Processing Analysis of Geometric Shapes Introduction Processing and

Processing & Analysis of Geometric Shapes Introduction Processing and Analysis of Geometric Shapes Department of Electrical Engineering – Technion Spring 2010 1

Processing & Analysis of Geometric Shapes Introduction 2 Introduction © Alexander & Michael Bronstein, 2006 -2009 © Michael Bronstein, 2010 tosca. cs. technion. ac. il/book 048921 Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010

Processing & Analysis of Geometric Shapes Introduction Dimensions of media 3

4 Processing & Analysis of Geometric Shapes Introduction Evolution of technology 1880 1900 1920 1940 Analog era 1960 1980 1990 Digital era 2000

Processing & Analysis of Geometric Shapes Introduction Human-machine-interfaces 5

Processing & Analysis of Geometric Shapes Introduction Medical imaging 6

Processing & Analysis of Geometric Shapes Introduction Graphics & animation 7

8 Processing & Analysis of Geometric Shapes Introduction Landscape “HORSE” Computer vision 2 D world Image processing Computer graphics Pattern recognition 3 D world Geometry processing

9 Processing & Analysis of Geometric Shapes Introduction Shapes vs images Geometry Parametrization Sampling Euclidean (flat) Global Uniform Cartesian Non-Euclidean (curved) Local “Uniform” is not well -defined

10 Processing & Analysis of Geometric Shapes Introduction Shapes vs images Representation Deformations Array of pixels Cloud of points, mesh, etc. Rotation, affine, projective, etc. Wealth of non-rigid deformations

Processing & Analysis of Geometric Shapes Introduction Non-rigid world from macro to nano Organs Nanomachines Proteins Microorganisms Animals 11

12 Processing & Analysis of Geometric Shapes Introduction Rock, paper, scissors Rock Scissors Paper

13 Processing & Analysis of Geometric Shapes Introduction Rock, paper, scissors Hands Rock Scissors Paper

Processing & Analysis of Geometric Shapes Introduction Similarity and correspondence Correspondence Similarity Structure Minimum Correspondence possiblequality correspondence = structure distortion similarity (distortion) 14

Processing & Analysis of Geometric Shapes Introduction Similarity and correspondence Similarity Transformation Invariant similarity Structure Minimum possible correspondence distortion 15

16 Processing & Analysis of Geometric Shapes Introduction Transformations Rigid Inelastic Topological Scale Missing data Elastic

17 Processing & Analysis of Geometric Shapes Introduction Topics ? Similarity Correspondence Partial similarity Symmetry Structural similarity Shape Analysis Structure

18 Processing & Analysis of Geometric Shapes Introduction Topics Metric spaces Embeddings “ ” “ Local features ” Geometric words & expressions Shape Representation

19 Processing & Analysis of Geometric Shapes Introduction Topics 1 2 + 1 2 = Calculus of shapes Editing Morphing Filtering Shape Synthesis & Processing

20 Processing & Analysis of Geometric Shapes Introduction Tools Metric and differential geometry Iterative closes point algorithms Fast marching 1 0. 5 0 -0. 5 -1 -1 Multidimensional scaling -0. 5 0 0. 5 1 Convex optimization Spectral methods

Processing & Analysis of Geometric Shapes Introduction Formalities n When? Thu 12: 30 -14: 30 n Where? Meyer 351 n Who? Dr. Michael Bronstein (mbron@cs. technion. ac. il) n Grading 50% homework 50% final project n Web http: //tosca. cs. technion. ac. il/book/course_technion 10. html n Facebook Numerical geometry of non-rigid shapes n Reference A. M. Bronstein, M. M. Bronstein, R. Kimmel, Numerical geometry of non-rigid shapes, Springer 2008 Various papers in the field 21