Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Abstract

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view -as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.Keywords Douglas-Rachford splitting · forward-backward splitting · Bregman methods · augmented Lagrangian method · alternating split Bregman algorithm · image denoising

Segmentation of images with separating layers by fuzzy c-means and convex optimization

“…For the above problem, the ADMM coincides with the alternating split Bregman method [13] and with the Douglas-Rachford splitting method applied to the dual problem of (11), see [8,9,11,27]. For any starting values and any γ > 0, the ADMM sequences {b (i) } and {y (i) } converge to some b andŷ, respectively.…”

“…For any starting values and any γ > 0, the ADMM sequences {b (i) } and {y (i) } converge to some b andŷ, respectively. The sequence {x (i) } converges to a solution of (11) if problem (11) has a unique solution, see, e.g., [27]. Note that 1 γb is a solution of the dual problem of (11) argmin…”

“…The minimizer v (i+1) can be obtained by the so-called coupled shrinkage of g v with threshold λγ, see [27]. Using (13) and v(j) T := (v x,1 (j), v y,1 (j), .…”

“…Then we use these prototypes together with the separating layer information in the data term of a convex optimization model which regularization term is the TV-functional. For solving the resulting convex optimization problem we propose an alternating direction method of multipliers (ADMM) which is for this problem identical with the alternating split Bregman algorithm [13] and the Douglas-RachfordSplitting algorithm [8,10,11,27]. We focus on a discrete model which uses matrix-vector notation based on tensor products of matrices.…”

Adaptive Proximal Point Algorithms for Total Variation Image Restoration

“…In recent years, many researchers [8,9,10,12,13,18,22,28,32] study the TV minimization via solving the following saddle-point problem:…”