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.KeywordsReference SignalImage EdgeMathematical MorphologyNonlinear PDEsReconstruction OpeningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.