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Given a p-dimensional nonnegative, integral vector α, this paper characterizes the convex hull of the set S of nonnegative, integral vectors x that is lexicographically less than or equal to α. To obtain a finite number of elements in S, the vectors x are restricted to be component-wise upper-bounded by an integral vector u. We show that a linear number of facets is sufficient to describe the convex hull. For the special case in which every entry of u takes the same value (n − 1) for some integer n ≥ 2, the convex hull of the set of n-ary vectors results. Our facets generalize the known family of cover inequalities for the n = 2 binary case. They allow for advances relative to both the modeling of integer variables using base-n expansions, and the solving of n-ary knapsack problems having weakly super-decreasing coefficients.

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Integer Linear Programming (ILP) formulations of Markov random fields (MRFs) models with global connectivity priors were investigated previously in computer vision, e.g., [1,2]. In these works, only Linear Programing (LP) relaxations [1,2] or simplified versions [3] of the problem were solved. This paper investigates the ILP of multi-label MRF with exact connectivity priors via a branch-and-cut method, which provably finds globally optimal solutions. The method enforces connectivity priors iteratively by a cutting plane method, and provides feasible solutions with a guarantee on sub-optimality even if we terminate it earlier. The proposed ILP can be applied as a post-processing method on top of any existing multi-label segmentation approach. As it provides globally optimal solution, it can be used off-line to generate ground-truth labeling, which serves as quality check for any fast on-line algorithm. Furthermore, it can be used to generate groundtruth proposals for weakly supervised segmentation.

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 correspond to images. We propose a novel Mixed Integer Program (MIP) formulation for the piecewise affine fitting problem, where binary variables determines 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 N P-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.