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 l 1-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.
KeywordsMarkov Random Field Interior Point Method Linear Programming Model Markov Random Field Model Pairwise Potential
Unable to display preview. Download preview PDF.
- 4.Sinop, A.K., Grady, L.: A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm. In: CVPR. IEEE (2007)Google Scholar
- 8.Jamriska, O., Sykora, D., Hornung, A.: Cache-efficient graph cuts on structured grids. In: IEEE CVPR, pp. 3673–3680 (2012)Google Scholar
- 9.Lempitsky, V.S., Kohli, P., Rother, C., Sharp, T.: Image segmentation with a bounding box prior. In: ICCV (2009)Google Scholar
- 11.Boykov, Y., Jolly, M.P.: Interactive graph cuts for optimal boundary & region segmentation of objects in n-d images. In: ICCV (2001)Google Scholar
- 13.Kappes, J.H., Andres, B., Hamprecht, F.A., Schnorr, C., Nowozin, S., Batra, D., Kim, S., Kausler, B.X., Lellmann, J., Komodakis, N.: et al.: A comparative study of modern inference techniques for discrete energy minimization problems. In: CVPR, pp. 1328–1335 (2013)Google Scholar
- 21.Gulshan, V., Rother, C., Criminisi, A., Blake, A., Zisserman, A.: Geodesic star convexity for interactive image segmentation. In: CVPR (2010)Google Scholar
- 23.Wang, P., Shen, C., van den Hengel, A.: A fast semidefinite approach to solving binary quadratic problems. In: CVPR (2013)Google Scholar
- 25.Yeung, S.K., Wu, T.P., Tang, C.K., Chan, T.F., Osher, S.J.: Normal estimation of a transparent object using a video. In: TPAMI (2014)Google Scholar
- 26.Yeung, S.K., Wu, T.P., Tang, C.K.: Extracting smooth and transparent layers from a single image. In: CVPR (2008)Google Scholar
- 27.Yeung, S.K., Wu, T.P., Tang, C.K., Chan, T.F., Osher, S.: Adequate reconstruction of transparent objects on a shoestring budget. In: CVPR (2011)Google Scholar