On Optimal 2-D Domain Segmentation Problem via Piecewise Smooth Approximation of a Selective Target Mapping

Abstract

In this paper we propose a new technique for the solution of the image segmentation problem which is based on the concept of a piecewise smooth approximation of some target functional. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences.

On Increasing of Resolution of Satellite Images via Their Fusion with Imagery at Higher Resolution

“…As for the rest terms in the cost functional J : Ξ → R, we have the following representation for their Gâteaux derivatives (for the proof and its substantiation we refer to [13,Section 3]). Proposition 6.1.…”

Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar

“…It can be shown that the standard norm in H −1 (Ω) is equivalent to the following one (for the details we refer to [24])…”

“…We begin with the following technical results (for the proof and their substantiation we refer to [24,Section 3]).…”

“…(6.10) Our next step is to define the structure of subdifferential for the functional E 1 given by the rule (6.1). To do so, we make use of the following result (for the proof we refer to [24,Proposition 4.4]).…”