A PDE approach to nonlinear image simplification via levelings and reconstruction filters

We present a nonlinear partial differential equation (PDE) that models the generation of a large class of advanced morphological filters, the levelings and the openings/closings by reconstruction. These types of filters are very useful in numerous image analysis and vision tasks ranging from enhancement, feature detection, image simplification, to segmentation. The developed PDE models these nonlinear filters as the limit of a controlled growth starting from an initial seed signal. This growth is of the multiscale dilation 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 this PDE, propose a discrete algorithm for its numerical solution and corresponding filter implementation, and provide insights via several experiments. Finally, we outline its use for improving the Gaussian scale-space by using the latter as initial seed to generate multiscale levelings that have a superior preservation of image edges and boundaries.