13 research outputs found
On the filtering effect of iterative regularization algorithms for linear least-squares problems
Many real-world applications are addressed through a linear least-squares
problem formulation, whose solution is calculated by means of an iterative
approach. A huge amount of studies has been carried out in the optimization
field to provide the fastest methods for the reconstruction of the solution,
involving choices of adaptive parameters and scaling matrices. However, in
presence of an ill-conditioned model and real data, the need of a regularized
solution instead of the least-squares one changed the point of view in favour
of iterative algorithms able to combine a fast execution with a stable
behaviour with respect to the restoration error. In this paper we want to
analyze some classical and recent gradient approaches for the linear
least-squares problem by looking at their way of filtering the singular values,
showing in particular the effects of scaling matrices and non-negative
constraints in recovering the correct filters of the solution
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given
Limited-memory scaled gradient projection methods for real-time image deconvolution in microscopy
Gradient projection methods have given rise to effective tools for image
deconvolution in several relevant areas, such as microscopy, medical imaging
and astronomy. Due to the large scale of the optimization problems arising
in nowadays imaging applications and to the growing request of real-time
reconstructions, an interesting challenge to be faced consists in designing
new acceleration techniques for the gradient schemes, able to preserve the
simplicity and low computational cost of each iteration. In this work we
propose an acceleration strategy for a state of the art scaled gradient
projection method for image deconvolution in microscopy. The acceleration
idea is derived by adapting a step-length selection rule, recently
introduced for limited-memory steepest descent methods in unconstrained
optimization, to the special constrained optimization framework arising in
image reconstruction. We describe how important issues related to the
generalization of the step-length rule to the imaging optimization problem
have been faced and we evaluate the improvements due to the acceleration
strategy by numerical experiments on large-scale image deconvolution problems
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model.
Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given
A new semi-blind deconvolution approach for Fourier-based image restoration: an application in astronomy
The aim of this paper is to develop a new optimization algorithm for the restoration of an image starting from samples of its Fourier Transform, when only partial information about the data frequencies is provided. The corresponding constrained optimization problem is approached with a cyclic block alternating scheme, in which projected gradient methods are used to find a regularized solution. Our algorithm is then applied to the imaging of high-energy radiation emitted during a solar flare through the analysis of the photon counts collected by the NASA RHESSI satellite. Numerical experiments on simulated data show that, both in presence and in absence of statistical noise, the proposed approach provides some improvements in the reconstructions
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
Constrained dogleg methods for nonlinear systems with simple bounds
We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problem