Graphics. Automatic Efficient to compute Smooth Low-distortion Defined for every point Aligns semantic features. Other disciplines

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1 Goal: Find a map between surfaces Blended Intrinsic Maps Vladimir G. Kim Yaron Lipman Thomas Funkhouser Princeton University Goal: Find a map between surfaces Automatic Efficient to compute Smooth Low-distortion Defined for every point Aligns semantic features Applications Graphics Texture transfer Morphing Parametric shape space Other disciplines Paleontology Medicine Praun et al Related Work Gromov-Hausdorff Surface Embedding Möbius Transformations Related Work Gromov-Hausdorff Surface Embedding Möbius 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.,

2 Related Work Gromov-Hausdorff Surface Embedding Möbius Transformations Related Work Gromov-Hausdorff Surface Embedding Möbius Transformations Finds best conformal map, or locally vote for. Lipman and Funkhouser Kim et al Lipman and Funkhouser Our Approach Blended Intrinsic Maps Weighted combination of intrinsic Our Approach Blended Intrinsic Maps Weighted combination of intrinsic 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 The Computational Pipeline Blended Intrinsic Maps Weighted combination of intrinsic 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 Generate consistent set of Find blending weights Blend 2

3 The Computational Pipeline Generate a set of candidate conformal by enumerating triplets of feature points Generate consistent set of Set of candidate Generate a set of candidate conformal by enumerating triplets of feature points Generate a set of candidate conformal by enumerating triplets of feature points Set of candidate Set of candidate Generate a set of candidate conformal by enumerating triplets of feature points Set of candidate Generating Conformal Maps Two triplets of points are used to find a Mobius transformation as a possible mapping. Mobius transformation are performed on a plane, not a general surface. Use Mid-edge uniformization. 3

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

5 Find blocks of low-distortion and mutually similar - should be zero if map i is inconsistent or distorted. Find blocks of low-distortion and mutually similar aps Candidate Ma Eigenanalysis Eigenanalysis aps Candidate Ma Top Eigenvalues aps Candidate Ma First Eigenvalue Correct Maps aps Candidate Ma Eigenanalysis Second Eigenvalue Symmetric Flip Consider 25% top eigenvectors (W) From each take the consistent W i > 0.75 max(w) Group into consistency groups G 1, G n 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. 5

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

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

8 Results Dataset Examples Evaluation Metric Predict the map for every point with a ground truth correspondence Evaluation Metric Comparison Evaluation Metric Predict the map for every point with a ground truth correspondence Measure geodesic distance between prediction and the ground truth 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 0 d <

9 Correspondence Rate Plot 0 d < d < d < d< d< Results Dataset Examples Evaluation Metric Comparison Comparison Gromov-Hausdorf Comparison Correspondence Plot 0 d < d < d < d< d< Heat Kernel Maps 1 Correspondence 2 Correspondences Mobius Voting Lipman and Funkhouser Möbius Voting GMDS Bronstein et al HKM 1 HKM 2 Ovsjannikov et al Conclusion Blending Intrinsic Maps Smooth Efficient to compute Outperforms other methods on benchmark dataset Code and Data: Future work Other (non-conformal) intrinsic : Partial Arbitrary genus surfaces Space of between surfaces Maps consistent across multiple surfaces Metric for comparing surfaces 9

10 Acknowledgments Data Giorgi et al.: SHREC 2007 Watertight Anguelov et al.: SCAPE Bronstein et al.: TOSCA Code: Ovsjanikov et al.: Heat Kernel Map Bronstein et al.: GMDS Funding: NSERC, NSF, AFOSR Intel, Adobe, Google Thank you! 10