Numerical minimization of a second-order functional for image segmentation

• The Blake–Zisserman functional is a second-order model for image segmentation.
• A variational approximation of the functional given by Ambrosio, Faina and March, is discretized.
• We propose an efficient block-coordinate descent method for the numerical minimization.
• Numerical experiments involve very different types of datasets, including digital surface models.
• Results show that, with the proposed method, the second order segmentation can be addressed in competitive time.

In this paper we address the numerical minimization of a variational approximation of the Blake–Zisserman functional given by Ambrosio, Faina and March. Our approach exploits a compact matricial formulation of the objective functional and its decomposition into quadratic sparse convex sub-problems. This structure is well suited for using a block-coordinate descent method that cyclically determines a descent direction with respect to a block of variables by few iterations of a preconditioned conjugate gradient algorithm. We prove that the computed search directions are gradient related and, with convenient step-sizes, we obtain that any limit point of the generated sequence is a stationary point of the objective functional. An extensive experimentation on different datasets including real and synthetic images and digital surface models, enables us to conclude that: (1) the numerical method has satisfying performance in terms of accuracy and computational time; (2) a minimizer of the proposed discrete functional preserves the expected good geometrical properties of the Blake–Zisserman functional, i.e., it is able to detect first and second order edge-boundaries in images and (3) the method allows the segmentation of large images.