Harmonic 3D Shape Matching

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Harmonic3D Shape Matching

• Michael KazhdanThomas Funkhouser
• Princeton University

2
Motivation

• Large databases of 3D models

Computer Graphics (Princeton 3D Search Engine)
Mechanical CAD (National Design Repository)
Molecular Biology (Audrey Sanderson)
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Goal

• Find 3D models with similar shapes

3D Model
ShapeDescriptor
Nearest Neighbor
Model Database
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Research Challenge

• Need shape descriptor that is
• Discriminating
• Concise to store
• Quick to compute
• Efficient to match

Nearest Neighbor
3D Model
ShapeDescriptor
Model Database
5
Research Challenge

• Finding a 3D shape descriptor that is
• Discriminating
• Concise to store
• Quick to compute
• Efficient to match

Nearest Neighbor
3D Model
ShapeDescriptor
Model Database
6
Research Challenge

• Finding a 3D shape descriptor that is
• Discriminating
• Concise to store
• Quick to compute
• Efficient to match

Nearest Neighbor
ShapeDescriptor
3D Model
Model Database
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Research Challenge

• Finding a 3D shape descriptor that is
• Discriminating
• Concise to store
• Quick to compute
• Efficient to match

Nearest Neighbor
3D Model
ShapeDescriptor
Model Database
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Research Challenge

• Finding a 3D shape descriptor that is
• Discriminating
• Concise to store
• Quick to compute
• Efficient to match

Many possible alignments
Nearest Neighbor
3D Model
ShapeDescriptor
Model Database
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3D Model Matching Approaches

• Search over all possible alignments
• Too slow for large database

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min
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3D Model Matching Approaches

• Search over all possible alignments
• Too slow for large database
• Normalize alignment (e.g., with moments)
• OK for translation and scale, not for rotation

PCA Aligned Models
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3D Model Matching Approaches

• Search over all possible alignments
• Too slow for large database
• Normalize alignment (e.g., with moments)
• OK for translation and scale, not for rotation
• Build alignment invariance into descriptor
• Previous methods not very discriminating

Shape Histograms Ankerst et al., 1999
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Outline

• Introduction
• Approach
• Implementation
• Experimental Results
• Conclusion and Future Work

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Our Approach

• Harmonic 3D shape descriptor
• Decompose 3D shapes into irreducible set of
  rotation independent components
• Store how much of the model resides in each
  component

3D Model
Rotation IndependentComponents
ShapeDescriptor
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Our Approach

• Harmonic 3D shape descriptor
• Decompose 3D shapes into irreducible set of
  rotation independent components
• Store how much of the model resides in each
  component

Concentric Spheres
3D Model
Rotation IndependentComponents
ShapeDescriptor
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Our Approach

• Harmonic 3D shape descriptor
• Decompose 3D shapes into irreducible set of
  rotation independent components
• Store how much of the model resides in each
  component

Frequency Decomposition
3D Model
Rotation IndependentComponents
ShapeDescriptor
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Our Approach

• Harmonic 3D shape descriptor
• Decompose 3D shapes into irreducible set of
  rotation independent components
• Store how much of the model resides in each
  component

Amplitudes
3D Model
Rotation IndependentComponents
ShapeDescriptor
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Outline

• Introduction
• Approach
• Implementation
• Experimental Results
• Conclusion and Future Work

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Voxelization

• Convert polygonal model to 3D voxel grid
• Rasterize surfaces (no solid reconstruction)
• Normalize for translation and scale

3D Model
3D Voxel Grid
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Spherical Decomposition

• Intersect with concentric spheres

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Frequency Decomposition

• Represent each spherical function as a sum of
  different frequencies

Frequency Components
Spherical Functions
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Fourier Analysis
CircularFunction
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Fourier Analysis






CircularFunction
Cosine/Sine Decomposition
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Fourier Analysis






CircularFunction

Constant
Frequency Decomposition
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Fourier Analysis







CircularFunction


Constant
1st Order
Frequency Decomposition
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Fourier Analysis







CircularFunction



Constant
1st Order
2nd Order
Frequency Decomposition
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Fourier Analysis







CircularFunction






Constant
1st Order
2nd Order
3rd Order
Frequency Decomposition
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Fourier Analysis
Amplitudes invariantto rotation







CircularFunction






Constant
1st Order
2nd Order
3rd Order
Frequency Decomposition
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Harmonic Analysis
SphericalFunction
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Harmonic Analysis






SphericalFunction
Harmonic Decomposition
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Harmonic Analysis






SphericalFunction






Constant
1st Order
2nd Order
3rd Order
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Building Shape Descriptor

• Store how much (L2-norm) of the shape resides
  in each frequency of each sphere

HarmonicShapeDescriptor
Amplitudes
Frequency Decomposition
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Matching

• Model similarity defined as L2-distance between
  their descriptors
• Bounds proximity of voxel gridsover all
  rotations

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Sim
,
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Outline

• Introduction
• Approach
• Implementation
• Experimental Results
• Conclusion and Future Work

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Princeton 3D Search Engine
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Retrieval Experiment

• Viewpoint household database1,890 models, 85
  classes

153 dining chairs
25 livingroom chairs
16 beds
12 dining tables
8 chests
28 bottles
39 vases
36 end tables
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Retrieval Results

• Precision-recall curve (mean for all queries)

1
0.8
0.6
Precision
0.4
3D Harmonics
0.2
Random
0
0
0.2
0.4
0.6
0.8
1
Recall
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Retrieval Results

• Precision versus recall (mean for all queries)

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3D Harmonics (Our Method) D2 Shape Distributions
Osada et al., 2001 Shape Histograms Ankerst,
1999 EGI Horn, 1984 Moments Elad et al.,
2001 Random
0.8
0.6
Precision
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Recall
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Retrieval Results

• Precision versus recall (mean for all queries)

1
3D Harmonics (Our Method) D2 Shape Distributions
Osada et al., 2001 Shape Histograms Ankerst,
1999 EGI Horn, 1984 Moments Elad et al.,
2001 Random
0.8
0.6
Precision
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Recall
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Retrieval Results

• Precision versus recall (mean for all queries)

1
3D Harmonics (Our Method) D2 Shape Distributions
Osada et al., 2001 Shape Histograms Ankerst,
1999 EGI Horn, 1984 Moments Elad et al.,
2001 Random
0.8
0.6
Precision
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Recall
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Retrieval Results

• Precision versus recall (mean for all queries)

1
3D Harmonics (Our Method) D2 Shape Distributions
Osada et al., 2001 Shape Histograms Ankerst,
1999 EGI Horn, 1984 Moments Elad et al.,
2001 Random
0.8
0.6
Precision
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Recall
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Summary and Conclusion

• Harmonic shape descriptor is a rotation invariant
  representation that is
• Discriminating (46-245 better than others
  tested)
• Concise to store (2048 bytes)
• Quick to compute (1-5 seconds)
• Efficient to match (0.45 seconds 20,000 model DB)

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Future Work

• Extensions
• Partial object matching?
• Other 3D shape functions?
• Other applications
• Molecular biology
• Medicine
• Paleontology
• Forensics

http//shape.cs.princeton.edu/
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Thank You

• Funding
• National Science Foundation
• Sloan Foundation
• People
• Bernard Chazelle, David Dobkin, David Jacobs,
  David Kazhdan, Allison Klein, Patrick Min, Szymon
  Rusinkiewicz, Peter Sarnak, Julianna Tymoczko

http//shape.cs.princeton.edu/
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Analysis

• The Harmonic Descriptor is not invertible
• Different spheres rotate independently
• Different orders rotate independently
• Rotations are not transitive within an order

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Analysis

• The Harmonic Descriptor is not invertible
• Different spheres rotate independently
• Different orders rotate independently
• Rotations are not transitive within an order

l0
l1
l2
l3
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Inter-Radial Coherence

• Force same orders on different spheres to rotate
  together by setting up the rotation invariant
  matrix Mk with
• where fk,j is the k-th order component of the
  restriction of f to the j-th radius.
• (The diagonal is precisely the collection of
  L2-norms.)

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Polygon Rasterization

• Rasterize using the Euclidean Distance Transform
  to measure how much models miss by

Polygonal Model
Voxel Model