Extensions of the Procrustes Method for the Optimal Superimposition of Landmarks

Superimposition methods for comparing configurations of landmarks in two or more specimens are reviewed. These methods show differences in shape among specimens as residuals after rotation, translation, and scaling them so that they align as well as possible. A new method is presented that generalizes Siegel and Benson's (1982) resistant-fit theta-rho analysis so that more than two objects can be compared at the same time. Both least-squares and resistant-fit approaches are generalized to allow for affine transformations (uniform shape change). The methods are compared, using artificial data and data on 18 landmarks on the wings of 127 species of North American mosquitoes. Graphical techniques .are also presented to help sum- marize the patterns of differences in shape among the objects being compared. (Morphometrics; resistant-fit; least-squares; theta-rho analysis; rotational fit; affine transformations.) An important problem in morphomet- is now easy to display a transformation grid rics is that of comparing configurations of that maps the configuration of landmarks landmarks in two or more specimens. of one organism exactly into those of Thompson (1917) suggested an elegant ap- another. proach, using "transformation grids," that An alternative approach to fitting a mod- depicts the overall form of one organism el that completely describes the differences as a distortion in the shape of a reference between two organisms is to fit a very sim- organism. The basic idea was to place a ple model that only takes into consider- Cartesian coordinate grid over the refer- ation global parameters such as differences ence organism and then distort the image in rotation, translation, and scale. Geo- of the organism (including the grid) in var- metrically, this corresponds to superim- ious ways until the form of the second or- posing one organism on top of another so ganism was achieved. The differences in that its landmarks align as well as possible shapes of the two organisms are shown by (in some sense) with the positions of the the deviations of the fitted grid (usually corresponding landmarks on the second. bent and stretched in various ways) from Differences in shape are then shown by the original simple square grid. Thompson differences in positions of corresponding (1917) sketched the grids subjectively landmarks. Shape differences between two without an explicit specification of which organisms are found by studying these re- landmarks were used. Not all landmarks siduals. These methods are the subject of shown in pairs of drawings are located ex- the present paper. actly where the superimposed grids would Sneath (1967) investigated the problem imply they should be. This means that the of finding the optimal translation, rota- grids should be more complex than those tion, and size change of one object in order shown in Thompson (1917) in order to ac- for it to be superimposed on another. The curately show the differences between two two objects were represented as sets of x,y- organisms. Bookstein (1978) developed the coordinates of landmarks. A least-squares method of biorthogonal grid analysis criterion was used to measure the goodness which quantifies Thompson's approach and of fit of one object to another. Gower (1971) makes it objective. But it is complex and further developed Sneath's (1967) method has not been applied very often. A recent and expressed the operations in terms of b~ak-through (Bookstein, 1989) is the use matrix algebra. Siegel and Benson (1982) of methods based on thin-plate splines. It made the important observation that a