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 ndings of our work in numerical examples.

In total variation denoising, one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion. Several approaches based on this idea have recently been shown to be very effective, particularly for denoising functions with discontinuities. This paper analyzes the convergence of an iterative method for solving such problems. The iterative method involves a "lagged diffusivity" approach in which a sequence of linear diffusion problems are solved. Global convergence in a finite-dimensional setting is established, and local convergence properties, including rates and their dependence on various parameters, are examined.

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In this paper. we propose an idea for reconsuucting 'blocky' conductivity profiles in electrical impedance tomography. By 'blocky' profiles, we mean functions that are piecewise constant, and hence have sharply defined edges. The method is based on selecting 3 conductivity distribution that has the least totll variation from all conductivities that are consistent with the measured data. We provide some motivation for this approach and formulate a computationally feasible problem for the linearized version of the impedance tomography problem. A simple gradient descent-type minimization algorithm. closely related to recent work on noise and blur removal in image processing via non-linear diffusion is described. The potential of the method is demonstrated in sevenl numerical experimentc.

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