Reprint of Solution of Ambrosio–Tortorelli model for image segmentation by generalized relaxation method

• We propose a new numerical approach for solving an image segmentation model.
• The Ambrosio–Tortorelli approximation of the Mumford–Shah model is considered.
• Finite-difference discretization of the Euler–Lagrange equations is proposed.
• Non-linear Gauss–Seidel coupled with linear geometric multigrid is used.
• Experiments show that the method is efficient for increasing image sizes.

Image segmentation addresses the problem to partition a given image into its constituent objects and then to identify the boundaries of the objects. This problem can be formulated in terms of a variational model aimed to find optimal approximations of a bounded function by piecewise-smooth functions, minimizing a given functional. The corresponding Euler–Lagrange equations are a set of two coupled elliptic partial differential equations with varying coefficients. Numerical solution of the above system often relies on alternating minimization techniques involving descent methods coupled with explicit or semi-implicit finite-difference discretization schemes, which are slowly convergent and poorly scalable with respect to image size. In this work we focus on generalized relaxation methods also coupled with multigrid linear solvers, when a finite-difference discretization is applied to the Euler–Lagrange equations of Ambrosio–Tortorelli model. We show that non-linear Gauss–Seidel, accelerated by inner linear iterations, is an effective method for large-scale image analysis as those arising from high-throughput screening platforms for stem cells targeted differentiation, where one of the main goal is segmentation of thousand of images to analyze cell colonies morphology.