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

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

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

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

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

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

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

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

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

10 Numerical geometry of non-rigid shapes Non-rigid correspondence ReferenceTransferred texture

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

12 Numerical geometry of non-rigid shapes Non-rigid correspondence ReferenceTransferred texture 12 Numerical geometry of non-rigid shapes Non-rigid correspondence

13 Numerical geometry of non-rigid shapes Non-rigid correspondence 13 ReferenceTransferred texture Numerical geometry of non-rigid shapes Non-rigid correspondence

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

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

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

17 Numerical geometry of non-rigid shapes Non-rigid correspondence Interpolation I N T E R P O L A T E D F R A M E S Temporal super-resolution: increase frame rate of 3D video by adding interpolated frames Interpolation of geometry and texture BBK, IEEE TVCG, 2006

18 Numerical geometry of non-rigid shapes Non-rigid correspondence Extrapolation Expression exaggeration: synthesize new expressions using a non-convex combination Interpolation of geometry and texture NEUTRALEXPRESSIONEXAGGERATED EXPRESSION BBK, IEEE TVCG, 2006

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

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

21 Numerical geometry of non-rigid shapes Non-rigid correspondence Metamorphing Convex combination between two different objects Morphing of geometry and texture SOURCETARGET BBK, IEEE TVCG, 2006

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