A Tensor-Based Algorithm for High-Order Graph Matching Olivier Duchenne, Francis Bach, Inso Kweon, Jean Ponce École Normale Supérieure, INRIA, KAIST Team.

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Presentation transcript:

A Tensor-Based Algorithm for High-Order Graph Matching Olivier Duchenne, Francis Bach, Inso Kweon, Jean Ponce École Normale Supérieure, INRIA, KAIST Team Willow

 Extension of [Leordeanu & Hebert] to the case of hypergraph. Contributions 2) Sparse output for the relaxed matching problem. 1) Tensor Formulation and Power Method

Generic Graph Matching Problem Previous Works

 NP-Hard combinatorial problem.  Wide and active literature existing on approximation algorithms. Greedy Relaxed formulation Convex-concave optimization Many others… [1] M. Zaslavskiy, F. Bach and J.-P. Vert. PAMI, [2] D. C. Schmidt and L. E. Druffel. JACM, [3] J. R. Ullmann. JACM, [4] L. P. Cordella, P. Foggia, C. Sansone, and M. Vento. ICIAP, [5] Shinji Umeyama. PAMI, [6] S.Gold and A.Rangarajan. PAMI, [7] Hong Fang Wang and Edwin R. Hancock. PR, [8] Terry Caelli and Serhiy Kosinov. PAMI [9] H.A. Almohamad and S.O.Duffuaa. PAMI, [10] C.Schellewald and C.Schnor. LNCS, Generic Graph Matching Previous Works

[1] M. Fishler and Elschlager. Computer, [2] Serge Belongie, Jitendra Malik and Jan Puzicha. NIPS Previous Works Graph Matching for Computer Vision (1)

[1] A. C. Berg, T. L. Berg, and J. Malik. CVPR 2005 [2] M. Leordeanu and M. Hebert. ICCV 2005 [3] T. Cour and J. Shi. NIPS Previous Works Graph Matching for Computer Vision (2)

= ? [1] Sivic & Zisserman [2] Lazebnik et al [3] Csurka et al Previous Works Why we need correspondance?  Graph Matching is an attempt to compare images when that relationship cannot be ignored.  Bag-Of-Word model works well when relationship between features is not important.

() () First order scoreSecond order score m1 m2 Previous Works Objective function [1] A. C. Berg, T. L. Berg, and J. Malik. CVPR [2] M. Leordeanu and M. Hebert. ICCV [3] T. Cour and J. Shi. NIPS 2006.

Hypergraph Matching Problem How graph matching enforces geometric consistency? [1] A. C. Berg, T. L. Berg, and J. Malik. CVPR 2005.

Hypergraph Matching Problem [1] A. C. Berg, T. L. Berg, and J. Malik. CVPR [2] M. Leordeanu and M. Hebert. ICCV How graph matching enforces geometric consistency?

Hypergraph Matching Problem How to improve it?

Hypergraph Matching Problem How to improve it?

A hyper-edge can link more than 2 nodes at the same time. [1] Ron Zass and Amnon Shashua. CVPR, 2008 Hypergraph Matching Problem Hypergraph Matching

Formulation Hypergraph Matching Problem

[1] M. Leordeanu and M. Hebert. ICCV Relaxation Hypergraph Matching Problem Each node is matched to at most one node. Constraints: Main Eigenvector Problem Relaxed constraints:

H Hypergraph Matching Hypergraph Matching Problem

 The power method can efficiently find the main eigenvector of a matrix.  It can efficiently use the sparsity of the Matrix.  It is very simple to implement. Power Method Hypergraph Matching Problem

[1] L. De Lathauwer, B. De Moor, and J. Vandewalle. SIAM J. Matrix Anal. Appl., 2000 [2] P. A. Regalia and E. Kofidis. ICASSP, 2000 Tensor Power Iteration Hypergraph Matching Problem  Converge to a local optimum (of the relaxed problem)  Always keep the full degree of the hypergraph (never marginalize it in a frist or second order)

Tensor Power Iteration Hypergraph Matching Problem  It is also possible to integrate cues of different orders.

Sparse Output

Implementation  Compute all descriptor triplets of image 2.  Same for a subsample of triplets of image 1.  Use Approximate Nearest Neighbor to find the closest triplets.  Compute the sparse tensor.  Do Tensor Power Iteration.  Projection to binary solution.

 We generate a cloud of random points.  Add some noise to the position of points.  Rotate it. Artificial cloud of point. Experiments

Accuracy depending on outliers number Experiments Our work Leordeanu & Hebert Zass & Shashua

Accuracy depending on scaling Experiments Our work Leordeanu & Hebert Zass & Shashua

CMU hotel dataset Experiments [1] CMU 'hotel' dataset:

Error on Hotel data set Experiments Our work Zass & Shashua Leordeanu & Hebert

Examples of Caltech 256 silouhettes matching Experiments

 Method for Hyper-Graph matching.  Tensor formulation  Power Iteration  Sparse output  Future Work Conclusion