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Blended Intrinsic MapsBlended Intrinsic MapsVladimir G.
KimVladimir G. Kim
Yaron LipmanYaron Lipman
Thomas FunkhouserThomas Funkhouser
Princeton University
Goal: Find a map between surfacesGoal: Find a map between
surfaces
Goal: Find a map between Goal: Find a map between
surfacessurfaces
AutomaticEfficient to computeSmoothLow-distortionDefined for
every pointAligns semantic features
ApplicationsApplications
GraphicsTexture transferMorphingParametric shape spaceParametric
shape space
Other disciplinesPaleontologyMedicine
Praun et al. 2001
Related WorkRelated Work
Gromov-HausdorffSurface EmbeddingMöbius Transformations
Finds a correspondence that minimizes the Hausdorff
distance.
Bronstein et al., 2006
Related WorkRelated Work
Gromov-HausdorffSurface EmbeddingMöbius Transformations
Maps surface points to feature space (HK), and finds NN.
Ovsjanikov et al., 2010
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Gromov-HausdorffSurface EmbeddingMöbius Transformations
Related WorkRelated Work
Finds “best” conformal map, or locally vote for maps.
Lipman and Funkhouser. 2009
Gromov-HausdorffSurface EmbeddingMöbius Transformations
Related WorkRelated Work
Lipman and Funkhouser. 2009Kim et al. 2010
Our ApproachOur Approach
Blended Intrinsic MapsWeighted combination of intrinsic maps
Distortion of m1 Distortion of m2 Distortion of m3
Blending Weights for m1, m2, and m3 Distortion of the Blended
Map
Our ApproachOur Approach
Blended Intrinsic MapsWeighted combination of intrinsic maps
Distortion of m1 Distortion of m2 Distortion of m3
Blending Weights for m1, m2, and m3 Distortion of the Blended
Map
Our ApproachOur Approach
Blended Intrinsic MapsWeighted combination of intrinsic maps
Distortion of m1 Distortion of m2 Distortion of m3
Blending Weights for m1, m2, and m3 Distortion of the Blended
Map
The Computational PipelineThe Computational Pipeline
Generate consistent set of maps
Find blending weights
Blend maps
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The Computational PipelineThe Computational Pipeline
Generate consistent set of maps
Generating Consistent MapsGenerating Consistent MapsGenerate a
set of candidate conformal maps by enumerating triplets of feature
points
Set of candidate maps
…
Generating Consistent MapsGenerating Consistent MapsGenerate a
set of candidate conformal maps by enumerating triplets of feature
points
…
Set of candidate maps
Generating Consistent MapsGenerating Consistent MapsGenerate a
set of candidate conformal maps by enumerating triplets of feature
points
…
Set of candidate maps
Generating Consistent MapsGenerating Consistent MapsGenerate a
set of candidate conformal maps by enumerating triplets of feature
points
…
Set of candidate maps
•• Two triplets of points are used to find a Two triplets of
points are used to find a MobiusMobius transformation as a possible
transformation as a possible mapping.mapping.
Generating Generating Conformal Conformal MapsMaps
•• MobiusMobius transformation are performed on a transformation
are performed on a plane, not a general surface.plane, not a
general surface.
•• Use MidUse Mid--edge edge uniformizationuniformization. .
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•• MidMid--edge edge uniformizationuniformization takes the
“midtakes the “mid--triangles” of the original surface.triangles”
of the original surface.
•• Flattens them out onto the Flattens them out onto the
comlexcomlex planeplaneusing harmonic functionsusing harmonic
functions
Generating Generating Conformal Conformal MapsMaps
using harmonic functions.using harmonic functions.•• The midThe
mid--edge mesh is easier toedge mesh is easier to
flatten since it is “less tight”.flatten since it is “less
tight”.•• Allows to approximate a mappingAllows to approximate a
mapping
of the original surface.of the original surface.
•• The The MobiusMobius transformation is:transformation is:••
When given When given 2 2 corresponding tripletscorresponding
triplets
Generating Generating Conformal Conformal MapsMaps
of points of points y,zy,z, parameters , parameters
a,b,c,da,b,c,d are are uniquely defined.uniquely defined.
•• MobiusMobius transformation are equivalent totransformation
are equivalent toStereographic projections:Stereographic
projections:–– Map from the plane to the sphere,Map from the plane
to the sphere,
Generating Generating Conformal Conformal MapsMaps
p p p ,p p p ,
–– Rotate & move the sphere,Rotate & move the sphere,––
Map back to the plane.Map back to the plane.
•• A A MobiusMobius transformation is transformation is
conformal (angle preserving).conformal (angle preserving).
•• IsometriesIsometries are a subset of all are a subset of all
conformal mapsconformal maps
Generating Generating Conformal Conformal MapsMaps
conformal maps.conformal maps.•• GenusGenus--zero surfaces
(surfaces zero surfaces (surfaces
without “holes”) can all be without “holes”) can all be
conformallyconformally mapped to the mapped to the
sphere.sphere.
Generating Consistent MapsGenerating Consistent MapsFind
consistent set(s) of candidate maps
…Set of consistentcandidate maps
Define a matrix S where every entry (i,j) indicates the
distortion of mi and mj and their pairwise similarity Si,j
Generating Consistent MapsGenerating Consistent Maps
Candidate Maps
Can
dida
te M
aps
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Find blocks of low-distortion and mutually similar maps
Generating Consistent MapsGenerating Consistent Maps
- should be zero if map i is inconsistent or distorted.
Candidate Maps
Can
dida
te M
aps
Generating Consistent MapsGenerating Consistent Maps
aps
Find blocks of low-distortion and mutually similar maps
Can
dida
te M
a
Candidate Maps
Candidate Maps
Can
dida
te M
aps
Generating Consistent MapsGenerating Consistent Maps
aps Top Eigenvalues
Eigenanalysis
Can
dida
te M
a
Candidate Maps
Eigenanalysis
Generating Consistent MapsGenerating Consistent Maps
aps FirstEigenvalue
Correct Maps
Can
dida
te M
a
Candidate Maps
Eigenanalysis
Generating Consistent MapsGenerating Consistent Maps
aps SecondEigenvalue
Symmetric Flip
Can
dida
te M
a
Candidate Maps
• Consider 25% top eigenvectors (W)• From each take the
consistent maps Wi > 0.75 max(W)• Group maps into consistency
groups G1,…Gn• Maps considered consistent if they have no
conflicting
Generating Consistent MapsGenerating Consistent Maps
correspondences: • Seems to me like this might make problems•
Very sensitive to the order of insertion• Probably not affected due
to specific order• Maybe could be solved by inserting
correspondence consistency into S matrix.
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• Choose best group Gj:• Calculate the blending-map for each
group:• Find map that minimizes the over-all distortion:
Generating Consistent MapsGenerating Consistent Maps The
Computational PipelineThe Computational Pipeline
Find blending weights
Finding Blending WeightsFinding Blending Weights
For every point pCompute a weight of each map mi at p
Candidate Map
Finding Blending WeightsFinding Blending Weights
For every point pCompute a weight of each map mi at p
We model the weight with deviation from isometry
Area distortion for conformal maps Candidate Map Blending
Weight
The Computational PipelineThe Computational Pipeline
Blend maps
Blending MapsBlending Maps
Input for each point p:An image mi(p) after applying each map
miA blending weight for
Blending Weights
Blended Map
A blending weight for each map
Output for each point:Weighted geodesic centroid of { mi(p)
}
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Blending MapsBlending Maps
Input for each point p:An image mi(p) after applying each map
miA blending weight for
Blending Weights
Blended Map
centroid
A blending weight for each map
Output for each point:Weighted geodesic centroid of { mi(p)
}
ResultsResults
Dataset
Examples
Evaluation Metric
Comparison
DatasetDataset
371 meshesGround Truth:
ResultsResults
Dataset
Examples
Evaluation Metric
Comparison
ExamplesExamples FailuresFailures
Stretched Symmetric flip
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ResultsResults
Dataset
Examples
Evaluation Metric
Comparison
Evaluation MetricEvaluation MetricPredict the map for every
point with a ground truth correspondence
Evaluation MetricEvaluation MetricPredict the map for every
point with a ground truth correspondence
Measure geodesic distance between prediction and the ground
truth
Evaluation MetricEvaluation MetricPredict the map for every
point with a ground truth correspondence
Measure geodesic distance between prediction and the ground
truth
Record fraction of points mapped within geodesic error
Correspondence Rate PlotCorrespondence Rate Plot Correspondence
Correspondence Rate Rate PlotPlot0 ≤ d < 0.05
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Correspondence Correspondence Rate Rate PlotPlot0 ≤ d < 0.05
0.2 ≤ d < ∞0.05 ≤ d < 0.1 0.1 ≤ d < 0.15 0.15 ≤ d <
0.2
ResultsResults
Dataset
Examples
Evaluation Metric
Comparison
ComparisonComparison
Gromov-Hausdorf
Heat Kernel Maps1 Correspondence
Mobius VotingLipman and Funkhouser. 2009
1 Correspondence2 Correspondences
Möbius Voting GMDSBronstein et al. 2006
Ovsjannikov et al. 2010HKM 1 HKM 2
Comparison Correspondence PlotComparison Correspondence Plot0 ≤
d < 0.05 0.2 ≤ d < ∞0.05 ≤ d < 0.1 0.1 ≤ d < 0.15 0.15
≤ d < 0.2
ConclusionConclusion
Blending Intrinsic MapsSmoothEfficient to computeOutperforms
other methodsOutperforms other methodson benchmark dataset
Code and
Data:http://www.cs.princeton.edu/~vk/CorrsCode/http://www.cs.princeton.edu/~vk/CorrsCode/
http://www.cs.princeton.edu/~vk/CorrsCode/Benchmark/http://www.cs.princeton.edu/~vk/CorrsCode/Benchmark/
Future workFuture work
Other (non-conformal) intrinsic maps:Partial mapsArbitrary genus
surfaces
Space of maps between surfacesMaps consistent across multiple
surfacesMetric for comparing surfaces
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AcknowledgmentsAcknowledgmentsData
Giorgi et al.: SHREC 2007 WatertightAnguelov et al.:
SCAPEBronstein et al.: TOSCA
C dCode:Ovsjanikov et al.: Heat Kernel MapBronstein et al.:
GMDS
Funding: NSERC, NSF, AFOSRIntel, Adobe, Google
Thank you!