3,855 research outputs found
An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems
We propose a new fast algorithm for solving one of the standard approaches to
ill-posed linear inverse problems (IPLIP), where a (possibly non-smooth)
regularizer is minimized under the constraint that the solution explains the
observations sufficiently well. Although the regularizer and constraint are
usually convex, several particular features of these problems (huge
dimensionality, non-smoothness) preclude the use of off-the-shelf optimization
tools and have stimulated a considerable amount of research. In this paper, we
propose a new efficient algorithm to handle one class of constrained problems
(often known as basis pursuit denoising) tailored to image recovery
applications. The proposed algorithm, which belongs to the family of augmented
Lagrangian methods, can be used to deal with a variety of imaging IPLIP,
including deconvolution and reconstruction from compressive observations (such
as MRI), using either total-variation or wavelet-based (or, more generally,
frame-based) regularization. The proposed algorithm is an instance of the
so-called "alternating direction method of multipliers", for which convergence
sufficient conditions are known; we show that these conditions are satisfied by
the proposed algorithm. Experiments on a set of image restoration and
reconstruction benchmark problems show that the proposed algorithm is a strong
contender for the state-of-the-art.Comment: 13 pages, 8 figure, 8 tables. Submitted to the IEEE Transactions on
Image Processin
Fast Image Recovery Using Variable Splitting and Constrained Optimization
We propose a new fast algorithm for solving one of the standard formulations
of image restoration and reconstruction which consists of an unconstrained
optimization problem where the objective includes an data-fidelity
term and a non-smooth regularizer. This formulation allows both wavelet-based
(with orthogonal or frame-based representations) regularization or
total-variation regularization. Our approach is based on a variable splitting
to obtain an equivalent constrained optimization formulation, which is then
addressed with an augmented Lagrangian method. The proposed algorithm is an
instance of the so-called "alternating direction method of multipliers", for
which convergence has been proved. Experiments on a set of image restoration
and reconstruction benchmark problems show that the proposed algorithm is
faster than the current state of the art methods.Comment: Submitted; 11 pages, 7 figures, 6 table
A new steplength selection for scaled gradient methods with application to image deblurring
Gradient methods are frequently used in large scale image deblurring problems
since they avoid the onerous computation of the Hessian matrix of the objective
function. Second order information is typically sought by a clever choice of
the steplength parameter defining the descent direction, as in the case of the
well-known Barzilai and Borwein rules. In a recent paper, a strategy for the
steplength selection approximating the inverse of some eigenvalues of the
Hessian matrix has been proposed for gradient methods applied to unconstrained
minimization problems. In the quadratic case, this approach is based on a
Lanczos process applied every m iterations to the matrix of the most recent m
back gradients but the idea can be extended to a general objective function. In
this paper we extend this rule to the case of scaled gradient projection
methods applied to non-negatively constrained minimization problems, and we
test the effectiveness of the proposed strategy in image deblurring problems in
both the presence and the absence of an explicit edge-preserving regularization
term
An Efficient Approach for Computing Optimal Low-Rank Regularized Inverse Matrices
Standard regularization methods that are used to compute solutions to
ill-posed inverse problems require knowledge of the forward model. In many
real-life applications, the forward model is not known, but training data is
readily available. In this paper, we develop a new framework that uses training
data, as a substitute for knowledge of the forward model, to compute an optimal
low-rank regularized inverse matrix directly, allowing for very fast
computation of a regularized solution. We consider a statistical framework
based on Bayes and empirical Bayes risk minimization to analyze theoretical
properties of the problem. We propose an efficient rank update approach for
computing an optimal low-rank regularized inverse matrix for various error
measures. Numerical experiments demonstrate the benefits and potential
applications of our approach to problems in signal and image processing.Comment: 24 pages, 11 figure
A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts
Recently, a fast alternating minimization algorithm for total variation image
deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM
J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of
a discrete Fourier transform-based alternating minimization algorithm with
periodic boundary conditions and in which two fast Fourier transforms (FFTs)
are required per iteration. In this paper, we propose an alternating
minimization algorithm for the continuous version of the total variation image
deblurring problem. We establish convergence of the proposed continuous
alternating minimization algorithm. The continuous setting is very useful to
have a unifying representation of the algorithm, independently of the discrete
approximation of the deconvolution problem, in particular concerning the
strategies for dealing with boundary artifacts. Indeed, an accurate restoration
of blurred and noisy images requires a proper treatment of the boundary. A
discrete version of our continuous alternating minimization algorithm is
obtained following two different strategies: the imposition of appropriate
boundary conditions and the enlargement of the domain. The first one is
computationally useful in the case of a symmetric blur, while the second one
can be efficiently applied for a nonsymmetric blur. Numerical tests show that
our algorithm generates higher quality images in comparable running times with
respect to the Fast Total Variation deconvolution algorithm
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