Building ‘3D growth curves’

The growth curves were trained from 5443 males and females aged between 0.05 and 88 years from the normative sample described above. Full computational details are given in Supplementary Information ², Section ², and the code is made available at https://github.com/harrymatthews50/3DGrowthCurves. Briefly, the 3D growth curves comprise a series of age and sex specific expected facial shapes and parameterizations of the variation around the expected shape. Facial shape was represented as a n (observations) x k variables matrix, where the k variables are x,y and z co-ordinates of the 7160 quasi-landmarks imposed on each face, after generalized Procrustes analysis was performed to remove non-shape related variation. To estimate the expected facial shape, we used locally weighted (kernel) linear regression^19,20. The local regression model was a partial-least-squares (PLS) regression of facial shape onto age²¹ and the weighting function was Gaussian (see Supplementary Information ², Sections ^2.1.2 and ^2.1.3). The weighting function had a variable width (standard deviation of the Gaussian) so as to accommodate the varying amounts of training data available at different ages while ensuring a representative model of normal variation (see Supplementary Information ², Section ^2.2). Essentially this models facial shape as a non-parametric, non-linear function of age as was done in Matthews et al.²².

We estimate normal facial variation for each age and sex within the residuals of a similarly weighted PLS-regression model with training faces recentered on the expected face for that age and sex (Supplementary Information ², Section ^2.1.4). The residuals of this regression model constitute age-adjusted displacements of each point on each face from the expected face, weighted according to the Gaussian weighting function. Variation at each point on the face was calculated as the root-mean-square (RMS) of the displacements at each point in the anterior–posterior; superior-inferior; and medial–lateral anatomical directions separately. RMS was also calculated along the direction normal to the surface of the expected face at each point, as well as the total magnitude of the displacement. RMS values were then scaled by the inverse of the sum of the weights. Hereafter we refer to these as ‘pointwise standard deviations’. See Supplementary Information ², Section ^2.1.5.1 for further details.

Variation at each age and sex was also modelled as separate statistical shape models (SSMs). A SSM comprises modes of variation (which are linear transformations of facial shape) and the expected variation along each mode. Modes of variation theoretically span all plausible realistic faces for a given age and sex and can be used to determine faces or facial regions that fall outside of this span. Each SSM was fitted using a singular value decomposition of the residuals described in the previous paragraph (Supplementary Information ², Section ^2.1.5.2), without column centering or standardization. For each SSM we retained the components explaining 98% of the variation.

As the normative training data cannot be made fully available, at https://github.com/harrymatthews50/3DGrowthCurves we provide models (SSMs and pointwise standard deviations) evaluated for every 0.3 years of age between 0.5 and 68.9 years for females and 0.5–51.8 years for males (see Supplementary Information ², Section ^2.2 for the rationale for the differing age ranges).