# Problem Formulation

Given a vector-valued image *f* of an arbitrary dimension with pixel *i **2* {1, ... ,N} consisting of vector *f **2* R^{M}, 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 A_{g}*, _{l},_{r}*), where

*g =*{1,2 , ..., G}, is defined according to a set of weights

*{w*

_{g},_{m}}, where

*m*

*2*{1, ..., M}. Channels with

*w*0 are included in the grouping and their weighted framelet coefficients are jointly considered via '

_{g},_{m}ф_{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

1

group LASSO [13]. We set *X _{gij}* = A

*’*if

^{2}gJ„)^{2}*Ur ф*0 or

*X*0 if otherwise. Here A is a constant that can be set independent of the weights.

_{gJ},_{r}=# Optimization

Problem (15) can be solved effectively using penalty decomposition (PD) [14]. Defining auxiliary variables *(v _{g},_{m}j,_{r})_{i}* :=

*w*this amounts to minimizing the following objective function with respect to

_{g},_{m}(Wi,_{r}u^{m})_{i},*u*and v := {v

_{g}

*,*

_{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., min_{v} *L^(u,* v). This is a group '_{0} problem and the solution can be obtained via hard-thresholding:

where

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., min_{u}L^(u, v), is for each *m*

where we have dropped the subscript *i* for notation simplicity. Note that since we have X)* _{l r} Wj_{r}Wi_{r}* =

*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 {w_{g},_{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} w _{g},_{m} =* e

^{K}[

^{(v}>

^{Vg)2_1}] if v> v

_{g}

*< 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 v

_{g}, there is a corresponding group of images with associated weights {w

_{g},

_{m}}. The weight is maximal at

*w*1 and is attenuated when

_{g;g}=*m Ф g.*To reduce computation costs, weights of images scanned at gradient directions deviating more than в from

*v*are set to 0, and the respective images are hence discarded from the group. We set

_{g}*в =*30°.