Computer Vision News - December 2016

32 Computer Vision News Research Research Where BV(Ω) is the set of functions over Ω with bounded variation. Consistent image recovery is equivalent to finding an element of Ω ( ; 1 , ⋯ , 2 ) . is called a shape if it is the union of a finite number of connected subsets of Ω. ( , ) denote the characteristic function of over Ω: For the consistent shape recovery problem, the authors limit the permissible solutions to the above shape characteristic functions. In this case, we call a shape image. Below is an example of a shape characteristic function and its associated 10×10 discrete image: Now, the consistent shape reconstruction problem is equivalent to finding a shape image = ( , ) ∈ Ω ( ; 1 , ⋯ , 2 ) . Solution is the minimizers of the following constraints: where Per( ) is the perimeter of . Problem ( P 0 ) is a variation non-convex problem prone to having many local minima. In the simplest scenario, having only a single pixel is a well-studied topic known as the Cheeger problem . There is already a rich literature regarding the existence, uniqueness properties and regularity of such sets for almost arbitrary kernels . To adapt the Cheeger problem for images with more than one pixel and gray- level values the authors define the reducibility constraint: ( ; 1 , ⋯ , 2 ) if A (the index set of active pixels) can be partitioned into K 1 and K 2 such that: In the paper, the authors prove that as long as the above reducibility constraint is met, the multi-constraint minimization problem is equivalent to the Cheeger problem.