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Non-unique solutions for a convex TV – L1 problem in image segmentation.
- Source :
- Applicable Analysis; Feb2020, Vol. 99 Issue 2, p232-248, 17p
- Publication Year :
- 2020
-
Abstract
- One important task in image segmentation is to find a region of interest, which is, in general, a solution of a nonlinear and nonconvex problem. The authors of Chan and Esedoglu (Aspects of total variation regularized L 1 function approximation. SIAM J. Appl. Math. 2005;65:1817–1837) proposed a convex T V − L 1 problem for finding such a region Σ and proved that when a binary input f is given, a solution u ∗ to the convex problem gives rise to other solutions 1 { u ∗ ≥ μ } for a.e. μ ∈ (0 , 1) from which they raised a question of whether or not u ∗ must be binary. The same is to ask if the two-phase Mumford–Shah model in image processing has a unique solution with a binary input. In this paper, we will discuss how to construct a non-binary solution that provides a negative answer to the question through a connection of two ideas, one from the two-phase Mumford–Shah model in image segmentation and the other from mean curvature motions discussed in some geometric problems (e.g. Alter, Caselles, Chambolle. A characterization of convex calibrable sets in R N . Math. Ann. 2005;322:329–366; Chambolle. An algorithm for mean curvature motion. Interfaces Free Boundaries 2004;6:195–218) revealing the nature of non-uniqueness in image segmentation. [ABSTRACT FROM AUTHOR]
- Subjects :
- IMAGE processing
CONVEX sets
NONLINEAR equations
CURVATURE
MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 00036811
- Volume :
- 99
- Issue :
- 2
- Database :
- Complementary Index
- Journal :
- Applicable Analysis
- Publication Type :
- Academic Journal
- Accession number :
- 140974079
- Full Text :
- https://doi.org/10.1080/00036811.2018.1489962