title: Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson.
abstract: The Computational Anatomy project is the morphome-scale study of shape and form, which we model as an orbit under diffeomorphic group action. Metric comparison calculates the geodesic length of the diffeomorphic flow connecting one form to another. Geodesic connection provides a positioning system for coordinatizing the forms and positioning their associated functional information. This article reviews progress since the Euler-Lagrange characterization of the geodesics a decade ago. Geodesic positioning is posed as a series of problems in Hamiltonian control, which emphasize the key reduction from the Eulerian momentum with dimension of the flow of the group, to the parametric coordinates appropriate to the dimension of the submanifolds being positioned. The Hamiltonian viewpoint provides important extensions of the core setting to new, object-informed positioning systems. Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series studies of growth and atrophy via shape splines.
authors.affiliation: Center of Imaging Science.|Department of Biomedical Engineering.|Kavli Neuroscience Discovery Institute, and.
authors.forename: Michael I
authors.initials: MI
authors.lastname: Miller
authors.affiliation: CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France; email: alain.trouve@cmla.ens-cachan.fr.
authors.forename: Alain
authors.initials: A
authors.lastname: Trouvé
authors.affiliation: Center of Imaging Science.|Department of Applied Mathematics, The John Hopkins University, Baltimore, Maryland 21218; email: mim@cis.jhu.edu , laurent.younes@jhu.edu.
authors.forename: Laurent
authors.initials: L
authors.lastname: Younes
mesh_terms: D000715:Anatomy / Q000379:methods* / Q000706:statistics & numerical data; D000818:Animals; D001698:Biomedical Engineering; D001921:Brain / Q000033:anatomy & histology; D019295:Computational Biology; D006352:Heart Ventricles / Q000033:anatomy & histology; D006801:Humans; D021621:Imaging, Three-Dimensional; D055641:Mathematical Concepts; D008675:Metamorphosis, Biological; D008953:Models, Anatomic; D008955:Models, Cardiovascular; D008959:Models, Neurological; D059906:Neuroimaging
keywords: brain mapping; diffeomorphisms; medical imaging; shape