1 Related Work Related Work Related Work Related Work - - PDF document

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Blended Intrinsic Maps Blended Intrinsic Maps

Vladimir G. Kim Vladimir G. Kim Yaron Lipman Yaron Lipman Thomas Funkhouser Thomas Funkhouser

Princeton University

Goal: Find a map between surfaces Goal: Find a map between surfaces Goal: Find a map between Goal: Find a map between surfaces surfaces Automatic Efficient to compute Smooth Low-distortion Defined for every point Aligns semantic features Applications Applications Graphics

Texture transfer Morphing Parametric shape space Parametric shape space

Other disciplines

Paleontology Medicine

Praun et al. 2001

Related Work Related Work Gromov-Hausdorff Surface Embedding Möbius Transformations Finds a correspondence

that minimizes the Hausdorff distance.

Bronstein et al., 2006

Related Work Related Work Gromov-Hausdorff Surface Embedding Möbius Transformations Maps surface points to

feature space (HK), and finds NN.

Ovsjanikov et al., 2010

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Gromov-Hausdorff Surface Embedding Möbius Transformations Related Work Related Work Finds “best” conformal

map, or locally vote for maps.

Lipman and Funkhouser. 2009

Gromov-Hausdorff Surface Embedding Möbius Transformations Related Work Related Work

Lipman and Funkhouser. 2009 Kim et al. 2010

Our Approach Our Approach Blended Intrinsic Maps

Weighted combination

• f intrinsic maps

Distortion of m1 Distortion of m2 Distortion of m3 Blending Weights for m1, m2, and m3 Distortion of the Blended Map

Our Approach Our Approach Blended Intrinsic Maps

Weighted combination

• f intrinsic maps

Distortion of m1 Distortion of m2 Distortion of m3 Blending Weights for m1, m2, and m3 Distortion of the Blended Map

Our Approach Our Approach Blended Intrinsic Maps

Weighted combination

• f 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 Pipeline The Computational Pipeline

Generate consistent set of maps Find blending weights Blend maps

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The Computational Pipeline The Computational Pipeline

Generate consistent set of maps

Generating Consistent Maps Generating Consistent Maps

Generate a set of candidate conformal maps by enumerating triplets of feature points Set of candidate maps …

Generating Consistent Maps Generating Consistent Maps

Generate a set of candidate conformal maps by enumerating triplets of feature points … Set of candidate maps

Generating Consistent Maps Generating Consistent Maps

Generate a set of candidate conformal maps by enumerating triplets of feature points … Set of candidate maps

Generating Consistent Maps Generating Consistent Maps

Generate 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 Mobius Mobius transformation as a possible transformation as a possible mapping. mapping. Generating Generating Conformal Conformal Maps Maps

• Mobius

Mobius transformation are performed on a transformation are performed on a plane, not a general surface. plane, not a general surface.

• Use Mid

Use Mid-

• edge

edge uniformization uniformization. .

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• Mid

Mid-

• edge

edge uniformization uniformization takes the “mid takes the “mid-

• triangles” of the original surface.

triangles” of the original surface.

• Flattens them out onto the

Flattens them out onto the comlex comlex plane plane using harmonic functions using harmonic functions Generating Generating Conformal Conformal Maps Maps using harmonic functions. using harmonic functions.

• The mid

The mid-

• edge mesh is easier to

edge mesh is easier to flatten since it is “less tight”. flatten since it is “less tight”.

• Allows to approximate a mapping

Allows to approximate a mapping

• f the original surface.
• f the original surface.
• The

The Mobius Mobius transformation is: transformation is:

• When given

When given 2 2 corresponding triplets corresponding triplets Generating Generating Conformal Conformal Maps Maps

• f points
• f points y,z

y,z, parameters , parameters a,b,c,d a,b,c,d are are uniquely defined. uniquely defined.

• Mobius

Mobius transformation are equivalent to transformation are equivalent to Stereographic projections: Stereographic projections:

– Map from the plane to the sphere,

Map from the plane to the sphere,

Generating Generating Conformal Conformal Maps Maps

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 Mobius Mobius transformation is transformation is conformal (angle preserving). conformal (angle preserving).

• Isometries

Isometries are a subset of all are a subset of all conformal maps conformal maps Generating Generating Conformal Conformal Maps Maps conformal maps. conformal maps.

• Genus

Genus-

• zero surfaces (surfaces

zero surfaces (surfaces without “holes”) can all be without “holes”) can all be conformally conformally mapped to the mapped to the sphere. sphere. Generating Consistent Maps Generating Consistent Maps

Find consistent set(s) of candidate maps … Set of consistent candidate 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 Maps Generating Consistent Maps

Candidate Maps Candidate Maps

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Find blocks of low-distortion and mutually similar maps

Generating Consistent Maps Generating Consistent Maps

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

Candidate Maps Candidate Maps

Generating Consistent Maps Generating Consistent Maps

aps

Find blocks of low-distortion and mutually similar maps

Candidate Ma Candidate Maps Candidate Maps Candidate Maps

Generating Consistent Maps Generating Consistent Maps

aps Top Eigenvalues

Eigenanalysis

Candidate Ma Candidate Maps

Eigenanalysis

Generating Consistent Maps Generating Consistent Maps

aps First Eigenvalue Correct Maps Candidate Ma Candidate Maps

Eigenanalysis

Generating Consistent Maps Generating Consistent Maps

aps Second Eigenvalue Symmetric Flip Candidate Ma 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 Maps Generating 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 Maps Generating Consistent Maps The Computational Pipeline The Computational Pipeline

Find blending weights

Finding Blending Weights Finding Blending Weights

For every point p

Compute a weight of

each map mi at p Candidate Map

Finding Blending Weights Finding Blending Weights

For every point p

Compute 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 Pipeline The Computational Pipeline

Blend maps

Blending Maps Blending Maps Input for each point p:

An image mi(p) after

applying each map mi

A 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 Maps Blending Maps Input for each point p:

An image mi(p) after

applying each map mi

A blending weight for

Blending Weights Blended Map

centroid

A blending weight for each map

Output for each point:

Weighted geodesic

centroid of { mi(p) }

Results Results Dataset Examples Evaluation Metric Comparison Dataset Dataset 371 meshes Ground Truth: Results Results Dataset Examples Evaluation Metric Comparison Examples Examples Failures Failures

Stretched Symmetric flip

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Results Results Dataset Examples Evaluation Metric Comparison Evaluation Metric Evaluation Metric Predict the map for every

point with a ground truth correspondence

Evaluation Metric Evaluation Metric Predict the map for every

point with a ground truth correspondence

Measure geodesic distance

between prediction and the ground truth

Evaluation Metric Evaluation Metric Predict 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 Plot Correspondence Rate Plot Correspondence Correspondence Rate Rate Plot Plot

0 ≤ d < 0.05

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Correspondence Correspondence Rate Rate Plot Plot

0 ≤ d < 0.05 0.2 ≤ d < ∞ 0.05 ≤ d < 0.1 0.1 ≤ d < 0.15 0.15 ≤ d < 0.2

Results Results Dataset Examples Evaluation Metric Comparison Comparison Comparison Gromov-Hausdorf Heat Kernel Maps

1 Correspondence

Mobius Voting Lipman and Funkhouser. 2009

1 Correspondence 2 Correspondences

Möbius Voting

GMDS Bronstein et al. 2006 Ovsjannikov et al. 2010 HKM 1 HKM 2

Comparison Correspondence Plot Comparison Correspondence Plot

0 ≤ d < 0.05 0.2 ≤ d < ∞ 0.05 ≤ d < 0.1 0.1 ≤ d < 0.15 0.15 ≤ d < 0.2

Conclusion Conclusion Blending Intrinsic Maps

Smooth Efficient to compute Outperforms other methods

Outperforms other methods

• n 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 work Future work Other (non-conformal) intrinsic maps:

Partial maps Arbitrary genus surfaces

Space of maps between surfaces

Maps consistent across multiple surfaces Metric for comparing surfaces

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Acknowledgments Acknowledgments

Data

Giorgi et al.: SHREC 2007 Watertight Anguelov et al.: SCAPE Bronstein et al.: TOSCA

C d

Code:

Ovsjanikov et al.: Heat Kernel Map Bronstein et al.: GMDS

Funding:

NSERC, NSF, AFOSR Intel, Adobe, Google

Thank you!