Comparing shapes, understanding evolution.

T he study of shape has intrigued some of the brightest minds of humanity, from Leonardo Da Vinci and Carl Friedrich Gauss to some of the top scientists of the modern era. The mathematics to analyze shapes are both beautiful and challenging, covering a variety of tools, from topology (1), to metric and differential geometry (2, 3), to statistics (4). The applications are very diverse and potentially life-changing, and they range from brain research to structural biology and archeology. At the core of this research area is shape comparison: determining how to match, compare, and compute the distance between pairs of shapes. This is the topic addressed by the work of Boyer et al. (5) in PNAS, with a unique interdisciplinary team of anthropologists, archaeologists, computer scientists, and mathematicians. Understanding why comparing shapes is so important is better illustrated with a number of examples. Let us start with brain research, where, for example, we are interested in investigating how the brain changes as we grow (6) or how a normal brain compares with that of a patient who has Alzheimer’s disease. In this case, the shapes can be obtained from MRI and can, for example, represent the gray-white matter boundary, thereby challenging us to find maps and distances between such 2D surfaces (7). In HIV research in structural biology, data can be obtained via cryotomography, and it is important to understand the underlying shape of the envelope glycoproteins that mediate virus binding to initiate infection and how this shape changes, for example, in the presence of antibodies, information critical for the development of a vaccine. Shape comparison and matching are fundamental to compute the conformations of such protein complexes (8). Boyer et al. (5) present unique applications in anthropology, archeology, and evolution, as discussed below. Let us not forget that our surrounding world is composed of shapes and that shape analysis is critical to navigate it as well as to develop automatic systems capable of emulating basic human performance, such as answering the simple question “Is this a chair?”. Comparing shapes is difficult because of the intrinsic complexity of shapes in nature (e.g., proteins, human brains) as well as the large variability encountered within shape classes. Although simple characteristics to compare shapes, such as volume, can already provide valuable information, more sophisticated features and distances are needed most often. These distances can be derived following the computation of a correspondence between the shapes: a map between points in the shapes being compared. Computing such correspondence, and, from it, distances, is the essence of the work of Boyer et al. (5) and of much of the literature in this area in recent years (finding correspondence has applications beyond shape comparison and is critical, for example, in morphing, as exemplified by the famous Michael Jackson musical video Black or White). Finding an appropriate correspondence between shapes is often addressed by considering a discrete set of landmarks or corresponding points or curves. This approach is common, for example, in Procrustes analysis, a form of statistical shape analysis that derives its name from the mythological Greek rogue who made his victims fit his bed by stretching their limbs or cutting them off, and has been very popular in brain matching (7). Some landmarks might be natural for some classes of shapes (e.g., tip of the nose for faces) but are not for others. Even if they are easy to define and universally acceptable by the corresponding community (which is often not the case), marking them requires having experts in the field or developing advanced computational techniques (which are often problematic by themselves); is very time-consuming, and thereby forbidden for large datasets; is subject to much subjectivity in their selection; and is also prone to errors and contamination. Avoiding landmarks altogether is thereby desirable, as done by Boyer et al. (5) based on a combination of beautiful and computable mathematical structures, including Monge–Kantorovitch mass transportation theory and conformal maps. The Monge–Kantorovitch theory follows from work of the French mathematician Gaspard Monge 3 centuries ago and the Soviet mathematician Leonid Kantorovitch in the past century, and it relates to the study of optimal transportation and resources allocation problems. Conformal maps are angle-preserving and are familiar to the readers because they are often used in cartography to map the round earth onto the plane. Fig. 1. Finding corresponding/matching points between shapes is a very challenging problem and is critical for a number of applications (A, with kind permission from Springer Science+Business Media: International Journal of Computer Vision, A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching, Vol 89, 2009, pp 266–286, A. M. Bronstein, M. M. Bronstein, M. Mahmoudi, R. Kimmel, and G. Sapiro, Fig. 11; B, manufactured by Plan Toys, Thailand). Author contributions: G.S. wrote the paper.