Symmetry Descriptors and 3D Shape Matching

1
Symmetry Descriptors and3D Shape Matching

• Michael Kazhdan
• Thomas Funkhouser
• Szymon Rusinkiewicz
• Princeton University

2
Motivation
Images courtesy ofCyberware, ATI, 3Dcafe

• Methods for acquiring and visualizing 3D models
  are becoming cheaper and 3D data is becoming more
  commonly available

Cyberware
3D Cafe
Cheap Scanners
World Wide Web
ATI
Fast Graphics Cards
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Motivation
Images courtesy ofStanford Utah

• With 3D models becoming ubiquitous, there has
  been a shift in research focus

Previous research has asked How do we construct
3D models?
Utah VW Bug
Utah Teapot
Stanford Bunny
Now we are asking How do we analyze 3D models?
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3D Model Analysis
Images courtesy of Ayellet Tal, Emil Praun,
Florida State and Viewpoint

• Recognition
• Matching
• Registration
• Classification
• Representation
• Compression
• Reconstruction
• Segmentation
• Feature detection
• Etc.

Handle
Cup
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Symmetry Detection

• Recognition
• Matching
• Registration
• Classification
• Representation
• Compression
• Reconstruction
• Segmentation
• Feature detection
• Etc.

Reflective
2-Fold
4-Fold
Axial
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Related Work

• Detect symmetry by comparing the model with its
  reflection/rotation

Reflective
2-Fold
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Binary Symmetry

• Find all perfect symmetries of a 2D model
• Represent a 2D model by a circular string S.
• Search for non-trivial repeating patterns in the
  concatenation (S?SS).

IEEE, 1985. Atallah The Visual Computer, 1985.
Wolter et al.
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Continuous Exhaustive Symmetry

• Measure of symmetry as distance to nearest
  symmetric model
• Compute the nearest k-fold symmetric model.
• Measure the distance between the initial model
  and the symmetric one.

Initial Model
Nearest 3-Fold Symmetric Model
Symmetry Distance
IWVF, 1994. Zabrodsky et al
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Circular Function Descriptors

• Replace discrete matching of circular string with
  correlation of circular function
• Represent a 2D model by a circular function S.
• Use the FFT to correlate the function with itself
  and find repeating patterns.

2D Model
PRL, 1995. Sun RTI, 1999. Sun et al.
Circular Function
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Circular Function Descriptors

• Replace discrete matching of circular string with
  correlation of circular function
• Represent a 2D model by a circular function S.
• Use the FFT to correlate the function with itself
  and find repeating patterns.

2D Model
PRL, 1995. Sun RTI, 1999. Sun et al.
Circular Function
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Our Approach
3D Shape
2-Fold Rotation
3-Fold Rotation
4-Fold Rotation
Reflection
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Our Approach
3D Shape
Shape Descriptor
EGI EXT REXT Sectors Sectors Shells GEDT Etc.
2-Fold Rotation
3-Fold Rotation
4-Fold Rotation
Reflection
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Outline

• Introduction
• Background
• Symmetry Descriptor
• Defining the descriptor
• Computing the descriptor
• Symmetry and Shape Matching
• Conclusion and Future Work

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Measure of Symmetry
90o
180o
270o

• Symmetry is defined by a group G that acts on the
  model/descriptor

180o
270o
90o
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Measure of Symmetry

• A model is symmetric if its descriptor is fixed
  by the action of G

p
p
180o
270o
90o
p
p
p
p
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Measure of Symmetry

• The measure of symmetry of a model is the
  distance to the nearest symmetric descriptor

f
h?Sym( f )
SymG( f )
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Measure of Symmetry

• The nearest symmetric descriptor is the average
  of the descriptors under the action of G

p

p

p
f
p
p

p
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Symmetry Descriptor

• For each type of symmetry, how much of the model
  has that symmetry?

p
p
p
,
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Symmetry Descriptor

• Represent the measures of all of the different
  symmetries of a 3D model

Symmetry Descriptors
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Outline

• Introduction
• Background
• Symmetry Descriptor
• Defining the descriptor
• Computing the descriptor
• Symmetry and Shape Matching
• Conclusion and Future Work

21
Computing Symmetry

• Computing symmetry distance requires comparing
  the descriptor with its rotations/reflections.

p

p

p
f
p
p

p
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Computing Symmetry

• Can compute the symmetry distance efficiently by
  pre-computing the correlation of the descriptor.

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Function Correlation

• Circular Functions
• Compute the Fourier transform
• Multiply frequency components
• Compute the inverse Fourier transform
• For O(n) function, takes O(n log(n)) time

• Spherical Functions
• Compute the spherical harmonic transform
• Multiply frequency components
• Compute the inverse Wigner-D transform
• For O(n2) / O(n3) function, takes ?O(n4) time

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L? Property (Rotation Invariant)
Model
Descriptor
2-Fold
3-Fold
Reflective
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Outline

• Introduction
• Background
• Symmetry Descriptor
• Symmetry and Shape Matching
• Symmetry Augmentation
• Experimental Results
• Conclusion and Future Work

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Shape Matching

• The shape of a model is independent of its
  alignment Want to compare models at their best
  alignment.

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Aligning for Rotation (PCA)

• Align the principal axes of the model with the
  coordinate axes.

PCA Alignment
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Rotation Invariance (Power Spectrum)






SphericalFunction
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Rotation Invariance (Power Spectrum)
Store how much (L2-norm) of the shape resides
in each frequency
Norms Invariantto Rotation






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Rotation Invariance (Power Spectrum)

• Power Spectrum
• Invariant to rotation
• Compact
• Less discriminating

Norms Invariantto Rotation






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Symmetry Augmentation
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Experimental Database

• Viewpoint household database1,890 models, 85
  classes

153 dining chairs
25 living-room chairs
16 beds
12 dining tables
8 chests
28 bottles
39 vases
36 end tables
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Retrieval Results
Power Spectrum
Symmetry Augmented Power Spectrum
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Retrieval Results

• Comparing the power spectrum representation, with
  and without symmetry augmentation, to PCA-aligned
  descriptors.

100
Spectrum Sym (28 floats)
Spectrum (16 floats)
PCA (240 floats)
Precision
50
0
0
50
100
Recall
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Retrieval Results

• Comparing the power spectrum representation, with
  and without symmetry augmentation, to PCA-aligned
  descriptors.

15
100
Spectrum Sym (28 floats)
Spectrum Sym (28 floats)
Spectrum (16 floats)
Spectrum (16 floats)
PCA (240 floats)
PCA (240 floats)
Precision
50
Improvement
Precision
0
50
100
0
0
50
100
-5
Recall
Recall
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Outline

• Introduction
• Background
• Symmetry Descriptor
• Symmetry and Shape Matching
• Conclusion and Future Work

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Symmetry Descriptor

• Provide a symmetry descriptor for identifying the
  symmetries of 3D models
• Defined the symmetry distance
• Giving a continuous measure of symmetry
• For all symmetries passing through the models
  center of mass
• That is efficient to compute

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Symmetry Augmentation

• Provide a method for augmenting existing shape
  descriptors with symmetry information
• Maintain a rotation invariant representation
• That is compact
• Without sacrificing retrieval performance

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Future Work

• Extending the Symmetry Descriptor
• Compute the measure of reflective symmetry for
  all planes of reflection (3D shape
  representation)
• Compute the measure of rotational symmetry for
  all axes of rotation (5D shape representation)?
• Applications to Shape Analysis
• Alignment
• Compression
• Reconstruction

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Thank You

• Funding
• National Science Foundation
• Spherical Harmonics
• Dan Rockmore and Peter Kostelec
• http//www.cs.dartmouth.edu/geelong/sphere
• http//www.cs.dartmouth.edu/geelong/soft
• Princeton Shape Matching
• Patrick Min and Phil Shilane
• http//shape.cs.princeton.edu