Finding maximum-weight consistent digitally convex regions

We describe an algorithm to find maximum-weight digitally convex regions in gray-scale images. After Christ et al. [CPS10] gave a system for consistent digital line systems, the question how to transfer the concept of convexity to digital regions naturally appeared. The regions computed using this definition are weight-maximal, the computational complexity of our algorithm amounts to O(n5 · log n). With additional input data constraining the possible solution, this bound can be improved to O(n3 · log n). We further describe a hybrid approximation algorithm that first generates suitable constraints by executing the O(n5 · log n)-algorithm on a downscaled version of the input image and then uses these constraints to achieve the O(n3 · log n) bound.