Recovery of Blocky Images from Noisy and Blurred Data

Abstract

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.

Blind motion deblurring using multiple images

“…Two steps are involved in the linearized Bregman iteration. The first step is to find an approximate solution (a least squares solution in our case) to the residual equation of the constraint in (13) to update the data. However, the updated data may not be sparse.…”

“…As a result, the support of the resulting blur kernel tends to be a disk or several isolated disks, instead of a continuous curvy camera trajectory. Also, for many images of nature scenes, TV-based regularization does not preserve the details and textures very well on the regions of complex structures due to the stair-casing effects (see [13,23]). …”

“…Therefore, if we choose a sufficiently large thresholding parameter µ, then the iteration (14) converges to a solution of (13). During the iterations of Algorithm 1, the intermediate results of the blur kernels are not accurate until the last few iterations.…”

“…More importantly, there are alignment errors among the observed images. Thus, to obtain a good deblurred image, one can still use (13), but need to stop it early when the error of the constraint in (13) satisfies…”

Some applications of BV functions in optimal controls and calculus of variations

“…Let us point out that besides some technical difficulties related to the space of BV -functions, the main theoretical and numerical difficulty relates to the lack of differentiability of the cost functional. The image reconstruction problem with BV -regularization was further investigated in Dobson and Santosa [5], Ito and Kunisch [8], Li and Santosa [9] and Rudin et al [14], for example. The contributions in [5], [9] and [14] using variations of the formulation in (CV) provide substantial evidence that BV -regularization can be a very effective numerical technique for image enhancement.…”