An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision

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AN EXPERIMENTAL COMPARISON OFMIN-CUT/MAX-FLOW ALGORITHMS

FORENERGY MINIMIZATION IN VISIONIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE

INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004

Yuri Boykov, Member, IEEE Vladimir Kolmogorov, Member, IEEE

Guan-Yu Liu

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Outline Introduction Overview Method

A : growth B : augmentation C : adoption

Experimental Results Part1 - introduce algorithm Part2 - results

Conclusion Reference Q & A

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Introduction(1/3)

GREIG et al. [15] were the first to discover that powerful min-cut/max-flow algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision.

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where L = is a labeling of image , is a data penalty function, is an interaction potential, and is a set of all pairs of neighboring pixels.

[15] D. Greig, B. Porteous, and A. Seheult, “Exact MaximumA Posteriori Estimation for Binary Images,” J. Royal StatisticalSoc., Series B, vol. 51, no. 2, pp. 271-279, 1989.

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Introduction(2/3)

Algorithm Dinic’s algorithm [14] Push-Relabel algorithm with the queue-based selection rule [12] Push-Relabel algorithm with the highest level selection rule [12]

[12] A.V. Goldberg and R.E. Tarjan, “A New Approach to theMaximum-Flow Problem,” J. ACM, vol. 35, no. 4, pp. 921-940,Oct. 1988.

[14] E.A. Dinic, “Algorithm for Solution of a Problem of Maximum Flow in Networks with Power Estimation,” Soviet Math. Dokl., vol. 11, pp. 1277-1280, 1970.

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Introduction(3/3)

This paper use Min-Cut/Max-Flow to find minimum energy. Image Restoration

○ Potts model○ Linear interaction

Stereo○ Pixel-Labeling stereo with the Potts model○ Stereo with occlusion○ Multi-camera scene reconstruction

Segmentation

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Overview(1/9)

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Overview(2/9)

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Source Sink

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Overview(3/9)

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Overview(5/9)

There are two basic way to search max-flow Breadth-first search(BFS) Push

A typical augmenting path algorithm stores information about the distribution of the current s→t flow f among the edges of G using a residual graph Gf. The topology of Gf is identical to G, but the capacity of an edge in Gf reflects the residual capacity of the same edge in G given the amount of flow already in the edge.

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Overview(6/9)

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Overview(9/9)

There are three steps on the proposed algorithm. A : growth B : augmentation C : adoption

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Method A : growth

search trees S and T grow until they touch giving an s→t path

B : augmentation the found path is augmented, search tree(s) break into forest(s)

C : adoption trees S and T are restored.

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Method.A(1/3)

At the growth stage, the search trees(s and t) expand. The active nodes explore adjacent nonsaturated edges and acquire new children from a set of free nodes. Active node Passive node Free node

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Method.A(2/3)

The growth stage terminates if an active node encounters a neighboring node that belongs to the opposite tree.

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Method.A(3/3)

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Method A : growth




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Method.B(1/3)

The augmentation stage augments the path found at the growth stage. Since we push through the largest flow possible, some edge(s) in the path become saturated. Orphan node

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Method.B(2/3)

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Method.B(3/3)

The nodes that become orphan nodes keep their original own parents. Change another parent Become free nodes

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Method A : growth




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Method.C(1/3)

The goal of the adoption stage is to restore the single-tree structure of sets S and T with roots in the source and the sink. Try to find a new valid parent for each orphan. A new parent should belong to the same set, S or T, as the orphan. If there is no qualifying parent, we remove the orphan from S or T and

make it a free node. Also declare all its former children orphans.

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Method.C(2/3)

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Method.C(3/3)

The stage terminates when no orphans are left and, thus, the search tree structures of S and T are restored. Since some orphan nodes in S and T may become free, the adoption stage results in contraction of these sets.

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Method.conclusion

After the adoption stage is completed, the algorithm returns to the growth stage. The algorithm terminates when the search trees S and T cannot grow (no active nodes) and the trees are separated by saturated edges. This implies that a maximum flow is achieved.

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Experimental Results.part1

A : Dinic’s algorithm

B : Push-Relabel algorithm with the queue-based

C : Push-Relabel algorithm with the highest level

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Experimental Results.B(1/1)

To push water from source to sink. Every node can store water temporally. Terminate when bottleneck.

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Experimental Results.C(1/1)

Improved queue-based Set height levels to avoid water flow back

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A : Image Restoration Potts model Linear interaction energy

B : Stereo Pixel-Labeling stereo with the potts model Stereo with occlusion Multi-camera scene reconstruction

C : Segmentation

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Experimental Results.A(1/4)

Potts model

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Linear interaction

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Reference [2] , [4] , [8] , [15] , [18] , [21] , [32] , [36] , [38]

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B : Stereo Pixel-Labeling stereo with the Potts model Stereo with occlusion Multi-camera scene reconstruction

C : Segmentation

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Potts model 1

Stereo with occlusion 1

Reference [4] , [7] , [23] , [24]

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Multi-camera scene reconstruction 1

○ 1

1○ The smoothness term is the sum of Potts energy terms over all cameras.

○ The visibility term is infinity if a configuration f violate the visibility constraint and zero otherwise.

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B : Stereo Pixel-Labeling stereo with the potts model Stereo with occlusion Multi-camera scene reconstruction

C : Segmentation

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Potts model

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Reference [3] , [5] , [28] , [29] , [33] , [35]

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Conclusions In most examples, our new min-cut/max-flow algorithm

worked 2-5 times faster than any of the other methods, including the push-relabel and the Dinic algorithms (which are known to outperform other min-cut/max-flow techniques).

However, our algorithm is not a clear outperformer when the complexity of underlying graphs is increased.

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Reference(1/4) [2] Y. Boykov and D. Huttenlocher, “A New Bayesian Framework for

Object Recognition,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. II, pp. 517-523, 1999.

[3] Y. Boykov and V. Kolmogorov, “Computing Geodesics and Minimal Surfaces via Graph Cuts,” Proc. Int’l Conf. Computer Vision, vol. I, pp. 26-33, 2003.

[4] Y. Boykov, O. Veksler, and R. Zabih, “Markov Random Fields with Efficient Approximations,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 648-655, 1998.

[5] Y. Boykov and G. Funka-Lea, “Optimal Object Extraction via Constrained Graph-Cuts,” Int’l J. Computer Vision, 2004. to appear.

[7] Y. Boykov and V. Kolmogorov, “An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision,” Proc. Int’l Workshop Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 359-374, Sept. 2001.

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Reference(2/4) [8] Y. Boykov, O. Veksler, and R. Zabih, “Fast Approximate Energy

Minimization via Graph Cuts,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 11, pp. 1222-1239, Nov. 2001.

[15] D. Greig, B. Porteous, and A. Seheult, “Exact Maximum A Posteriori Estimation for Binary Images,” J. Royal Statistical Soc., Series B, vol. 51, no. 2, pp. 271-279, 1989.

[18] H. Ishikawa and D. Geiger, “Segmentation by Grouping Junctions,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 125-131, 1998.

[21] J. Kim, J.W. Fisher III, A. Tsai, C. Wible, A.S. Willsky, and W.M. Wells III, “Incorporating Spatial Priors into an Information Theoretic Approach for f MRI Data Analysis,” Medical Image Computing and Computer-Assisted Intervention, pp. 62-71, 2000.

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Reference(3/4) [23] V. Kolmogorov and R. Zabih, “Computing Visual Correspondence

with Occlusions via Graph Cuts,” Proc. Int’l Conf. Computer Vision, July 2001.

[24] V. Kolmogorov and R. Zabih, “Multi-Camera Scene Reconstruction via Graph Cuts,” Proc. Seventh European Conf. Computer Vision, vol. III, pp. 82-96, May 2002.

[28] S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer Verlag 2003.

[29] S. J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag, 2002.

[32] S. Roy and I. Cox, “A Maximum-Flow Formulation of the N-camera Stereo Correspondence Problem,” IEEE Proc. Int’l Conf. Computer Vision, pp. 492-499, 1998.

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Reference(4/4) [33] G. Sapiro, Geometric Partial Differential Equations and Image

Analysis. Cambridge Univ. Press, 2001. [35] J.A. Sethian, Level Set Methods and Fast Marching Methods.

Cambridge Univ. Press, 1999. [36] D. Snow, P. Viola, and R. Zabih, “Exact Voxel Occupancy with

Graph Cuts,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 345-352, 2000.

[38] B. Thirion, B. Bascle, V. Ramesh, and N. Navab, “Fusion of Color, Shading and Boundary Information for Factory Pipe Segmentation,” IEEE Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 349-356, 2000.

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Q & A

An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision

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