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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
2Processing & Analysis of Geometric Shapes Introduction to Geometry
Raffaello Santi, School of Athens, Vatican
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Distances
Euclidean Manhattan Geodesic
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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
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Metric balls
Euclidean ball L1 ball L∞∞∞∞ ball
� Open ball:
� Closed ball:
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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
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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
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Examples of metrics
Euclidean Path length
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Length spaces
Path
Path length , e.g. measured as time it takes to travel along the path
Length metric
is called a length space
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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
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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
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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
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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
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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
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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
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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
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Bi-Lipschitz continuity
A function is called bi-Lipschitz continuous if
there exists a constant such that
for all
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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
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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
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Euclidean isometries
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Euclidean isometries
Rotation Translation Reflection
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Geodesic isometries
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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
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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:
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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)
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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
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Almost isometries
� Almost isometry is a map satisfying
� Distortion is the maximum absolute change of the metric
� Almost isometry is not necessarily bijective
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Almost isometries
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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
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Shape
metric space
Similarity
Distance between metric
spaces and .
Invariance
isometry w.r.t.
≈≈≈≈
Shapes as metric spaces
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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)