05/12/04

 

 

           General Pattern Theory

          All science tries to discover patterns whether it is in nature or in the man made world. Often the pattern structure is

far from obvious and it takes intellectual curiosity coupled with insight in the subject matter to discover the true pattern.

          A simple example is the following picture

 

                    What does it mean? It looks like the shards of a broken urn, but the pieces seem to have internal structure. Let us blow up the picture to see the inside:

        

But it is difficult to guess what the shards represent. In this case we have started from a known pattern and we show it by

bringing the shards together:               (move cursor to picture to activate movie )

Surprise!

 

 

 

or blown up:

 

a crude photo of the author: Ulf Grenander.

 General Pattern Theory assumes that:

1) patterns are formed from generators (primitives) like the shards above  - an atomistic approach

2) generators are joined together so that they fit like the shards above according to a specified criterion

3) generators are joined as controlled by a graph, for example

 

 

 

 

 

 

                                            

 

 

 

4) the resulting configuration is deformed by random diffeomorphisms, for example the following time sequence of

deformations  (move cursor to picture to activate movie )

Using pattern theoretic constructs one can perform pattern inference of various types; some examples to be given below

but many more in

MAIN REFERENCE: Ulf Grenander (1993) General Pattern Theory, Oxford University Press

SECONDARY REFERENCE ( like Cliff Notes): Ulf Grenander (1996) Elements of Pattern Theory, Johns Hopkins University Press

Computational Anatomy

               A remarkable application of pattern theory is the emerging field Computational Anatomy, in which generators are the anatomical

components to which random transformations are applied. With templates from strata of patient populations the transformations are selected using

Bayesian techniques to fit observed images but with smoothness constraints. Some differential geometric techniques are based on analogies with

 fluid mechanics, leading to trajectories in homogeneous spaces.

                 One version starts from an energy functional

where the first integral expresses the deviation between the deformed (observed) image and the template transformed via

 a diffeomorphism; the second integral measures the smoothness of the diffeomorphism in terms of a linear differential

operator L, something like the Laplacian. Minimizing the energy functional we arrive at a mapping that will be used for

identifying anatomical components in the observed image by mapping the ones in the given template.

           An example from brain imaging:                            (move cursor onto picture to activate the film clip)

 

 

 

 

 

 

 

The left panel means the template, the middle one the patient image and the the right one is the resulting, deformed image

as it evolves. After that horizontal cuts are shown. The images are actually 3D but we display them in only 2D.

REFERENCES: U. Grenander and M. I. Miller (1998): Computational Anatomy: An Emerging Discipline, Quarterly of Applied

Mathematics

and many papers in www.cis.jhu.edu

 Patterns in Natural Scenes

           Scenes in nature, like forests, seascapes, deserts, have obviously some structure but how can one describe it in mathematical form?

 Due to their intrinsic variability the representation ought to take the form of a stochastic process. The first one would try is the Gaussian one,

but it was observed that the marginal distribution of the pixel values, or of simple linear functionals of pixel values had much longer tails.  Also,

it was observed by Mumford and others that, for example the gradient of the image, typically has a cusp in the middle. What is the explanation?

               Taking a pattern theoretic approach with trees (in forest scenes) as generators and with simple probabilistic

assumptions  it was shown that, miraculously, there seemed to be a universal law governing a large class of natural scenes

although they differ a lot.  More precisely, we have the analytical characterization of natural scenes:

Graphically this means, dotted curves empirical, full drawn curves according to Theorem:

      

REFERENCE: U. Grenander and A. Srivastava (2001) Probability Models for Natural Images, IEEE PAMI

                J. Huang, A. Lee and D. Mumford (2000) Statstics of Range Images, IEEE Conf.  Comp. Vision and Pattern Rec.

                and with consequences for image understanding (inference):

                 U. Grenander (2003) Towards a Theory of Natural Scenes, www.dam.brown.edu/ptg

 Patterns of Thought

                      In contrast to the other projects described on this home page the Patterns of Thought study is not anchored in empirical solid facts.

 It is entirely speculative based only on introspection and commons sense ideas about the mind. But we offer it with no excuses for its lack of

experimental validity: on the contrary the author is convinced that neurophysiology in due time will  connect to a mind theory with some

similarities to the following  - this will happen but who knows when.

                       In the study GOLEM we try to represent human thinking in a probabilistic set up that includes emotions and non-logical thought

of various kinds. Starting from ideas - primitive thought elements - we combine them into configurations - thoughts - binding them together  by

semantic constraints modified by usage frequencies. The mind appears as a boiling cauldron of thought fragments, only some of which

reach the level of consciousnes: thought chatter.

 

                    GOLEM has not yet reached maturity and will need much development before its behavior is sufficiently

anthropomorphic.

REFERENCE Patterns of Thought, in www.dam.brown.edu/ptg

 

Patterns in Growth

                   There is a huge literature on the mathematical modeling of biological growth and we are suggesting an alternative that seems

promising but has not yet been tested on real data. The emphasis is on allometric growth, in particular situations where the anatomy is affected

by gene manipulations. We use the following setup:

 

The diffeomorphisms in the hemi-group represent cell changes: mitosis, growth and decay as well as their effects on the other cells in the

organism. Starting with a curvilinear coordinate system in the initial organism it will evolve as time increases and we get a biologically

meaningful coordinate system in the spirit of d'Arcy Thompson. It can look like:

 

 

REFERENCE Patterns of Growth, in www.dam.brown.edu/ptg

Family

Wife: Paj

Children: Sven, Angela, Charlotte

Grandchildren: Alexander, Ariana, Nikolas, Tatiana, Annika, Anders

Adopted: Rufsan

 

 

Hobby

Mathematics

 

Contact

A reader who wishes to know more about pattern theory just write to

ulf-grenander@cox.net

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

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This site was last updated 05/12/04