We study shapes of facial surfaces for the purpose of face recognition. The main idea is to 1) represent surfaces by unions of level curves, called facial curves, of the depth function and 2) compare shapes of surfaces implicitly using shapes of facial curves. The latter is performed using a differential geometric approach that computes geodesic lengths between closed curves on a shape manifold. These ideas are demonstrated using a nearest-neighbor classifier on two 3D face databases: Florida State University and Notre Dame, highlighting a good recognition performance.

International audienceA statistical analysis of shapes of facial surfaces can play an important role in biometric authentication and other applications. The main difficulty in developing such an analysis comes from the lack of a canonical system to represent and compare all facial surfaces. This paper suggests a specific, yet natural, coordinate system on facial surfaces, that enables comparisons of their shapes. Here a facial surface is represented as an indexed collection of closed curves, called facial curves, that are level curves of a surface distance function from the tip of the nose. Defining the space of all such representations of face, this paper studies its differential geometry and endows it with a Riemannian metric. It presents numerical techniques for computing geodesic paths between facial surfaces in that space. This Riemannian framework is then used to: (i) compute distances between faces to quantify differences in their shapes, (ii) find optimal deformations between faces, and (iii) define and compute average of a given set of faces. Experimental results generated using laser-scanned faces are presented to demonstrate these ideas

Given data points p0, . . . , pN on a closed submanifold M of R n and time instants 0 = t0 < t1 < . . . < tN = 1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as "regular" as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M . The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in R n and on the unit sphere.

Insights into potential differences among the bony labyrinths of Plio-Pleistocene hominins may inform their evolutionary histories and sensory ecologies. We use four recently-discovered bony labyrinths from the site of Kromdraai to significantly expand the sample for Paranthropus robustus. Diffeomorphometry, which provides detailed information about cochlear shape, reveals size-independent differences in cochlear shape between P. robustus and Australopithecus africanus that exceed those among modern humans and the African apes. The cochlea of P. robustus is distinctive and relatively invariant, whereas cochlear shape in A. africanus is more variable, resembles that of early Homo, and shows a degree of morphological polymorphism comparable to that evinced by modern species. The curvature of the P. robustus cochlea is uniquely derived and is consistent with enhanced sensitivity to low-frequency sounds. Combined with evidence for selection, our findings suggest that sound perception shaped distinct ecological adaptations among southern African early hominins.

Sex differences in behavioral and neural characteristics can be caused by cultural influences but also by sex-based differences in neurophysiological and sensorimotor features. Since signal-response systems influence decision-making, cooperative and collaborative behaviors, the anatomical or physiological bases for any sex-based difference in sensory mechanisms are important to explore. Here, we use uniform scaling and nonparametric representations of the human cochlea, the main organ of hearing that imprints its adult-like morphology within the petrosal bone from birth. We observe a sex-differentiated torsion along the 3D cochlear curve in samples of 94 adults and 22 juvenile skeletons from cross-cultural contexts. The cochlear sexual dimorphism measured in our study allows sex assessment from the human skeleton with a mean accuracy ranging from 0.91 to 0.93 throughout life. We conclude that the human cochlea is sex-typed from an early post-natal age. This, for the first time, allows nondestructive sex determination of juveniles’ skeletal remains in which the biomolecules are too degraded for study but in which the petrosal is preserved, one of the most common bone within archaeological assemblages. Our observed sex-typed cochlear shape from birth is likely associated with complex evolutionary processes in modern humans for reasons not yet fully understood.

Morphometric assessments of the dentition have played significant roles in hypotheses relating to taxonomic diversity among extinct hominins. In this regard, emphasis has been placed on the statistical appraisal of intraspecific variation to identify morphological criteria that convey maximum discriminatory power. Three-dimensional geometric morphometric (3D GM) approaches that utilize landmarks and semi-landmarks to quantify shape variation have enjoyed increasingly popular use over the past twenty-five years in assessments of the outer enamel surface (OES) and enamel-dentine junction (EDJ) of fossil molars. Recently developed diffeomorphic surface matching (DSM) methods that model the deformation between shapes have drastically reduced if not altogether eliminated potential methodological inconsistencies associated with the a priori identification of landmarks and delineation of semi-landmarks. As such, DSM has the potential to better capture the geometric details that describe tooth shape by accounting for both homologous and non-homologous (i.e., discrete) features, and permitting the statistical determination of geometric correspondence. We compare the discriminatory power of 3D GM and DSM in the evaluation of the OES and EDJ of mandibular permanent molars attributed to Australopithecus africanus, Paranthropus robustus and early Homo sp. from the sites of Sterkfontein and Swartkrans. For all three molars, classification and clustering scores demonstrate that DSM performs better at separating the A. africanus and P. robustus samples than does 3D GM. The EDJ provided the best results. Paranthropus robustus evinces greater morphological variability than A. africanus. The DSM assessment of the early Homo molar from Swartkrans reveals its distinctiveness from either australopith sample, and the "unknown" specimen from Sterkfontein (Stw 151) is notably more similar to Homo than to A. africanus.

This paper studies the problem of analyzing variability in shapes of facial surfaces using a Riemannian framework, a fundamental approach that allows for joint matchings, comparisons, and deformations of faces under a chosen metric. The starting point is to impose a curvilinear coordinate system, named the Darcyan coordinate system, on facial surfaces; it is based on the level curves of the surface distance function measured from the tip of the nose. Each facial surface is now represented as an indexed collection of these level curves. The task of finding optimal deformations, or geodesic paths, between facial surfaces reduces to that of finding geodesics between level curves, which is accomplished using the theory of elastic shape analysis of 3D curves. Elastic framework allows for nonlinear matching between curves and between points across curves. The resulting geodesics provide optimal elastic deformations between faces and an elastic metric for comparing facial shapes. We demonstrate this idea using examples from FSU face database.

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