1 Numerical geometry of non-rigid shapes Introduction Introduction Alexander Bronstein, Michael Bronstein © 2008 All rights reserved. Web: tosca.cs.technion.ac.il - Do you see that cloud, that’s almost in shape like a camel? - By the mass, and ’t is like a camel, indeed. - Methinks, it is like a weasel. - It is backed like a weasel. - Or, like a whale? - Very like a whale. Shakespeare, Hamlet

2 Numerical geometry of non-rigid shapes Introduction Welcome to non-rigid world

3 Numerical geometry of non-rigid shapes Introduction Non-rigid shapes everywhere Articulated shapes Volumetric medical data Computer graphics models Two-dimensional shapes

4 Numerical geometry of non-rigid shapes Introduction Rock Paper Scissors Rock-Scissors-Paper Roccia, carta, forbici

5 Numerical geometry of non-rigid shapes Introduction Hands Rock Paper Scissors Rock-Scissors-Paper

6 Numerical geometry of non-rigid shapes Introduction Deformation-invariant similarity Define a class of deformations Find properties of the shape which are invariant under the class of deformations and discriminative (uniquely describe the shape) Define a shape distance based on these properties

7 Numerical geometry of non-rigid shapes Introduction Invariance Rigid motion Elastic deformation Inelastic deformation

8 Numerical geometry of non-rigid shapes Introduction Similarity and correspondence CORRESPONDENCESIMILARITY

9 Numerical geometry of non-rigid shapes Introduction Analysis and synthesis Elephant image: courtesy M. Kilian and H. Pottmann SYNTHESISANALYSIS

10 Numerical geometry of non-rigid shapes Introduction Landscape “HORSE” Image processingGeometry processing Pattern recognition Computer vision Computer graphics 2D world3D world

11 Numerical geometry of non-rigid shapes Introduction Thorny path Via dolorosa

12 Numerical geometry of non-rigid shapes Introduction Course in a nutshell Analysis and synthesis of non-rigid shapes Archetype problems: shape similarity and correspondence Metric geometry as a common denominator Tools from geometry, algebra, optimization, numerical analysis, statistics, and data analysis. Practical numerical methods MATLAB tutorials Applications in computer vision, pattern recognition, computer graphics, and geometry processing

13 Numerical geometry of non-rigid shapes Introduction Material Excerpts from the book On paper Online tosca.cs.technion.ac.il/book Problems Solutions Lecture slides Software Links Tutorials Data