Optimal Approximation by Piecewise Constant Functions

Abstract

In their seminal 1989 paper [8], D. Mumford and J. Shah proposed a variational approach to image segmentation in Computer Vision Theory, and studied in particular the following problem: Given an open rectangle R ⊂ℝ², a function g continuous on the closure $\overline 4$ of R, and a positive coefficient ν, find a finite set Г = {γ₁,…, γ_n} of C ² arcs contained in $\overline 4$, meeting each other only at their end-points, and minimizing the following functional

$$ E\left( \Gamma \right) = \sum\limits_{i = 1}^N {\iint_{{R_i}} {{{\left| {{a_i} - g\left( {x,y} \right)} \right|}^2}dxdy + v \times length\left( \Gamma \right)}} $$

where R ₁,…, R _N denote the connected components of R\Γ, a _i is the average of g on R ₁, i.e. $ai = \iint {Ri(x,y)}dxdy$ , and length(Γ) is the sum of the lengths of the arcs γ_j.