Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts-a review. TPAMI 29, 1274–1279 (2007)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. TPAMI 23, 1222–1239 (2001)
Grady, L.: Random walks for image segmentation. TPAMI 28, 1768–1783 (2006)
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)
Li, H., Shen, C.: Interactive color image segmentation with linear programming. Machine Vision and Applications 21, 403–412 (2010)
Bhusnurmath, A., Taylor, C.J.: Graph cuts via l_1 norm minimization. TPAMI 30, 1866–1871 (2008)
Derigs, U., Meier, W.: Implementing goldberg’s max-flow-algorithm — a computational investigation. Zeitschrift für Operations Research 33, 383–403 (1989)
Jamriska, O., Sykora, D., Hornung, A.: Cache-efficient graph cuts on structured grids. In: IEEE CVPR, pp. 3673–3680 (2012)
Lempitsky, V.S., Kohli, P., Rother, C., Sharp, T.: Image segmentation with a bounding box prior. In: ICCV (2009)
Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? TPAMI 26, 147–159 (2004)
Boykov, Y., Jolly, M.P.: Interactive graph cuts for optimal boundary & region segmentation of objects in n-d images. In: ICCV (2001)
Komodakis, N., Paragios, N., Tziritas, G.: Mrf energy minimization and beyond via dual decomposition. TPAMI 33, 531–552 (2011)
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)
Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. IJCV 1, 321–331 (1988)
Chan, T., Vese, L.: Active contours without edges. TIP 10, 266–277 (2001)
Ye, Y.: An o(n 3 l) potential reduction algorithm for linear programming. Mathematical Programming 50, 239–258 (1991)
Megiddo, N.: Linear programming in linear time when the dimension is fixed. Journal of the ACM (JACM) 31, 114–127 (1984)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60, 259–268 (1992)
Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 136–152. Springer, Heidelberg (2005)
Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. Journal of Global Optimization 1, 15–22 (1991)
Gulshan, V., Rother, C., Criminisi, A., Blake, A., Zisserman, A.: Geodesic star convexity for interactive image segmentation. In: CVPR (2010)
Levinshtein, A., Stere, A., Kutulakos, K.N., Fleet, D.J., Dickinson, S.J., Siddiqi, K.: Turbopixels: Fast superpixels using geometric flows. TPAMI 31, 2290–2297 (2009)
Wang, P., Shen, C., van den Hengel, A.: A fast semidefinite approach to solving binary quadratic problems. In: CVPR (2013)
Wu, T.P., Yeung, S.K., Jia, J., Tang, C.K., Medioni, G.G.: A closed-form solution to tensor voting: Theory and applications. TPAMI 34, 1482–1495 (2012)
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)
Yeung, S.K., Wu, T.P., Tang, C.K.: Extracting smooth and transparent layers from a single image. In: CVPR (2008)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Wang, J., Yeung, SK. (2015). A Compact Linear Programming Relaxation for Binary Sub-modular MRF. In: Tai, XC., Bae, E., Chan, T.F., Lysaker, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2015. Lecture Notes in Computer Science, vol 8932. Springer, Cham. https://doi.org/10.1007/978-3-319-14612-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-14612-6_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14611-9
Online ISBN: 978-3-319-14612-6
eBook Packages: Computer ScienceComputer Science (R0)