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 )
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
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
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
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
Children: Sven, Angela, Charlotte
Grandchildren: Alexander, Ariana, Nikolas, Tatiana, Annika, Anders
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This site was last updated 05/12/04