where denotes element-wise product. and come from two least-squares minimization problems:

Here, is a diagonal operator composed of the elements of , is a diagonal operator composed of the elements of . Note that in equations 10-12, , , and denote vectorized 2D matrices. Equations 11 and 12 can be solved using shaping regularization with a local-smoothness constraint:

where is a smoothing operator, and are two parameters controlling the physical dimensionality and enabling fast convergence when inversions for and expressed in equations 13 and 14 are implemented iteratively. and can be chosen as and .

2020-02-10