Image segmentation by variational and elliptic boundary value problems
Yang Wang · 1991
The purpose of this thesis is to study the image segmentation problems introduced by David Mumford and Jayant Shah, both theoretically and numerically. Given any image which is represented by a function $f(x,y)$ defined in a domain $D$, the segmentation problem involves mimimizing the Mumford-Shah energy functional$$E(u,\Gamma) = \mu\sp2 \int\sb{D}(u - f)\sp2 dx dy + {\sum\limits\sb{i}}\int\sb{\Omega{\sb i}}\vert abla u\vert\sp2 dx dy + u\sp2\vert\Gamma\vert,$$where $\mu, u > 0$ are constants, $\Gamma$ is a finite union of simple curves in the domain $D$ that divides $D$ into some connected components $\Omega\sb{i}$'s, for each $i$ the function $u(x,y) \in W\sp{1,2}(\Omega\sb{i}),$ and $\vert\Gamma\vert$ is the total length of $\Gamma.$ This thesis will provide a proof for the existence of solutions in the $\mu\to 0$ case. It will also provide studies on a problem which is slightly modified from the $\mu\to 0$ case. Two dynamic programming algorithms are provided for solving the one dimensional segmentation problem. And finally, the thesis will provide some estimates of solutions in the $\mu > 0$ case.