A Geometric Flow Approach for Region-based Image Segmentation-theoretical Analysis

• Published in: Acta Mathematicae Applicatae Sinica Vol. 34; no. 1; pp. 65 - 76
• Main Authors: Jing, Zhu-cui, Ye, Juntao, Xu, Guo-liang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2018; Springer Nature B.V
• Edition: English series
• Subjects: Applications of Mathematics; Basis functions; Calculus of variations; Energy conservation; Evolution; Image segmentation; Math Applications in Computer Science; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical; Well posed problems; direct method; Bi-cubic B-spline; gradient flow; image segmentation; 00A05
• ISSN: 0168-9673; 1618-3932
• DOI: 10.1007/s10255-018-0723-4

In this paper, we analyze the well-posedness of an image segmentation model. The main idea of that segmentation model is to minimize one energy functional by evolving a given piecewise constant image towards the image to be segmented. The evolution is controlled by a serial of mappings, which can be represented by B-spline basis functions. The evolution terminates when the energy is below a given threshold. We prove that the correspondence between two images in the segmentation model is an injective and surjective mapping under appropriate conditions. We further prove that the solution of the segmentation model exists using the direct method in the calculus of variations. These results provide the theoretical support for that segmentation model.