Calculus of Variations and Geometric Measure Theory

T. Boccellari - F. Tomarelli

Generic uniqueness of minimizer for Blake & Zisserman functional

created by tomarelli on 08 Feb 2013
modified on 03 Jul 2013


Published Paper

Inserted: 8 feb 2013
Last Updated: 3 jul 2013

Journal: Revista Matematica Complutense
Volume: 26
Number: 2
Pages: 361-408
Year: 2013
Doi: 10.1007/s13163-012-0103-1

AMS class: 35A02; 35Q31; 49K05; 46N60.

Links: link


Blake-Zisserman functional $F^g_{α,β}$ achieves a finite minimum for any pair of real numbers $α, β$ such that $0<β≤α≤2β$ and any $g∈L^2(0,1)$. Uniqueness of minimizer does not hold in general. Nevertheless, in the 1D case uniqueness of minimizer is a generic property for $F^g_{α,β}$ in the sense that it holds true for almost all gray levels data $g$ and contrast parameters α, β: we prove that, whenever $α/β∉ℚ$, the minimizer is unique for any $g$ belonging to a dense $G_δ$ set of $L^2(0,1)$ dependent on $α$ and $β.$

Keywords: calculus of variations, Image segmentation, Euler equations, generic uniqueness