Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

Computational anatomy: an emerging discipline


Authors: Ulf Grenander and Michael I. Miller
Journal: Quart. Appl. Math. 56 (1998), 617-694
MSC: Primary 92C50; Secondary 00A69, 92B05
DOI: https://doi.org/10.1090/qam/1668732
MathSciNet review: MR1668732
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Abstract: This paper studies mathematical methods in the emerging new discipline of Computational Anatomy. Herein we formalize the Brown/Washington University model of anatomy following the global pattern theory introduced in [1, 2], in which anatomies are represented as deformable templates, collections of 0, 1, 2, 3-dimensional manifolds. Typical structure is carried by the template with the variabilities accommodated via the application of random transformations to the background manifolds. The anatomical model is a quadruple $ \left( \Omega , H, I, P \right)$, the background space $ \Omega \dot = {U_\alpha }{M_\alpha }$ of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background space $ {H} : \Omega \leftrightarrow \Omega $, the space of idealized medical imagery $ I$, and $ P$ the family of probability measures on $ H$. The group of diffeomorphic transformations $ H$ is chosen to be rich enough so that a large family of shapes may be generated with the topologies of the template maintained. For normal anatomy one deformable template is studied, with $ \left( \Omega , H, I \right)$ corresponding to a homogeneous space [3], in that it can be completely generated from one of its elements, $ I = {HI_{temp}}, {I_{temp}} \in I$. For disease, a family of templates $ {U_\alpha }I_{temp}^\alpha $ are introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spaces $ {I_{total}} = {U_\alpha }\left( {I^\alpha }, {H^\alpha } \right)$.


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DOI: https://doi.org/10.1090/qam/1668732
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