Piecewise affine functions are widely used to approximate nonlinear and discontinuous functions. However, most, if not all existing models, only deal with fitting a continuous function. In this paper, we investigate the problem of fitting a discontinuous piecewise affine function to a given function defined on an arbitrary subset of an integer lattice, where no restriction on the partition of the domain is enforced (i.e., its geometric shape can be nonconvex). This is useful for segmentation and denoising when the given function corresponds to a mapping from pixels of a bitmap image to their color depth values. We propose a novel Mixed Integer Program (MIP) formulation for the piecewise affine fitting problem, where binary edge variables determine the boundary between two partitions of the function domain. To obtain a consistent partitioning (e.g., image segmentation), we include multicut constraints in the formulation. The resulting problem is \(\mathcal {NP}\)-hard, and two techniques are introduced to improve the computation. One is to adopt a cutting plane method to add the exponentially many multicut inequalities on-the-fly. The other is to provide initial feasible solutions using a tailored heuristic algorithm. We show that the MIP formulation on grid graphs is approximate, while on king’s graph, it is exact under certain circumstances. We conduct initial experiments on synthetic images as well as real depth images, and discuss the advantages and drawbacks of the two models.

分段仿射函数被广泛用于逼近非线性和不连续函数。但是，大多数（如果不是全部）现有模型只能处理拟合连续函数。在本文中，我们研究了将不连续的分段仿射函数拟合到在整数晶格的任意子集上定义的给定函数的问题，其中对域的分区没有限制（即，其几何形状可以是非凸的） 。当给定函数对应于从位图图像的像素到其色深值的映射时，这对于分段和去噪很有用。针对分段仿射拟合问题，我们提出了一种新颖的混合整数程序（MIP）公式，其中二进制边缘变量确定了函数域的两个分区之间的边界。为了获得一致的分区（例如，图像分割），我们在公式中包括多切约束。产生的问题是\（\ mathcal {NP} \）- hard，并引入了两种技术来改进计算。一种是采用切平面方法来动态地指数增加许多多重切割不等式。另一种方法是使用量身定制的启发式算法提供初始可行的解决方案。我们证明网格图上的MIP公式是近似的，而国王图上的MIP公式在某些情况下是精确的。我们对合成图像和真实深度图像进行了初步实验，并讨论了这两种模型的优缺点。