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Pattern TheoryFrom representation to inference$
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Ulf Grenander and Michael I. Miller

Print publication date: 2006

Print ISBN-13: 9780198505709

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780198505709.001.0001

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Manifolds, Active Models, And Deformable Templates

Manifolds, Active Models, And Deformable Templates

(p.214) 8 Manifolds, Active Models, And Deformable Templates
Pattern Theory

Ulf Grenander

Michael I. Miller

Oxford University Press

To study shape we introduce manifolds and submanifolds examined in the continuum as the generators. Transformations are constructed which are built from the matrix groups and infinite products. This gives rise to many of the widely used structural models in image analysis often termed active models, essentially the deformable templates. These deformations are studied as both diffeomorphisms as well as immersions. A calculus is introduced based on transport theory for activating these deformable shapes by taking variations with respect to the matrix groups parameterizing them. Segmentation based on activating these manifolds is examined based on Gaussian random fields and variations with respect to the parameterizations.

Keywords:   Brownian motion, Fourier transform, Gram-Schmidt orthogonalization, Hoffman brain phantom, Laplacian circulant boundary, PIE phantom, Wiener process

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