blended intrinsic mapsdcor/graphics/adv-slides/blended... · 2012. 5. 10. · generating generating...

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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



Maps surface points to feature space (HK), and finds NN. Ovsjanikov et al., 2010

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Finds “best” conformal map, or locally vote for maps.

Lipman and Funkhouser. 2009



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









The Computational PipelineThe Computational Pipeline

Generate consistent set of maps

Find blending weights

Blend maps

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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

…


…



…



…


•• 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


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


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,


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


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


- should be zero if map i is inconsistent or distorted.

Candidate Maps

Can

dida

te M

aps


aps

Find blocks of low-distortion and mutually similar maps

Can

dida

te M

a

Candidate Maps

Candidate Maps

Can

dida

te M

aps


aps Top Eigenvalues

Eigenanalysis

Can

dida

te M

a

Candidate Maps

Eigenanalysis


aps FirstEigenvalue

Correct Maps

Can

dida

te M

a

Candidate Maps

Eigenanalysis


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


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


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


Measure geodesic distance between prediction and the ground truth


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!

blended intrinsic mapsdcor/graphics/adv-slides/blended... · 2012. 5. 10. · generating generating...

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