VARIATIONAL IMAGE SEGMENTATION USING BOUNDARY FUNCTIONS

A general variational framework for image approximation and segmentati on is introduced. By using a continuous ''line-process'' to represent edge boundaries, it is possible to formulate a variational theory of i mage segmentation and approximation in which the boundary function has a simple explicit form in terms of the approximation function. At the same time, this variational framework is general enough to include th e most commonly used objective functions. Application is made to Mumfo rd-Shah type functionals as well as those considered by Geman and othe rs. Employing arbitrary L-p norms to measure smoothness and approximat ion allows the user to alternate between a least squares approach and one based on total variation, depending on the needs of a particular i mage. Since the optimal boundary function that minimizes the associate d objective functional for a given approximation function can be found explicitly, the objective functional can be expressed in a reduced fo rm that depends only on the approximating function. From this a partia l differential equation (PDE) descent method, aimed at minimizing the objective functional, is derived. The method is fast and produces exce llent results as illustrated by a number of real and synthetic image p roblems.