# Symmetries of a Mathematical Model for Deformation Noise in Realistic Biometric Contexts

Wednesday, April 5, 2006 - 10:30am - 11:30am

EE/CS 3-180

Fred Bookstein (University of Washington)

By the realistic biometric context of my title, I mean

an investigation of well-calibrated images from

a moderately large sample of organisms in order to evaluate

some nontrivial hypothesis about systematic form-factors

(e.g., a group difference). One common approach to

such problems today is geometric morphometrics, a short name

for

the multivariate statistics of landmark location data.

The core formalism here, which handles data schemes that

mix discrete points, curves, and surfaces, applies otherwise

conventional linear statistical modeling strategies to

representatives of equivalence classes of these

schemes under similarity transformations or relabeling maps.

As this tradition has matured, algorithmic successes involving

statistical

manipulations and the associated diagrams have directed our

community's attention away from a serious underlying problem:

Most biological processes operate not on the submanifolds of

the data

structure but in the embedding space in-between. In that

context

constructs such as diffeomorphism, shape distance, and image

energy

are mainly metaphors, however visually compelling, that may

have no particular scientific authority when some actual

biometrical hypothesis is being seriously weighed.

Instead of phrasing this as a problem in the representation of

a signal, it may be useful to recast the problem as that of a

suitable

model for noise (so that signal becomes, in effect, whatever

patterns rise

above the amplitude of the noise). The Gaussian model of

conventional statistics

can be derived as an expression of the symmetries of a

plausible physical model

(the Maxwell distribution in statistical mechanics), and it

would be

nice if some equally compelling symmetries could be invoked to

help us formulate

biologically meaningful noise models for deformations.

We have had initial success with a new model of

self-similar isotropic noise

borrowed from the field of stochastic geometry. In this

approach, a deformation

is construed not as a deterministic mapping but as a

distribution of mappings given by an intrinsic random process

such that the plausibility of a meaningful focal structural

finding is the same

regardless of physical scale. Simulations instantiating this

process are graphically quite compelling--their selfsimilarity

comes

as a considerable (and counterintuitive) surprise--and yet as

a

tool of data analysis, for teasing out interesting regions

within an

extended data set, the symmetries (and their breaking, which

constitutes

the signal being sought) seem quite promising.

My talk will review the core of geometric morphometrics

as it

is practiced today, sketch the deep difficulties that arise in

even

the most compelling biological applications, and then

introduce the

formalisms that, I claim, sometimes permit a systematic

circumvention of

these problems when the context is one of a statistical data

analysis of a serious scientific hypothesis.

This work is joint with K. V. Mardia.

an investigation of well-calibrated images from

a moderately large sample of organisms in order to evaluate

some nontrivial hypothesis about systematic form-factors

(e.g., a group difference). One common approach to

such problems today is geometric morphometrics, a short name

for

the multivariate statistics of landmark location data.

The core formalism here, which handles data schemes that

mix discrete points, curves, and surfaces, applies otherwise

conventional linear statistical modeling strategies to

representatives of equivalence classes of these

schemes under similarity transformations or relabeling maps.

As this tradition has matured, algorithmic successes involving

statistical

manipulations and the associated diagrams have directed our

community's attention away from a serious underlying problem:

Most biological processes operate not on the submanifolds of

the data

structure but in the embedding space in-between. In that

context

constructs such as diffeomorphism, shape distance, and image

energy

are mainly metaphors, however visually compelling, that may

have no particular scientific authority when some actual

biometrical hypothesis is being seriously weighed.

Instead of phrasing this as a problem in the representation of

a signal, it may be useful to recast the problem as that of a

suitable

model for noise (so that signal becomes, in effect, whatever

patterns rise

above the amplitude of the noise). The Gaussian model of

conventional statistics

can be derived as an expression of the symmetries of a

plausible physical model

(the Maxwell distribution in statistical mechanics), and it

would be

nice if some equally compelling symmetries could be invoked to

help us formulate

biologically meaningful noise models for deformations.

We have had initial success with a new model of

self-similar isotropic noise

borrowed from the field of stochastic geometry. In this

approach, a deformation

is construed not as a deterministic mapping but as a

distribution of mappings given by an intrinsic random process

such that the plausibility of a meaningful focal structural

finding is the same

regardless of physical scale. Simulations instantiating this

process are graphically quite compelling--their selfsimilarity

comes

as a considerable (and counterintuitive) surprise--and yet as

a

tool of data analysis, for teasing out interesting regions

within an

extended data set, the symmetries (and their breaking, which

constitutes

the signal being sought) seem quite promising.

My talk will review the core of geometric morphometrics

as it

is practiced today, sketch the deep difficulties that arise in

even

the most compelling biological applications, and then

introduce the

formalisms that, I claim, sometimes permit a systematic

circumvention of

these problems when the context is one of a statistical data

analysis of a serious scientific hypothesis.

This work is joint with K. V. Mardia.

MSC Code:

32A07

Keywords: