Optimal Approximation by Piecewise Constant Functions

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,…, γn} 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 γj.