The Emerging Field of Computational Anatomy
Michael Miller
Johns Hopkins University
Center for Imaging Science
Computational Anatomy (CA) is the mathematical study of anatomy, viewed as an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) acting on anatomical exemplars. The observable images are produced by medical imaging devices. CA examines three principal components: (i) construction of anatomical submanifolds; (ii) comparison of anatomical manifolds through estimation of the underlying diffeomorphisms that define their shape or geometry; and (iii) generation of probability laws describing anatomical variation on images for inference and disease classification within anatomical models.
We review recent advances, focusing primarily on metric comparison of anatomical coordinates. The Euler equations are discussed in the context of both the static metric matching problem and the growth problem. Recent results concerning the normal equations of motion for momentum are presented, highlighting the relationship between conservation of momentum, geodesic shooting based on the initial vector field at the identity, and representations of growth and photometric variation.
An example is provided through the study of growth of the murine hippocampal surface using Large Deformation Diffeomorphic Metric Mapping (LDDMM). Surface displacements are color-coded, and growth directions are visualized using three-dimensional glyphs. Although the results reflect not only biological growth but also variations due to landmark placement, they suggest that hippocampal growth is not spatially uniform. In particular, the ventral hippocampus exhibits greater tissue displacement during development than the dorsal region.