A Compact Linear Programming Relaxation for Binary Sub-modular MRF

Direct linear programming (LP) solution to binary sub-modular MRF energy has recently been promoted because i) the solution is identical to the solution by graph cuts, ii) LP is naturally parallelizable and iii) it is flexible in incorporation of constraints. Nevertheless, the conventional LP relaxation for MRF incurs a large number of auxiliary variables and constraints, resulting in expensive consumption in memory and computation. In this work, we propose to approximate the solution of the conventional LP at a significantly smaller complexity by solving a novel compact LP model. We further establish a tightenable approximation bound between our LP model and the conventional LP model. Our LP model is obtained by linearizing a novel l1-norm energy derived from the Cholesky factorization of the quadratic form of the MRF energy, and it contains significantly fewer variables and constraints compared to the conventional LP relaxation. We also show that our model is closely related to the total-variation minimization problem, and it can therefore preserve the discontinuities in the labels. The latter property is very desirable in most of the imaging and vision applications. In the experiments, our method achieves similarly satisfactory results compared to the conventional LP, yet it requires significantly smaller computation cost.