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Shape Matching and Anisotropy Michael Kazhdan, Thomas Funkhouser, and Szymon Rusinkiewicz Princeton...
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Transcript of Shape Matching and Anisotropy Michael Kazhdan, Thomas Funkhouser, and Szymon Rusinkiewicz Princeton...
Shape Matching and Anisotropy
Shape Matching and Anisotropy
Michael Kazhdan, Thomas Funkhouser, and Szymon
Rusinkiewicz
Princeton University
Michael Kazhdan, Thomas Funkhouser, and Szymon
Rusinkiewicz
Princeton University
Motivation
3D data is becoming more commonly available
Someday 3D models will be as common as images are today
Someday 3D models will be as common as images are today
Cheap Scanners World Wide Web3D CafeCyberware
Fast Graphics Cards
ATI
Images courtesy ofCyberware, ATI, & 3Dcafe
Motivation
When 3D models are ubiquitous, there will be a shift in research focus
Future research will ask:“How do we find 3D models?”
Future research will ask:“How do we find 3D models?”
Utah VW Bug Utah Teapot Stanford Bunny
Images courtesy ofStanford & Utah
Previous research has asked:“How do we construct 3D models?”
Previous research has asked:“How do we construct 3D models?”
Challenge
Given: A database of 3D models and a query model
Find: The k database models most similar to the query
Images courtesy ofGoogle & Princeton
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.
3D Query
Database ModelsBest Match(es)
Sorted Models
Sort by proximityQuery
comparison
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.
2 ii qppnpn
qnqn
pn-1
qn-1qn-1q3q3
q2q1q1
p3
p2
p1
Similarity defined as distance between models Establishing correspondences is difficult and slow
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, BeslShape 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
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
Q
Q M2M2
Q
M1
21 MQMQ ?
Match models in two steps:
1. Factor out low-frequency alignment of the models
2. Match the aligned models
Define similarity by combininglow-frequency alignment infowith high-frequency difference
Our Approach
Isotropic ModelsAnisotropy
Outline
Introduction
Aligning Anisotropic Scales• Related Work• Anisotropy Normalization• Convergence Properties
Shape Matching
Conclusion and Future Work
Point Set Alignment
Given point sets P={p1,…,pn} and Q={q1,…qn}, what is the optimal alignment A minimizing:
n
iii qAp
1
2)(
P Q
Original
[Horn, 1987]
Point Sets (Translation)
Translate so that the center is at the origin:
n
ii
n
ii qandp
11
0 0
A model can be aligned for translation independent of what it will be compared against
P Q
Original
[Horn, 1987]
Translated
P’ Q’
Point Sets (Isotropic Scale)
Scale so that mean variance from center is equal to 1:
n
ii
n
ii q
nandp
n 1
2
1
21
1 1
1
A model can be aligned for isotropic scaleindependent of what it will be compared against
P Q
Original Translated
P’ Q’ P” Q”
Scaled
[Horn, 1987]
Point Sets (Anisotropic Scale)
Scale so that the variance in every direction equal to 1: 1 1
1
1
2
vpvn
n
ii
A model can be aligned for anisotropic scaleindependent of what it will be compared against
Anisotropic Models Isotropic Models
Unit Variance
Unit Variance in Every Direction
Covariance Matrix is Identity (Covariance Ellipse is a Sphere)
Anisotropic Model
Isotropic Model
Initial Point Set
Rescaled Point Set
Covariance Ellipse
Covariance Ellipse
For point sets, transform by inverse square root of the covariance matrix
From Points to Surfaces
Points samples from a surface become isotropic but the sampled surface does not.
Point Set Model Surface Model
From Points to Surfaces
Uniform samples do not stay uniform…
Initial Point Set Isotropic Point Set
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
Convergence of Iteration
Provably convergent Show that in the worst case smallest eigenvalue doesn’t
get smaller and largest one doesn’t get larger. Use the triangle inequality to show that at least one of the
eigenvalues has to change. In practice, converges very quickly
0
1
2
0 2 4 6 8 10Iterations
RM
S E
rro
r
Max
Average
Tested on 1890 Viewpoint models
Outline
Introduction
Aligning Anisotropic Scales
Shape Matching Extending Shape Descriptors Experimental Results
Conclusion and Future Work
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
Initial ModelInitial Model
Isotropic ModelIsotropic Model
Rescaling EllipseRescaling Ellipse
DescriptorDescriptor
EigenvaluesEigenvalues
New DescriptorNew Descriptor
Factored Matching
Isotropic Models Anisotropy
Parameterized family of shape metrics, as a function of anisotropy importance .Parameterized family of shape metrics, as a function of anisotropy importance .
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/
Example Query
Results Without Anisotropy Factorization
Results With Anisotropy Factorization (=3)
Query
1 2 3 4
8765
1 2 3 4
8765
Gaussian EDT, Funkhouser et al. 2003
Retrieval Results (=3)
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 Invariant 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%
Rotation Varying Descriptors
* Spherical Power Spectrum Representation
Outline
Introduction
Aligning Anisotropic Scales
Shape Matching
Conclusion and Future Work
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
Initial ModelInitial Model
Isotropic ModelIsotropic Model
Rescaling EllipseRescaling Ellipse
DescriptorDescriptor
EigenvaluesEigenvalues New DescriptorNew Descriptor
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
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
Future Work
Address cross-class anisotropy variance Factor out higher order transformations
1 2 3 4
5 6 7 8
1 2 3 4
5 6 7 8
Without Anisotropy Factorization
With Anisotropy Factorization
1 2 3 4
5 6 7 8
Without Anisotropy Factorization
1 2 3 4
5 6 7 8
With Anisotropy Factorization
Future Work
Address cross-class anisotropy variance Factor out higher order transformations
TranslationAnisotropic
Scale ?Isotropic
Scale Rotation
Thank You
FundingNational Science Foundation
Source Code Dan Rockmore and Peter Kostelec
http://www.cs.dartmouth.edu/~geelong/spherehttp://www.cs.dartmouth.edu/~geelong/soft
DatabasesViewpoint Data Labs, Cacheforce, De Espona Infografica
http://www.viewpoint.comhttp://www.cacheforce.comhttp://www.deespona.com
Princeton Shape Matching GroupPatrick Min and Phil Shilane
http://shape.cs.princeton.edu
Thank You