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...said be "geometrically similar". In other words, they have isometry. For example, a cube will have its length:width:height ratios the same for smaller as well as larger sizes. When one or more dimensions of a shape change disproportionately with size, that object is said to scale allometrically. Allometery has been shown to apply to people and animals (McMahon, 1975; Ross and Ward, 1981; Biewener, 1992; Brown and West, 2000; Lindstedt and Schaeffer, 2002). A similar relation is to be expected even for body parts such as feet. A person with twice the average foot length is not anticipated to have double the foot width.
The objective of our study is to understand the shape structure of an object of interest by fitting the observed data to various models. In the study, the measurements of each object are taken corresponding to a fixed coordinate system at a set of well-defined locations on the surface of the object. As the coordinate systems of various objects may differ from each other, a transformation is needed to align the observed data to the same orientation before fitting data to the object shape. Although the applications that we have in mind are for measuring two-dimensional and three-dimensional objects, it should be pointed out that the methodology presented here is general and it applies to objects in any dimension. In the remainder of this section, we will introduce the basic model and how it is related to shape data. Section 2 discusses a general algorithm for estimating the parameters of the model. Section 3 addresses statistical inference and model selection. The proposed methods are illustrated with a real-life foot shape example in Section 4.
Equation (1) gives a simple model that describes the relationship between the observed data and the ideal measurements of the fixed locations of the object:
[y.sub.ij] = [[mu].sub.i] + [A.sub.i][x.sub.j] + [[epsilon].sub.ij]. (1)
where [y.sub.ij], j = 1, 2,...,m, i = 1,2,...,n, represents the three-dimensional vector of measurements of the jth location for the ith subject. The quantities [x.sub.j], j = 1, 2,...,m, are the vectors of ideal measurements at the m locations after proper scaling and alignment, which is the same for each object. The quantities [A.sub.i] and [[mu].sub.i] are the scale and translation functions that transform the ideal vector ([x.sub.j]) to a corresponding vector for the ith object ([[mu].sub.i] + [A.sub.i][x.sub.j]) according to its size and coordinate system. Finally, [[epsilon].sub.ij] represents the deviation of the actual vector measurements ([y.sub.ij]) from the transformed vector caused by deformation and measurement errors. As these two sources of error are related to the size of the ith object as well as the jth location of the measurement, it is assumed that [[epsilon].sub.ij] follows a multivariate normal distribution with zero mean and covariance matrix [A'.sub.i][[SIGMA].sub.j][A.sub.i], where denotes the matrix transpose. To interpret this covariance structure, it is helpful to consider the following transformed model:
[A.sub.i.sup.-1] [y.sub.ij] = [A.sub.i.sup.-1][[mu].sub.i] + [x.sub.j] + [[epsilon]*.sub.ij], (2)
where [[epsilon]*.sub.ij] = [A.sup.-1] [[epsilon].sub.ij] has a covariance matrix [[SIGMA].sub.j]. It follows from this model that the error of the observed data after an appropriate scale transformation ([A.sub.i.sup.-1]) depends only on the location of the measurement but not on the object-to-object variation, i.e., [[SIGMA].sub.j] depends on j only. The object-to-object variation is assumed to be proportional to the object size and is modeled by the scale function [A.sub.i] as described by the covariance matrix of [[epsilon].sub.ij], [A'.sub.i][[SIGMA].sub.j][A.sub.i]. This type of...
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