Title: Frame Dictionaries in Medical Biophysics and Modeling Directionality by Means of Probabilistic Frame Potentials
Abstract: We aim to identify and use interrelations between applied mathematics, multi-spectral imaging, and gene regulatory networks:
1) Biological function is most likely not just reflected in a few dominant genes, but in many complex interactions within gene regulatory networks. Due to the functional biological robustness of any organism against genetic and environmental variations, it seems reasonable to believe that the gene regulatory network in embryonic development is in a form of equilibrium state.
2) Multi-spectral imaging is a growing field in biomedical optics. Spectral classes in retinal imaging are almost never orthogonal to each other. Thus, qualitative and quantitative analysis requires basis-like systems that are not orthogonal.
3) Finite unit norm tight frames (FUNTFs) generalize orthonormal bases by allowing for redundancy. They are exactly the minimizers of the frame potential. Choosing i.i.d. random points from the uniform probability distribution on the sphere is known to approximate a FUNTF.
The analysis of both, microarray gene expression data in embryonic development and multi-spectral retinal image sets, involves frame theory. We first recall a classification approach for multi-spectral image sets based on the frame potential. From our perspective, it is a discretization of a continuous scheme that must still be developed. To shed some light on such a continuous scheme, we introduce a novel probabilistic frame potential and characterize its minimizers among all probability distributions on the sphere. The probabilistic frame potential is then used to significantly weaken the requirements on the random choice of points to obtain an approximate FUNTF: we allow for any distribution that minimizes the probabilistic frame potential and we remove the requirement that the points have to be identically distributed.
Based on the random choice of points on the sphere, we finally aim to join frame theory with directional statistics. The Bingham test is a common statistical test to reject the hypothesis of directional uniformity in a dataset. We characterize sample distributions that would lead to failure of rejection although the sample is clearly not uniformly distributed in a directional sense. Such Bingham-Alternatives are characterized in terms of probabilistic tight frames on the sphere. We finally introduce and test a novel directional equilibrium paradigm for the analysis of microarray gene expression data and test the proposed scheme on a mouse model of Coloboma, a common human genetic defect in retinal development.
Section on Medical Biophysics, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda
Norbert Wiener Center for Harmonic Analysis and Applications, Department of Mathematics, University of Maryland, College Park