Discrete minimum ratio curves and surfaces

Graph cuts have proven useful for image segmentation and for volumetric reconstruction in multiple view stereo. However, solutions are biased: the cost function tends to favour either a short boundary (in 2D) or a boundary with a small area (in 3D). This bias can be avoided by instead minimising the cut ratio, which normalises the cost by a measure of the boundary size. This paper uses ideas from discrete differential geometry to develop a linear programming formulation for finding a minimum ratio cut in arbitrary dimension, which allows constraints on the solution to be specified in a natural manner, and which admits an efficient and globally optimal solution. Results are shown for 2D segmentation and for 3D volumetric reconstruction.