Numerical geometry of nonrigid shapes Nonrigid correspondence Alexander

Numerical geometry of non-rigid shapes Non-rigid correspondence Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved 1

Numerical geometry of non-rigid shapes Non-rigid correspondence 2 Correspondence problems n Given two objects and , find a mapping copying features to corresponding similar features n Not always well-defined semantically n Aesthetic rather than geometric considerations often apply n Yet, if objects are sufficiently similar (nearly isometric), correspondence is likely to have a geometric meaning

Numerical geometry of non-rigid shapes Non-rigid correspondence One-dimensional intuition n A closed curve has an arc-length parametrization n Arc-length parametrization is unique up to choice of starting point and direction n Correspondence of curves: bring and to arc-length parametrization n Find the isometry aligning features 3

Numerical geometry of non-rigid shapes Non-rigid correspondence One-dimensional intuition 4

Numerical geometry of non-rigid shapes Non-rigid correspondence 5 Intrinsic parametrization n Bad news: no equivalent of the canonical arc-length parametrization for surfaces n We can still find an intrinsic parametrization n Given and its bending parametrization find a method to compute a such that n Intrinsic parametrization gives a correspondence between shapes

Numerical geometry of non-rigid shapes Non-rigid correspondence Euclidean embedding n Embedding defined up to congruence in n Requires alignment n Inaccuracies due to embedding error 6

7 Numerical geometry of non-rigid shapes Non-rigid correspondence Minimum distortion correspondence: a map n Correspondence relates “similar parts to similar parts” n The correspondence is defined up to self-isometries of and n Isometry groups are trivial, unless the shapes have symmetries BBK, IEEE TVCG, 2007

Numerical geometry of non-rigid shapes Non-rigid correspondence Minimum distortion correspondence n If shapes are symmetric, minimum distortion correspondence is not unique n Intrinsic information is insufficient to select any of them n Adding extrinsic information (e. g. orientation) can resolve ambiguity n Photometric information can be added as well 8

9 Numerical geometry of non-rigid shapes Non-rigid correspondence Texture transfer Problem: transfer texture from to n GMDS provides a natural correspondence between n Define new texture on BBK, IEEE TVCG, 2006 and

Numerical geometry of non-rigid shapes Non-rigid correspondence Reference Transferred texture 10

Numerical geometry of non-rigid shapes Non-rigid correspondence Virtual makeup n A “virtual mask” following the facial deformations BBK, IEEE TVCG, 2006 11

Numerical geometry of non-rigid shapes Non-rigid correspondence Reference Transferred texture 12

Numerical geometry of non-rigid shapes Non-rigid correspondence Reference Transferred texture 13

Numerical geometry of non-rigid shapes Non-rigid correspondence Calculus of non-rigid shapes n Extrinsic geometry can also be manipulated n Correspondence makes affine combination of shapes well-defined n Establishes a calculus of shapes BBK, IEEE TVCG, 2006 14

Numerical geometry of non-rigid shapes Non-rigid correspondence Calculus of non-rigid shapes Interpolation Extrapolation n Abstract manifold of shape deformations n Shape animation: trajectory n Minimum-distortion correspondence allows creating a (locally) linear space, in which shapes are represented as vectors BBK, IEEE TVCG, 2006 15

Numerical geometry of non-rigid shapes Non-rigid correspondence Calculus of non-rigid shapes CORRESPONDENCE Extrinsic geometry Texture n Extrinsic coordinates and texture interpolation BBK, IEEE TVCG, 2006 16

17 Numerical geometry of non-rigid shapes Non-rigid correspondence Interpolation INTERPOLATED 0 0. 25 0. 5 FRAMES 0. 75 1 n Temporal super-resolution: increase frame rate of 3 D video by adding interpolated frames n Interpolation of geometry and texture BBK, IEEE TVCG, 2006

18 Numerical geometry of non-rigid shapes Non-rigid correspondence Extrapolation NEUTRAL EXPRESSION EXAGGERATED EXPRESSION 1 1. 5 0 n Expression exaggeration: synthesize new expressions using a non-convex combination n Interpolation of geometry and texture BBK, IEEE TVCG, 2006

Numerical geometry of non-rigid shapes Non-rigid correspondence NEARLY-ISOMETRIC NON-ISOMETRIC 19

Numerical geometry of non-rigid shapes Non-rigid correspondence Texture substitution ALICE BBK, IEEE TVCG, 2006 BOB ALICE’S TEXTURE BOB’S GEOMETRY 20

21 Numerical geometry of non-rigid shapes Non-rigid correspondence Metamorphing SOURCE 0 TARGET 0. 25 0. 75 n Convex combination between two different objects n Morphing of geometry and texture BBK, IEEE TVCG, 2006 1

Numerical geometry of non-rigid shapes Non-rigid correspondence Conclusions so far… n GMDS can be used to find non-rigid correspondence n Correspondence allows to establish calculus of shapes 22