A scaled gradient projection method for constrained image deblurring

Abstract

Abstract. A class of scaled gradient projection methods for optimization problems with simple constraints is considered. These iterative algorithms can be useful in variational approaches to image deblurring that lead to minimize convex nonlinear functions subject to nonnegativity constraints and, in some cases, to an additional flux conservation constraint. A special gradient projection method is introduced that exploits effective scaling strategies and steplength updating rules, appropriately designed for improving the convergence rate. We give convergence results for this scheme and we evaluate its effectiveness by means of an extensive computational study on the minimization problems arising from the maximum likelihood approach to image deblurring. Comparisons with the standard expectation maximization algorithm and with other iterative regularization schemes are also reported to show the computational gain provided by the proposed method.

On the constrained minimization of smooth Kurdyka—Łojasiewicz functions with the scaled gradient projection method

“…There exists a subsequence {x (k j ) } j∈N andx such that x (k j ) →x and Ψ(x (k j ) ) → Ψ(x), as j → ∞. In our case, condition H3 is assured by the continuity of Ψ in Ω and the fact that x (k) ∈ Ω, for every k ∈ N. Indeed H3 is needed in [2] only to ensure the stationarity of the limit pointx, which has already been proved for SGP in [4]. Throughout the entire section, {x (k) } k∈N will denote the sequence generated by SGP.…”

“…Proof: The stationarity ofx has been proved in [4]. Since Ψ is a KL function, it satisfies the KL property at each point of Ω and, in particular, atx.…”

“…We also remark that, unlike the schemes presented e.g. [4]. Convergence of the sequence to a minimum point of (1) has recently been proved for convex objective functions by choosing suitable adaptive bounds for the eigenvalues of the scaling matrices [8].…”

“…The scaled gradient projection algorithm (SGP) [4] belongs to the class of first-order methods designed to solve any constrained optimization problem of the form…”

Limited-memory scaled gradient projection methods for real-time image deconvolution in microscopy

“…In [3] numerical evidence has been also provided indicating remarkable gain in the convergence rate over the classical BarzilaiBorwein (BB) step-length rule [4]. Since in the last years promising image reconstruction algorithms have been designed by exploiting BB-based rules within gradient methods [5,6,7,8,9,10,11], it is worthwhile to investigate if useful acceleration can be achieved with the new step-length selection idea. In particular, we focus on the algorithm for image deconvolution in microscopy provided by the Scaled Gradient Projection (SGP) method recently developed in [12], that can be appropriately modified for managing the step-length rule proposed in [3].…”