9 research outputs found
New convergence results for the scaled gradient projection method
The aim of this paper is to deepen the convergence analysis of the scaled
gradient projection (SGP) method, proposed by Bonettini et al. in a recent
paper for constrained smooth optimization. The main feature of SGP is the
presence of a variable scaling matrix multiplying the gradient, which may
change at each iteration. In the last few years, an extensive numerical
experimentation showed that SGP equipped with a suitable choice of the scaling
matrix is a very effective tool for solving large scale variational problems
arising in image and signal processing. In spite of the very reliable numerical
results observed, only a weak, though very general, convergence theorem is
provided, establishing that any limit point of the sequence generated by SGP is
stationary. Here, under the only assumption that the objective function is
convex and that a solution exists, we prove that the sequence generated by SGP
converges to a minimum point, if the scaling matrices sequence satisfies a
simple and implementable condition. Moreover, assuming that the gradient of the
objective function is Lipschitz continuous, we are also able to prove the
O(1/k) convergence rate with respect to the objective function values. Finally,
we present the results of a numerical experience on some relevant image
restoration problems, showing that the proposed scaling matrix selection rule
performs well also from the computational point of view
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
Accelerated gradient methods for the X-ray imaging of solar flares
In this paper we present new optimization strategies for the reconstruction
of X-ray images of solar flares by means of the data collected by the Reuven
Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The imaging concept of
the satellite is based of rotating modulation collimator instruments, which
allow the use of both Fourier imaging approaches and reconstruction techniques
based on the straightforward inversion of the modulated count profiles.
Although in the last decade a greater attention has been devoted to the former
strategies due to their very limited computational cost, here we consider the
latter model and investigate the effectiveness of different accelerated
gradient methods for the solution of the corresponding constrained minimization
problem. Moreover, regularization is introduced through either an early
stopping of the iterative procedure, or a Tikhonov term added to the
discrepancy function, by means of a discrepancy principle accounting for the
Poisson nature of the noise affecting the data
Variable metric inexact line-search based methods for nonsmooth optimization
We develop a new proximal-gradient method for minimizing the sum of a
differentiable, possibly nonconvex, function plus a convex, possibly non
differentiable, function. The key features of the proposed method are the
definition of a suitable descent direction, based on the proximal operator
associated to the convex part of the objective function, and an Armijo-like
rule to determine the step size along this direction ensuring the sufficient
decrease of the objective function. In this frame, we especially address the
possibility of adopting a metric which may change at each iteration and an
inexact computation of the proximal point defining the descent direction. For
the more general nonconvex case, we prove that all limit points of the iterates
sequence are stationary, while for convex objective functions we prove the
convergence of the whole sequence to a minimizer, under the assumption that a
minimizer exists. In the latter case, assuming also that the gradient of the
smooth part of the objective function is Lipschitz, we also give a convergence
rate estimate, showing the O(1/k) complexity with respect to the function
values. We also discuss verifiable sufficient conditions for the inexact
proximal point and we present the results of a numerical experience on a convex
total variation based image restoration problem, showing that the proposed
approach is competitive with another state-of-the-art method
Inexact Bregman iteration with an application to Poisson data reconstruction
This work deals with the solution of image restoration problems by an
iterative regularization method based on the Bregman iteration. Any iteration of this
scheme requires to exactly compute the minimizer of a function. However, in some
image reconstruction applications, it is either impossible or extremely expensive to
obtain exact solutions of these subproblems. In this paper, we propose an inexact
version of the iterative procedure, where the inexactness in the inner subproblem
solution is controlled by a criterion that preserves the convergence of the Bregman
iteration and its features in image restoration problems. In particular, the method
allows to obtain accurate reconstructions also when only an overestimation of the
regularization parameter is known. The introduction of the inexactness in the iterative
scheme allows to address image reconstruction problems from data corrupted by
Poisson noise, exploiting the recent advances about specialized algorithms for the
numerical minimization of the generalized Kullback–Leibler divergence combined with
a regularization term. The results of several numerical experiments enable to evaluat
A comparison of edge-preserving approaches for differential interference contrast microscopy
In this paper we address the problem of estimating the phase from color images acquired with differential-interference-contrast microscopy. In particular, we consider the nonlinear and nonconvex optimization problem obtained by regularizing a least-squares-like discrepancy term with an edge-preserving functional, given by either the hypersurface potential or the total variation one. We investigate the analytical properties of the resulting objective functions, proving the existence of minimum points, and we propose effective optimization tools able to obtain in both the smooth and the nonsmooth case accurate reconstructions with a reduced computational demand
Scaling techniques for gradient projection-type methods in astronomical image deblurring
The aim of this paper is to present a computational study on scaling techniques in gradient projectiontype
(GP-type) methods for deblurring of astronomical images corrupted by Poisson noise. In this
case, the imaging problem is formulated as a non-negatively constrained minimization problem in
which the objective function is the sum of a fit-to-data term, the Kullback\u2013Leibler divergence, and a
Tikhonov regularization term. The considered GP-type methods are formulated by a common iteration
formula, where the scaling matrix and the step-length parameter characterize the different algorithms.
Within this formulation, both first-order and Newton-like methods are analysed, with particular attention
to those implementation features and behaviours relevant for the image restoration problem. The
numerical experiments show that suited scaling strategies can enable the GP methods to quickly
approximate accurate reconstructions and then are useful for designing effective image deblurring
algorithms
Scaling techniques for gradient projection-type methods in astronomical image deblurring
The aim of this paper is to present a computational study on scaling techniques in gradient projection-type (GP-type)methods for deblurring of astronomical images corrupted by Poisson noise. In this case, the imaging problem is formulated as a non-negatively constrained minimization problem in which the objective function is the sum of a fit-to-data term, the Kullback–Leibler divergence, and a Tikhonov regularization term. The considered GP-type methods are formulated by a common iteration formula, where the scaling matrix and the step-length parameter characterize the different algorithms. Within this formulation, both first-order and Newton-like methods are analysed, with particular attention to those implementation features and behaviours relevant for the image restoration problem. The numerical experiments show that suited scaling strategies can enable the GP methods to quickly approximate accurate reconstructions and then are useful for designing effective image deblurring algorithms