Shape Gradient Based Non-Parametric Mumford-Shah Segmentation Without Level Sets
Shafeequdheen P, Jyotiranjan Nayak, Vijayakrishna Rowthu
TL;DR
The paper addresses 2D image segmentation by avoiding level-set methods and evolving a polygonal boundary via the shape gradient of the Mumford–Shah energy $E(\Omega)$. It decomposes the energy into $E_1(\Omega)$, $E_2(\Omega)$, and $E_3(\Gamma)$ with definitions for the region means $\mu(\Omega)$ and $\mu(\Omega^c)$, guiding variance minimization inside/outside the boundary. A shape-derivative analysis yields $\delta E=\delta E_1+\delta E_2+\delta E_3$ and a normal boundary velocity; the polygonal boundary is updated by gradient descent with step $dt$ and periodic resampling. Experiments on grayscale and color images (RGB and LAB) show robust variance reduction, convergence, and improved boundary fidelity, with LAB offering particularly strong performance due to perceptual uniformity. The method provides a topology-preserving, computationally efficient alternative to level-set segmentation for real-world imaging tasks.
Abstract
A non parametric, level set free method is proposed for detecting image boundaries using the shape gradient of the Mumford Shah energy for segmentation. Minimizing the variance in pixel intensities inside and outside a boundary set of points is the primary pursuit. The boundary set as a polygon of points rather than a parametric form or a level set, evolves under the guidance of a shape gradient of the Mumford Shah piece wise constant segments model. Iteratively updating through the gradient descent method. The proposed method has been tested on various images, demonstrating its effectiveness in capturing intricate and narrow boundaries texture images.