Continuous and discrete approaches to morphological image analysis with applications: PDEs, curve evolution, and distance transforms

Analysis of images at multiple scales is necessary because real-life images contain features of all sizes. It is better to remove the unnecessary detail and process the images at different scales of interest separately. Traditionally, linear filters have been used to smooth out small-scale features to obtain a series of filtered images, but linear filters blur and shift the edges of the large features that they do not remove. Recently, different edge-preserving smoothing approaches, based on mathematical morphology and nonlinear diffusion have been proposed. Such multiscale morphological filtering operations can be modeled with curve evolution and can be solved using the distance transforms. Many other image analysis problems that are related to the eikonal equation of optics can also be solved with distance transforms.;While there are many discrete approaches for computing the distance transforms, a continuous approach has emerged recently. This continuous approach models the distance transform as wavefront propagation. The model is then analyzed using partial differential equations which can be solved very accurately using numerical methods. Even though the traditional discrete algorithms are not as accurate as the newly proposed algorithms based on continuous modeling and partial differential equations (PDEs), they are still useful because the algorithms based on PDEs are usually complex and slow. Hence, it is useful to study both of these approaches.;In this thesis, we optimize the discrete distance transforms and present new methodologies for efficient implementation of PDE-based algorithms. We apply these algorithms to some useful image analysis problems and compare the new solutions with previously available solutions. Specifically, we have found new optimal discrete distance transforms under various optimization criteria and have developed methods for their faster implementations. We have developed and analyzed the computationally intensive PDE-based algorithms and have proposed some new implementation schemes for their efficient implementation using queue-type data structures. The new algorithms have been applied to the problems of multiscale image analysis, shape recovery, gridless halftoning, ray tracing, and image segmentation.