In this research we address the denoising problem, and specifically,
retrieving signals corrupted by additive white Gaussian noise. A classic and
well-known framework for handling this task is the Basis-Pursuit Denoising
(BPDN). Despite being frequently deployed in many applications, the BPDN
problem has proven hard to solve using classic iterative algorithms, which
exhibit slow convergence rate. In a recent work, a new iterative-shrinkage
method called Parallel Coordinate Descent (PCD) was devised. We provide herein
a convergence analysis of the PCD algorithm, and also introduce a form of the
regularization function, which permits analytical solution to the coordinate optimization.
Several other recent works, which considered the deblurring problem in a
Bayesian methodology, also obtained iterative-shrinkage algorithms. We show
that these three methods are essentially equivalent, and the unified method is
termed Separable Surrogate Functionals (SSF). We provide a convergence analysis
for SSF as well, and draw a connection between this family and the PCD method.
To further accelerate PCD and SSF, we merge them into a recently developed
sequential subspace optimization technique (SESOP), with almost no additional
complexity. A thorough numerical comparison, both with synthetic data and with
real images, is presented. The advantage of our combined PCD-SESOP method is
clearly demonstrated.
In this research we have also generalized the BPDN, so as to take into
account the inter-dependencies between representation coefficients. Several
advanced methods that model these dependencies were developed recently, and
were shown to yield significant improvement over simple shrinkage. However,
these methods operate on the transform domain error rather than on the image
domain one, errors which are in general entirely different for redundant
transforms. Our new method combines the image domain error with the transform
domain dependency structure, resulting in a general objective function,
applicable for any wavelet-like transform. We focus here on the Contourlet
Transform (CT), a relatively new transform designed to sparsely represent
images. Our method shows state-of-the-art results, thus providing a more advanced
tool for image restoration.
