Numerical minimization of a second-order functional for image segmentation

Abstract

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

A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

“…We solve equations (7) and (8) using a multigrid method [25,28]. Note that we use the Gauss-Seidel iterative method in multigrid process combined with Newton's approximation to compute the nonlinear term in equation (8). Figures 4(a) and 4(b) show the errors for the LSS and CN scheme, respectively.…”

“…Space. Next, we consider the twodimensional version of equations (7) and (8) to the CH equation on Ω � (0, 1) × (0, 1). Straightforward extensions of the LSS and CN schemes are as follows:…”

“…where Ω ⊂ R d (d � 1, 2, 3) is a bounded domain, ϕ(x, t) is a compositional field, F(ϕ) � 0.25(ϕ 2 − 1) 2 , and ε is a positive constant. e CH equation was proposed for a model of the spinodal decomposition in a binary mixture and has been used to model many scientific phenomena such as topology optimization [4], phase separation [5][6][7], image processing [8], two-phase fluid flows [9,10], crystal model [11], tumor growth [12,13], and microstructure formations (see [14] for the basic principles and practical applications and [3] for the physical, mathematical, and numerical derivations of the CH equation).…”

A parallel approach for image segmentation by numerical minimization of a second-order functional

“…In [1], the numerical minimization of F (s, z, u) is obtained by a version of block coordinate descent algorithm [6], especially tailored to exploit the features of the functional. Starting from an initial vector x 0 = (s 0 , z 0 , u 0 ), the basic idea of the method, named BCDA, is to cyclically determine for each block variable (s, z or u) a descent direction d by few iterations of a preconditioned conjugate gradient (PCG) method applied to the quadratic and strongly convex subproblem in this block with the other block variables fixed.…”

Smoothing for signals with discontinuities using higher order Mumford–Shah models

“…They penalize the deviation from a piecewise polynomial instead of the deviation from a piecewise constant function. The multivariate discrete higher order Mumford-Shah and Potts models are particularly interesting in image processing for edge preserving smoothing of images with locally linear or polynomial trends; for example, second order methods are applied for piecewise approximation and segmentation of images [18,65,77,78] or regularization of flow fields [24,76]. The univariate discrete models are particularly interesting for smoothing time series with discontinuities; examples with biological applications are for instance [48,57].…”