Diffusion MRI is a powerful neuroimaging technique due to its unique ability to extract microstructural information by utilizing restricted and hindered diffusion to probe compartments that are much smaller than the voxel size. One important goal of diffusion MRI is to estimate axonal orientations, tracing of which will allow one to gauge connectivity between brain regions and will provide in vivo information on white matter pathways for neuroscience studies involving development, aging, and disorders [1-5]. In order to capture orientation information, the brain has to be scanned using a range of diffusion-sensitizing gradient directions that are ideally distributed uniformly on the unit sphere.
As shown in Fig. 1, DW images that are scanned with similar gradient directions share a lot of commonalities. However, these commonalities diminish very quickly if the difference between the gradient directions increases. As can also be seen from the figure, the images are typically very noisy and can benefit greatly from denoising. Denoising performance can be improved by borrowing information between images scanned at similar gradient directions; however, images scanned at a very different direction have to be avoided in this process to reduce artifacts.
In this paper, we take advantage of the correlation between DW images scanned with neighboring gradient directions in a group '0 minimization denoising framework that is based on tight wavelet frames. The power of tight wavelet frames lies in their ability to sparsely approximate piecewise smooth functions and the existence of fast decomposition and reconstruction algorithms associated with them. In contrast, total-variation (TV) based methods are effective in restoring images that are piecewise constant, e.g., binary or cartoon-like images. They will, however, cause staircasing effects in images that are not piecewise constant .
Instead of the more conventional ' regularization, which has been shown in the theory of compressed sensing  to produce sparse solutions, we opted to use '0 regularization. In , both iterative soft and hard thresholding algorithms were adopted and the latter was found to achieve better image quality. In , wavelet frame based '0 regularization also shows better edge-preserving quality compared
Fig. 1 Diffusion-weighted images scanned at different gradient directions. The left and middle images were scanned with similar gradient directions. The right image was scanned at a nearly perpendicular gradient direction with respect to the reference with the conventionall1 regularization. In contrast to previous works, we propose a group version of 'o minimization to take advantage of the correlation between DW images.
Evaluations performed using synthetic data with noncentral chi noise distribution as well as real data with repeated scans indicate that the proposed method is superior to its 'i counterpart and non-local means denoising.