Post on 09-Feb-2016
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Rotation Invariant Spherical Harmonic Representation of
3D Shape DescriptorsMichael KazhdanMichael Kazhdan
ThThoomas Funkhousermas FunkhouserSzymon RusinkiewiczSzymon RusinkiewiczPrinceton UniversityPrinceton University
Motivation
Large databases of 3D modelsLarge databases of 3D models
Mechanical CAD(National Design Repository)
Molecular Biology(Audrey Sanderson)
Computer Graphics(Princeton 3D Search Engine)
Retrieval Approach
3D Model ShapeDescriptor
Model Database
Nearest Neighbor
Shape Unchanged by Rotation
=
Problem
Many shape descriptors are functions that Many shape descriptors are functions that rotate with the shaperotate with the shape
Extended Gaussian Image[Horn ’84]
Spherical Attribute Image[Ikeuchi ’95]
Shape Histogram [Ankerst ’99]
Spherical Extent Function[Vranic ’00]
Reflective Symmetry Descriptor [Kazhdan ’02]
Gaussian EDT[Funkhouser ’03]
Goal
Compute similarity of shape descriptors Compute similarity of shape descriptors independent of rotationindependent of rotation
- = ?
Brute Force Approach
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-
-
-
min (rotation)- =
Impractical for databasesImpractical for databases
Normalization
Use PCA to place models into a canonical Use PCA to place models into a canonical coordinate framecoordinate frame
Covariance MatrixComputation
Principal Axis Alignment
Normalization
Doesn’t always workDoesn’t always work• Only second order informationOnly second order information
Our Approach
Eliminate rotation dependence in spherical Eliminate rotation dependence in spherical and 3D descriptorsand 3D descriptors
Shape Descriptor
EGI [Horn ’84]SAI [Ikeuchi ’95]EXT [Vranic ’00]RSD [Kazhdan ’02]EDT [Funkhouser ’03]etc.
Shape Descriptor
Our Approach
Eliminate rotation dependence in spherical Eliminate rotation dependence in spherical and 3D descriptorsand 3D descriptors
Rotation Invariant
Representation
Shape Descriptor
Outline
IntroductionIntroductionBackgroundBackgroundHarmonic RepresentationHarmonic RepresentationPropertiesPropertiesExperimental ResultsExperimental ResultsConclusion and Future WorkConclusion and Future Work
Key Idea
Obtain rotation invariant representation by Obtain rotation invariant representation by storing amplitude and eliminating phasestoring amplitude and eliminating phase
+ + + +…=
[Lo 1989]
[Burel 1995]
Fourier Descriptors
CircularFunction
Fourier Descriptors
+ + += + …
Cosine/Sine Decomposition
CircularFunction
Fourier Descriptors
=
+ + +
Constant
= + …
Frequency Decomposition
CircularFunction
Fourier Descriptors
=
+ + +
+
Constant 1st Order
= + …+
Frequency Decomposition
CircularFunction
Fourier Descriptors
=
+ + +
+ +
Constant 1st Order 2nd Order
= + …+
Frequency Decomposition
CircularFunction
Fourier Descriptors
=
+ + +
+ + +
Constant 1st Order 2nd Order 3rd Order
= + …
+ …
+
Frequency Decomposition
CircularFunction
+ + + + …+
Fourier Descriptors
= + + +
Constant 1st Order 2nd Order 3rd Order
+ …
Frequency Decomposition
=Amplitudes invariant
to rotationCircularFunction
Harmonic Representation
SphericalFunction
Harmonic Representation
=
SphericalFunction
+ + + + …
Harmonic Decomposition
Harmonic Representation
=
SphericalFunction
+ + + + …
+ + + +…=
Constant 1st Order 2nd Order 3rd Order
Harmonic Representation
+ + + +…=
Norms Invariantto Rotation
Store “how much” Store “how much” (L(L22-norm) of the -norm) of the shape resides in shape resides in each frequencyeach frequency
3D Function (Voxel Grid)
Restrict to concentric spheresRestrict to concentric spheres
3D Function (Voxel Grid)
=
=
=
+
+
+ +
+
+ +
+
+
+
+
+
Compute harmonic representation of each Compute harmonic representation of each sphere independentlysphere independently
3D Function (Voxel Grid)
Combine harmonic representationsCombine harmonic representations
Radius
Frequency
Matching
LL22-difference of harmonic representations…-difference of harmonic representations…
Harmonic Representation Harmonic Representation
-2
Matching
min (rotations)
--2 2
… … bounds proximity of descriptors over all bounds proximity of descriptors over all rotationsrotations
Outline
IntroductionIntroductionBackgroundBackgroundHarmonic RepresentationHarmonic RepresentationPropertiesPropertiesExperimental ResultsExperimental ResultsConclusion and Future WorkConclusion and Future Work
Advantages
The harmonic representations is:The harmonic representations is:•Rotation invariantRotation invariant•Multi-resolutionMulti-resolution•CompactCompact•DiscriminatingDiscriminating
Compact
…
……
Compact
……
……
Compact
……
……
Compact
……
……
Compact
……
……
Information Loss
•Intra-frequency information lossIntra-frequency information loss•Cross-frequency information lossCross-frequency information loss•Cross-radial information lossCross-radial information loss
Information Loss (Spherical Descriptor)
•Intra-frequency information lossIntra-frequency information loss•Cross-frequency information lossCross-frequency information loss
Information Loss (Spherical Descriptor)
•Intra-frequency information lossIntra-frequency information loss•Cross-frequency information lossCross-frequency information loss
+
+
22.5o90o
=
=
Information Loss (3D Descriptor)
•Cross-radial information lossCross-radial information loss
Outline
IntroductionIntroductionBackgroundBackgroundHarmonic RepresentationHarmonic RepresentationPropertiesPropertiesExperimental ResultsExperimental ResultsConclusion and Future WorkConclusion and Future Work
Shape DescriptorsExtended Gaussian Image
Horn 1984
Spherical Extent Function Vranic 2000
Shape Histogram Ankerst 1999
Gaussian EDT Funkhouser 2003
Experimental Database
Viewpoint “household” databaseViewpoint “household” database1,890 models, 85 classes1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
Gaussian EDT ResultsPCA-Normalized Results
Harmonic Representation Results
Query
11 22 33 44 55
66 77 88 99 1010
11 22 33 44 55
66 77 88 99 1010
Gaussian EDT Results
Precision vs. RecallPrecision vs. Recall
50% 100%
100%
0%
50%
Recall
Prec
isio
n
0%
HarmonicsPCA
Retrieval ResultsHarmonicsPCA
HarmonicsPCA
HarmonicsPCA
HarmonicsPCA
EGI
EDTEXT
SECT
50% 100%
100%
0%
50%
RecallPr
ecis
ion
0%
50% 100%
100%
0%
50%
Recall
Prec
isio
n
0%50% 100%
100%
0%
50%
Recall
Prec
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0%
50% 100%
100%
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50%
Recall
Prec
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•EGI: Extended Gaussian Image
•SECT: Shape Histogram (Sectors)
•EXT: Spherical Extent Function
•EDT: Gaussian Euclidean Distance Transform
Retrieval ResultsHarmonicsPCA
EGIHarmonicsPCA
SECT
50% 100%
100%
0%
50%
RecallPr
ecis
ion
0%
HarmonicsPCA
EDT
50% 100%
100%
0%
50%
Recall
Prec
isio
n
0%
HarmonicsPCA
EXT
50% 100%
100%
0%
50%
Recall
Prec
isio
n
0%
50% 100%
100%
0%
50%
Recall
Prec
isio
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0%
•EGI: Extended Gaussian Image
•SECT: Shape Histogram (Sectors)
•EXT: Spherical Extent Function
•EDT: Gaussian Euclidean Distance Transform
Exhaustive Gaussian EDT Results
HarmonicPCA
min L2
100%
50%
0%0% 50% 100%
Recall
Prec
isio
n
Gaussian EDT -
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min (rotation)
Summary and Conclusion
Provide a rotation invariant representation of Provide a rotation invariant representation of shape descriptors that:shape descriptors that:• Eliminates PCA dependenceEliminates PCA dependence• Gives better matching performanceGives better matching performance• Is more compactIs more compact• Is a multi-resolution representationIs a multi-resolution representation
Future Work
Managing Information LossManaging Information Loss• Obtain cross radial information for 3D descriptorsObtain cross radial information for 3D descriptors• Obtain cross frequency informationObtain cross frequency information• Get finer resolution of rotation invariance within Get finer resolution of rotation invariance within
frequenciesfrequencies
More GenerallyMore Generally• Consider new shape descriptorsConsider new shape descriptors
Thank YouFundingFunding
National Science FoundationNational Science Foundation
Sloan FoundationSloan Foundation
Spherical HarmonicsSpherical HarmonicsDan Rockmore and Peter KostelecDan Rockmore and Peter Kostelec
SpharmonicKit:SpharmonicKit:
http://www.cs.dartmouth.edu/~geelong/spherehttp://www.cs.dartmouth.edu/~geelong/sphere3D Shape Matching3D Shape Matching
Patrick Min, Alex Halderman, Phil Shilane, David Jacobs, Joyce ChenPatrick Min, Alex Halderman, Phil Shilane, David Jacobs, Joyce Chen
Princeton 3D Model Search Engine:Princeton 3D Model Search Engine:
http://shape.cs.princeton.eduhttp://shape.cs.princeton.edu