Optimal Approximation by Piecewise Constant Functions
Italo Tamanini · Birkhäuser Basel eBooks · 1996
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.