The Fshape Framework for the Variability Analysis of Functional Shapes

This article introduces a full mathematical and numerical framework for treating functional shapes (or fshapes) following the landmarks of shape spaces and shape analysis. Functional shapes can be described as signal functions supported on varying geometrical supports. Analyzing variability of fshap...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 17; no. 2; pp. 287 - 357
Main Authors: Charlier, B., Charon, N., Trouvé, A.
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2017
Springer Nature B.V
Springer Verlag
Subjects:
Applications of Mathematics
Attachment
Bundles
Computational Geometry
Computer Science
Computer Vision and Pattern Recognition
Differential Geometry
Economics
Engineering Sciences
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Optimization
Populations
Signal and Image Processing
Transformations
Utilities
Large deformation models
68U10
Shape analysis
Varifolds
58E50
Metamorphoses
Signals on manifolds
58B32
68U05
49M25
Atlas estimation algorithms
49Q20
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary:This article introduces a full mathematical and numerical framework for treating functional shapes (or fshapes) following the landmarks of shape spaces and shape analysis. Functional shapes can be described as signal functions supported on varying geometrical supports. Analyzing variability of fshapes’ ensembles requires the modeling and quantification of joint variations in geometry and signal, which have been treated separately in previous approaches. Instead, building on the ideas of shape spaces for purely geometrical objects, we propose the extended concept of fshape bundles and define Riemannian metrics for fshape metamorphoses to model geometric-functional transformations within these bundles. We also generalize previous works on data attachment terms based on the notion of varifolds and demonstrate the utility of these distances. Based on these, we propose variational formulations of the atlas estimation problem on populations of fshapes and prove existence of solutions for the different models. The second part of the article examines thoroughly the numerical implementation of the tangential simplified metamorphosis model by detailing discrete expressions for the metrics and gradients and proposing an optimization scheme for the atlas estimation problem. We present a few results of the methodology on a synthetic dataset as well as on a population of retinal membranes with thickness maps.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-015-9288-2