Shape%20Matching%20and%20Anisotropy

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Shape Matching and Anisotropy

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

2
Motivation
Images courtesy ofCyberware, ATI, 3Dcafe

• 3D data is becoming more commonly available

Cyberware
3D Cafe
Cheap Scanners
World Wide Web
ATI
Fast Graphics Cards
Someday 3D models will be as common as images
are today
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Motivation
Images courtesy ofStanford Utah

• When 3D models are ubiquitous, there will be a
  shift in research focus

Previous research has asked How do we construct
3D models?
Utah VW Bug
Utah Teapot
Stanford Bunny
Future research will ask How do we find 3D
models?
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Challenge
Images courtesy ofGoogle Princeton

• Given A database of 3D models and a query model
• Find The k database models most similar to the
  query

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Approach

• To retrieve the nearest k models
• Compute the distance between the query and every
  database model.
• Sort the database models by proximity.
• Return the first k matches.

Sort by proximity
Query comparison
3D Query
Best Match(es)
Database Models
Sorted Models
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Comparing 3D Models

• Direct Approach
• Establish pair-wise correspondences between
  points on the surfaces of the two models.
• Define the distance between the models as the
  distance between corresponding points.

• Similarity defined as distance between models
• Establishing correspondences is difficult and
  slow

p1
qn
q1
pn
q2
p2
qn-1
q3
pn-1
p3
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Comparing 3D Models

• Practical Approach
• Represent each 3D model by a shape descriptor.
• Define the distance between two models as the
  distance between their shape descriptors.

• Shape Descriptors
• Extended Gaussian Images, Horn
• Complex Extended Gaussian Images, Kang et al.
• Spherical Attribute Images, Delingette et al.
• Crease Histograms, Besl
• Shape Histograms, Ankerst et al.
• Shape Distributions, Osada et al.
• Spherical Extent Functions, Vranic et al.
• Gaussian EDTs, Funkhouser et al.
• Symmetry Descriptors, Kazhdan et al.

• Approximates distance between models
• Correspondences are implicit
• Comparison is easy and fast

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Observation

• It is not enough to consider the distance between
  two 3D surfaces
• We also need to consider how the surfaces
  transform into each other.

M1
M1
Q
?
Q
M2
M2
Q
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Our Approach

• Match models in two steps
• Factor out low-frequency alignment of the models
• Match the aligned models
• Define similarity by combining
• low-frequency alignment info
• with high-frequency difference

Anisotropy
Isotropic Models
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Outline

• Introduction
• Aligning Anisotropic Scales
• Related Work
• Anisotropy Normalization
• Convergence Properties
• Shape Matching
• Conclusion and Future Work

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Point Set Alignment
Horn, 1987

• Given point sets Pp1,,pn and Qq1,qn, what
  is the optimal alignment A minimizing

P
Q
Original
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Point Sets (Translation)
Horn, 1987

• Translate so that the center is at the origin

P
Q
P
Q
Original
Translated
A model can be aligned for translation
independent of what it will be compared against
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Point Sets (Isotropic Scale)
Horn, 1987

• Scale so that mean variance from center is equal
  to 1

P
Q
P
Q
P
Q
Original
Translated
Scaled
A model can be aligned for isotropic
scale independent of what it will be compared
against
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Point Sets (Anisotropic Scale)

• Scale so that the variance in every direction
  equal to 1

Anisotropic Models
Isotropic Models
A model can be aligned for anisotropic
scale independent of what it will be compared
against
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Unit Variance
Unit Variance in Every Direction
Covariance Matrix is Identity (Covariance Ellipse
is a Sphere)
Initial Point Set
Covariance Ellipse
Isotropic Model
For point sets, transform by inverse square root
of the covariance matrix
Covariance Ellipse
Rescaled Point Set
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From Points to Surfaces

• Points samples from a surface become isotropic
  but the sampled surface does not.

Point Set Model
Surface Model
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From Points to Surfaces

• Uniform samples do not stay uniform

Initial Point Set
Isotropic Point Set
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From Points to Surfaces

• Uniform samples do not stay uniform
• But the model gets more isotropic.

Iteratively rescale to get models that are
progressively more isotropic
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Convergence of Iteration

• Provably convergent
• Show that in the worst case smallest eigenvalue
  doesnt get smaller and largest one doesnt get
  larger.
• Use the triangle inequality to show that at least
  one of the eigenvalues has to change.
• In practice, converges very quickly

Tested on 1890 Viewpoint models
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Outline

• Introduction
• Aligning Anisotropic Scales
• Shape Matching
• Extending Shape Descriptors
• Experimental Results
• Conclusion and Future Work

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Product Descriptor

• For any shape descriptor, we define a new shape
  descriptor that is the product of
• The descriptor of the isotropic model, and
• The anisotropic scales

Isotropic Model
Descriptor
Initial Model
New Descriptor
Eigenvalues
Rescaling Ellipse
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Factored Matching

• Parameterized family of shape metrics, as a
  function of anisotropy importance ?.

?
Isotropic Models
Anisotropy
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Experimental Database

• Princeton Shape Benchmark 900 models, 90 classes

14 biplanes
50 human bipeds
7 dogs
17 fish
16 swords
6 skulls
15 desk chairs
13 electric guitars
http//shape.cs.princeton.edu/benchmark/
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Example Query
Results Without Anisotropy Factorization
1
2
3
4
8
7
6
5
Results With Anisotropy Factorization (?3)
1
2
3
4
Query
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7
6
5
Gaussian EDT, Funkhouser et al. 2003
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Retrieval Results (?3)
Rotation Invariant Descriptors
Descriptor Dim Improvement
SHELLS 1D 63
D2 1D 36
EGI 1D 64
CEGI 1D 28
Sectors 1D 31
EXT 1D 39
REXT 2D 16
Voxel 2D 23
Sectors Shells 2D 35
Gaussian EDT 2D 4
Rotation Varying Descriptors
Descriptor Dim Improvement
EGI 2D 29
CEGI 2D 20
Sectors 2D 5
EXT 2D 7
REXT 3D 4
Voxel 3D 1
Sectors Shells 3D 7
Gaussian EDT 3D 4
Spherical Power Spectrum Representation
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Outline

• Introduction
• Aligning Anisotropic Scales
• Shape Matching
• Conclusion and Future Work

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Conclusion

• Presented and iterative approach for transforming
  anisotropic models into isotropic ones
• Provides a method for factoring shape matching
• Improves matching for all descriptors
• Facilitates registration of models

Isotropic Model
Descriptor
Initial Model
New Descriptor
Eigenvalues
Rescaling Ellipse
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Conclusion

• Presented and iterative approach for transforming
  anisotropic models into isotropic ones
• Provides a method for factoring shape matching
• Gives rise to improved matching retrieval results
• Facilitates registration of models

Anisotropic Models
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Conclusion

• Presented and iterative approach for transforming
  anisotropic models into isotropic ones
• Provides a method for factoring shape matching
• Gives rise to improved matching retrieval results
• Facilitates registration of models

Isotropic Models
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Future Work

• Address cross-class anisotropy variance
• Factor out higher order transformations

Without Anisotropy Factorization
Without Anisotropy Factorization
1
2
3
4
5
6
7
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With Anisotropy Factorization
With Anisotropy Factorization
1
2
3
4
5
6
7
8
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Future Work

• Address cross-class anisotropy variance
• Factor out higher order transformations

Anisotropic Scale
Isotropic Scale
?
Translation
Rotation
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Thank You

• Funding
• National Science Foundation
• Source Code
• Dan Rockmore and Peter Kostelec
• http//www.cs.dartmouth.edu/geelong/sphere
• http//www.cs.dartmouth.edu/geelong/soft
• Databases
• Viewpoint Data Labs, Cacheforce, De Espona
  Infografica
• http//www.viewpoint.com
• http//www.cacheforce.com
• http//www.deespona.com
• Princeton Shape Matching Group
• Patrick Min and Phil Shilane
• http//shape.cs.princeton.edu

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Thank You
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