Classification as a partitioning problem

A partition configuration

 

• Let be an open domain subset of with smooth boundary, and let represent the observed data function. Let be the region defined as
  
  
  
  We note , and let the interface between and be
  
  
  
  We note the one-dimensional Hausdorff measure of verifying
  
  
  

• The classification model we consider for an image u_odefined over , is a set defined by (1) and satisfying :
  
  
  CONDITION A : is a partition of : 
  
  
  

  CONDITION B :  The partition is a classification of the observed data u₀ and takes into account the Gaussian distribution property of the classes (data term) :
  
  
  

  CONDITION C :  The partition is regular in the sense that the sum of the length of interfaces is minimum :