J. Shen. Gamma-Convergence Approximation to Piecewise Constant Mumford-Shah Segmentation. International Conference on Advanced Concepts for Intelligent Vision Systems, 2005.

Shen discusses an approximation to the Mumford-Shah segmentation model [1]:

$h(O, x_+, c_-, u^{(0)}) ) \int_O (u^{(0)}(x) - c_+)^2 dx + \int_{\Omega\setminus O} (u^{(0)}(x) - c_-)^2 dx + \lambda \text{per}(O)$

where $u^{(0)} : \Omega \rightarrow [0,1]$ is an image (for simplicity, let $u^{(0)}$ be a grayscale image) and $O \subset \Omega$ refers to the foreground. Furthermore, $\lambda$ is a regularization parameter and $c_+$/$c_-$ is the average foreground/background intensity. The perimeter $\text{per}(O)$ of $O$ is defined using the total variation of its characteristic function:

$\text{per}(O) := |\chi_O|_{BV(\Omega)}$.

Obviously, minimization over the set $O$ is problematic. One of the most popular approaches to this problem is the work by Chan and Vese [2] who apply the level set framework to minimize over a $3$-dimensional curve instead of the set $O$. Instead, Shen derives a $\Gamma$-convergence formulation of the reduced Mumford-Shah model:

$h_\epsilon (u; c_+, c_-, u^{(0)}, \lambda) = \lambda \underbrace{\int_\Omega \left(9 \epsilon \|\nabla u(x)\|_2^2+ \frac{(1 - u(x)^2)^2}{64 \epsilon}\right) dx}_{= f_\epsilon(u)}$
$+ \int_\Omega \left(\frac{1 + u(x)}{2}\right)^2 (u^{(0)} (x) - c_+)^2 dx + \int_\Omega\left(\frac{1 - u(x)}{2}\right)^2 (u^{(0)} - c_-)^2 dx$

As discussed by Shen, $f_\epsilon(u)$ is used to approximate the perimeter $\text{per}(O)$. To minimize Equation (1), Shen uses an alternative scheme, i.e. for fixed $c_+$ and $c_-$ minimize $h_\epsilon$ and for fixed $u$ compute

$c_+ = \frac{\int_\Omega (1 + u(x))^2 u^{(0)}(x) dx}{\int_\Omega (1 + u(x))^2 dx}$;

$c_- = \frac{\int_\Omega (1 - u(x))^2 u^{(0)}(x) dx}{\int_\Omega (1 - u(x))^2 dx}$.

  • [1] D. Mumford, J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. on Pure and Applied Mathematics, 42, 1989.
  • [2] L. A. Vese, T. F. Chan. A multiphase level set framework for image segmentation using the mumford and shah model. International Journal of Computer Vision, 50, 2002.
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