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

April 5, 2006
  • Special domains
  • 32A07
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.