Advances in Piecewise Smooth Image Reconstruction

Abstract

Advances and new insights into algorithms for piecewise smooth image reconstruction are presented. Such algorithms fit a piecewise smooth function to image data without prior knowledge of the number of regions or the location of region boundaries in the best fitting function. This is a difficult model selection problem since the number of parameters of possible solutions varies widely.The approach followed in this work was proposed by Yvan Leclerc. It uses the Minimum Description Length principle to make the reconstruction problem wellposed-the best fitting function yields the shortest encoding of the image data. In order to derive a code length formula, the class of functions is restricted to piecewise polynomial. The resulting optimization problem may have many local minima, and a good initial approximation is required in order to find acceptable solutions. Good initial approximations may be generated at the cost of solving a sequence of related optimization problems, as prescribed by a continuation method.Several problems with this approach are identified and addressed. First, success or failure of the continuation method is found to be sensitive to the choice of objective function parameters. Second, the optimization method used in prior work may fail to converge, and, third, it converges too slowly to be useful in many vision applications.I address the first problem in three different ways. First, a revised continuation method is less sensitive to parameter choice. Second, I show how to move control over success or failure from the objective function parameters to the continuation method.Third, a new objective function is derived which includes one parameter instead of the two parameters used in prior work. Experimental results show that all measures improve robustness with respect to parameter choice.