In ordinary language the shape of a geometric or pictorial object is described by measurements (for instance, angles or ratios of distances or areas) that do not change when the object is moved, rotated, or enlarged or reduced on the page or in your hands. The translations, rotations, and changes of scale that we are ignoring constitute the similarity group of transformations of the plane. When the "objects" are point sets like that in Figure 4.1, it turns out to be useful to say that their shape simply is the set of all point sets that "have the same shape."

outlines are 26-point polygons of semilandmarks as explained in the text.

We need a distance measure for shapes defined in this way. If we were talking just about sets of labeled points, a reasonable formula for squared distance would be the usual Pythagorean sum of squared distances between corresponding points over the list. It is reasonable to define shape distance as the minimum of these sums of squares over the classes that are involved in the definition of "shape"—over the classes of point sets generated by the similarity group that the notion of shape explicitly disregards. The squared shape distance between one point set A and another point set B might then be taken as the minimum summed squared Euclidean distances between the points of A and the corresponding points in point sets C as C ranges over the whole set of shapes having the same shape as B.

For this definition to make sense, we have to fix the scale of A. The mathematics of all this is most elegant if the sum of squares of the points of A around their center of gravity is constrained to be exactly 1. (The square root of this sum of squares is usually called centroid size.) The resulting squared Procrustes distance is proportional to the sum of the areas of the circles in Figure 4.2. A small adjustment of the definition is required to make it symmetric in A and B, a property that one would reasonably expect anything called a distance to have. Notation is easiest if one uses complex numbers. Write the two sets of landmarks, each one centered at (0, 0) and of centroid size unity, as vectors of complex numbers Zi with Szi = 0 and SziZi = 1. (The overbar is the operation of complex conjugation, reflection in the x-

axis.) Then the Procrustes distance between z and z' is arccos |2z¡z'|, where |w| is the complex modulus VwW. The arccosine is taken in radians; Procrustes distance is a dimensionless number between 0 and p/2. For an extended and rigorous development of this non-Euclidean geometry, see Dryden and Mardia (1998).

To average ordinary numbers, you add them up and then divide by their count. Because we can't add up shapes or divide, we borrow instead a different characterization of the ordinary average: It is also the least-squares fit to those numbers, the quantity about which they have the least sum of squared distances. Since we already have a distance between shapes, we inherit a notion of average in this way as soon as we have an algorithm for minimizing that sum of squares. That turns out not to be too difficult. One picks a tentative average, fits every form of the sample to it, substitutes the average of the fits for the previous guess at the average, and iterates. For biologically realistic data sets, this algorithm is guaranteed to converge to a shape that has the minimum summed squared distances to the original forms—just how you'd want the average shape to behave. Formally, in the complex notation introduced a few lines above, this average shape is the first principal component of the sample summed Hermitian outer product Szz', where zzl is the k X k matrix whose ijth entry is zizj (k is the number of landmarks). For the data set here, with k = 26 landmarks, the result—the sample Procrustes average—is displayed at the left in Figure 4.3.

When any or all of the points subject to this sort of analysis are semilandmarks (the "sliding" points introduced above), an additional step is inserted in this algorithm, whereby these points are simultaneously slid along their several estimated tangent directions at the same time the forms are fitted to the emerging average one by one. (In effect, we are free to continue minimizing sums of squared distances over all the other outline lines shown. Third row left: The centroids are superposed, and then one form is rotated over the other so that the sum of squared distances between corresponding landmarks is a minimum. Third row right: With the construction lines erased, the squared Procrustes distance between the pair of forms is that sum of squared distances. It is proportional to the total area of the circles. Bottom row: The Procrustes computation of average shape. Left: Emergence of an average form (solid circles) from the alternation of two-form superpositions, as above, with averages of the resulting superposed locations (big black dots). Right: Resulting sample scatter of linearized shapes, made up of five separate scatters of shape coordinates. The information in this scatter is the domain of all multivariate manipulations of landmark data, such as the group mean comparisons in this chapter.

Figure 4.3. Procrustes analysis of the data set in Figure 4.1. Left: Procrustes average shape, full sample. Right: Scatter of Procrustes fits of each specimen to the average shape. In this and all subsequent Procrustes plots, axes are in units of Pro crustes distance, which is dimensionless. There is an apparently outlying arc at the lower left here, corresponding to the form in the third row, fifth column of Figure 4.1; its presence does not affect the multivariate findings reported.

Figure 4.3. Procrustes analysis of the data set in Figure 4.1. Left: Procrustes average shape, full sample. Right: Scatter of Procrustes fits of each specimen to the average shape. In this and all subsequent Procrustes plots, axes are in units of Pro crustes distance, which is dimensionless. There is an apparently outlying arc at the lower left here, corresponding to the form in the third row, fifth column of Figure 4.1; its presence does not affect the multivariate findings reported.

points that these ''might have been'' while continuing adequately to represent the same outlines.) All the semi-landmarks slide at once so as to minimize their joint bending energy (see below) with respect to their tentative average, just as the rest of the Procrustes algebra ends up minimizing the joint Procrustes distance of all specimens from the average. The slipping step is a straightforward linear matrix operation in the coordinates of each configuration. For formulas and details, see Bookstein (1997c).

After we have computed the average, we can put each individual shape down over the average using the similarity transformation that made the sum of squares from the average a minimum in its particular case. Continuing the notation above, this superposition is (approximately) z! z(SzA)/|SzA|, where A is the Procrustes average just computed. For the data set here, there results the diagram at the right Figure 4.3. These points, the Procrustes shape coordinates of our sample, describe the variation of the whole set of shapes around the average in terms of variations ''at'' the component points separately. Any analyses that follow this construction of coordinates are of shape only. The information about scale that was sequestered when centroid size was standardized remains available for group comparisons (here, the groups differ by about 3% in average centroid size, which is not significant) or for studies of allometry (cor relation of size with shape), group differences, and the like. We will not need these additional analyses in either of the examples here, but they have not been precluded.

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