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Approximate Correspondences in High Dimensions Approximate Correspondences in High Dimensions Kristen Grauman Kristen Grauman 1,2 1,2 and Trevor Darrell and Trevor Darrell 1 1 1 1 CSAIL, Massachusetts Institute of Technology CSAIL, Massachusetts Institute of Technology 2 2 Department of Computer Sciences, University of Texas-Austin Department of Computer Sciences, University of Texas-Austin The Vocabulary-Guided Pyramid The Vocabulary-Guided Pyramid Match Match No explicit search for matches! Hierarchical k-means over corpus of features Record diameters of the irregularly shaped cells Uniformly shaped bins result in decreased matching accuracy for high-dimensional features… Tune pyramid partitions to the feature distribution Uniform bins Vocabulary-guided bins Results Results 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) Data-dependent pyramid structure allows more gradual distance ranges. The Pyramid Match The Pyramid Match [Grauman and Darrell, ICCV 2005] set of features histogram pyramid Number of new matches at level i counted by difference in histogram intersections across levels Weight according to bin size Problem Problem Our approach Our approach • 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 The correspondence between sets of local feature vectors is often a good measure of similarity, but it is computationally expensive. Accuracy of existing matching approximations declines linearly with the feature dimension. flak es cool col d snow ice ski • Approximate partial matching • Linear-time match • Mercer kernel • Accurate for feature dimensions > 100 Optimal partial match Optimal partial match time In time, approximate the optimal partial matching cost: use multi-resolution histograms to count matches that are possible within a discrete set of distances. Pyramid match cost: Vocabulary-guided (VG) pyramid match cost: Number of matches in bin i,j’s children Number of matches in bin i,j Number of new matches for j th bin at i th level Weighting options: admits a Mercer kernel diameter of cell i,j Explicit correspondence fields are more accurate and faster to compute. VG pyramid structure stored once in Histograms stored sparsely in entries Inserting point sets into histograms adds time Match time still only Improved object recognition when used as a kernel in an SVM. Future work Future work Sub-linear time PM hashing (ongoing) Distortion bounds for the VG-PM? Learning weights on pyramid bins Beyond geometric vocabularies Pyramid matching method Mean recognition rate/class (d=128/d=10) Time/match (s) (d=128/d=10) Vocabulary-guided bins Uniform bins 99.0 / 97.7 64.9 / 96.5 6.1e-4/6.2e-4 1.5e-3 / 5.7e-4 (Caltech-4 data set, Harris and MSER-detected SIFT features) input-specific upper bound

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

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

Approximate Correspondences in High DimensionsApproximate Correspondences in High DimensionsKristen GraumanKristen Grauman1,2 1,2 and Trevor Darrelland Trevor Darrell11

11CSAIL, Massachusetts Institute of TechnologyCSAIL, Massachusetts Institute of Technology22Department of Computer Sciences, University of Texas-AustinDepartment of Computer Sciences, University of Texas-Austin

The Vocabulary-Guided Pyramid MatchThe Vocabulary-Guided Pyramid Match

No explicit search for matches!

• Hierarchical k-means over corpus of features• Record diameters of the irregularly shaped cells

Uniformly shaped bins result in decreased matching accuracy for high-dimensional features…

Tune pyramid partitions to the feature distribution

Uniform bins Vocabulary-guided bins

ResultsResultsVG 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)

Data-dependent pyramid structure allows more gradual distance ranges.

The Pyramid MatchThe Pyramid Match[Grauman and Darrell, ICCV 2005]

set of features → histogram pyramid

Number of new matches at level i counted by difference in histogram intersections across levels

Weight according to bin size

ProblemProblem

Our approachOur approach• 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

The correspondence between sets of local feature vectors is often a good measure of similarity, but it is computationally expensive.

Accuracy of existing matching approximations declines linearly with the feature dimension.

flakescool

cold

snow

iceski

• Approximate partial matching

• Linear-time match• Mercer kernel• Accurate for feature

dimensions > 100

Optimal partial matchOptimal partial match

time

In time, approximate the optimal partial matching cost: use multi-resolution histograms to count matches that are possible within a discrete set of distances.

Pyramid match cost:

Vocabulary-guided (VG) pyramid match cost:

Number of matches in bin i,j’s childrenNumber of matches in bin i,j

Number of new matches for jth bin at ith level

Weighting options:

admits a Mercer kernel

diameter of cell i,j

Explicit correspondence fields are more accurate and faster to compute.

• VG pyramid structure stored once in • Histograms stored sparsely in entries• Inserting point sets into histograms adds time• Match time still only

Improved object recognition when used as a kernel in an SVM.

Future workFuture work• Sub-linear time PM hashing (ongoing)• Distortion bounds for the VG-PM?

• Learning weights on pyramid bins• Beyond geometric vocabularies

Pyramid matching method Mean recognition rate/class (d=128/d=10) Time/match (s) (d=128/d=10)

Vocabulary-guided bins

Uniform bins

99.0 / 97.7

64.9 / 96.5

6.1e-4/6.2e-4

1.5e-3 / 5.7e-4

(Caltech-4 data set, Harris and MSER-detected SIFT features)

input-specific upper bound