probabilistic fingerprints for shapes niloy j. mitraleonidas guibas joachim giesenmark pauly...
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Probabilistic Fingerprints for Shapes
Niloy J. Mitra Leonidas Guibas Joachim Giesen Mark Pauly
Stanford University MPII Saarbrücken ETH Zurich
Introduction
• Shape Analysis and Comparison• shape retrieval, shape clustering, feature selection,
correspondence, compression, re-use, etc
•Question: Are two shapes similar?
≈ ?
Introduction
• More general: Are two shapes similar in parts?• relative size of overlap region partially matching under
rigid motion
• scan alignment
• context-based editing
• shape recognition, etc.
• Efficient tests require compact signatures• database query
• network setting
• fast pre-filtering, etc.
Background
• Methods for global registration• Gelfand, Mitra, Guibas and Pottmann, Robust Global Registration, SGP 2005
• Li and Guskov, Multi-scale Features for Approximate Alignment of Point-based Surfaces, SGP 2005
• Huber and Hebert, Fully Automatic Registration of Multiple 3D Data Sets, CVBVS 2001
• Global shape descriptors• Kazhdan, Funkhouser and Rusinkiewicz, Rotation Invariant Spherical
Harmonic Representation of 3D Shape Descriptors, SGP 2003
• Osada, Funkhouser, Chazelle and Dobkin, Shape Distributions, ACM TOG 2002
• Reuter and Wolter, Laplace-Spectra as fingerprints for shape matching, SPM 2005
Background
• Geometric Hashing• Wolfson, Rigoutsos. Geometric Hashing: An Overview, IEEE Computational
Science and Engineering, 4(4), 1997
• Gal and Cohen-Or, Salient geometric features for partial shape matching and similarity, ACM TOG 2006
• File matching• Broder, Glassman, Manasse, Zweig. Syntactic Clustering of the Web, World
Wide Web Conference, 1997
• Broder, On the Resemblance and Containment of Documents, Sequences 1997
• Schleimer, Wilkerson and Alex Aiken, Winnowing: local algorithms for document fingerprinting, Sigmod, ’03
Probabilistic Fingerprints
• Function such that• Given two shapes S1 and S2, with high probability
• if f(S1) ≠ f(S2) then S1 and S2 are dissimilar
• if f(S1) = f(S2) then S1 and S2 are similar
• f is efficiently computable
• compact, i.e.,
• output sensitive
• localized (partial matching)
• robust to sampling and articulated motion
Pre-Processing
Input Sample
• Uniform random sample • guarantee δ-coverage
• avoid arbitrarily dense sampling [Turk 92]
such that
Sample
Pre-Processing
Shingles
• Local surface patches• intersection with ρ-balls
• create sufficient overlap for robust signature estimation, i.e.,
Pre-Processing
Shingles Signatures
• Local signatures should be invariant to• rigid transforms
• sampling & local perturbations
• Examples: Spin images, shape histograms, integral descriptors, etc.
Pre-Processing
DescriptorsSignatures
• Optional: Compressed descriptors• e.g., Rabin’s hashing
• Signature set • multi-set of points in high-dimensional space
• spatial relation of shingles not preserved
Probabilistic Fingerprint
• Estimate of resemblance
• Example: m = 3
8 0 32 54 76 91 1011 1213 14
8 03 25 4769 1 101112 1314
8 03 25 47 6 91 1011 12 1314
Probabilistic Fingerprint
• Estimate of resemblance
8 0 32 54 76 91 1011 1213 14
80 32 54 76 91 10 11 12 13 14
80 32 54 76 91 10 11 12 13 14
Probabilistic Fingerprint
• Estimate of resemblance
8 0 32 54 76 91 1011 1213 14
80 32 54 76 91 10 11 12 13 14
80 32 54 76 91 10 11 12 13 14
Probabilistic Fingerprint
• Estimate of resemblance
80 32 54 76 91 10 11 12 13 14
80 32 54 76 91 10 11 12 13 14
8 03 25 4769 1 101112 1314
Probabilistic Fingerprint
• Estimate of resemblance
8 03 25 4769 1 101112 1314
80 32 54 76 91 10 11 12 13 14
80 32 54 76 91 10 11 12 13 14
Probabilistic Fingerprint
• Estimate of resemblance
80 32 54 76 91 10 11 12 13 14
80 32 54 76 91 10 11 12 13 14
8 03 25 47 6 91 1011 12 1314
Probabilistic Fingerprint
• Estimate of resemblance
80 32 54 76 91 10 11 12 13 14
80 32 54 76 91 10 11 12 13 14
8 03 25 47 6 91 1011 12 1314
Pre-Processing
FingerprintDescriptors
• Probabilistic Fingerprint• reduce using min-hashing
• based on random permutations of universe
• set of ‘random experts’ consistent for all models
Min-Hashing
• Feature selection by random experts• reduces set comparison to element-wise
comparison
• estimate resemblance using m permutations = perform m coin tosses to estimate bias of coin
• Analysis• probabilistic bounds using Markov inequality &
strong Chernoff bound
• relates size of the fingerprint to confidence in estimated resemblance
Data Reduction
Shingles Signatures Descriptors Fingerprint
quantization min hashing
set size remains constant
100k 100k 100k 1k
set reduction
Applications
• Multiple scans• greedy alignment
using priority queue
• fingerprint matching determines score
• advanced alignment method for verification
• merging fingerprints requires no re-computation
Statistics
• Pre-processing time in seconds:
• Query time: ~ 15 msec on average
• Fingerprint size ~10kb
model #vts. uniform sampl.
spin image
Rabin hash
min-hash
skull 54k 0.8 7.5 0.05 4.5
Caesar 65k 1.4 7.3 0.08 10.3
bunny 121k 1.8 13.8 0.04 2.9
horse 8k 0.7 5.7 0.05 7.3
Remarks & Insights
• Resemblance defined as set operation on signature sets → quantization is crucial
• Random experts effectively extract consistent set of features → requires no explicit correspondence
• Fingerprints do not preserve spatial relation of shingles → false positives are possible
• Few parameters that are easy to tune
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