# An introduction to the geometry and topology of point cloud data

Document Sample

```					An introduction to the geometry and
topology of point cloud data
Peter Bubenik

p.bubenik@csuohio.edu

http://academic.csuohio.edu/bubenik p/

Cleveland State University
Department of Mathematics
September 19, 2005

An introduction to the geometry and topology of point cloud data – p. 1/1
1. Point Cloud Data
Motivation: Given a set of points that “looks like a circle”

We would like to be able to say so mathematically.

An introduction to the geometry and topology of point cloud data – p. 2/1
2. The Rips complex
X4

X1

X3

X2

An introduction to the geometry and topology of point cloud data – p. 3/1
2. The Rips complex
X4

r2
r4
X1

X3               r3
r1
X2

An introduction to the geometry and topology of point cloud data – p. 3/1
The Rips complex
X4

r2
r4
X1

X3              r3
r1
X2

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4          r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 4/1
3. Homology of Simplices
X4

Consider the following simplicial                                                        X1
complex ∆:

X3
X2
It consists of
four 0-simplices: X1 , X2 , X3 , X4 ,
ﬁve 1-simplicies: {X1 , X2 }, {X1 , X3 }, {X2 , X3 }, {X1 , X4 },
{X3 , X4 },
and one 2-simplex: {X1 , X2 , X3 }.

An introduction to the geometry and topology of point cloud data – p. 5/1
Homology of Simplicies
X4

Sums of n-simplicies are called                                                            X1
n-chains.

X3
X2

the boundary of {X1 , X2 , X3 } is
{X1 , X2 } + {X2 , X3 } + {X3 , X1 }
the boundary of {Xi , Xj } is Xj − Xi ,
the boundary of {Xi } is 0

An introduction to the geometry and topology of point cloud data – p. 6/1
Homology of Simplicies
X4
By linearity this deﬁnes the
boundary on all n-chains.
X1
Cycles are n-chains with
boundary equal to zero.              X3
X2

For example {X1 , X2 } + {X2 , X3 } + {X3 , X1 } is a cycle and
so is X1 .

One can check that boundaries are always cycles.

An introduction to the geometry and topology of point cloud data – p. 7/1
Homology of Simplicies

The homology Hn (∆) is the quotient of the cycles modulo
the boundaries.

The Betti number Bn (∆) is the dimension of Hn (∆).

B0 (∆) is the number of connected components of ∆.
B1 (∆) is the number of holes in ∆.
B2 (∆) is the number of voids in ∆.

An introduction to the geometry and topology of point cloud data – p. 8/1
Homology of Simplicies
X4

X1

X3
X2

In our example B0 (∆) = 1, B1 (∆) = 1,
and all higher Betti numbers are zero.

An introduction to the geometry and topology of point cloud data – p. 9/1
4. Persistent homology

Assume we have a simplicial complex that changes as we
vary some parameter r.

The homology that persists as r changes is called
persistent homology.

We can record how the Betti numbers change as r changes
using Betti barcodes.

We illustrate this using the Rips complex on our earlier
example.

An introduction to the geometry and topology of point cloud data – p. 10/1
The Rips complex
X4

r2
r4
X1

X3              r3
r1
X2

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
The Rips complex
X4

r2
r4
X1

X3                    r3
r1
X2

r2 + r3

0         r1 r2 r3    r4           r1 + r3 π

An introduction to the geometry and topology of point cloud data – p. 11/1
Betti 0-barcode

0   r1 r2 r3   r4     r1 + r3

An introduction to the geometry and topology of point cloud data – p. 12/1
Betti 1-barcode

0   r1 r2 r3   r4     r1 + r3

An introduction to the geometry and topology of point cloud data – p. 13/1

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 30 posted: 11/21/2008 language: English pages: 30
How are you planning on using Docstoc?