An Elasticity-Based Covariance Analysis of Shapes

We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is...

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Published in:	International journal of computer vision Vol. 92; no. 3; pp. 281 - 295
Main Authors:	Rumpf, Martin, Wirth, Benedikt
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.05.2011 Springer Springer Nature B.V
Subjects:	Analysis of covariance Applied sciences Artificial Intelligence Computer Imaging Computer Science Computer science; control theory; systems Contours Covariance Data processing. List processing. Character string processing Deformation Elasticity Energy Exact sciences and technology Image Processing and Computer Vision Linearization Mathematical models Memory organisation. Data processing Nonlinearity Pattern Recognition Pattern Recognition and Graphics Pattern recognition. Digital image processing. Computational geometry Principal component analysis Principal components analysis Software Vector space Vision Shape analysis Covariance metric Non-rigid registration Phase field approximation Finite element discretization Nonlinear elasticity Principal components Geometrical shape Factor analysis Statistical analysis Image processing Covariance analysis Strain energy Form defect Covariancemetric Modeling Finite element method Image analysis Covariance Morphology Morphological analysis Metric Hessian matrices Principal component analysis Linearization
ISSN:	0920-5691, 1573-1405
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Abstract	We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric—an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L 2 -type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D.
AbstractList	We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric--an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard [L.sup.2]-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D. We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric—an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L 2 -type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D. We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric--an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L ^sup 2^-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D.[PUBLICATION ABSTRACT] We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric--an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L 2-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D. We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric--an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard [L.sup.2]-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D. Keywords Shape analysis * Principal components * Non-rigid registration * Nonlinear elasticity * Covariance metric * Phase field approximation * Finite element discretization
Audience	Academic
Author	Rumpf, Martin Wirth, Benedikt
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Cites_doi	10.1007/BF01052971 10.1002/cpa.3160300502 10.1016/j.neuroimage.2004.07.015 10.1007/BF02344682 10.1016/S0031-3203(03)00056-6 10.1146/annurev.bioeng.4.092101.125733 10.1007/BF02392977 10.1007/978-3-540-39903-2_94 10.1007/978-3-642-03641-5_22 10.1007/978-3-540-75759-7_66 10.1112/blms/16.2.81 10.1002/cpa.3160420503 10.1006/cviu.1995.1004 10.1109/83.902291 10.1007/11566489_50 10.1007/978-3-540-39701-4_9 10.1002/cpa.3160430805 10.1007/s10208-003-0094-x 10.1137/080738337 10.1007/978-3-642-02256-2_59 10.1088/0031-9155/50/24/009 10.1023/A:1011161132514 10.1137/S0036139995287685 10.1007/s10851-009-0156-z 10.1016/j.neuroimage.2004.07.068 10.1109/34.765654 10.1007/11889762_9 10.1007/s10208-004-0145-y 10.1109/VLSM.2001.938895 10.1145/1276377.1276457 10.1017/S0308210500026378 10.1109/SSBI.2002.1233989 10.1109/CVPR.2003.1211342 10.1109/ICASSP.2006.1661428 10.1109/CVPR.2008.4587747 10.1017/S030821050002014X
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Keywords	Shape analysis Covariance metric Non-rigid registration Phase field approximation Finite element discretization Nonlinear elasticity Principal components Geometrical shape Factor analysis Statistical analysis Image processing Covariance analysis Strain energy Form defect Covariancemetric Modeling Finite element method Image analysis Covariance Morphology Morphological analysis Metric Hessian matrices Principal component analysis Linearization
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Snippet	We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear...
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SubjectTerms	Analysis of covariance Applied sciences Artificial Intelligence Computer Imaging Computer Science Computer science; control theory; systems Contours Covariance Data processing. List processing. Character string processing Deformation Elasticity Energy Exact sciences and technology Image Processing and Computer Vision Linearization Mathematical models Memory organisation. Data processing Nonlinearity Pattern Recognition Pattern Recognition and Graphics Pattern recognition. Digital image processing. Computational geometry Principal component analysis Principal components analysis Software Vector space Vision
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Title	An Elasticity-Based Covariance Analysis of Shapes
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