Solving MRFs with Higher-Order Smoothness Priors Using Hierarchical Gradient Nodes

  • Dongjin Kwon
  • Kyong Joon Lee
  • Il Dong Yun
  • Sang Uk Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6492)


In this paper, we propose a new method for solving the Markov random field (MRF) energies with higher-order smoothness priors. The main idea of the proposed method is a graph conversion which decomposes higher-order cliques as hierarchical auxiliary nodes. For a special class of smoothness priors which can be formulated as gradient-based potentials, we introduce an efficient representation of an auxiliary node called a gradient node. We denote a graph converted using gradient nodes as a hierarchical gradient node (HGN) graph. Given a label set \(\mathcal{L}\), the computational complexity of message passings of HGN graphs are reduced to \(\mathcal{O}(|\mathcal{L}|^2)\) from exponential complexity of a conventional factor graph representation. Moreover, as the HGN graph can integrate multiple orders of the smoothness priors inside its hierarchical structure, this method provides a way to combine different smoothness orders naturally in MRF frameworks. For optimizing HGN graphs, we apply the tree-reweighted (TRW) message passing which outperforms the belief propagation. In experiments, we show the efficiency of the proposed method on the 1D signal reconstructions and demonstrate the performance of the proposed method in three applications: image denoising, sub-pixel stereo matching and nonrigid image registration.


Message Passing Markov Random Field Image Denoising Factor Graph Label Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Weiss, Y., Freeman, W.T.: On the Optimality of Solutions of the Max-Product Belief-Propagation Algorithm in Arbitrary Graphs. IEEE Trans. Inf. Theory 47, 736–744 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Yedidia, J.S., Freeman, W.T., Weiss, Y.: Constructing Free-Energy Approximations and Generalized Belief Propagation Algorithms. IEEE Trans. Inf. Theory 51, 2282–2312 (2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    Boykov, Y., Veksler, O., Zabih, R.: Fast Approximate Energy Minimization via Graph Cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23, 1222–1239 (2001)CrossRefGoogle Scholar
  4. 4.
    Boykov, Y., Jolly, M.P.: Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in N-D Images. In: ICCV (2001)Google Scholar
  5. 5.
    Glocker, B., Komodakis, N., Paragios, N., Tziritas, G., Navab, N.: Inter and Intra-modal Deformable Registration: Continuous Deformations Meet Efficient Optimal Linear Programming. In: IPMI (2007)Google Scholar
  6. 6.
    Kwon, D., Lee, K.J., Yun, I.D., Lee, S.U.: Nonrigid Image Registration Using Dynamic Higher-Order MRF Model. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part I. LNCS, vol. 5302, pp. 373–386. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Roth, S., Black, M.J.: Fields of Experts. Int. J. Comput. Vision 82, 205–229 (2009)CrossRefGoogle Scholar
  8. 8.
    Yang, Q., Wang, L., Yang, R., Stewénius, H., Nistér, D.: Stereo Matching with Color-Weighted Correlation, Hierarchical Belief Propagation, and Occlusion Handling. IEEE Trans. Pattern Anal. Mach. Intell. 31, 492–504 (2009)CrossRefGoogle Scholar
  9. 9.
    Kohli, P., Kumar, M.P., Torr, P.H.S.: P3 & Beyond: Solving Energies with Higher Order Cliques. In: CVPR (2007)Google Scholar
  10. 10.
    Kohli, P., Ladický, L., Torr, P.H.S.: Robust Higher Order Potentials for Enforcing Label Consistency. In: CVPR (2008)Google Scholar
  11. 11.
    Rother, C., Kohli, P., Feng, W., Jia, J.: Minimizing Sparse Higher Order Energy Functions of Discrete Variables. In: CVPR (2009)Google Scholar
  12. 12.
    Ishikawa, H.: Higher-Order Clique Reduction in Binary Graph Cut. In: CVPR (2009)Google Scholar
  13. 13.
    Liao, L., Fox, D., Kautz, H.: Location-based Activity Recognition. In: NIPS (2005)Google Scholar
  14. 14.
    Potetz, B.: Efficient Belief Propagation for Vision Using Linear Constraint Nodes. In: CVPR (2007)Google Scholar
  15. 15.
    Komodakis, N., Paragios, N.: Beyond Pairwise Energies: Efficient Optimization for Higher-order MRFs. In: CVPR (2009)Google Scholar
  16. 16.
    Komodakis, N., Paragios, N., Tziritas, G.: MRF Optimization via Dual Decomposition: Message-Passing Revisited. In: ICCV (2007)Google Scholar
  17. 17.
    Kschischang, F.R., Frey, B.J., Loeliger, H.A.: Factor Graphs and the Sum-Product Algorithm. IEEE Trans. Inf. Theory 47, 498–519 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Wainwright, M.J., Jaakkola, T., Willsky, A.S.: MAP Estimation Via Agreement on Trees: Message-Passing and Linear Programming. IEEE Trans. Inf. Theory 51, 3697–3717 (2005)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kolmogorov, V.: Convergent Tree-Reweighted Message Passing for Energy Minimization. IEEE Trans. Pattern Anal. Mach. Intell. 28, 1568–1583 (2006)CrossRefGoogle Scholar
  20. 20.
    Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., Tappen, M., Rother, C.: A Comparative Study of Energy Minimization Methods for Markov Random Fields. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 16–29. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Birchfield, S., Tomasi, C.: A Pixel Dissimilarity Measure That Is Insensitive to Image Sampling. IEEE Trans. Pattern Anal. Mach. Intell. 20, 401–406 (1998)CrossRefGoogle Scholar
  22. 22.
    Middlebury stereo datasets,

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dongjin Kwon
    • 1
  • Kyong Joon Lee
    • 1
  • Il Dong Yun
    • 2
  • Sang Uk Lee
    • 1
  1. 1.School of EECS, ASRISeoul Nat’l Univ.SeoulKorea
  2. 2.School of EIEHankuk Univ. of F. S.YonginKorea

Personalised recommendations