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Analysis of shape data: From landmarks to elastic curves

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Abstract Proliferation of high‐resolution imaging data in recent years has led to substantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two‐dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark‐based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape‐preserving transformation of a curve. The transition to the curve‐based approach moves the mathematical setting of shape analysis from finite‐dimensional non‐Euclidean spaces to infinite‐dimensional ones. We discuss some of the challenges associated with the infinite‐dimensionality of the shape space, and illustrate the use of geometry‐based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state‐of‐the‐art in the field. This article is categorized under:   Image and Spatial Data < Data: Types and Structure Computational Mathematics < Applications of Computational Statistics
Two examples of shape distances and deformations for mouse vertebrae. (a) Shape 1. (b) Shape 2 prior to optimal reparameterization. (c) Shape 2 after optimal reparameterization. The colored points correspond to the same parameter value across the three shapes (they are not landmarks). (d) Path of minimal shape deformation and the shape distance
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(a) Different similarity transformations applied to the same mouse vertebra. Note that all have the same exact shape. (b) Pictorial representation of an orbit of a curve
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A mouse vertebra curve β reparameterized using a diffeomorphism of , γ (represented as an angle [0, 2π)), resulting in βγ. The sampling of points according to the curve's parameterization is shown in red
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Four different representations of a mouse vertebra: (a) grayscale image, (b) color‐labeled landmarks, (c) curve, and (d) curve with color‐labeled landmarks
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Applications of statistical shape analysis include (a) leaf shape classification, (b) facial recognition, (c) anthropology, (d) tumor shape modeling, (e) clustering of diffusion tensor magnetic resonance imaging fibers, (f) military defense, and (g) gait recognition
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Six randomly generated mouse vertebrae shapes using a principal component analysis‐based Gaussian model
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Left: Mean shape of the sample of 76 mouse vertebrae. Right: Three principal directions of shape variability, with the mean shape in red, and the three shapes to its right and left representing shapes that are one, two and three standard deviations from the mean
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Comparison of elastic and nonelastic (based on arc‐length parameterizations) shape deformations between two mouse vertebrae
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Statistical and Graphical Methods of Data Analysis > Nonparametric Methods
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