Approximate Correspondences in High Dimensions Kristen Grauman 1,2 and Trevor Darrell 1

1. Number of matches in bin i,j Number of new matches for jth bin at ith level In time, approximate the optimal partial matching cost: use multi-resolution histograms to count matches that are possible within a discrete set of distances. diameter of cell i,j Vocabulary-guided bins Uniform bins Pyramid matching method 6.1e-4/6.2e-4 1.5e-3 / 5.7e-4 Time/match (s) (d=128/d=10) 99.0 / 97.7 64.9 / 96.5 Mean recognition rate/class (d=128/d=10) (Caltech-4 data set, Harris and MSER-detected SIFT features) Uniform bins Vocabulary-guided bins • VG pyramid structure stored once in • Histograms stored sparsely in entries • Inserting point sets into histograms adds time • Match time still only set of features → histogram pyramid Approximate Correspondences in High Dimensions Kristen Grauman1,2 and Trevor Darrell1 1CSAIL, Massachusetts Institute of Technology 2Department of Computer Sciences, University of Texas-Austin Results Problem The correspondence between sets of local feature vectors is often a good measure of similarity, but it is computationally expensive. VG pyramids’ matching scores consistently highly correlated with the optimal matching, even for high dimensional features. (ETH-80 image data, SIFT features, k=10, L=5, results from 10 runs) flakes snow cool ice ski cold No explicit search for matches! Accuracy of existing matching approximations declines linearly with the feature dimension. The Vocabulary-Guided Pyramid Match Our approach Data-dependent pyramid structure allows more gradual distance ranges. • Form multi-resolution decomposition of the feature space to efficiently count “implicit” matches without directly comparing features • Exploit structure in feature space when placing partitions in order to fully leverage their grouping power • Approximate partial matching • Linear-time match • Mercer kernel • Accurate for feature dimensions > 100 Uniformly shaped bins result in decreased matching accuracy for high-dimensional features… Tune pyramid partitions to the feature distribution Explicit correspondence fields are more accurate and faster to compute. • Hierarchical k-means over corpus of features • Record diameters of the irregularly shaped cells Optimal partial match Vocabulary-guided (VG) pyramid match cost: time Number of matches in bin i,j’s children The Pyramid Match [Grauman and Darrell, ICCV 2005] Improved object recognition when used as a kernel in an SVM. Weighting options: input-specific upper bound admits a Mercer kernel Pyramid match cost: Future work • Learning weights on pyramid bins • Beyond geometric vocabularies • Sub-linear time PM hashing (ongoing) • Distortion bounds for the VG-PM? Number of new matches at level i counted by difference in histogram intersections across levels Weight according to bin size