Shape and the information in medical images: A decade of the morphometric synthesis

In the past decade there have been unprecedented theoretical and practical advances in morphometrics, the multivariate statistics of object shape. In this approach, shapes are defined as equivalence classes of discrete point-sets under the operation of the Euclidean similarity group, Three basic ideas underlie scientific analysis of these constructs, First, a distance (Procrustes distance) between every pair of shapes is characterized by a least-squares formula. This metric makes the set of shapes into a Riemannian manifold (Kendall's shape space) and leads immediately to a sturdy algorithm for averaging samples of shapes, Second, a complete linear multivariate analysis of the equivalence classes goes forward after normal projection of these equivalence classes onto the tangent plane to the manifold in the vicinity of the average shape. Third, effects on shape, such as comparisons of group means, may be visualized by the thin-plate spline, an interpolant that also, indirectly, supplies an orthonormal basis for the tangent space. The methods apply quite generally to representations of scenes in terms of corresponding (landmark) points; many extend to the additional information in smooth curves of characteristic shape. This article reviews the three main features of the present toolkit, the underlying statistical and geometric models, and the extensions, some already demonstrated and some currently under construction, that reach out to the additional information content of biomedical images in common clinical or scientific applications. The tasks for which these tools are suited include standardizing scenes against Euclidean similarity transformations or shear transformations, detecting, testing, encoding, or visualizing patterns of shape variation within or between groups, and averaging images after shape variation is normalized. The optimal protocols the new methods provide for many formal problems of image understanding should be considered for versions of the same problems in diverse other application domains. (C) 1997 Academic Press.