Nonlinear PDEs and Numerical Algorithms for Modeling Levelings and Reconstruction Filters

Abstract

In this paper we develop partial differential equations (PDEs) that model the generation of a large class of morphological filters, the levelings and the openings/closings by reconstruction. These types of filters are very useful in numerous image analysis and vision tasks rang- ing from enhancement, to geometric feature detection, to segmentation. The developed PDEs are nonlinear functions of the first spatial deriva- tives and model these nonlinear filters as the limit of a controlled growth starting from an initial seed signal. This growth is of the multiscale di- lation or erosion type and the controlling mechanism is a switch that reverses the growth when the difference between the current evolution and a reference signal switches signs. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution and corresponding filter implementation, and provide insights via several ex- periments. Finally, we outline the use of these PDEs for improving the Gaussian scale-space by using the latter as initial seed to generate mul- tiscale levelings that have a superior preservation of image edges and boundaries.