Download ppt - tosca

1
Numerical geometry of non-rigid shapes Mathematical background
Mathematical background
Tutorial 1
© Maks Ovsjanikov, Alex & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009
2
Numerical geometry of non-rigid shapes Mathematical background
Metric balls
 Open ball:
 Closed ball:
Euclidean ball
L1 ball
L ball
3
Numerical geometry of non-rigid shapes Mathematical background
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
A set, whose compliment is open is called closed
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
Numerical geometry of non-rigid shapes Mathematical background
4
Topological spaces
A set
together with a set
 Empty set and
of subsets of
form a topological space if
are both in
 Union of any collection of sets in
is also in
 Intersection of a finite number of sets in
is also in
is called a topology on
The sets in
are called open sets
The metric induces a topology through the definition of open sets
5
Numerical geometry of non-rigid shapes Mathematical background
Connectedness
The space
is connected if it cannot be divided into two disjoint nonempty
closed sets, and disconnected otherwise
Connected
Stronger property: path connectedness
Disconnected
6
Numerical geometry of non-rigid shapes Mathematical background
Compactness
The space
is compact if any open
covering
has a finite subcovering
Finite
Infinite
For a subset of Euclidean space, compact = closed and bounded (finite
diameter)
7
Numerical geometry of non-rigid shapes Mathematical background
Convergence
A sequence
converges to
for any open set
exists
containing
such that for all
Topological definition
(denoted
for all
) if
exists
such that
for all
Metric definition
8
Numerical geometry of non-rigid shapes Mathematical background
Continuity
A function
is called continuous if
for any open set
, preimage
for all
for all
is also open.
Topological definition
exists
s.t.
satisfying
it follows that
Metric definition
Numerical geometry of non-rigid shapes Mathematical background
9
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
10
Numerical geometry of non-rigid shapes Mathematical background
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
Numerical geometry of non-rigid shapes Mathematical background
11
Topology of Latin alphabet
abde
opq
h
f
c sklm
n r t uz
vwx y
homeomorphic to
homeomorphic to
i j
homeomorphic to
Numerical geometry of non-rigid shapes Mathematical background
12
Lipschitz continuity
A function
is called Lipschitz continuous if there
exists a constant
for all
such that
. 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
Numerical geometry of non-rigid shapes Mathematical background
Bi-Lipschitz continuity
A function
there exists a constant
for all
is called bi-Lipschitz continuous if
such that
13
14
Numerical geometry of non-rigid shapes Mathematical background
Examples of Lipschitz continuity
0
1
Continuous,
not Lipschitz on [0,1]
0
1
Lipschitz on [0,1]
0
1
Bi-Lipschitz on [0,1]
Numerical geometry of non-rigid shapes Mathematical background
15
Isometries
A bi-Lipschitz function with
is called distance-preserving or an
isometric embedding
A bijective distance-preserving function is called isometry
Isometries copy metric geometries – two isometric spaces are equivalent
from the point of view of metric geometry
Numerical geometry of non-rigid shapes Mathematical background
Dilation
Maximum relative change of distances by a function is called dilation
Dilation is the Lipschitz constant of the function
Almost isometry has
16
Numerical geometry of non-rigid shapes Mathematical background
17
Distortion
Maximum absolute change of distances by a function is called distortion
Almost isometry has
18
Numerical geometry of non-rigid shapes Mathematical background
Groups
A set
with a binary operation
is called a group if the
following properties hold:
 Closure:
for all
 Associativity:
 Identity element:
 Inverse element: for any
for all
such that
,
for all
such that
Numerical geometry of non-rigid shapes Mathematical background
19
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:
20
Numerical geometry of non-rigid shapes Mathematical background
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
 Closure
is a group
is a self-isometry for all
 Associativity from definition of function composition
 Identity element
 Inverse element
(exists because isometries are bijective)
21
Numerical geometry of non-rigid shapes Mathematical background
Isometry groups
A
B
AB C A
AA
C
Trivial group
(asymmetric)
CB
CB
Cyclic group
(reflection)
C AC B
CAB B
Permutation group
(reflection+rotation)
22
Numerical geometry of non-rigid shapes Mathematical background
Symmetry in Nature
Butterfly
(reflection)
Diamond
Snowflake
(dihedral)