Page 1

Foundation of re-normalized

synergetics:

issues of computability and

complexity

Milan Jovovic

Page 2

Modeling approach based on free energy and

distortion energy

Near linear model - aims for the simplest

explanations

Estimation of dynamical parameters of

clustering by statistical inference

Multi-spectral decomposition, in hierarchy of

scales

Application: scale analysis of complex systems

Analysis of signal distortion

by multi-scale decomposition

Page 3

Introduction (1 of 2)

Cluster parameters:

• Selected spatial window: Wr

• Computed cluster vector within Wr

Statistical inference defines PDF, with the associated

distortion energies, F and V

Energy functions are generally multi-dimensional and

non-convex

Non-linear map defines dynamical scale-space

clustering

Clustering is important optimization problem

c

Page 4

Introduction (2 of 2)

Page 5

Model of signal distortion:

- definitions

Distortion measure:

1. d = z2= (Cx-X)2 + (Cy-Y)2

2. d = z2=

[

It

eg. in TSP

eg. in 3D video communication

2]

vI

Partition functions:

,

2

W

r

z

rZ

Distortion energies -

free energy, and variance:

W

r

PvxdV

,

PDF:

,

2

Z

r

P

z

,log

1

,

rZvF

Page 6

Scale-space computing

Series of convex min/max of free energy F

brings in eq. up-scale melting & down-scale cooling:

2

,

1

0

dVF

0

dV

rZ

) 1 (

.

,

F

PIc

c

F

c

r

W

Evolution scheme – path integrals:

Way to move through the scale-space ?

Page 7

Motion through the scale-space:

- wave equation

,vF

The same potential level difference the equilibrium point moves by (2)

and (3)

12

Vv grad

(1)

(2)

) 2 (

v

V

v

) 3 (

F

S

S

d

v

V

vd

F

dU

0

V

F

2

2

2

Page 8

Cluster Bindings

Motion binding:

0

1

1

2

2

2

2

2

1

1

1

v

F

v

F

v

v

F

v

F

v

Determinant of the map:

1

.

2

2

2

2

2

1

1

2

2

2

2

2

2

2

1

1

2

2

v

F

v

F

v

F

v

F

λλ

D

Criteria of splitting a cluster at the “wave collapse”:

Spatial coherency of information:

Information content wrt the uncertanty relation:

Coupled domains of computation:

v

V

V

vG where

Wv

Wv

vdvGO

r

r

S

2

2

,

0

0

,

2

1

V

V

vG

2

2

,

1

,2

vCov

Page 9

Scalable coding

Coupled data structure of the hierarchy of

binary images

Efficient coding, control, data transfer

Parallelization: computing and control by

parallel computing architectures

(v4, W4) (v3, W3)

(v2, W2)

(v1, W1)

(v0, W0)

c3(v3, W3)

c2(v2, W2)

c0(v0, W0)

c1(v1, W1)

Page 10

Focus on computability and complexity –

relationship to statistical physics

o Computing paradigm assumes:

o Motion via scale-space wave information propagation, and

o Uncertainty relation wrt the information content of a cluster

o What makes it, therefore, polynomial in complexity (ref. 2)?

o Unique statistical description, although chaotic motion possible

o No strange attractors due to the conservative motion

Within this description: multi-scale decomposition of the information

content into clusters

Coupling of the energy exchange – synergetics

Coupled manifolds spanning the content of the information clusters

Page 11

Counting dimensions

Bringing in resonance system of 2 clusters

(ref. 2)

System of 3 clusters is much more complex !

3D spectral components

3D cluster covariance

3D coupled cluster covariance

results in 3 coupled clusters 3D manifolds

dynamical scale parameter β

= 10

operators div and rot

o System of 3 clusters at the “wave resonance”:

12

21

21

FF

FF

F

0

Page 12

Summary presentation of current work

Images: multi-spectral decomposition and clusters

coupling, spectral signature recognition

Movements: trajectory analysis, learning, coding and

control by scale-space computing

Bio/chemical informatics: data-mining and knowledge

discovery

Scalable data decomposition: coding, control, and

transmission

Synchronous computing scheme: upscale melting &

downscale cooling

Parallel computing implementation

Page 13

Scale singularity of data sets is used in detecting rain

patterns

Still images decomposition

Page 14

Sequence of 2 images: 2 clusters

decomposition

2D ball expansion expansion and diagonal

Page 15

Sequences – different intervals: 2 clusters decompos.

Vortex sequence 2. images

sequence 4. images sequence 7. images