## Thinplate splines for the relation of two shapes

Thus, we have a significant shape difference in the nonuniform subspace between the averages of these two samples of callosal outlines. From the representation in Figure 4.4, however, it is difficult to say in exactly what feature(s) this difference might be expressed. To render shape differences legible, we turn to an idea as old as the invention of artistic perspective in the Renaissance. We can show the changes of all the points in Figure 4.3 as one coherent graphical display by imagining one of the averages—say, the normals (the dots)—to have been put down on ordinary square graph paper. Call it the starting shape. We deform the paper so that the dots now fall directly over the other set of points, the set constituting the target shape. At the left, Figure 4.7 shows what happens to the grid.

Naturally, it matters what deformation one uses. Part of the modern efflorescence of modern morphometric methodology depends on the mathematical properties of one particular choice, the thin-plate spline, that minimizes yet another sum of squares. In this context, we are minimizing the summed squared second derivatives (integrated over the whole plane) of the map in the figure—something like the summed squared deviations of the shapes of the little squares from the shapes of their neighbors and thus a measure of local information in the mapping. It is quite remarkable that the function we want can be written out in full in elementary notation. Let the landmarks of the form on which the grid is squared be denoted Pi; i = 1,..., k, in one single image,