Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification

In this work, we first address the problem of simultaneous image segmentati on and smoothing by approaching the Mumford-Shah paradigm from a curve evol ution perspective. In particular, we let a set of deformable contours defin e the boundaries between regions in an image where we model the data via pi ecewise smooth functions and employ a gradient flow to evolve these contour s. Each gradient step involves solving an optimal estimation problem for th e data within each region, connecting curve evolution and the Mumford-Shah functional with the theory of boundary-value stochastic processes. The resu lting active contour model offers a tractable implementation of the origina l Mumford-Shah model (i.e., without resorting to elliptic approximations wh ich have traditionally been favored for greater ease in implementation) to simultaneously segment and smoothly reconstruct the data within a given ima ge in a coupled manner. Various implementations of this algorithm are intro duced to increase its speed of convergence. We also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions. Next, by generalizing the data fidelity term of the original Mumford-Shah functional to incorporate a spa tially varying penalty, we extend our method to problems in which data qual ity varies across the image and to images in which sets of pixel measuremen ts are missing. This more general model leads us to a novel PDE-based appro ach for simultaneous image magnification, segmentation, and smoothing, ther eby extending the traditional applications of the Mumford-Shah functional w hich only considers simultaneous segmentation and smoothing.