# Methods

This section describes our methods for regularized dictionary learning, strongly sparse fitting, and compressed sensing multishell HARDI directly from k-space data.

## Dictionary Learning for Multishell HARDI

Using a HARDI image dataset, we collect multishell signals at voxels in regions that have a single tract passing through them, e.g., corpus callosum and lateral corticospinal tract, where the strongly anisotropic diffusion leads to higher contrast- to-noise ratios in the signal. To factor out arbitrary differences in (1) imaging coordinate-frame origins and poses across individuals, (2) orientations of a specific tract across individuals, and (3) orientations of different tracts within an individual, we reorient each diffusion signal to align it with a fixed diffusion signal that models prolate-tensor diffusion along a fixed direction ([0,0,1]^{T}). The alignment entails interpolation, on the spherical domain of the diffusion-signal function, using Barycentric coordinates and geodesic distances on the sphere S^{2}. We align two realvalued functions, i.e., diffusion signals, defined on the spherical domain of gradient directions. The underlying deformation is rotation. The registration gives aligned signals defined on the same gradient directions (for each shell). We divide each aligned signal by the voxel value in the corresponding *b0* image.

Because diffusion signal values are modeled as non-negative real [20], we learn atoms with non-negative values from magnitude HARDI images. We constrain coefficients to be non-negative real to ensure that the fits lie in the same space as the data. For multishell diffusion signals, shells with larger *b* values lead to drastic reductions in SNR. To counter the noise, during dictionary learning, we propose to enforce a *smoothness prior* on the atom over the spherical domain of gradient directions. We use a robust penalty to reduce noise while preserving contrast over the spherical domain.

Let the HARDI acquisition employ *S* shells, with shell s comprising *N _{s}* gradient directions

*fg*R

_{sn}e^{3}:

*g*1^1

_{sn}_{2}=_{1}. In a training dataset of

*I*aligned multishell diffusion signals, let the i-th signal be

*f'*. Let

*f*R>

_{n}e_{0}be the signal value from shell s for gradient direction

*g*Let the dictionary matrix

_{sn}.*D*comprise

*K*multishell diffusion signals (atoms)

*d*as columns. Let

^{k}*d*R>

^{k}sn e_{0}be the atom value from shell s for gradient direction

*g*Let

_{sn}.*c'*be the vector of non-negative coefficients used to represent

*f'.*We propose the optimal dictionary as the solution to

where *H*(-,1) is the Huber loss function [12] (a smooth approximation to the *L ^{1}* penalty) with parameter 1,

*a > 0*controls the sparsity prior strength,

*> 0*controls the strength of the robust smoothness prior

*H(-, -)*for shell

*s*of each atom’s diffusion signal,

*w*[0,1] weights the roughness penalty for the deviation between atom values between directions

_{smn}e*gsn*and

*gsm*;

*wsmn*:= exp(-0.5| arccos(g

_{S}n,

*g*)|

_{sm}^{2}/(ж/12)

^{2}).

We use iterative optimization to alternatingly optimize atoms *d ^{k}* and coefficient vectors c

*',*while fixing the other, in each case solving a convex optimization problem. We use K-means to initialize

*d*to the

^{k}*K*cluster centers. Fixing atoms

*{d*the optimization problem for each coefficient vector

^{k}}f_{=1},*C*has a quadratic objective function and a linear (positive) constraint; the global minimum has a closed form. Fixing coefficient vectors

*{C*}I

_{= 1}, because we choose

*H(? ,8)*as convex, the optimization problem for each atom

*d*is convex; we find the global minimum via projected gradient descent with adaptive step size. To the learned dictionary, we add constant atoms d

^{k}^{K+s}, one for each shell s, to better model isotropic diffusion in the fluid and gray matter.