in different contexts.

3.4. Theoretical Insights from Sampling Theory

In the modern q-space literature, it is relatively rare for authors to discuss fundamental sampling theory concepts like the Nyquist rate, aliasing, and resolution, even though such concepts should be fundamental for Fourier-based EAP reconstruction methods like diffusion spectrum imaging (DSI) (Wedeen et al., 2005) and radial DSI (Baete et al., 2016). When these concepts are discussed, they are frequently examined empirically rather than theoretically (Lacerda et al., 2016; Tian et al., 2016), and popular methods for DSI sampling design are often based on extensive empirical testing (Kuo et al., 2008, 2013) instead of leveraging well-established insights from sampling theory. In this regard, q-space sampling design methods are very different from modern MRI k-space sampling design methods (which rely heavily on theoretical concepts like the point-spread function (Liang and Lauterbur, 2000)), even though the underlying Fourier sampling and reconstruction problems are nearly identical between the different cases. In this subsection, we observe that a theoretical analysis using ERFs helps to bridge this longstanding gap, and allows new insights into the various trade-offs associated with different sampling approaches.

For the sake of concreteness, we considered GQI-based EAP reconstruction (Yeh et al., 2010) using several Cartesian and multi-shell sampling schemes that cover a range of different sampling densities (with higher density characterized by smaller distance between nearby q-space sampling locations) and different amounts of q-space coverage (with larger coverage characterized by higher maximum b-values), as illustrated in Fig. 10. The Cartesian sampling schemes were designed using either the 515-sample (for the high sampling density case) or 203-sample (for the other two cases) DSI sampling patterns described by Kuo et al. (2008), with appropriate scaling to achieve a maximum b-value of 4000 s/mm² (for the moderate q-space coverage cases) or 10000 s/mm² (for large q-space coverage). The multi-shell schemes were designed with 3 shells with 68 samples per shell using generalized multi-shell electrostatic repulsion (Caruyer et al., 2013), with b-values of 1300, 2700, and 4000 s/mm² (for moderate q-space coverage) or 3300, 7000, and 10000 s/mm² (for large q-space coverage). In all cases, the diffusion times were set to Δ = 21.8ms and δ = 12.9 ms. In addition to sampling patterns, Fig. 10 also shows ERFs along with corresponding EAP reconstructions from simulated multi-tensor data with two diffusion tensors crossing at a 75° angle.

Figure 10.

(Top row) Various Cartesian and multi-shell q-space sampling schemes with different sampling densities and different amounts of q-space coverage. Note that in the Cartesian case, the sampling points are spaced evenly in q-space, which results in parabolic distribution of the sampling points when represented using b-values. For the Cartesian case, we have only shown the sampling locations where the z-component of the b vector is zero to assist in visualization. For the multi-shell case, we have rotated each sampling point to the nearest point in the b_x-b_y plane to assist in visualization. (Middle row) ERFs g(r′, r) for GQI-based EAP estimation, shown as a function of x and z (i.e., x = [x, 0, z]^T) for r′ = [0, 0, 0]^T μm. (Bottom row) EAPs estimated from simulated noiseless multi-tensor data with two diffusion tensors crossing at a 75° angle.

The behavior we observe is that the mainlobe width of the ERF and the resolution of the estimated EAP seems to be primarily influenced by the extent of q-space coverage, with smaller FWHMs and higher resolution when larger b-values are measured. There is not a substantial difference in mainlobe width between Cartesian and multi-shell sampling when the amount of q-space coverage is matched. On the other hand, the sidelobe behavior is heavily influenced by the sampling density, with smaller sidelobes when the sampling density is higher. There is also a substantial difference in aliasing structure between Cartesian sampling versus multi-shell sampling, with Cartesian sampling producing highly coherent aliasing artifacts if data is not sampled at the Nyquist rate, while multi-shell sampling produces incoherent artifacts. All of these features are easily predicted by standard k-space sampling theory for Cartesian and radial sampling trajectories (Liang and Lauterbur, 2000), though are not usually discussed or analyzed in a sampling-theoretic context for diffusion MRI.

When looking at these results, it may be tempting to try and decide which of these q-space sampling schemes is optimal. However, it is important to point out that this kind of question is impossible to answer definiteively without additional context. Specifically, the ERF is a tool that can be used to inform a practitioner about the behavior of different sampling schemes, but the ERF does not have any ability to infer which EAP features are most important for a given application. As a concrete example, the low-density large-coverage Cartesian sampling results from Fig. 10 can provide an excellent representation of the EAP if a practitioner knows to discard all of the unwanted aliases. On the other hand, if a practitioner blindly used the estimated EAP to generate an ODF estimate using Eq. (3), the presence of aliasing would cause the ODF estimate to be heavily biased (Lacerda et al., 2016). While it is possible to use the ERF to select between different sampling patterns, it is generally necessary to have specific optimality criteria in mind (e.g., as was illustrated when comparing the specific angular resolution criterion between single-shell and multi-shell acquisition in the previous subsection).

4. Discussion and Conclusions

This work described a new EAP-based theoretical framework for understanding the behavior of linear diffusion estimation schemes, particularly with respect to resolution, signal leakage, and noise sensitivity. Since this framework makes minimal assumptions (i.e., it only relies on the standard Fourier model of q-space data), it is broadly applicable to a wide range of different diffusion acquisition, estimation, and modeling methods.

We also showed a variety of illustrative applications for such theoretical characterizations in the contexts of choosing between different estimation methods and choosing between different q-space sampling patterns. However, it is worth reemphasizing that our goal in each of these cases was only to be illustrative and not to be exhaustive – there are many interesting facets that we have left unexplored, and which provide potentially promising directions for future research. Nevertheless, we believe that even our simple examples have already produced new insights that have not been previously accessible, and also provide new opportunities for ERF-based optimization (based on mainlobe widths, sidelobe levels, etc.) and design of linear diffusion estimation methods. These types of approaches have not been previously possible to the best of our knowledge, aside from our preliminary work on CSA ODF estimation (Varadarajan and Haldar, 2015b). However, it is also worth mentioning that there is not a single way of measuring mainlobe widths and sidelobe levels that is universally accepted within the signal processing literature (Harris, 1978), such that the choice of optimal methods for quantifying the “goodness” of an ERF remains an interesting open problem.

On a related topic, using the ERF to select between two different alternative schemes can require some nuance when the difference between the two schemes is relatively small. For example, if a given estimation method has an ERF FWHM that is 0.002 microns smaller than that of another estimation method, does that really mean that the first method is always better? Similar nuances have existed in the signal processing literature for decades. For example, there are many different window functions that have similar FWHM values that represent slightly different tradeoffs in resolution, signal leakage, and noise characteristics (Harris, 1978). The choice between these windows window is nontrivial in practice, and will ultimately depend on more detailed contextual information about the goals of the specific application. This is exactly the same type of nuance that will exist whenever trying to choose between similar ERFs. Nevertheless, we believe that measures like the FWHM are useful for comparing different approaches in cases where there are substantial FWHM differences. And in cases where the FWHM differences are small, knowing that the FWHM is similar is still useful information. In all settings, we believe that having access to theoretical information about estimator behavior is substantially better than having no theoretical information at all.

In practical scenarios where the EAP modeling assumptions of Eq. (1) do not perfectly hold (e.g., in the case of finite-width gradient pulses or in the case of physiological motion), our ERF-based characterization will only provide an approximation of the true characteristics. We expect our theoretical characterization to degrade gracefully in such cases, in much the same way that q-space data processing methods are still routinely applied to data that does not perfectly fit the q-space model. However, it should be noted that a thorough exploration of such cases requires a more detailed model of diffusion physics than the one considered in this paper, since the EAP alone is no longer sufficient to describe the diffusion encoding process. While our modeling framework can potentially be generalized to handle a more complicated diffusion model that could capture such effects, this kind of extension is beyond the scope of this paper.

It is also worth noting that an interesting feature of our proposed theoretical framework is that it enables a theoretical analysis of shift invariance in a way that was not previously possible. Furthermore, during the course of our investigation, we discovered that many linear diffusion estimation systems are not shift-invariant in the way that we may have originally anticipated. This observation has a few consequences. From an ERF perspective, this observation means that it may sometimes be worthwhile to compute multiple complementary ERFs that explore different regions of the estimation space. In practice, we have found that calculating the ERF is not very computationally expensive, meaning that the need to perform multiple ERF evaluations is not prohibitive. For example, for our basic unoptimized MATLAB-based implementation and considering EAP estimation using 3D-SHORE with the HCP q-space sampling protocol, it takes an Intel Core i7 3.4 GHz CPU roughly 6.3 seconds to evaluate the ERF at 10⁶ different r values. However, there are also cases where the ERF characteristics demonstrate some form of invariance. For illustration, supplementary Figs. S2 and S3 analyze the rotation invariance of 3D-SHORE EAP and CSA ODF estimation from data sampled with the HCP protocol. These figures show that the ERFs computed for different orientations are approximately invariant to rotation in this case, as may be expected since the spherical harmonic representation (used in the 3D-SHORE basis functions) is robust to rotation, and the q-space data for this case is sampled on uniformly-sampled shells with relatively high sampling density. Cases like this that possess approximate orientation invariance are convenient, since they obviate the need to evaluate ERF characteristics at many different orientations. Finally, from the perspective of an estimator designer, the fact that many of the current estimators do not seem to possess shift-invariance may be seen as an opportunity to develop better estimators, as has previously been done in other fields like positron emission tomography reconstruction (Qi and Leahy, 2000; Stayman and Fessler, 2000) or X-ray computed tomography reconstruction (Cho and Fessler, 2015).

In conclusion, we expect the theoretical tools proposed in this paper to be useful in a wide variety of diffusion MRI contexts. In addition, since there are many other MRI-based imaging experiments that use a similar linear data acquisition model with linear parameter estimation (e.g., this is especially true for certain kinds of perfusion/flow imaging, which can also be formulated in terms of EAPs (Callaghan, 1991)), we believe that the general approach we have taken is easily expanded to other quantitative MRI modalities.

Appendix A. Linear Basis Function Expansion Methods

As mentioned in Section 2.1, there are a variety of linear basis expansion methods (Tournier et al., 2004; Anderson, 2005; Hess et al., 2006; Descoteaux et al., 2007, 2009; Assemlal et al., 2009; Cheng et al., 2010; Descoteaux et al., 2011; Hosseinbor et al., 2013; Özarslan et al., 2013; Haldar and Leahy, 2013b, a; Varadarajan and Haldar, 2015b; Ning et al., 2015b; Zucchelli et al., 2016; Fick et al., 2016) that, after some manipulation, can be expressed in the form of Eq. (2). This appendix describes how to perform these manipulations.

A common assumption of the linear basis expansion methods is that, in the absence of noise, the noiseless diffusion signal E(q) can be expressed as a weighted sum of K basis functions Φ_k(q) according to

$$E(\mathbf{q})=\sum _{k=1}^K{c}_k{\mathrm{\Phi }}_k(\mathbf{q}),$$ (A.1)

for some appropriate coefficients c_k. The basis functions Φ_k(q) are frequently chosen as combinations of the spherical harmonic basis to describe the angular component of the diffusion signal and polynomial bases to describe the radial component of the diffusion decay. Basis function representations are powerful, since the infinite dimensional function E(q) has been represented by a K-dimensional model which could potentially be estimated from as few as K different q-space measurements. Additionally, with suitably-chosen basis functions, these methods can enable fast compuational methods for evaluating the EAP parameters defined by the integral formulas from Eqs. (3)–(5).

Estimation of the coefficients c_k is frequently performed by solving a regularized least-squares optimization problem of the form

$$\widehat{\mathbf{c}}=arg\underset{\mathbf{c}\in {ℝ}^K}{min}\underset{\text{Data}\text{Consistency}}{\underbrace{{\Vert \mathbf{Hc}-\mathbf{e}\Vert }_2^2}}+\underset{\text{Regularization}}{\underbrace{\lambda {\Vert \mathbf{Dc}\Vert }_2^2}},$$ (A.2)

where e ∈ ℝ^M is the vector of E(q_m) values, c ∈ ℝK is the vector of c_k values, the matrix H ∈ ℝ^M×K has entries [H]_mk = Φ_k(q_m), the matrix D is often used to regularize the solution and reduce the ill-posedness of the matrix inversion (e.g., choosing D as a Laplace-Beltrami operator will reduce noise amplification by encouraging the solution to be smooth), and λ is the regularization parameter that determines the strength of the regularization constraint. If the nullspaces of H and $\sqrt{\lambda }\mathbf{D}$ have trivial intersection, then the optimal solution to Eq. (A.2) is

$$\widehat{\mathbf{c}}=\mathbf{Ge},$$ (A.3)

where the matrix G ∈ ℝ^K×M is given by

$$\mathbf{G}\triangleq {\left({\mathbf{H}}^T\mathbf{H}+\lambda {\mathbf{D}}^T\mathbf{D}\right)}^{-1}{\mathbf{H}}^T.$$ (A.4)

Once the c_k coefficients have been estimated, it is common to use Eq. (A.1) to generate an estimate of the measured signal Ê(q) at every point q, and then apply analytic transformations to Ê(q) to obtain estimates of the EAP parameters of interest. For example, it is straightforward to estimate the EAP using the inverse Fourier transform according to

$$\begin{gathered} {\widehat{P}}_{\mathrm{\Delta }}(\mathbf{r})=\int \widehat{E}(\mathbf{q})e^{i2\pi {\mathbf{q}}^T\mathbf{r}}d\mathbf{q}=\sum _{k=1}^K{c}_k\int {\mathrm{\Phi }}_k(\mathbf{q})e^{i2\pi {\mathbf{q}}^T\mathbf{r}}d\mathbf{q} \\ =\sum _{k=1}^K{\widehat{c}}_k{\mathrm{\Psi }}_k(\mathbf{r})=\sum _{m=1}^M\left(\sum _{k=1}^K{\mathrm{\Psi }}_k(\mathbf{r}){[\mathbf{G}]}_{km}\right)E({\mathbf{q}}_m), \end{gathered}$$ (A.5)

where we have used the fact from Eq. (A.3) that

$${\widehat{c}}_k=\sum _{m=1}^M{[\mathbf{G}]}_{km}E({\mathbf{q}}_m)$$ (A.6)

and have used Ψ_k(r) to denote the inverse Fourier transform of Φ_k(q). Note that, as expected, Eq. (A.5) has exactly the desired linear form of Eq. (2) with θ = r and $g_m(\mathbf{r})={\sum }_{k=1}^K{\mathrm{\Psi }}_k(\mathbf{r}){[\mathbf{G}]}_{km}$. When estimating other EAP-derived parameters, a common approach is to apply the integral formulas from Eqs. (3)–(5) to the estimated EAP from Eq. (A.5). For example, the CSA ODF from Eq. (4) can be estimated as

$$\begin{gathered} O\widehat{D}{F}_{\mathit{CSA}}(\mathbf{u})={\int }_0^{\infty }{\widehat{P}}_{\mathrm{\Delta }}(\alpha \mathbf{u}){\alpha }^{2}d\alpha \\ ={\int }_0^{\infty }\sum _{k=1}^K{\widehat{c}}_k{\mathrm{\Psi }}_k(\alpha \mathbf{u}){\alpha }^{2}d\alpha \\ =\sum _{m=1}^M\left(\sum _{k=1}^K{B}_k(\mathbf{u}){[\mathbf{G}]}_{km}\right)E({\mathbf{q}}_m), \end{gathered}$$ (A.7)

with

$$B_k(\mathbf{u})\triangleq {\int }_0^{\infty }{\mathrm{\Psi }}_k(\alpha \mathbf{u}){\alpha }^{2}d\alpha .$$ (A.8)

Again, Eq. (A.7) has the desired linear form of Eq. (2), with θ = u and $g_m(\mathbf{u})={\sum }_{k=1}^K{B}_k(\mathbf{u}){[\mathbf{G}]}_{km}$. Linear estimators in the form of Eq. (2) for estimating other EAP-derived parameters like those defined in Eqs. (3)–(5) can be obtained in similar ways.