Learning Discontinuous Piecewise Affine Fitting Functions using Mixed Integer Programming for Segmentation and Denoising

Piecewise affine functions are widely used to approximate nonlinear and discontinuous functions. However, most, if not all existing models only deal with fitting continuous functions. In this paper, we investigate the problem of fitting a discontinuous piecewise affine function to given data that lie in an orthogonal grid, where no restriction on the partition is enforced (i.e., its geometric shape can be nonconvex). This is useful for segmentation and denoising when data corresponding to images. We propose a novel Mixed Integer Program (MIP) formulation for the piecewise affine fitting problem, where binary variables determine the location of break-points. To obtain consistent partitions (i.e. image segmentation), we include multi-cut constraints in the formulation. Since the resulting problem is $\mathcal{NP}$-hard, two techniques are introduced to improve the computation. One is to add facet-defining inequalities to the formulation and the other to provide initial integer solutions using a special heuristic algorithm. We conduct extensive experiments by some synthetic images as well as real depth images, and the results demonstrate the feasibility of our model.