Problem Formulation

Table of Contents:

Given a vector-valued image f of an arbitrary dimension with pixel i 2 {1, ... ,N} consisting of vector f 2 RM, we are interested in restoring its denoised counterpart u by solving the following problem:

The regularization term is in fact a sum of G regularization terms, each of which grouping a set of images. The gth grouping (with associated tuning parameter Ag,l,r), where g = {1,2 , ..., G}, is defined according to a set of weights {wg,m}, where m 2 {1, ..., M}. Channels with wg,m ф 0 are included in the grouping and their weighted framelet coefficients are jointly considered via '2-norm for penalization. The different groupings can possibly overlap, implying each image can be at the

same time considered in different groups. This is in similar spirit as the overlapped


group LASSO [13]. We set Xgij = A 2gJ„)2 if Ur ф 0 or XgJ,r = 0 if otherwise. Here A is a constant that can be set independent of the weights.


Problem (15) can be solved effectively using penalty decomposition (PD) [14]. Defining auxiliary variables (vg,mj,r)i := wg,m(Wi,rum)i, this amounts to minimizing the following objective function with respect to u and v := {vg ,m;l,r}:

In PD, we (1) alternate between solving for u and v using block coordinate descent (BCD). Once this converges, we (2) increase д > 0 by a multiplicative factor that is greater than 1 and repeat step (1). This is repeated until increasing д does not result in further changes to the solution [14].

First Subproblem

We solve for v in the first problem, i.e., minv L^(u, v). This is a group '0 problem and the solution can be obtained via hard-thresholding:


An 'i version of the algorithm can be obtained by using soft-thresholding instead. Second Subproblem

By taking the partial derivative with respect to u(m), the solution to the second subproblem, i.e., minuL^(u, v), is for each m

where we have dropped the subscript i for notation simplicity. Note that since we have X)l r WjrWir = I, the the problem can be simplified to become

Solving the above equation for u(m) is trivial and involves only simple division.

Setting the Weights

In setting the weights {wg,m}, we note that the weights should decay with the dissimilarity between gradient directions associated with a pair of diffusion- weighted images. To reflect this, we let G = M and set for g, m 2 {1,..., M} wg,m = eK[(v>Vg)2_1] if v> vg < cos(9) or 0 otherwise, where к > 0 is a parameter that determines the rate of decay of the weight. The exponential function is in fact modified from the probability density function of the Watson distribution [15] with concentration parameter к. Essentially, this implies that for the gth diffusion- weighted image acquired at gradient direction vg, there is a corresponding group of images with associated weights {wg,m}. The weight is maximal at wg;g = 1 and is attenuated when m Ф g. To reduce computation costs, weights of images scanned at gradient directions deviating more than в from vg are set to 0, and the respective images are hence discarded from the group. We set в = 30°.

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