345 research outputs found
BM3D Frames and Variational Image Deblurring
A family of the Block Matching 3-D (BM3D) algorithms for various imaging
problems has been recently proposed within the framework of nonlocal patch-wise
image modeling [1], [2]. In this paper we construct analysis and synthesis
frames, formalizing the BM3D image modeling and use these frames to develop
novel iterative deblurring algorithms. We consider two different formulations
of the deblurring problem: one given by minimization of the single objective
function and another based on the Nash equilibrium balance of two objective
functions. The latter results in an algorithm where the denoising and
deblurring operations are decoupled. The convergence of the developed
algorithms is proved. Simulation experiments show that the decoupled algorithm
derived from the Nash equilibrium formulation demonstrates the best numerical
and visual results and shows superiority with respect to the state of the art
in the field, confirming a valuable potential of BM3D-frames as an advanced
image modeling tool.Comment: Submitted to IEEE Transactions on Image Processing on May 18, 2011.
implementation of the proposed algorithm is available as part of the BM3D
package at http://www.cs.tut.fi/~foi/GCF-BM3
Convexity in source separation: Models, geometry, and algorithms
Source separation or demixing is the process of extracting multiple
components entangled within a signal. Contemporary signal processing presents a
host of difficult source separation problems, from interference cancellation to
background subtraction, blind deconvolution, and even dictionary learning.
Despite the recent progress in each of these applications, advances in
high-throughput sensor technology place demixing algorithms under pressure to
accommodate extremely high-dimensional signals, separate an ever larger number
of sources, and cope with more sophisticated signal and mixing models. These
difficulties are exacerbated by the need for real-time action in automated
decision-making systems.
Recent advances in convex optimization provide a simple framework for
efficiently solving numerous difficult demixing problems. This article provides
an overview of the emerging field, explains the theory that governs the
underlying procedures, and surveys algorithms that solve them efficiently. We
aim to equip practitioners with a toolkit for constructing their own demixing
algorithms that work, as well as concrete intuition for why they work
Transformed Schatten-1 Iterative Thresholding Algorithms for Low Rank Matrix Completion
We study a non-convex low-rank promoting penalty function, the transformed
Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty,
as a matrix quasi-norm defined on its singular values, interpolates the rank
and the nuclear norm through a nonnegative parameter a. We consider the
unconstrained TS1 regularized low-rank matrix recovery problem and develop a
fixed point representation for its global minimizer. The TS1 thresholding
functions are in closed analytical form for all parameter values. The TS1
threshold values differ in subcritical (supercritical) parameter regime where
the TS1 threshold functions are continuous (discontinuous). We propose TS1
iterative thresholding algorithms and compare them with some state-of-the-art
algorithms on matrix completion test problems. For problems with known rank, a
fully adaptive TS1 iterative thresholding algorithm consistently performs the
best under different conditions with ground truth matrix being multivariate
Gaussian at varying covariance. For problems with unknown rank, TS1 algorithms
with an additional rank estimation procedure approach the level of IRucL-q
which is an iterative reweighted algorithm, non-convex in nature and best in
performance
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
We consider linear inverse problems where the solution is assumed to have a
sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that
replacing the usual quadratic regularizing penalties by weighted l^p-penalties
on the coefficients of such expansions, with 1 < or = p < or =2, still
regularizes the problem. If p < 2, regularized solutions of such l^p-penalized
problems will have sparser expansions, with respect to the basis under
consideration. To compute the corresponding regularized solutions we propose an
iterative algorithm that amounts to a Landweber iteration with thresholding (or
nonlinear shrinkage) applied at each iteration step. We prove that this
algorithm converges in norm. We also review some potential applications of this
method.Comment: 30 pages, 3 figures; this is version 2 - changes with respect to v1:
small correction in proof (but not statement of) lemma 3.15; description of
Besov spaces in intro and app A clarified (and corrected); smaller pointsize
(making 30 instead of 38 pages
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