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Statistical methods in computational anatomy

Department of Electrical Engineering, Washington University, St Louis, Missouri, USA, mim{at}cis.wustl.edu

Ayananshu Banerjee

Department of Electrical Engineering, Washington University, St Louis, Missouri, USA

Gary Christensen

Department of Electrical Engineering and Computer Engineering, The University of Iowa, Iowa City, IA 52242, USA

Sarang Joshi

Department of Electrical Engineering, Washington University, St Louis, Missouri, USA

Navin Khaneja

Department of Electrical Engineering, Washington University, St Louis, Missouri, USA

Ulf Grenander

Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

Larissa Matejic

Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

This paper reviews recent developments by the Washington/Brown groups for the study of anatomical shape in the emerging new discipline of computational anatomy. Parametric representations of anatomical variation for computational anatomy are reviewed, restricted to the assumption of small deformations. The generation of covariance operators for probabilistic measures of anatomical variation on coordinatized submanifolds is formulated as an empirical procedure. Populations of brains are mapped to common coordinate systems, from which template coordinate systems are constructed which are closest to the population of anatomies in a minimum distance sense. Variation of several one-, two and three-dimensional manifolds, i.e. sulci, surfaces and brain volumes are examined via Gaussian measures with mean and covariances estimated directly from maps of templates to targets. Methods are presented for estimating the covariances of vector fields from a family of empirically generated maps, posed as generalized spectrum estimation indexed over the submanifolds. Covariance estimation is made parametric, analogous to autoregressive modelling, by introducing small deformation linear operators for constraining the spectrum of the fields.

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