David C. Dobson

Recovery of blocky images from noisy and blurred data

The purpose of this investigation is to understand situations under which an enhancement method succeeds in recovering an image from data which are noisy and blurred. The method in question is due to Rudin and Osher. The method selects, from a class of feasible images, one that has the least total variation.Our investigation is limited to images which have small total variation. We call such images “blocky” as they are commonly piecewise constant (or nearly so) in grey-level values. The image enhancement is applied to three types of problems, each one leading to an optimization problem. The optimization problems are analyzed in order to understand the conditions under which they can be expected to succeed in reconstructing the desired blocky images. We illustrate the main findings of our work in numerical examples.

Mathematical studies in rigorous grating theory

We consider the diffraction of a time-harmonic wave incident upon a grating (or periodic) structure. We study mathematical issues that arise in the direct modeling, inverse, and optimal design problems. Particular attention is paid to the variational approach and to finite-element methods. For the direct problem various results on existence, uniqueness, and numerical approximations of solutions are presented. Convergence properties of the variational method and sensitivity to TM polarization are examined. Our recent research on inverse diffraction problems and optimal design problems is also discussed.

The time-harmonic maxwell equations in a doubly periodic structure☆☆☆

Abstract Consider the diffraction of a beam of particles in R 3 when the dielectric coefficient is a constant e1 above a surface S and a (different) constant e2 below S, and the magnetic permeability is constant μ throughout R 3. S is assumed to be a doubly periodic surface, say z = f(x, y) with f(x + mL1, y + nL2) = f(x, y) for integers m, n. The existence and uniqueness of a solution satisfying a “radiation condition” at infinity is reduced to a system of Fredholm equations. Thus, for all but a discrete set of es there exists a unique solution.

Convergence of a reconstruction method for the inverse conductivity problem

The inverse conductivity problem is that of reconstructing a spatially varying isotropic conductivity in the interior of some region by means of steady-state measurements taken at the boundary. Reconstruction schemes including least-squares type minimization methods have been widely studied and implemented, but convergence analysis has been largely ignored. This paper establishes the convergence of a well-known least-squares minimization scheme—the Levenberg–Marquardt method—on a regularized formulation of the inverse conductivity problem.

ANALYSIS OF REGULARIZED TOTAL VARIATION PENALTY METHODS FOR DENOISING

The problem of recovering images with sharp edges by total variation denoising has recently received considerable attention in image processing. Numerical difficulties in implementing this nonlinear filter technique are partly due to the fact that it involves the stable evaluations of unbounded operators. To overcome that difficulty we propose to approximate the evaluation of the unbounded operator by a stable approximation. A convergence analysis for this regularized approach is presented.

On the scattering by a biperiodic structure

Consider scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure separates the whole space into three regions: above and below the structure the medium is assumed to be homogeneous. Inside the structure, the medium is assumed to be de ned by a bounded measurable dielectric coe cient. Given the structure and a timeharmonic electromagnetic plane wave incident on the structure, the scattering (di raction) problem is to predict the eld distributions away from the structure. In this note, the problem is reduced to a bounded domain and solved by a variational method. The main result establishes existence and uniqueness of the weak solutions in W .

Resolution and stability analysis of an inverse problem in electrical impedance tomography: dependence on the input current patterns

Electrical impedance tomography is a procedure by which one finds the conductivity distribution inside a domain from measurements of voltages and currents at the boundary. This work addresses the i...

Phase reconstruction via nonlinear least-squares

Consider the problem of reconstructing the phase phi of a complex-valued function fei phi , given knowledge of the magnitude mod f mod and the magnitude of the Fourier transform mod (fei phi )V-product mod . The author considers the formulation as a least-squares minimization problem. It is shown that the linearized problem is ill posed. Also, surprisingly, the gradient of the least-squares objective functional is not Frechet differentiable. A regularization is introduced which restores differentiability and also counteracts instability. It is shown how a certain implementation of Newtons method can be used to solve the regularized least-squares problem efficiently, and that the method converges locally, almost quadratically. Numerical examples are given with an application to diffractive optics.

Regular ArticleAn Efficient Method for Band Structure Calculations in 3D Photonic Crystals

A method for computing band structures for three-dimensional photonic crystals is described. The method combines a mixed finite element discretization on a uniform grid with a fast Fourier transform preconditioner and a preconditioned subspace iteration algorithm. Numerical examples illustrating the behavior of the method are presented.

Second harmonic generation in nonlinear optical films

Second harmonic generation, an important phenomenon in nonlinear optics, is modeled in this work. The model is derived from a nonlinear system of Maxwell’s equations, which overcomes the known shortcomings of some commonly used models in the literature. Existence and uniqueness of solutions are established by a combination of a variational approach and the contraction mapping principle. Some numerical results are also presented.