A Convex Relaxation of the Ambrosio--Tortorelli Elliptic Functionals for the Mumford-Shah Functional
In this paper, we revisit the phase-field approximation of Ambrosio and Tortorelli for the Mumford--Shah functional. We then propose a convex relaxation for it to attempt to compute globally optimal solutions rather than solving the nonconvex functional directly, which is the main contribution of this paper. Inspired by McCormick's seminal work on factorable nonconvex problems, we split a nonconvex product term that appears in the Ambrosio--Tortorelli elliptic functionals in a way that a typical alternating gradient method guarantees a globally optimal solution without completely removing coupling effects. Furthermore, not only do we provide a fruitful analysis of the proposed relaxation but also demonstrate the capacity of our relaxation in numerous experiments that show convincing results compared to a naive extension of the McCormick relaxation and its quadratic variant. Indeed, we believe the proposed relaxation and the idea behind would open up a possibility for convexifying a new class of functions in the context of energy minimization for computer vision.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Medial Axis Aware Learning of Signed Distance Functions
We propose a novel variational method to compute a highly accurate global signed distance function (SDF) to a given point cloud. To this end, the jump set of the gradient of the SDF, which coincides with the medial axis …
Point CloudsTyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure
We address structured covariance estimation in elliptical distributions by assuming that the covariance is a priori known to belong to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the Gen…
Lifting Vectorial Variational Problems: A Natural Formulation based on Geometric Measure Theory and Discrete Exterior Calculus
Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of …
Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus
Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of…
Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the …