Augmentation of linear facial anthropometrics through modern morphometrics: a facial convexity example


Dr Peter Claes
Melbourne Dental School
The University of Melbourne
Melbourne VIC 3010


Background:  The facial region has traditionally been quantified using linear anthropometrics. These are well established in dentistry, but require expertise to be used effectively. The aim of this study was to augment the utility of linear anthropometrics by applying them in conjunction with modern 3-D morphometrics.

Methods:  Facial images of 75 males and 94 females aged 18–25 years with self-reported Caucasian ancestry were used. An anthropometric mask was applied to establish corresponding quasi-landmarks on the images in the dataset. A statistical face-space, encoding shape covariation, was established. The facial median plane was extracted facilitating both manual and automated indication of commonly used midline landmarks. From both indications, facial convexity angles were calculated and compared. The angles were related to the face-space using a regression based pathway enabling the visualization of facial form associated with convexity variation.

Results:  Good agreement between the manual and automated angles was found (Pearson correlation: 0.9478–0.9474, Dahlberg root mean squared error: 1.15°–1.24°). The population mean angle was 166.59°–166.29° (SD 5.09°–5.2°) for males–females. The angle-pathway provided valuable feedback.

Conclusions:  Linear facial anthropometrics can be extended when used in combination with a face-space derived from 3-D scans and the exploration of property pathways inferred in a statistically verifiable way.

Abbreviations and acronyms:

anthropometric mask


conventional metrical approach




principal component


principal component analysis


root mean squared error


The capacity to quantify facial form is important to most dental disciplines, in particular orthodontics, oral and maxillofacial surgery, as well as paediatric dentistry and prosthodontics. The facial skeleton forms a hard-tissue foundation that the soft-tissue overlays and this determines the visual appearance of the face.1 The importance of facial appearance outcomes in the treatment planning of individuals with cranio-dental dysmorphologies is gaining attention. For example, in the ‘face-airway-bite’ (FAB) paradigm,2 it is suggested that facial harmony is the primary objective whilst achieving a functional airway and bite.3 To achieve this paradigm shift, a means to determine facial harmony encompassing the overall soft-tissue form that can be visualized and interpreted by individuals has become the new ideal upon which treatment planning in craniofacial reconstruction is performed.4

Facial harmony can be defined as a qualitative feature encompassing both the aesthetics and function of the craniofacial complex. Conventionally, the craniofacial complex has been quantified and measured using craniofacial anthropometrics on cephalometric radiographs and 2-D photography. This has been important in the initial assessment, subsequent treatment planning process and audit of outcomes. The measurements involved, irrespective of the imaging modality used, are typical linear measurements like distances, ratios and angles between indicated landmarks.5 Many of these linear anthropometrics are well established and have been used successfully in clinical practice. Furthermore, new and alternative measurements on 3-D facial imaging, facilitating an efficient means to extract facial anthropometrics, are an active topic of research.6,7

The use of linear measurements to quantify form is commonly referred to as the conventional metrical approach (CMA)8 in morphometrics, which is the general science behind the study of form and form variations. However, when such specific measurements are used individually, they oversimplify the 3-D craniofacial complex and when they are used collectively, they are often difficult to interpret.9,10 Recent advances in morphometrics to deal with this critique have been both conceptual and technological. For example, in modern geometric morphometrics,11 actual landmark locations in 3-D are employed as against selected measurements between them. Furthermore, typical variability over similarly shaped objects is encoded in shape-spaces using a statistical shape analysis.12 Finally, recent increases in computer power combined with improved mathematical algorithms have allowed large numbers of landmarks to be used to represent the associated form,13 thereby providing a more complete representation of the facial region. For example, an anthropometric mask (AM) comprising a spatially dense set of facial anthropometric quasi-landmarks applied to a set of 3-D facial manifolds can generate a statistical shape-space that describes the harmonic covariance in complete facial form of any given population sample. Systematic variation in facial harmonies can be explored in the shape-space using any chosen metadata, e.g. commonly used facial anthropometric variables.

In this study, we investigate the utility of augmenting linear facial anthropometrics with techniques derived from modern morphometrics to enhance and automate their extraction and to investigate their variance in a ‘normal population’. Furthermore, patterns of 3-D facial form associated with variance in extracted facial convexity anthropometric variables are explored using a statistical face shape-space, or face-space for short, derived from the same ‘normal population’.

Materials and Methods


Ethics approval for ‘The characterization of three-dimensional facial profile in young adult Western Australians’ was granted from the Princess Margaret Hospital for Children (PMH) Ethics Committee (PMHEC 1443/EP) in Perth, Western Australia and ‘Establishment of identity from quantitative analysis of facial characteristics (digital 3-D facial modelling)’ was granted from The University of Melbourne, Human Research Ethics Committee (HESC 050550.1) in Melbourne, Victoria.

Three-dimensional facial images of healthy young adults with informed consent between the ages of 18–25 and of self-reported European ancestry were made available from a library of facial scans comprising The Western Australian Three-Dimensional Facial Reference Range for Children and Adolescence. Subjects completed a questionnaire on relevant health history and population affinity. Subjects were sitting in an upright position and were instructed to display a neutral facial expression in their natural head position when scanned. The 3-D facial images consist of a spatially dense set of 3-D points connected to form a wireframe made of polygons representing the facial surface as a 3-D manifold. Exclusion was made on the basis of any self-reported prior surgery or the diagnoses of a syndromic condition known to have any manifestations in the craniofacial region. The study cohort of 3-D manifolds consisted of images of 75 males and 94 females. The male and female cohorts were treated as separate populations. The precision and repeatability of the 3dMDface™ (two pod) System had been previously tested and validated by Aldridge et al.14 with sub-millimetre resolutions.

Anthropometric mask and mapping

An AM was fitted to each 3-D image in the study cohort. The mask was constructed using spatially dense and uniformly sampled (equally distanced at ∼2 mm) points on an existing averaged facial manifold covering the facial area of interest. The AM was mapped onto the 3-D facial images equivalent to indicating conventional landmarks. Because of the dense nature of the mask (∼10.000 points), manual indication of the points was impracticable and therefore an automated mapping strategy was required. The strategy used15 has been validated in terms of accuracy and consistency with the sub-millimetre resolutions required for clinical practice. The mapping strategy is akin to fitting an elastic mask onto a solid facial statue through a geometry-driven mapping of geometrical or anatomically corresponding features onto each other. Initially, the mask was roughly aligned. Then, by allowing iteratively more flexibility in the elasticity of the mask, initially larger, then progressively more local and more subtle differences were accommodated. This process was continued until the mask fitted the manifold and defined the underlying facial structure using the standardized and predetermined template points. The resulting dense set of points mapped in a quasi-anatomical manner provided a dense set of automated corresponding quasi-landmark indications over all of the 169 facial manifolds in the cohort.

Convexity angle

Godt et al.7 determined that the convexity angle most suitable for determining skeletal class was the angle subtended by Nasion-Subnasale-Pogonion N’SnPog’ landmarks, as defined by Zylinski et al.1 and illustrated in Fig 1. This angle lies in the mid-sagittal plane, excludes the nose and provides an anterior-posterior assessment of the maxilla and the mandible. Manual indication of the angle was originally performed on lateral photographs7 that provide a silhouette of the apparent midline. However, with the advent of 3-D imaging, the midline has to be determined from the 3-D manifold. To aid the manual indication of the angle on 3-D manifolds, the midline was first extracted automatically16 and then projected onto the facial manifold. Subsequently, the same landmarks used by Godt et al.7 were indicated on the midline.

Figure 1.

 Illustration of the facial convexity angle as defined by Zylinski et al.1 indicated on a generic 3-D facial manifold.

An automated angle indication was obtained via the AM and mapping strategy. A single angle indication using the midline extraction was performed on the AM. The landmarks defining the N’SnPog’ angle were incorporated in the set of landmarks on the AM and were automatically indicated on all the facial 3-D manifolds after mapping.

The N’SnPog’ angles determined from manual and automated indications were compared using a root mean squared error (RMSE) according to Dahlberg17 to score the degree of difference. A score of similarity was generated using a Pearson’s correlation coefficient. Summary statistics (mean, standard deviation and range) of male and female automated angle distributions were compared to an adult American Caucasian male1 and an adult Italian Caucasian male population18 using box-and-whisker plots.

Face-space and angle-pathway

A statistical facial shape-space of the fully landmarked facial manifolds was constructed using principal component analysis (PCA) after a generalized Procrustes superimposition of the dense landmark locations. PCA enabled the description of the maximum amount of shape variation while using the minimum number of variables. As such, it redescribed the original dataset in terms of the observed variations that are independent of each other. All principal components (PC) described a direction of facial variation independent of all other PCs in the dataset. The most prominent spread of variation within the dataset was always extracted by the first PC, and the second widest variation by the second PC. This was entirely dependent on the distribution of the dataset being studied.

The crucial disadvantage of PCA for studying a specific variation of interest was the inability to control the type of variation extracted by each PC. A solution to this problem was found19 using a property pathway which is the trajectory through the face-space defining the covariance in facial form with the variance in any given property.20 In other words, a property pathway is the combination of PCs that cumulatively expresses the direction of variation caused by any property of interest. Defining the facial convexity angle as a ‘property’ enabled the construction of an angle-pathway permitting the identification, quantification and exploration of angle-related covariations on the full 3-D facial manifold.


There was an extremely strong correlation (Pearson correlation coefficient 0.95) in N’SnPog’ angles between those extracted from manual indication and those automated for both males and females. The RMSE between the manual and automated angle extractions for males and females was 1.15° and 1.24°, respectively. These outcomes validated the utility of automated extraction of midline based facial anthropometric convexity angles throughout the remainder of the results.

Population summary statistics of the distributions of N’SnPog’ angles for males and females separately are given in Fig 2. Males had a mean value of 166.59° (SD 5.09; range 152.94°–178.99°) and females had a mean value of 166.29° (SD 5.21; range 155.52°–176.87°). Both distributions were very similar and comparable to other population studies as anticipated.

Figure 2.

 Box-and-whisker diagram of the adult Caucasian Western Australian population N’SnPog’ in relation to N’SnPog’ angles found in previous studies.

Systematic facial variation in the male face-space is illustrated in Fig 3. Synthetic faces (Fig 3 I1, I2 and I3) were generated from the face-space to illustrate facial harmonic variation at different loci within the face space. A series of faces representing the two first modes of variation, PC1 and PC2 are also given (Fig 3 PC1 and PC2). The first principal component (PC1) characterizes facial variation between fuller and thinner faces. The second (PC2) characterizes a spectrum from long to broad facial forms. Additional modes of variation exist, each characterizing specific facial spectra of variation observed within the population.

Figure 3.

 Face-space of the male cohort illustrated using 3-D facial manifold visualizations of the average (A), three random faces (I1, I2 and I3) and the principal components (PC1 and PC2), respectively.

The covariation between the convexity angle (N’SnPog’) and facial shape is encoded in an angle-pathway and this relationship is depicted in Figs 4 and 5. The magnitude of facial covariation with 1° angle variation is made explicit by visualizing a greyscale distance map (mm scale) on an average face. Differences in facial form are depicted in a series of synthetic faces generated at sequential locations along the angle-pathway for the average face of males (Fig 4C) and females (Fig 5C), ranging from 150 to 180 in steps of 10°. The direction of local facial differences associated for the same angle range is depicted in a vector field for males (Fig 4B) and females (Fig 5B).

Figure 4.

 Illustration of the angle-pathway for the male cohort. (A) Male N’SnPog’ magnitude map in mm (greyscale ranges from 0 (black) to 0.45 mm (white)) expressing the facial relation with an increase by 1° in angle value. (B) Male N’SnPog’ vector map expressing the direction of change with changing angle values from 150° to 180°. (C) Male N’SnPog’ angle-pathway visualization by actual changing or morphing of the average male face.

Figure 5.

 Illustration of the angle-pathway for the female cohort. (A) Female N’SnPog’ magnitude map in mm (greyscale ranges from 0 (black) to 0.45 mm (white)) expressing the facial relation with an increase by 1° in angle value. (B) Female N’SnPog’ vector map expressing the direction of change with changing angle values from 150° to 180°. (C) Female N’SnPog’ angle-pathway visualization by actual changing or morphing the average female face.

An increase in the N’SnPog’ angle for males generated facial shape with greater concavity and less convexity as anticipated. The majority of significant facial covariation was found in descending order of magnitude from the lower third to the middle third and then the upper third of the face. Lower facial third angle covariation is related to the chin that experiences an anterior and upward translation. The middle facial third angle covariation was related to the region around the maxilla and philtrum experiencing a posterior translation. The upper facial third angle covariation consisted in an anterior and slightly downwards translation of the forehead.

In females, similar patterns were found but with some differences. Typically, the upper third of the face changes is lower in magnitude when compared to males with differences in the vertical dimension. The middle third of the face varies similarly to males. The lower third of the face demonstrates a greater degree of difference than seen in males; this includes variance in the mandibular border as well as the chin.


This study presents proof of the concept that conventional linear facial anthropometric measurements used to quantify the facial complex can be complemented by modern morphometric techniques with available 3-D facial imaging. The development of automated means to retrieve a facial convexity angle measurement, and to then be able to quantify whole surface facial harmonic relationships with this commonly used anthropometric parameter, was used as an example of the applicability of these emerging technologies.

The demonstrated validity of the automated determination of facial midline and the facial convexity angle from a 3-D facial image provides an efficient means to extract commonly used midline anthropometric measures. This was made possible by an anthropometric mask and mapping algorithm to provide a dense set of anatomically corresponding quasi-landmarks. Coded into this mask mapping strategy is the automated indication of defined landmarks needed for the anthropometric variables to be calculated. The RMSE between the manual and automated angle extraction protocols of 1.15°–1.124° (males–females) were of the same magnitude to the RMSE of 1.07° reported by Godt et al.7 between different manual indications at separate time points on 2-D photography of profiles. This, together with the high correlation between manual and automated angle determinations, justified the automated angle extractions used in this study. Furthermore, the successful use of the anthropometric mask with its dense corresponding quasi-landmarks suggests the possibility that the extraction of other commonly used facial anthropometric variables in a highly automated manner is entirely feasible. Such an advance would find utility in both clinical practice and research as automated extraction of variables would heavily reduce the manual labour and time involved increasing their efficiency. Furthermore, measurements taken consistently and without any subjective user-intervention allow for the easy exchange of data and findings between practitioners and researchers facilitating evidence based treatment planning in the future.

The efficiency of facial convexity angle extraction for research purposes is evident in the ability to generate summary statistics for the Western Australian Caucasian population which were found to be similar to distributions reported on American and Italian populations. The growing number of libraries of 3-D facial scans worldwide, coupled with the technology reported here, will facilitate efficient means to make anthropometric comparisons between populations and provide localized reference ranges.

Through the utility of modern morphometrics, using spatially dense quasi-landmarks defined by the AM and the use of PCA, ‘normal’ facial variation within a population was encoded in a face-space. This facilitated the definition of the covariance of facial form with the variance in convexity angle and confirmed a long-standing principle assumption in orthodontic practice that facial anthropometric measurements correlate with 3-D harmonies in facial form. Modelling an anthropometric variable, such as the convexity angle as a ‘property’, the subsequent angle-pathway within the face-space would also allow systematic ‘morphing’ of a facial manifold (as in Figs 4C and 5C) according to the properties of harmonic covariance learned from a reference population. The pattern and extent of variance can be visualized in a range of formats (e.g. Figs 4 and 5) to map magnitude, and vectors encoded into a distance map or vector field providing detailed and novel information on the relationship between facial convexity angle and 3-D facial form.

Linear facial anthropometric measurements are well established and a common tool in craniofacial clinical practice. However, they require a significant amount of training and expertise to be used and understood successfully. The novel tools provided and explored in this study demonstrated that facial anthropometrics can be extracted indirectly and in an automated manner with the application of 3-D facial imaging and tools evolved from modern morphometrics. Furthermore, the capacity to define and visualize whole surface facial harmonic covariance with commonly used anthropometric variables is unique and provides access to additional information of facial form, thereby improving their use.

The appearance of the face and the harmonious disposition of features within it are important treatment outcomes. However, in the past such characteristics were difficult to quantify or predict at the treatment planning stage. Other measures derived from dental casts or radiographic images are used as best available surrogates from which facial form and composition still have to be inferred subjectively. Therefore, a more complete objective and quantitative prediction of facial form is an imperative in patient assessment and treatment planning, particularly as patient expectations rise. The integration of conventional anthropometrics with modern geometric morphometrics may fulfil these expectations. The generation and utility of face-spaces such as those used in this work to define variance in facial harmony deserves further exploration.


This work has been supported by an undergraduate research grant 2009/10 ‘Craniofacial anthropometry of region specific populations’ from the Australian Dental Research Foundation Inc, by the Australian Research Council (ARC) grant DP0772650 and by funding provided by the Princess Margaret Hospital Foundation, in Perth, Western Australia. Thanks are expressed to the participants who provided images and data for the study.