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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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Metrics Spaces for the Infinite Dimensional Diffeomorphisms

Metrics Spaces for the Infinite Dimensional Diffeomorphisms

Chapter:
(p.332) 11 Metrics Spaces for the Infinite Dimensional Diffeomorphisms
Source:
Pattern Theory
Author(s):

Ulf Grenander

Michael I. Miller

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198505709.003.0012

In this chapter the metric space structure of shape is developed by studying the action of the infinite dimensional diffeomorphisms on the coordinate systems of shape. Riemannian manifolds allow us to developmetric distances between the groupelements. We examine the natural analog of the finite dimensional matrix groups corresponding to the infinite dimensional diffeomorphisms which are generated as flows of ordinary differential equations.We explore the construction of the metric structure of these diffeomorphisms and develop many of the properties which hold for the finite dimensional matrix groups in this infinite dimensional setting.

Keywords:   affine motions, elastic matching, facial expressions, geodesic shooting, inexact matching, left invariance, photometric variability, scaling, viscous matching

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