The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features

1. The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features Kristen Grauman Trevor Darrell MIT

2. Sets of features

3. invariant region descriptors local shape features examples under varying conditions Sets of features

4. Problem How to build a discriminative classifier using the set representation? • Kernel-based methods (e.g. SVM) are appealing for efficiency and generalization power… • But what is an appropriate kernel? • Each instance is unordered set of vectors • Varying number of vectors per instance

5. Compute pair-wise similarity between all vectors in each set • Wallraven et al., Lyu, Boughhorbel et al. • General family of algebraic functions combining local (vector) kernels • Shashua & Hazan High complexity Existing set kernels • Fit(parametric) model to each set, compare with distance over models Kondor & Jebara, Moreno et al., Lafferty & Lebanon, Cuturi & Vert, Wolf & Shashua Restrictive assumptions Ignoring set statistics

6. Partial matching for sets of features Compare sets by computing a partialmatching between their features. Robust to clutter, segmentation errors, occlusion…

7. optimal partial matching Pyramid match

8. Pyramid match overview Pyramid match kernel measures similarity of a partial matching between two sets: • Place multi-dimensional, multi-resolution grid over point sets • Consider points matched at finest resolution where they fall into same grid cell • Approximate similarity between matched points with worst case similarity at given level No explicit search for matches!

9. Number of newly matched pairs at level i Measure of difficulty of a match at level i Pyramid match kernel Approximate partial match similarity

10. , Histogram pyramid: level i has bins of size 2i Feature extraction

11. Counting matches Histogram intersection

12. matches at this level matches at previous level Difference in histogram intersections across levels counts number ofnew pairs matched Counting new matches Histogram intersection

13. histogram pyramids number of newly matched pairs at level i measure of difficulty of a match at level i Pyramid match kernel • Weights inversely proportional to bin size • Normalize kernel values to avoid favoring large sets

14. Efficiency For sets with m features of dimension d, and pyramids with L levels, computational complexity of Pyramid match kernel: Existing set kernel approaches: or

15. Example pyramid match Level 0

16. Example pyramid match Level 1

17. Example pyramid match Level 2

18. Example pyramid match pyramid match optimal match

19. Approximation of the optimal partial matching [Indyk & Thaper] Matching output Trial number (sorted by optimal distance) 100 sets with 2D points, cardinalities vary between 5 and 100

20. Building a classifier • Train SVM by computing kernel values between all labeled training examples • Classify novel examples by computing kernel values against support vectors • One-versus-all for multi-class classification Convergence is guaranteed since pyramid match kernel is positive-definite.

21. Object recognition results • ETH-80 database 8 object classes • Features: • Harris detector • PCA-SIFT descriptor, d=10 Eichhorn and Chapelle 2004

22. Object recognition results • Caltech objects database 101 object classes • Features: • SIFT detector • PCA-SIFT descriptor, d=10 • 30 training images / class • 43% recognition rate (1% chance performance) • 0.002 seconds per match

23. Localization • Inspect intersections to obtain correspondences between features • Higher confidence correspondences at finer resolution levels target observation

24. Pyramid match regression • Pose estimation from contour features • Train SVR with CG data • Features: shape context histograms

25. Summary: Pyramid match kernel optimal partial matching between sets of features difficulty of a match at level i number of new matches at level i

26. Summary: Pyramid match kernel • A new similarity measure based on implicit correspondences that approximates the optimal partial matching • linear time complexity • no independence assumption • model-free • insensitive to clutter • positive-definite function • fast, effective object recognition

27. Future work • Geometric constraints • Fast search of large databases with the pyramid match for image retrieval • Use as a filter for a slower, explicit correspondence method • Alternative feature types and classification domains