# DAVIDSTUTZ

I am looking for full-time (applied) research opportunities in industry, involving (trustworthy and robust) machine learning or (3D) computer vision, starting early 2022. Check out my CV and get in touch on LinkedIn!
29thAPRIL2016

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 :

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

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  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

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

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

•  D. Mumford, J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. on Pure and Applied Mathematics, 42, 1989.
•  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.

What is your opinion on the summarized work? Or do you know related work that is of interest? Let me know your thoughts in the comments below: