  This paper presents a fast approach for penalized least squares (LS)
regression problems using a 2D Gaussian Markov random field (GMRF) prior. More
precisely, the computation of the proximity operator of the LS criterion
regularized by different GMRF potentials is formulated as solving a
Sylvester-like matrix equation. By exploiting the structural properties of
GMRFs, this matrix equation is solved columnwise in an analytical way. The
proposed algorithm can be embedded into a wide range of proximal algorithms to
solve LS regression problems including a convex penalty. Experiments carried
out in the case of a constrained LS regression problem arising in a
multichannel image processing application, provide evidence that an alternating
direction method of multipliers performs quite efficiently in this context.
