Inverting dedevelopment: geometric singularity theory in embryology

Fred L. Bookstein; Bradley R. Smith

doi:10.1117/12.402451

4 October 2000 Inverting dedevelopment: geometric singularity theory in embryology

Fred L. Bookstein, Bradley R. Smith

Proceedings Volume 4121, Mathematical Modeling, Estimation, and Imaging; (2000) https://doi.org/10.1117/12.402451
Event: International Symposium on Optical Science and Technology, 2000, San Diego, CA, United States

Abstract

The diffeomorphism model so useful in the biomathematics of normal morphological variability and disease is inappropriate for applications in embryogenesis, where whole coordinate patches are created out of single points. For this application we need a suitable algebra for the creation of something from nothing in a carefully organized geometry: a formalism for parameterizing discrete nondifferentiabilities of invertible functions on R^k, k $GTR 1. One easy way to begin is via the inverse of the development map - call it the dedevelopment map, the deformation backwards in time. Extrapolated, this map will inevitably have singularities at which its derivative is zero. When the dedevelopment map is inverted to face forward in time, the singularities become appropriately isolated infinities of derivative. We have recently introduced growth visualizations via extrapolations to the isolated singularities at which only one directional derivative is zero. Maps inverse to these create new coordinate patches directionally rather than radically. The most generic singularity that suits this purpose is the crease f(x,y) equals (x,x²y+y³), which has already been applied in morphometrics for the description of focal morphogenetic phenomena. We apply it to embryogenesis in the form of its analytic inverse, and demonstrate its power using a priceless new data set of mouse embryos imaged in 3D by micro-MR with voxels smaller than 100micrometers ³.

Citation Download Citation

Fred L. Bookstein and Bradley R. Smith "Inverting dedevelopment: geometric singularity theory in embryology", Proc. SPIE 4121, Mathematical Modeling, Estimation, and Imaging, (4 October 2000); https://doi.org/10.1117/12.402451

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