In this work, we approach the classic Mumford-Shah problem from a curve evolution perspective. In particular, we let a given family of curves define the boundaries between regions in an image within which the data are modeled by piecewise smooth functions plus noise as in the standard Mumford-Shah functional. The gradient descent equation of this functional is then used to evolve the curve. Each gradient descent step involves solving a corresponding optimal estimation problem, which connects the Mumford-Shah functional, and our curve evolution implementation with the theory of boundary-value stochastic processes. The resulting active contour model, therefore, inherits the attractive ability of the Mumford-Shah technique to generate, in a coupled manner, both a smooth reconstruction of the image and segmentation as well. We demonstrate applications of our method to problems in which data quality is spatially varying and to problems in which sets of pixel measurements are missing. Finally, we demonstrate a hierarchical implementation of our model, which leads to a fast and efficient algorithm capable of dealing with important image features such as triple points.