The Statistical Theory of Shape

Small, Christopher G.

doi:10.1007/978-1-4612-4032-7

Cited by 371 publications

(320 citation statements)

References 36 publications

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“…We would like to note that our usage of the terms shape and pre-shape space differs from that employed in [42,68,119]. In the terminology of [68] a pre-shape space is the space of labelled landmarks modulo translations and scalings and the shape space additionally quotients out rotations as well.…”

Section: Spaces Of Interestmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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Section: Spaces Of Interestmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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“…Further, GM methods are firmly grounded in a statistical theory for how shape is defined and how patterns of shape variation may be analyzed (Kendall, 1984 and1985;Small, 1996). Shape can be defined as a point in a k × p dimensional space (k dimensions, p number of landmarks), the basic assumption of 'Kendall's shape space' (Goodall, 1991).…”

Section: Conformation Scoring and Imagingmentioning

confidence: 99%

The use of novel phenotyping methods for validation of equine conformation scoring results

Druml

Dobretsberger

Brem

2015

Animal

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In this experiment, which is based on a cohort of 44 Lipizzan mares from the Austrian state stud farm of Piber, we present new statistical techniques for the analysis of shape and equine conformation using image data. In addition, we examined which strategies and procedures of image processing techniques led to a successful interpretation of the traits implemented in horse breeding programs. A total of 246 two-dimensional anatomical and somatometric landmarks were digitized from standardized photographs, and the variation of shape has been analyzed by the use of generalized orthogonal least-squares Procrustes (generalized Procrustes analysis (GPA)) procedures. The resulting shape variables have been regressed on the results from linear type trait classifications. In addition, the rating scores of six conformation classifiers were tested for agreement, yielding an interrater correlation (inter-class correlation) ranging from 0.41 to 0.68, respectively, a κ coefficient ranging from 0.16 to 0.53. From the 12 linear type traits assessed on a valuating scale, only the type-related traits (type, breed-type and harmony) revealed significant (P < 0.05) results in the regression analysis of shape variables on linear type traits. The other nine traits were characterized by a lower agreement between classifiers and did not result in a significant 'shape regression'. Finally, the 'horse shape space' defined by shape variables resulting from GPA procedures offered the possibility to assist in trait definition and in the evaluation of ratings, and it is an adequate biological and objective scale to human perception of conformation, which is expressed in numerical data only.

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“…Furthermore, landmark-based shape analysis can reduce the problem at hand to multivariate data analysis where existing statistical tools can be used with minor adjustments. Kendall (1984) is one of the pioneers of landmark analysis, and many others (including Dryden and Mardia (1998) and Small (1996)) have extended these concepts and developed statistical techniques on landmark shape spaces (Bookstein, 1986;Dryden and Mardia, 1992). However, representing an outline by a finite set of landmark points has its limitations, including:…”

Section: Introductionmentioning

confidence: 99%

Landmark-Constrained Elastic Shape Analysis of Planar Curves

Strait

Kurtek

Bartha

et al. 2017

Journal of the American Statistical Association

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Various approaches to statistical shape analysis exist in current literature. They mainly differ in the representations, metrics and/or methods for alignment of shapes. One such approach is based on landmarks, i.e., mathematically or structurally meaningful points, which ignores the remaining outline information. Elastic shape analysis, a more recent approach, attempts to fix this by using a special functional representation of the parametrically-defined outline in order to perform shape registration, and subsequent statistical analyses. However, the lack of landmark identification can lead to unnatural alignment, particularly in biological and medical applications, where certain features are crucial to shape structure, comparison, and modeling. The main contribution of this work is the definition of a joint landmark-constrained elastic statistical shape analysis framework. We treat landmark points as constraints in the full shape analysis process. Thus, we inherit benefits of both methods: the landmarks help disambiguate shape alignment when the fully automatic elastic shape analysis framework produces unsatisfactory solutions. We provide standard statistical tools on the landmark-constrained shape space including mean and covariance calculation, classification, clustering, and tangent principal component analysis (PCA). We demonstrate the benefits of the proposed framework on complex shapes from the MPEG-7 dataset and two real data examples: mice T2 vertebrae and Hawaiian Drosophila fly wings.

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The Statistical Theory of Shape

Cited by 371 publications

References 36 publications

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

The use of novel phenotyping methods for validation of equine conformation scoring results

Landmark-Constrained Elastic Shape Analysis of Planar Curves

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