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# Statistics of shape

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Abstract This article reviews various statistical methods that are available for the analysis of the shapes of images or objects. Statistical shape analysis is too large a topic to be reviewed in full. So, this review concentrates on shape analysis by means of landmarks, and in particular on the representation of landmark shapes on manifolds that was proposed by D. G. Kendall. The use of Kendall shape analysis on landmarks is briefly contrasted with alternative ways of representing and analyzing shapes. WIREs Comp Stat 2011 3 428–433 DOI: 10.1002/wics.173 This article is categorized under: Applications of Computational Statistics > Computational Mathematics

Landmarks chosen by an expert on the fifth cervical vertebra of a gorilla.

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The statistical problem of discriminating the shapes of cervical vertebrae between two subspecies of gorilla. Left: Gorilla gorilla beringei (GGB) cervical vertebrae. Right: Gorilla gorilla gorilla (GGG) cervical vertebrae. The line segments displayed between landmarks are drawn for visual comparison purposes and are not part of the data.

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Kendall's sphere of triangle shapes. Each point on the sphere corresponds to a unique triangle shape, which is an equivalent class under similarity transformations. On this sphere, the set of isosceles triangles with two equal sides meeting at a given vertex (landmark) forms a great circle of the sphere intersecting the ‘north and south poles’ as displayed. There are three such great circles forming lines of ‘longitude’ and meeting at angles of 60° to each other. The ‘poles’ themselves correspond to the two equilateral triangles distinguished only by the ordering of their vertices. Halfway between the poles, the ‘equator’ corresponds to the set of a degenerate triangles, whose vertices are collinear.

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Schematic diagram of size‐and‐shape space. (The reader is cautioned not to interpret the figure too literally.)

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Iron Age brooches with their shapes encoded using four landmarks.

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