1Processing & Analysis of Geometric Shapes Introduction

Processing and Analysisof Geometric Shapes

Department of Electrical Engineering – TechnionSpring 2010

2Processing & Analysis of Geometric Shapes Introduction

Introduction

© Alexander & Michael Bronstein, 2006-2009© Michael Bronstein, 2010tosca.cs.technion.ac.il/book

048921 Advanced topics in visionProcessing and Analysis of Geometric Shapes

EE Technion, Spring 2010

3Processing & Analysis of Geometric Shapes Introduction

Dimensions of media

4Processing & Analysis of Geometric Shapes Introduction

1880 1900 1920 1960 1980 20001940 1990

Evolution of technology

Analog era Digital era

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Human-machine-interfaces

6Processing & Analysis of Geometric Shapes Introduction

Medical imaging

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Graphics & animation

8Processing & Analysis of Geometric Shapes Introduction

Landscape

“HORSE”

Image processing Geometry processing

Pattern recognition

Computervision

Computergraphics2D world 3D world

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Shapes vs images

Geometry SamplingParametrization

Euclidean (flat)

Non-Euclidean (curved)

Global

Local “Uniform” is not well-defined

Uniform Cartesian

10Processing & Analysis of Geometric Shapes Introduction

Shapes vs images

Representation Deformations

Array of pixels

Cloud of points, mesh, etc, etc.

Rotation, affine, projective, etc.

Wealth of non-rigid deformations

11Processing & Analysis of Geometric Shapes Introduction

Non-rigid world from macro to nano

Animals

Organs

Micro-organisms

ProteinsNano-machines

12Processing & Analysis of Geometric Shapes Introduction

Rock

Paper

Scissors

Rock, paper, scissors

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Hands

Rock

Paper

Scissors

Rock, paper, scissors

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Similarity and correspondence

Structure Structure

Correspondence

Correspondence quality = structure similarity

(distortion)

Similarity

Minimum possible correspondence distortion

15Processing & Analysis of Geometric Shapes Introduction

Similarity and correspondence

Structure Structure

Similarity

Minimum possible correspondence distortion

Transformation

Invariant similarity

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Elastic

Rigid

Transformations

Inelastic Topological

Scale

Missing data

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Similarity Partial similarity

Topics

?

Symmetry

Structure Structural similarity

Correspondence

Shape Analysis

18Processing & Analysis of Geometric Shapes Introduction

Topics

“ ”

“ ”

Metric spaces Embeddings

Local features Geometric words & expressions

Shape Representation

19Processing & Analysis of Geometric Shapes Introduction

Topics

Calculus of shapes

=+1 2

1 2

Morphing

Editing Filtering

Shape Synthesis & Processing

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Tools

Metric and differential geometry

Fast marching

Multidimensional scaling

Iterative closes point algorithms

Spectral methodsConvex optimization

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21Processing & Analysis of Geometric Shapes Introduction

Formalities

When? Thu 12:30-14:30

Where? Meyer 351

Who? Dr. Michael Bronstein (mbron@cs.technion.ac.il)

Grading 50% homework

50% final project

Web http://tosca.cs.technion.ac.il/book/course_technion10.html

Facebook Numerical geometry of non-rigid shapes

Reference A. M. Bronstein, M. M. Bronstein,

R. Kimmel, Numerical geometry of non-rigid

shapes, Springer 2008

Various papers in the field