The Pyramid Match Kernel Discriminative Classification with Sets

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

Sets of features

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

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

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 • 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 Restrictive assumptions High Ignoring set complexity statistics

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

Pyramid match optimal partial matching

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!

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

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

Counting matches Histogram intersection

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

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

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

Example pyramid match Level 0

Example pyramid match Level 1

Example pyramid match Level 2

Example pyramid match optimal match

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

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-classification Convergence is guaranteed since pyramid match kernel is positive-definite.

Object recognition results • ETH-80 database 8 object classes • Features: – Harris detector – PCA-SIFT descriptor, d=10 Kernel Complexity Recognition rate Match [Wallraven et al. ] 84% Bhattacharyya affinity [Kondor & Jebara] 85% Pyramid match 84% Eichhorn and Chapelle 2004

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

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

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

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

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

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