Non-unique solutions for a convex TV – L¹ problem in image segmentation

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 $TV-L^1$ problem for finding such a region Σ and proved that when a binary input f is given, a solution $u_{\ast }$ to the convex problem gives rise to other solutions $1_{\{u_{\ast }\ge \mu \}}$ for a.e. $\mu \in (0,1)$ from which they raised a question of whether or not $u_{\ast }$ 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 ${\mathbb{R}}^N$. 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.

KEYWORDS:

• Convex optimization
• image segmentation
• mean curvature motion

AMS SUBJECT CLASSIFICATIONS:

• 90C25
• 49J40
• 68U10

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Y. Kim  http://orcid.org/0000-0003-2679-0090

Additional information

Funding

This work was supported partially by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2014R1A1A1002667) and partially by Ulsan National Institute of Science and Technology(UNIST) (1.140074.01).