Fig 1.
These sculptures demonstrate the homology provided by shared topographic features, and the morphological variations that make individuals distinctive. Each sculpture is a ‘model’—a simplification, an abstraction, that addresses only selected, salient properties of an object at the omission of others. (A) Head of a Satyr, Roman, 2nd Century AD, Uffizi Gallery; (B) Busto di Cosimo II de’ Medici, Tommaso Fedeli, 1624, Uffizi Gallery; (C) Futur ou Une jeune femme anglaise, Fernand Khnopff, 1898, Musee D’Orsay. Photos by Kent A. Stevens.
Fig 2.
Modeling feature attributes by blendshape deformers.
The smooth ridge in (A) is a Catmull-Clark ‘subdivision surface’ [51] created by the recursive subdivision of a simple polygonal mesh of ‘control vertices’. The vertices of the polygonal mesh are shown in yellow. The white vertices on the surface are the interpolated counterparts of the control vertices upon the surface shown in white. The smooth surface is manipulated indirectly by adjusting these control vertices, much as the shape of a Bézier curve is adjusted by shifting its control points. In this way, discrete attributes such as the height and width of the ridge can be implemented by shifting select control vertices. Here, blendshape deformers are used to modify the positions of control vertices. For example, ridge height can be controlled by a deformer that interpolates (with some coefficient α) between two homologous meshes that represent two height extremes (B). Likewise, ridge width is modified by a second blendshape deformer that interpolates between the meshes in (C) by some coefficient β. The results of the two deformations can then be combined (D) for various α and β. Since the two deformers create displacements in perpendicular directions, they do not create ‘blendshape interference’ (see text). The position of the ridge could be added as another independent attribute by a third blendshape that shifts all control points associated with the ridge. For further discussion of subdivision surfaces and blendshape deformers, see [51–58].
Table 1.
Attributes of facial features.
Thirty-six facial features are tabulated below, each with associated attributes, for a total of 71 attributes. Widths are mediolateral; lengths and heights are superoinferior; and protrusions and depths are anteroposterior; x, y, and z are positions in these same axes respectively. Throughout this paper, we systematically refer to these anatomical terms by the abbreviations listed here, such as “FOR_protrusion” rather than the more cumbersome “anteroposterior protrusion of the forehead”.
Fig 3.
Examples of nose variation created by combinations of 18 nose attributes.
Each attribute produces a local deformation restricted to one of three orthogonal orientations. In combination they create a large space of linear combinations of those nose shapes.
Fig 4.
A parametric model with 71 attributes permits a large ‘face space’.
To appreciate the space of possible faces, if each attribute were very conservatively assumed to support only four perceptually-distinct values (i.e., a just noticeable difference of roughly 0.25 in the normalized range from 0.0 to 1.0), the model would permit 471 (i.e., 1042) combinations. See also S3 Movie.
Fig 5.
Each face was modeled parametrically using the TFM to create a close approximation of its corresponding photogrammetric scan (see also S4 Movie).
Fig 6.
Each digital scan was landmarked with 43 conventional landmarks, which includes some new landmarks (2, 14/29, 23/38, 24/39, and 28/43) that provide coverage across areas that are modeled in the TFM. This face is a rendered TFM model.
Fig 7.
Evaluation of mean fit between model and scan.
The photogrammetric averages for each group of 20 individuals are shown in (A) for reference. The models in (B) are the corresponding computed averages of the individual models in each group. The heatmaps in (C) show the mean disparity between each model and its corresponding digital scan. The rightmost model in (B) and its heatmap in (C) are averages based on all 80 models. Table 2 shows that more than 80% of the sample points across the models were separated by less than 1 mm.
Table 2.
Percentage of sample points of the model within a given distance of the original scan (millimeters).
Table 3.
The first ten eigenvalues for PCAs of the Procrustes and TFM analyses.
Fig 8.
Bivariate plots comparing principal component scores for the Procrustes and TFM datasets.
Scores from Procrustes (left) and TFM (right) data are shown for principal components 1 and 2 (top row), and 1 and 3 (bottom row). PCA of the Procrustes data segregates ancestry groups along PC1 fairly well. The sexes appear best differentiated along PC3, but less clearly than along TFM PC 2. For the TFM, PCs 1 and 2 essentially distinguish the two geographic ancestry groups and the two sexes, respectively, whereas all groups largely overlap in PC3. Open/closed circles = Female/Male; red/blue = EUR/EAS.
Table 4.
Attribute comparison of 40 European versus 40 East Asian models (pooled sex).
Geographic ancestry groups were found to differ in mean shape based on Wilks’s Lambda. Those individual attributes that were significantly different based on univariate t-tests with Bonferroni correction are listed. Attributes are sorted in order of decreasing difference between means (Δ) as quantified in terms of standard deviations.
Table 5.
Attribute comparison of 40 female versus 40 male (pooled geographic ancestry).
Biological sexes were found to differ in mean shape based on Wilks’s Lambda. Those individual attributes that were significantly different based on univariate t-tests with Bonferroni correction are listed. Attributes are sorted in order of decreasing difference between means (Δ) as quantified in terms of standard deviations.
Fig 9.
Visualization of four warps of the Procrustes data.
The two geographic ancestry groups are shown in the upper row and the two sexes in the lower row. The magnitude heatmaps have different scales (red corresponds to 9 mm for ancestry differences and 3 mm for sex differences). The orientation of displacement is encoded in the ternary maps with the same convention as used elsewhere in this study.
Fig 10.
Visualization of differences based on the TFM dataset.
Mean differences are shown for geographic ancestry group (top row), sex (middle row), and size (bottom row) based on the TFM dataset. The first two columns show the mean face associated with each factor. The third column shows the magnitude of difference with the same heatmap scale, where red corresponds to 35 mm (see key). Since differences in ancestry and sex are more subtle than the size differences, the fourth column shows their relative magnitudes scaled to the maximum displacement for each case (10.8, 9.7, and 35.0 mm for ancestry, sex, and size, respectively). The last column shows displacement orientation.
Table 6.
Attributes most strongly correlated with size as measured by meanVertexRadius.
Size was found to have a significant relationship with shape based on Wilks’s Lambda. Those individual attributes that were significantly correlated with size with Bonferroni correction are listed. Attributes are sorted in order of decreasing correlation coefficient (r2).
Fig 11.
Differences between EAS and EUR on a per-region basis.
Displacement magnitude is shown in the upper set of heatmaps, and orientation is shown in the lower set of ternary maps. The magnitude and orientation maps labeled ‘All’ are from the top row of Fig 10.
Fig 12.
Visualization of sexual dimorphism (upper) and size (lower) on a per-region basis.
Displacement magnitude is shown in the upper set of heatmaps, and orientation is shown in the lower set of ternary maps. The magnitude and orientation maps labeled ‘All’ are from the second and third rows of Fig 10.
Table 7.
Angles between group and size vectors.
Angles near 90° or 270° degrees indicate statistical independence, while 0° or 180° indicate perfect correlation, values between these indicate degrees of correlation. Values above the diagonal are using TFM data, whereas those below are from Procrustes data. ‘Anc’ = geographic ancestry, ‘Sex’ = biological sex, ‘MVR’ = meanVertexRadius, and ‘CS’ = centroid size.
Fig 13.
Exaggerating shape differences.
The difference between two models can be exaggerated by increasing, for each of their 71 attributes, the difference between the value of that attribute relative to their mean by a multiplicative factor (2.0 in this demonstration). If, for example, a given attribute has values 0.3 and 0.5, they differ by ±0.1 relative to their mean, and with that difference doubled, the exaggerated values would be 0.2 and 0.6. Averaged models for EAS and EUR are shown in (A), each the average of 20 individual models (10 female and 10 male each). The corresponding exaggerated models are shown in (B). Likewise, (C) shows averaged models for females and males, each the average of 20 individual models (10 EAS and 10 EUR each), and their corresponding exaggerations are shown in (D). See also the corresponding animations S5 Movie (EAS-EUR) for (A), S6 Movie (exaggerations of EAS-EUR) for (B), S7 Movie (female-male) for (C), and S8 Movie (exaggerations of female-male) for (D).