Introduction to geometry - PROJECT...

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1Processing & Analysis of Geometric Shapes Introduction to Geometry

Introduction to geometryThe Greek way

© 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


Raffaello Santi, School of Athens, Vatican


Distances

Euclidean Manhattan Geodesic


Metric

A function satisfying for all

� Non-negativity:

� Indiscernability: if and only if

� Symmetry:

� Triangle inequality:

is called a metric space

A

B

CAB ≤ BC + AC


Metric balls

Euclidean ball L1 ball L∞∞∞∞ ball

� Open ball:

� Closed ball:


Topology

A set is open if for any there exists such that

� Empty set is open

� Union of any number of open sets is open

� Finite intersection of open sets is open

Collection of all open sets in is called topology

The metric induces a topology through the definition of open sets

Topology can be defined independently of a metric through an axiomatic

definition of an open set

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Connectedness

Connected Disconnected

The space is connected if it cannot be divided into two disjoint nonempty

open sets, and disconnected otherwise

Stronger property: path connectedness

Processing & Analysis of Geometric Shapes Introduction to Geometry

Compactness

The space is compact if any open

covering

has a finite subcovering

For a subset of Euclidean space, compact = closed and bounded (finite

diameter)

InfiniteFinite


Convergence

Topological definition Metric definition

for any open set containing

exists such that for all

for all exists such that

for all

A sequence converges to (denoted ) if


Examples of metrics

Euclidean Path length


Length spaces

Path

Path length , e.g. measured as time it takes to travel along the path

Length metric

is called a length space


Restricted vs. intrinsic metric

Restricted metric Intrinsic metric

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Induced metric

Path length is approximated as sum of lengths of line segments

Can induce another length metric?

of which the path consists, measured using Euclidean metric

The Euclidean metric induces a length metric


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


Convexity

A subset of a metric space is convex if the restricted and

the induced metrics coincide

Non-convex Convex

A convex set contains all the geodesics


Continuity

Topological definition Metric definition

for any open set , preimage

is also open.

for all exists s.t.

for all satisfying

it follows that

A function is called continuous if


Properties of continuous functions

� Map limits to limits, i.e., if , then

� Map open sets to open sets

� Map compact sets to compact sets

� Map connected sets to connected sets

Continuity is a local property: a function can be continuous at one point and

discontinuous at another


Homeomorphisms

A bijective (one-to-one and onto)

continuous function with a continuous

inverse is called a homeomorphism

Homeomorphisms copy topology –

homeomorphic spaces are topologically

equivalent

Torus and cup are homeomorphic

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Topology of Latin alphabet

a b d eo p q

c f h kn r s

i j

l mt u

v w x y z

homeomorphic to homeomorphic to

homeomorphic to


Lipschitz continuity

A function is called Lipschitz continuous if there

exists a constant such that

for all . The smallest possible is called Lipschitz constant

Lipschitz continuous function does not change the distance between any pair

of points by more than times

Lipschitz continuity is a global property

For a differentiable function


Bi-Lipschitz continuity

A function is called bi-Lipschitz continuous if

there exists a constant such that

for all


Examples of Lipschitz continuity

Continuous,

not Lipschitz on [0,1]

Bi-Lipschitz on [0,1]Lipschitz on [0,1]

0 1 0 1 0 1


Isometries

� Two metric spaces and are equivalent if there exists a

distance-preserving map (isometry) satisfying

� Such and are called isometric, denoted

� Isometries copy metric geometries – isometric spaces are equivalent

from the point of view of metric geometry


Euclidean isometries

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Euclidean isometries

Rotation Translation Reflection


Geodesic isometries


Groups

A set with a binary operation is called a group if the

following properties hold:

� Closure: for all

� Associativity: for all

� Identity element: such that for all

� Inverse element: for any , such that


Examples of groups

Integers with addition operation

� Closure: sum of two integers is an integer

� Associativity:

� Identity element:

� Inverse element:

Non-zero real numbers with multiplication operation

� Closure: product of two non-zero real numbers is a non-zero real number

� Associativity:

� Identity element:

� Inverse element:


Self-sometries

A function is called a self-isometry if

for all

Set of all self-isometries of is denoted by

with the function composition operation is a group

� Closure is a self-isometry for all

� Associativity from definition of function composition

� Identity element

� Inverse element (exists because isometries are bijective)


Isometry groups

A

B C

A

B C

A

B CC B AC

B

A

C

B

Cyclic group (reflection)

Permutation group(reflection+rotation)

Trivial group(asymmetric)

A A

BC

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Symmetry in Nature

Snowflake(dihedral)

Butterfly(reflection) Diamond


Almost isometries

� Almost isometry is a map satisfying

� Distortion is the maximum absolute change of the metric

� Almost isometry is not necessarily bijective


Almost isometries


ε ε ε ε-isometries

A function is an

Isometry -isometry

� Distance preserving

� Bijective (one-to-one and on)

� -distance preserving

� -surjective

� Continuous � Not necessarily continuous


Shape

metric space

Similarity

Distance between metric

spaces and .

Invariance

isometry w.r.t.

≈≈≈≈

Shapes as metric spaces


Similarity as metric

Shape space

~Human and monkey

are ε-similar

Human is twice more similar

to monkey than to dog

Two deformations of a

human are equivalent

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Recap

� Metric is a generic notion of distance/dissimilarity

� Metric induces topology

� Continuous maps preserve topology

� Isometric maps preserve metric and topology

� Almost isometric maps preserve neither

� Shapes as metric spaces (metric is invariant structure)

� Shape spaces (metric is a notion of shape similarity)

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