1 Numerical geometry of nonrigid shapes Metric geometry

1 Numerical geometry of non-rigid shapes Metric geometry Lecture 1 © Alexander & Michael Bronstein tosca. cs. technion. ac. il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

Numerical geometry of non-rigid shapes Metric geometry Raffaello Santi, School of Athens, Vatican 2

3 Numerical geometry of non-rigid shapes Metric geometry Distances Euclidean Manhattan Geodesic

4 Numerical geometry of non-rigid shapes Metric geometry Metric A function satisfying for all n Non-negativity: n Indiscernability: if and only if n Symmetry: n Triangle inequality: B is called a metric space A C AB BC + AC

5 Numerical geometry of non-rigid shapes Metric geometry Shapes as metric spaces SHAPE SIMILARITY metric space Distance between metric spaces and INVARIANCE isometry w. r. t. .

Numerical geometry of non-rigid shapes Metric geometry Similarity as metric Two deformations of a human are equivalent ~ Human and monkey are -similar Human is twice more similar to monkey than to dog Shape space 6

Numerical geometry of non-rigid shapes Metric geometry Examples of metrics Euclidean Path length 7

Numerical geometry of non-rigid shapes Metric geometry Length spaces Path length , e. g. measured as time it takes to travel along the path Length metric is called a length space 8

Numerical geometry of non-rigid shapes Metric geometry Restricted vs. intrinsic metric Restricted metric Intrinsic metric 9

Numerical geometry of non-rigid shapes Metric geometry Induced metric Path length is approximated as sum of lengths of line segments of which the path consists, measured using Euclidean metric The Euclidean metric induces a length metric Can induce another length metric? 10

11 Numerical geometry of non-rigid shapes Metric geometry Completeness is called complete if between any there exists a path such that Complete Incomplete In a complete length space, The shortest path realizing the length metric is called a geodesic and the corresponding length metric is called the geodesic metric

12 Numerical geometry of non-rigid shapes Metric geometry Convexity A subset of a metric space is convex if the restricted and the induced metrics coincide Non-convex A convex set contains all the geodesics Convex

13 Numerical geometry of non-rigid shapes Metric geometry Isometries n Two metric spaces and are equivalent if there exists a distance-preserving map (isometry) n Such and satisfying are called isometric, denoted n Isometries copy metric geometries – isometric spaces are equivalent from the point of view of metric geometry

Numerical geometry of non-rigid shapes Metric geometry Euclidean isometries 14

Numerical geometry of non-rigid shapes Metric geometry Euclidean isometries Rotation Translation Reflection 15

Numerical geometry of non-rigid shapes Metric geometry Geodesic isometries 16

17 Numerical geometry of non-rigid shapes Metric geometry Almost isometries n Almost isometry is a map satisfying n Distortion is the maximum absolute change of the metric n Almost isometry is not necessarily bijective

Numerical geometry of non-rigid shapes Metric geometry Almost isometries 18

19 Numerical geometry of non-rigid shapes Metric geometry -isometries A function is an Isometry -isometry n Distance preserving n -distance preserving n Bijective (one-to-one and on) n -surjective n Continuous n Not necessarily continuous