We evaluate the proposed framework on simulated and human brain HARDI images. The parameters A, a, , and у need to be tuned empirically using cross-validation.

Regularized Dictionary Learning for Brain HARDI While we learn a multishell dictionary for compressed sensing later, we first show the utility of the smoothness prior via single-shell learning. From the brain dataset, we obtained 1000 high- variance diffusion signals for b = 1000 s/mm^{2} as “clean” and learned a dictionary of ten atoms from that (two examples in Fig. 1a); we show the diffusion signal values, colormapped, at the corresponding gradient directions on a unit hemisphere. We then used the corresponding 1000 diffusion signals from the shell with b = 3000 s/mm^{2}, corrupted it with noise, and learned two dictionaries, one using the smoothness prior and one without. Atoms learned without the smoothness prior undesirably exhibited more noise/random perturbations (examples in Fig. 1b) than atoms learned with the prior (examples in Fig. 1c). The relative root mean squared error (RRMSE) between the fitted diffusion signals a and the reference signal b is II^{я} — b||/||b||. The regularized atoms, expectedly, produced better fits to noisy data, which were qualitatively smoother (example in Fig. 2c, e) and quantitatively having

Fig. 1 Regularized dictionary learning. Two example atoms learned: (a) using clean data at b = 1000 s/mm^{2} (considered as baseline), (b) using noisy data at b = 3000 s/mm^{2}, without smoothness prior (p_{s} = 0), and (c) using noisy data at b = 3000 s/mm^{2}, with smoothness prior (S_{s} > 0) (proposed). Dot locations = gradient direction vectors; colors = diffusion signal values

Fig. 2 Regularized dictionary learning. Example fit to noisy data at b = 3000 s/mm^{2} using a dictionary learned: (a) using clean data (reference), (b) using noisy data, without smoothness prior (fi_{s} = 0), and (c) using noisy data, with smoothness prior (fi_{s} > 0) (proposed). (d)-(e) Residuals for (b)-(c), respectively

Fig. 3 Dictionary modeling of human brain HARDI. (a) Baseline (high quality) GFA image, averaged over shells. (b) GFA of the image resulting from fitting the dictionary to the baseline image. (c) Magnitude of the difference between per-voxel GFA in image (a) and (b). (d) Per voxel RRMSEs between baseline and dictionary-fitted diffusion signals

about half the RRMSE, as compared to the atoms learned without regularization (example in Fig. 2b, d).

Figure 3 evaluates the utility of the proposed dictionary learning framework in modeling brain HARDI. The GFA of the baseline HARDI signals (Fig. 3a) and the GFA of the dictionary-fitted HARDI signal (Fig. 3b) are very similar, with differences (Fig. 3c) almost always being less than half percent or less. The per-

Fig. 4 Compressed sensing simulated HARDI. (a1)-(a2) Ground truth diffusion signals, for 2 shells, in simulated phantom. We simulated 16X undersampling in gradient directions and SNR-7 noise (see text). (b) Proposed method’s performance with different weights X for the dictionary and wavelet prior. (c1)-(c2) True signal (zoomed in 2 x 2 patch boxed red ? in (a)-(b)), for both shells. Reconstructions with: (d1)-(d2) dictionary and wavelet, X = 0.6, RRMSE 0.02 (proposed) which is near perfect; (e1)-(e2) sliding window, RRMSE 0.44; (f1)-(f2) dictionary, X = 0, RRMSE 0.17; (g1)-(g2) wavelet (3D overcomplete), X = 1 (RRMSE 0.94); (h1)-(h2) dictionary and TV, optimized X (RRMSE 0.05)

voxel RRMSEs (Fig. 3d) between the dictionary fit and the baseline image are almost always less than 1%.

Compressed Sensing Simulated HARDI We use a simulated phantom; 16 x 16 voxel matrix, 81 gradient directions, 2 shells) comprising 2 crossing tracts, 1 horizontal and 1 vertical (Fig.4(a1), (a2)). We simulate 16X undersampling in gradient directions, acquiring only 5 of the 81 directions, per shell, spread roughly uniformly over the hemisphere that resulted in small magnitudes of the inner-product between pairs of selected directions. We introduce independent and identically distributed zero-mean complex Gaussian noise of variance ct^{2} in k- space such that the SNR, defined as the largest signal magnitude among all DW images divided by ct, equals 7 that mimics a realistic clinical acquisition scenario. The proposed framework gives results [Fig.4(d1), (d2)] that are near perfect, as compared to other approaches. The best performance occurs when both multishelldictionary- and wavelet-based models are used, i.e., X roughly midway between 0 and 1 (Fig. 4c). The simple sliding window reconstruction [Fig. 4(e1), (e2)], where we replace the missing k-space data by the acquired data for the same location in k- space but in the nearest direction in the same shell, fails to enforce any regularity on the reconstruction in space or directions. The dictionary model alone gives results with poor spatial regularity [Fig.4(f1), (f2)]. An analytical wavelet basis jointly for space and directions/shells does poorer because the proposed approach uses a dictionary that is adapted to multishell diffusion signals [Fig.4(g1), (g2)]. The dictionary model with a spatial TV prior cannot model multiscale spatial regularity, unlike wavelet transforms [Fig.4(h1), (h2)]; a spatial TV prior can be easily incorporated in our framework by replacing the wavelet transform with a linear transform that takes one-sided spatial derivatives along each dimension of the image.

Compressed Sensing Brain HARDI We used ten fully-sampled high-quality 2-shell HARDI images with 90 gradient directions, treated as baselines, and performed retrospective undersampling and noise corruption (mimicking SNR 7). We performed 6X undersampling of directions (15 out of 90 directions acquired roughly uniformly over the hemisphere, for each shell) and 1.3X undersampling of k-space (undersampled frequency encodes), corresponding to a reduction in scan time by a factor slightly larger than 6. The reconstructions (Figs. 5 and 6) from our approach of using the learned multishell dictionary model combined with a spatial regularity model (wavelet or TV) are of significantly higher quality than the other

Fig. 5 Compressed sensing brain HARDI (slice I). (a) Baseline (high quality) GFA images, averaged over shells. We simulated 6X undersampling of directions and 1.3X undersampling of k-space for each direction acquired, introducing noise to get SNR 7 (see text). GFA images of reconstructions with: (b) dictionary and wavelet, X = 0.7, RRMSE 0.12 (proposed); (c) dictionary and TV with optimized X, RRMSE 0.12; (d) dictionary, X = 0, RRMSE 0.23; (e) sliding window, RRMSE 0.23; (f) wavelet (3D overcomplete), X = 1, RRMSE 0.91

Fig. 6 Compressed sensing brain HARDI (slice II). (a) Baseline (high quality) GFA images, averaged over shells. We simulated 6X undersampling of directions and 1.3X undersampling of k-space for each acquired direction, introducing noise to get SNR 7 (see text). GFA images of reconstructions with (b) dictionary and wavelet, X = 0.7, RRMSE 0.12 (proposed); (c) dictionary and TV with optimized X, RRMSE 0.12; (d) dictionary, X = 0, RRMSE 0.23; (e) sliding window, RRMSE 0.22; (f) wavelet (3D overcomplete), X = 1, RRMSE 0.9

Fig. 7 Compressed sensing brain HARDI (slice I). Images of per-voxel RRMSE (multishell) for reconstructions in Fig. 5 with (a) dictionary and wavelet, RRMSE 0.12 (proposed); (b) dictionary and TV, RRMSE 0.12; (c) dictionary, RRMSE 0.23; (d) sliding window, RRMSE 0.23

approaches. The approach using the dictionary alone failed to reproduce spatial regularity in the reconstructed image, and instead produced noisy reconstructions with about twice as much RRMSE. The sliding window approach lead to a similar lack of regularity and much larger RRMSE. The approach using the wavelet frame only, coupled in space and directions, performed the worst because, compared to the dictionary, the wavelets are unable to effectively model the regularity for this class of signals. The images of RMS errors (Figs. 7 and 8) clearly show that our coupled

Fig. 8 Compressed sensing brain HARDI (slice II). Images of per-voxel RRMSE (multishell) for reconstructions with (a) dictionary and wavelet, RRMSE 0.12 (proposed); (b) dictionary and TV, RRMSE 0.12; (c) dictionary, RRMSE 0.23; (d) sliding window, RRMSE 0.22

Fig. 9 Compressed sensing brain HARDI (region I: corpus callosum) (a) Baseline diffusion signals, for b = 1000 s/mm^{2}. Reconstructed diffusion signals with (b) dictionary and wavelet, (c) dictionary and TV, (d) dictionary, (e) sliding window, (f) wavelet (3D overcomplete)

approach, dictionary in addition to spatial regularization (wavelet or TV), yields the lowest errors.

We now show the reconstruction in two regions: a 3 x 3 voxel patch in the corpus callosum in Fig. 9, and another 3 x 3 voxel patch in a fiber-crossing region of the corpus callosum and the lateral corticospinal tract in Fig. 10. In the corpus callosum (Fig. 9), while the dictionary-only reconstruction underfits to the data, the wavelet- only and sliding-window reconstructions are noisy and erroneous. In the crossing

Fig. 10 Compressed sensing brain HARDI (region II: fiber crossing) (a) Baseline diffusion signals, for b = 1000 s/mm^{2}. Reconstructed diffusion signals with (b) dictionary and wavelet, (c) dictionary and TV, (d) dictionary, (e) sliding window, (f) wavelet (3D overcomplete)

Fig. 11 Box plots of RRMSEs for ten randomly selected coronal brain MRI slices

region (Fig. 10), the dictionary-only reconstruction has modified the directions of the individual tracts; at each voxel, the blue area shifts from the bottom right to the bottom left. Our approach of using the dictionary coupled with spatial regularization (wavelet or TV) gives the best results. We experimented with ten different coronal slices from brain HARDI images and found (Fig. 11) that our approach of using the dictionary along with a spatial prior (wavelet or TV) outperforms other approaches.