Curve Evolution, Boundary-Value Stochastic Processes, the Mumford-Shah Problem, and Missing Data Applications

We present an estimation-theoretic approach to curve evolution for the Mumford-Shah problem. By viewing an active contour as the set of discontinuities in the Mumford-Shah problem, we may use the corresponding functional to determine gradient descent evolution equations to deform the active contour. In each gradient descent step, we solve a corresponding optimal estimation problem, connecting the Mumford-Shah functional and curve evolution with the theory of boundary-value stochastic processes. In employing the Mumford-Shah functional, our active contour model inherits its attractive ability to generate, in a coupled manner, both a smooth reconstruction and a segmentation of the image. Next, by generalizing the data fidelity term of the original Mumford-Shah functional to incorporate a spatially varying penalty, we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing. This more general model leads us to a novel PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford-Shah functional which only considers simultaneous segmentation and smoothing.