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Goal: Find a map between surfaces Goal: Find a map between surfaces Blended Intrinsic Maps Blended Intrinsic Maps Vladimir G. Kim Vladimir G. Kim Yaron Lipman Yaron Lipman Thomas Funkhouser Thomas Funkhouser Princeton University Goal:

  1. Goal: Find a map between surfaces Goal: Find a map between surfaces 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 Goal: Find a map between surfaces surfaces Applications Applications � Automatic � Graphics � Efficient to compute � Texture transfer � Morphing � Smooth � Parametric shape space � Parametric shape space � Low-distortion � Defined for every point � Other disciplines � Aligns semantic features � Paleontology Praun et al. 2001 � Medicine Related Work Related Work Related Work Related Work � Gromov-Hausdorff � Gromov-Hausdorff � Surface Embedding � Surface Embedding � Möbius Transformations � Möbius Transformations � Finds a correspondence � Maps surface points to that minimizes the feature space (HK), and Ovsjanikov et al., 2010 Hausdorff distance. finds NN. Bronstein et al., 2006 1

  2. Related Work Related Work Related Work Related Work � Gromov-Hausdorff � Gromov-Hausdorff � Surface Embedding � Surface Embedding � Möbius Transformations � Möbius Transformations � Finds “best” conformal map, or locally vote for maps. Lipman and Funkhouser. 2009 Lipman and Funkhouser. 2009 Kim et al. 2010 Our Approach Our Approach Our Approach Our Approach � Blended Intrinsic Maps � Blended Intrinsic Maps � Weighted combination � Weighted combination of intrinsic maps of intrinsic maps Distortion of m 1 Distortion of m 2 Distortion of m 3 Distortion of m 1 Distortion of m 2 Distortion of m 3 Blending Weights for m 1, m 2 , and m 3 Distortion of the Blended Map Blending Weights for m 1, m 2 , and m 3 Distortion of the Blended Map Our Approach Our Approach The Computational Pipeline The Computational Pipeline � Blended Intrinsic Maps � Weighted combination of intrinsic maps Distortion of m 1 Distortion of m 2 Distortion of m 3 Generate Find Blend consistent blending maps set of maps weights Blending Weights for m 1, m 2 , and m 3 Distortion of the Blended Map 2

  3. The Computational Pipeline The Computational Pipeline Generating Consistent Maps Generating Consistent Maps Generate a set of candidate conformal maps by enumerating triplets of feature points … Generate Set of consistent candidate maps set of maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Generate a set of candidate conformal maps Generate a set of candidate conformal maps by enumerating triplets of feature points by enumerating triplets of feature points … … Set of Set of candidate candidate maps maps Generating Consistent Maps Generating Consistent Maps Generating Generating Conformal Conformal Maps Maps Generate a set of candidate conformal maps • Two triplets of points are used to find a Two triplets of points are used to find a by enumerating triplets of feature points Mobius Mobius transformation as a possible transformation as a possible mapping. mapping. • 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. . Set of candidate maps 3

  4. Generating Generating Conformal Conformal Maps Maps Generating Conformal Generating Conformal Maps Maps • Mid Mid- -edge edge uniformization uniformization takes the “mid takes the “mid- - • The triangles” of the original surface. triangles” of the original surface. The Mobius Mobius transformation is: transformation is: • Flattens them out onto the Flattens them out onto the comlex comlex plane plane • When given When given 2 2 corresponding triplets corresponding triplets using harmonic functions. using harmonic functions. using harmonic functions using harmonic functions of points of points y,z y,z , parameters , parameters a,b,c,d a,b,c,d are are • The mid The mid- -edge mesh is easier to edge mesh is easier to uniquely defined. uniquely defined. flatten since it is “less tight”. flatten since it is “less tight”. • Allows to approximate a mapping Allows to approximate a mapping of the original surface. of the original surface. Generating Generating Conformal Conformal Maps Maps Generating Generating Conformal Conformal Maps Maps • A • Mobius A Mobius Mobius transformation is transformation is Mobius transformation are equivalent to transformation are equivalent to conformal (angle preserving). conformal (angle preserving). Stereographic projections: Stereographic projections: • Isometries Isometries are a subset of all are a subset of all – Map from the plane to the sphere, Map from the plane to the sphere, p p p p p p , , conformal maps conformal maps. conformal maps conformal maps. – Rotate & move the sphere, Rotate & move the sphere, • Genus Genus- -zero surfaces (surfaces zero surfaces (surfaces without “holes”) can all be without “holes”) can all be – Map back to the plane. Map back to the plane. conformally conformally mapped to the mapped to the sphere. sphere. Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Define a matrix S where every entry (i,j) indicates the Find consistent set(s) of candidate maps distortion of m i and m j and their pairwise similarity S i,j Candidate Maps … Set of consistent candidate maps Candidate Maps 4

  5. Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Find blocks of low-distortion and mutually similar maps Find blocks of low-distortion and mutually similar maps - should be zero if map i is inconsistent or distorted. aps Candidate Ma Candidate Maps Candidate Maps Candidate Maps Candidate Maps Candidate Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Eigenanalysis Eigenanalysis Correct Maps Top First aps aps Eigenvalues Eigenvalue Candidate Ma Candidate Ma Candidate Maps Candidate Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Generating Consistent Maps Eigenanalysis • Consider 25% top eigenvectors (W) • From each take the consistent maps W i > 0.75 max(W) • Group maps into consistency groups G 1 ,…G n Symmetric Flip Second • Maps considered consistent if they have no conflicting aps Eigenvalue Candidate Ma correspondences: • Seems to me like this might make problems • Very sensitive to the order of insertion • Probably not affected due to specific order Candidate Maps • Maybe could be solved by inserting correspondence consistency into S matrix. 5

  6. Generating Consistent Maps Generating Consistent Maps The Computational Pipeline The Computational Pipeline • Choose best group G j : • Calculate the blending-map for each group: • Find map that minimizes the over-all distortion: Find blending weights Finding Blending Weights Finding Blending Weights Finding Blending Weights Finding Blending Weights � For every point p � For every point p � Compute a weight of � Compute a weight of each map m i at p each map m i at p � We model the weight with deviation from isometry � Area distortion for conformal maps Candidate Map Candidate Map Blending Weight The Computational Pipeline The Computational Pipeline Blending Maps Blending Maps � Input for each point p: � An image m i (p) after applying each map m i � A blending weight for A blending weight for each map Blending Weights � Output for each point: � Weighted geodesic centroid of { m i (p) } Blend maps Blended Map 6

  7. Blending Maps Blending Maps Results Results � Input for each point p: � Dataset � An image m i (p) after applying each map m i � Examples � A blending weight for A blending weight for each map Blending Weights � Evaluation Metric � Output for each point: centroid � Weighted geodesic � Comparison centroid of { m i (p) } Blended Map Dataset Dataset Results Results � 371 meshes � Dataset � Ground Truth: � Examples � Evaluation Metric � Comparison Examples Examples Failures Failures Stretched Symmetric flip 7

  8. Results Results Evaluation Metric Evaluation Metric � Predict the map for every � Dataset point with a ground truth correspondence � Examples � Evaluation Metric � Comparison Evaluation Metric Evaluation Metric Evaluation Metric Evaluation Metric � Predict the map for every � Predict the map for every point with a ground truth point with a ground truth correspondence correspondence � Measure geodesic distance � Measure geodesic distance between prediction and the between prediction and the ground truth 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 8

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