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Regularized Dictionary Learning with Robust Sparsity Fitting for Compressed Sensing Multishell HARDI

Kratika Gupta, Deepali Adlakha, Vishal Agarwal, and Suyash P. Awate

Abstract This paper presents a new compressed sensing framework for multishell HARDI. Unlike methods that model diffusion signals using analytical bases, we learn a dictionary of multishell diffusion signals, with a proposed regularization term to handle low signal-to-noise ratios at high b values. We combine the dictionary model for diffusion signals together with a multiscale (wavelet-based) spatial model on images for compressed sensing. To control overfltting of the dictionary to tracts with unknown orientations, we use a strong non-sparsity penalty that behaves close to the desirable L0 pseudo-norm. Our framework allows undersampling gradient directions, shells, and k-space. The results show improved reconstructions from our framework, over the state of the art.

Introduction and Related Work

Multishell high angular resolution diffusion imaging (HARDI) [1, 20] acquires diffusion weighted (DW) magnetic resonance (MR) images using a large number of gradient directions over multiple shells (i.e., b values). Multishell HARDI combines (1) higher signal-to-noise ratio (SNR) at lower b values with (2) the greater ability to resolve tract directions and narrow-angle crossings at larger b values [19], at the cost of scan time.

We propose a novel compressed sensing framework that allows undersampling gradient directions, shells, and k-space. The key approach for speeding up HARDI scans is, indeed, undersampling the set of gradient directions and the results in

Suyash P. Awate thanks funding via IIT Bombay Seed Grant 14IRCCSG010. All work done at IIT Bombay.

K. Gupta • V. Agarwal • S.P. Awate (H)

Computer Science and Engineering Department, Indian Institute of Technology (IIT) Bombay,

Mumbai, India

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D. Adlakha

Ads Team, Facebook, London, UK © Springer International Publishing AG 2017

A. Fuster et al. (eds.), Computational Diffusion MRI, Mathematics

and Visualization, DOI 10.1007/978-3-319-54130-3_3

the paper focus on that practical scenario. Nevertheless, the framework is general and allows to explore reconstructions involving undersampling in shells or k-space. In some DW MRI applications involving a small number of diffusion-encoding directions, k-space undersampling can be preferred, as shown for diffusion tensor MRI in [13]; our framework is applicable, in principle, to such scenarios as well. Unlike methods [6,7, 10,15,18] that model diffusion signals using analytical bases (e.g., spherical harmonics or ridgelets) for each shell independently, we propose to learn a dictionary of multishell diffusion signals, exploiting correlations between multiple shells. In addition to the dictionary, we use an overcomplete wavelet frame for multiscale spatial regularization; others works use total-variation (TV) regularization [15, 17] or no spatial regularization [2, 11]. Our approach is similar to the one in [3] for dynamic MRI.

Some works [2, 4, 11, 14] use dictionaries based on tensor or parametric models to reconstruct diffusion signals from single-shell HARDI with undersampled directions. Some methods [9, 11] use positivity constraints on the dictionary atoms through nonnegative sparse coding, which relates to our approach. To handle unknown tract orientations in practice, while [11] expands the dictionary to include rotated atoms at the risk of overfltting, [2,4] explicitly optimize each atom’s rotation at a high computational cost and the risk of local optima with corrupted data. In contrast, we (1) fit the dictionary to arbitrarily oriented tracts while controlling overfltting at low computational cost by modifying the non-sparsity penalty to give sparser dictionary fits and (2) reconstruct multishell HARDI directly from undersampled noisy k-space data.

In this paper, we propose a new method for learning a dictionary of multishell diffusion signals, employing a spherical-domain regularization on the estimated atoms to counter the low SNR at higher b values. Our formulation leads to a convex optimization problem in each variable (atoms or coefficients), which can be solved efficiently. We propose a new framework for compressed sensing that employs the learned dictionary together with multiscale spatial regularity using wavelets. While the learned dictionary is expanded to handle arbitrary tract orientations, we control overfitting by a strong non-sparsity penalty that behaves close to the desirable L0 pseudo-norm [8] and leads to an efficient majorization minimization (MM) algorithm performing convex optimization iteratively. Our compressed sensing framework reconstructs DW MR images directly from undersampled noisy k- space data. In this way, the framework allows exploration of application-specific acquisition optimization regarding the choice of (1) number of shells, (2) gradient directions in each shell, and (3) undersampling in q-space and k-space. Results show the advantages of our framework over the state of the art.

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