Superresolution reconstruction produces a high-resolution image from a set of low-resolution images. Previous iterative methods for superresolution had not adequately addressed the computational and numerical issues for this ill-conditioned and typically underdetermined large scale problem. We propose efficient block circulant preconditioners for solving the Tikhonov-regularized superresolution...

We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are...

This paper presents a systematic and computable method for choosing the regularization parameter appearing in Tikhonov-type regularization based on non-quadratic regularizers. First, we extend the notion of the L-curve, originally defined for quadratically regularized problems, to the case of non-quadratic functions. We then associate the optimal value of the regularization parameter for these ...

We consider the inverse problem of reconstructing boundary curve a cavity embedded in bounded domain. The is formulated two dimensions for wave equation. combine Laguerre transform with integral equation method and we reduce to system equations. propose an iterative scheme that linearizes using Fréchet derivative forward operator. application special quadrature rules results ill-conditioned lin...

In this paper, several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition methods, for solving the Cauchy problem associated to the Helmholtz equation are developed and compared. Regularizing stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are an...

We propose a regularization method for solving ill-posed problems, under the assumption that the solutions are piecewise constant functions with unknown level sets and unknown level values. A level set framework is established for the inverse problem and a Tikhonov regularization approach is proposed. Existence of generalized minimizers for the Tikhonov functional is proven. Moreover, we establ...

This paper studies the regularization of constrained Maximum Likelihood iterative algorithms applied to incompatible ill-posed linear inverse problems. Specifically we introduce a novel stopping rule which defines a regularization algorithm for the Iterative Space Reconstruction Algorithm in the case of Least-Squares minimization. Further we show that the same rule regularizes the Expectation M...

A constrained least squares problem in a Hilbert space H is considered. The standard Tikhonov regularization method is used. In the case where the set of the constraints is the nonempty intersection of a finite collection of closed convex subsets of H , an iterative algorithm is designed. The resulting sequence is shown to converge strongly to the unique solution of the regularized problem. The...