Comparing semi-landmarking approaches for analyzing 3D cranial morphology

Abstract

Objectives:

Increased use of three-dimensional imaging data has led to a need for methods capable of capturing rich shape descriptions. Semi-landmarks have been demonstrated to increase shape information but placement in 3D can be time consuming, computationally expensive, or may introduce artifacts. This study implements and compares three strategies to more densely sample a 3D image surface.

Materials and methods:

Three dense sampling strategies: patch, patch-TPS, and pseudo-landmark sampling, are implemented to analyze skulls from three species of great apes. To evaluate the shape information added by each strategy, the semi or pseudo-landmarks are used to estimate a transform between an individual and the population average template. The average mean root squared error between the transformed mesh and the template is used to quantify the success of the transform.

Results:

The landmark sets generated by each method result in estimates of the template that on average were comparable or exceeded the accuracy of using manual landmarks alone. The patch method demonstrates the most sensitivity to noise and missing data, resulting in outliers with large deviations in the mean shape estimates. Patch-TPS and pseudo-landmarking provide more robust performance in the presence of noise and variability in the dataset.

Conclusions:

Each landmarking strategy was capable of producing shape estimations of the population average templates that were generally comparable to manual landmarks alone while greatly increasing the density of the shape information. This study highlights the potential trade-offs between correspondence of the semi-landmark points, consistent point spacing, sample coverage, repeatability, and computational time.

Introduction

Quantitative assessment of morphological variation using landmark coordinates placed on three-dimensional images requires the ability to locate homologous points across images. Gold-standard methods generally rely on an expert to manually place landmarks at locations that are considered ‘biologically homologous’ (Adams et al., 2013; Bookstein, 1997b). The shape information captured using these anatomical landmarks is limited by the number of these landmark points available, often resulting in a sparse representation of the anatomy. Regions that pose challenges to reliable landmark identification, such as smooth surfaces, poorly defined boundaries between tissues, or large morphological differences, can contain biologically relevant variability that traditional landmark analysis may not be sufficient to capture (Watanabe, 2018).

Semi-landmarks are widely used to supplement the information provided by manually placed-landmark points (Bookstein, 1997a; Gunz et al., 2005; Gunz & Mitteroecker, 2013). The biological homology of the semi-landmark points is not guaranteed as is the goal when using traditional landmarks, but by relaxing the requirement for correspondence, shape information can be obtained between landmark points that is not possible to access using manual landmarks alone. Pseudo-landmarks, which in this paper we use to refer to points placed automatically on an image surface with no relationship to manually placed landmarks, further relaxes the requirement for homology by placing points at densely sampled points that correspond across images (Boyer et al., 2015).

All methods for placing semi-landmark and pseudo-landmark points pose tradeoffs between several factors including: improving the correspondence of the points across images, providing regular samples, sample coverage, repeatability, and computational time (Bardua et al., 2019; Gonzalez et al., 2016; Gunz et al., 2005). When selecting a semi-landmarking method, the impact of each of these factors should be considered for the specific application and dataset.

While many detailed descriptions of the mathematical basis of semi-landmark analysis have been published and toolkits for statistical analysis such as the R packages Morpho (S Schlager, 2016) and geomorph (Adams & Otárola-Castillo, 2013) are freely available, there are comparably fewer resources for generating large numbers of surface points for 3D data. In this study, we have implemented and tested three general-purpose automated semi-landmarking strategies for 3D surface data and have applied each to analyze cranial morphology in three species of great apes, using the open-source biomedical visualization platform 3D Slicer. The first method, patch-based semi-landmarking, projects semi-landmarks to a mesh surface from triangular patches constructed from the manual landmark points. The landmarks are applied to each specimen independently, preserving the geometric interpretation of each semi-landmark. The second method, patch-TPS, generates a semi-landmark set for a single template mesh using the patch-based method. These semi-landmarks are transferred to each sample in a dataset using a thin-plate spline (TPS) transform followed by projection along the template surface normal vectors. The third method, pseudo-landmark sampling, generates a set of points from a template model regularly sampled at arbitrary locations with no geometric relationship to the original landmarks. The points are projected to the external surface of the model under the assumption of a spherical topology and a spatial filter removes coincident points to enforce a user-specified minimum distance between points. The pseudo-landmarks placed on the template mesh are projected to each sample using a TPS transform and projection along the normal vectors of the template.

The semi-landmarks produced by these methods can be further optimized by sliding on the image surface to minimize either the bending energy or the Procrustes distance between all the samples and the mean shape. Sliding semi-landmarks has been demonstrated to reduce the artifacts that may be introduced by the use of arbitrary equidistant samples (Gunz et al., 2005; Gunz & Mitteroecker, 2013). In this study, we have analyzed the semi-landmarks directly to evaluate the methods for automated initial point placement and highlight the strengths and drawbacks of each approach in a typical evolutionary developmental biology focused geometric morphometrics study in which multiple species are represented.

Materials and methods

Images

DICOM stacks of three species of great apes, Pan troglodytes (N=11), Gorilla gorilla (N=22), and Pongo pygmaeus (N=18), housed in the skeletal collection of the National Museum of Natural History (NMNH) were acquired from the Anthropology Division of the NMNH. No new specimens were collected for this study. All data comes from vouchered specimens that were previously collected and digitized by the Smithsonian Institution. As such no ethical approval is required. DICOM stacks were converted into volumes and reviewed for completeness of the cranial features. The list of specimens included in the analysis and the associated manual set of landmarks used in the study can be found as supplemental online material (SOM). Manual landmarks on these specimens are collected by CD for a previous study using 3D Slicer (Davis & Maga, 2018). All methods described below are publicly available as part of SlicerMorph, a 3D morphometrics and landmark based shape analysis extension to the open-source 3D visualization program 3D Slicer (Fedorov et al., 2012; Kikinis et al., 2014). The source code for all analysis in this work is available from the SlicerMorph repository, a 3D geometric morphometrics extension to the functionality of the 3D Slicer (https://github.com/SlicerMorph/SlicerMorph).

Patch-based semi-landmarking (patch)

In the patch approach (Figure 1), the user specifies three landmarks already digitized on the specimen to define the boundaries of the triangular region from which a patch of semi-landmarks will be generated on the surface. The triangle is the smallest shape configuration that can be used to represent a patch. Any other polygonal outline can be broken down into a series of triangles, giving the user maximum flexibility in selecting the regions covered. For each patch, a template triangular grid with a user-specified number of semi-landmark points is registered to the vertices of the bounding triangle using a thin-plate-spline deformation. The vertices of the triangular sampling grid are then projected to the surface of the specimen using the following algorithm:

Figure 1:

Patch placement of semi-landmarks. First, a triangular patch with regular sampling is defined (a). The triangle patch is warped to the image by creating a thin plate transform between each triangle endpoint and three manual landmark points from the mesh. The points are then projected from the warped triangle patch to the image surface using the surface normal from the three landmark points (b). Individual patches covering the extent of the area defined by the manual landmarks are connected to form the final patch placed landmark set (c).

1. Smooth the surface normal vectors to reduce the impact of noise (e.g. holes, bumps) on the mesh surface. The template model is smoothed using Laplacian smoothing to adjust the coordinate locations and “relax” the mesh, while preserving the number and connectivity of vertices. The smoothed surface normal vectors are extracted for each manual landmark point.

2. Estimate the orientation of the model surface underlying each triangular patch to set the projection vector direction. The surface orientation is estimated by taking the average of the surface normal vectors at the three manual landmark points defining the patch.

3. For each sampling point on the triangular patch, cast a ray in the direction of the projection vector. The length of the ray is constrained by the average distance between the vertices of the triangle. The final intersection between the ray and the surface mesh is selected as the projected point.

4. If there is no intersection, reverse the direction of the projection vector ray and select the first intersection.

5. In the case that no intersection was found, select the closest mesh point to the triangular sampling point.

When all triangular patches of sampled points have been projected onto the image surface, the grids are merged into a single set of landmarks. The merging step prevents overlap between adjacent triangles and improves coverage. The triangular patches of semi-landmarks are merged according to the following steps:

1. Identify each unique triangle edge in the grid

2. Place a uniformly sampled line between the edge endpoints. The sampling rate is specified by the user.

3. Project sample points from the line onto the image surface.

4. Add manual landmark points to the final set.

This method provides a repeatable way to generate points on a sample without prior knowledge of the mean shape of other specimens in the dataset. Each semi-landmark point has a known geometric relationship to the manual landmark points that define the particular patch. The coverage of the semilandmarks is dependent on the availability of manual landmarks in the region of interest. The placement of each patch of semi-landmarks is dependent on the underlying geometry, due to the assumption of a single underlying curvature at each grid location. Noise in the data, sharp curves or edges in sampled regions can result in placement errors, such as sampling of an interior surface.

Patch-based semi-landmarks applied through Thin-Plate Splines (patch-TPS)

In the patch-TPS approach, a single template image (whether a synthetic or a representative specimen from the study sample) is used to place the triangular grids and semi-landmark points as described in the previous section. These landmarks are then transferred to every specimen in a dataset using the following steps:

1. Warp the subject to the template using a thin-plate-spline transformation defined by the manual landmark points from the template and specimen models.

2. For each semi-landmark point, cast a ray from the semi-landmark point on the template in the direction of the normal vector. Select the final intersection with the warped subject mesh as the intersection point.

3. If there is no intersection, reverse the normal vector and select the first intersection with the warped subject mesh.

4. In the case no intersection is found, select the closest mesh point on the warped subject mesh.

This method provides an improvement in robustness over the previous method, particularly if the template image is a synthetic population average. The initial semi-landmarks are placed on a template that is typically smooth and the placement has been validated. The TPS transform (Bookstein, 1989) estimates a mapping function between two surfaces with corresponding points. The transform is interpolated between corresponding points by minimizing the bending energy required to deform the subject to the target surface, resulting in the smoothest possible transformation. When applied to each specimen in the first step, this transform has the effect of removing some of the size and shape variation before placement. Projecting the points along the template surface normal vectors at each semilandmark point provides an improvement over the surface normal estimation from the normal vectors at the patch vertices of each specimen. The patch-TPS method is limited to the extent of the manual landmarks and unlike the patch method, the placement of points on a single new subject requires the template image and corresponding landmarks and may necessitate recalculating the population average used for the template image.

Pseudo-landmark sampling

In the pseudo-landmark sampling method (Figure 2), a dense set of pseudo-landmarks are generated on a single template, which can be a synthetic image or a representative specimen from the study sample. Points are transferred to individual samples using the TPS-based projection method described above. The pseudo-landmarks are initialized by subsampling the template model to enforce a user-specified minimum sampling distance. The subsampled points are regularly distributed across the surface at arbitrary positions and include points from the interior and exterior surface of the mesh. To remove points placed on interior structures that will not be included in the analysis, each point is projected onto the most external model surface using the following steps:

Figure 2:

Pseudo-landmark sampling. A plane of symmetry is placed by the user on the Gorilla gorilla template model, using the midline manual landmarks as a guide (a). The template model is downsampled to produce a simplified mesh with the number of vertices determined by the user-defined minimum sampling distance (b). Each vertex of the simplified mesh is projected to the most external model surface along a ray in the direction of the surface normal at that point (c). In the final step, spatial filtering is applied to enforce the minimum sampling distance on the projected points and the landmarks are symmetrized by cropping the pointset with the plane of symmetry, reflecting the remaining points across the plane, and projecting the points back to the model surface (c).

1. cast a ray from the point on the subsampled model in the direction of the normal vector at that location. Select the final intersection with the warped subject mesh as the intersection point.

2. If there is no intersection, reverse the normal vector and select the first intersection with the warped subject mesh.

3. In the case where no intersection is found, select the closest mesh point on the subject mesh as the intersection point.

As a final post-processing step, a spatial filter is applied to remove duplicate points within the user-specified sampling distance to eliminate points that may have been projected to the same location on the model surface. The pseudo-landmarks points are then transferred from the template to each individual using the TPS-based projection method.

An option is provided to symmetrize the pseudo-landmark set across a plane of symmetry placed by the user. If this step is included, we recommend that a symmetric template model be used, as identifying a plane of symmetry on a non-symmetric specimen is challenging and can produce variable results due to the assumption of symmetry in this algorithm. To produce the symmetric point set, the pseudo-landmarks are cropped using the plane of symmetry and the remaining points are mirrored across the plane. Each mirrored point is projected to the image surface along a projection vector calculated by mirroring the surface normal vector from the corresponding point on the opposite side of the plane. Points that are within the sampling distance of the symmetry plane are merged into a single point. This support for symmetrizing the pseudo-landmarks supports analysis of bilateral symmetry and allows the user to enforce an equal number of points placed on each side of the symmetry plane.

Since the pseudo-landmarks generated by this method are initially placed on a single template image, the points can be edited on the model before transfer. Additional points can be added to regions of interest or be removed from structures that are highly variable or missing in some images (e.g. teeth).

The pseudo-landmark sampling method provides an improvement in sampling coverage for datasets with geometry that can be approximated by a spherical geometry. This is a particular advantage over the patch-based methods when the coverage of the manual landmarks is limited or non-existent in an area of interest. One important tradeoff for this improvement in coverage is that the geometric interpretation of the semi-landmarks is no longer directly related to the landmark positions and the position of the semi-landmarks on the template image is arbitrary.

Analysis

All analysis was done using the SlicerMorph extension of 3D Slicer. Sample data, including 3D meshes, and the scripts used to run these experiments are available in the SlicerMorph Github repository (https://github.com/SlicerMorph/SlicerMorph).

Estimating the template specimen

While semi and pseudo-landmarks provide improvements in sample coverage and consistency, evaluating the shape information provided by these points is a challenging task. The goal of including these dense samples is to capture variation outside of the manual landmarks, which are traditionally treated as the “ground truth” of morphological difference. To address this, we propose an evaluation that treats the template model points as missing information that is estimated from an individual specimen model and the landmarks produced by each method, as outlined in Figure 3. For each specimen, a TPS transformation is defined by the manual landmarks on the specimen and those on the template model. TPS is a standard method of estimating the value of a spatial property at unsampled locations from a series of sampled locations. In this case, the known point correspondences from the manual landmarks represent the sampled points in a dense, 3D transform that maps each point on the specimen model to a point on the template model. When this transform is applied to the specimen model, it will approximate the template model, with a better approximation resulting in higher degree of similarity between models. In the case where the transform between the two models is fully described by the landmark set, or where all point correspondences are correctly identified, the estimated model would be identical to the template model. Our hypothesis is that denser sampling with a larger number of correctly corresponding points will provide a better approximation of the template and decrease the distance between the models.

Figure 3:

The evaluation workflow for determining the quality of shape information produced by the semi and pseudo-landmarks compares their ability to estimate the template model using each subject and the corresponding landmark points. For each specimen in a group, a TPS transform is defined by the landmark points on the specimen and template models. The specimen is warped to the template model, producing an estimation that is compared to the actual template using the nearest point distance, a standard mesh similarity metric. A landmark set with more complete shape information is expected to produce a better estimate of the template model, so the error between the estimated and actual template is used to assess quality of the shape information added by each semi and pseudo-landmark set.

To estimate the similarity of the estimated and template models, a pointwise mesh difference is evaluated by calculating the nearest point distance at each mesh vertex on the template model. The quality of the TPS approximation was scored by the average root mean square error (RMSE) in the region defined by the extent of the manually placed facial landmark points, as reported in Table 1. An example of this evaluation process for one specimen is shown in Figure 4. Transforming each specimen into the template space could introduce some bias as the template is typically a synthetic image and not generated directly from the CT scans, which is considered to be the gold standard. However, the transform to the template space allows for the influence of noise or missing data to be reduced and facilitates the identification of a consistent facial region with an identical number of points.

Table 1.

Average RMSE and standard deviation (SD) between the group average mesh and the individual approximation of the mean mesh for each semi-landmarking method. All errors are reported in millimeters.

Species	N	Manual landmarks (mm)	Patch semi-landmarks (mm)	Patch-TPS semi-landmarks (mm)	Pseudo-landmark sampling (mm)
Gorilla gorilla	22	1.267 (SD=0.177)	1.177 (SD=0.162)	1.076 (SD=0.521)	0.741 (SD=0.097)
Pan troglodytes	11	0.805 (SD=0.095)	2.675 (SD=2.010)	0.787 (SD=0.061)	0.525 (SD=0.071)
Pongo pygmaeus	18	0.957 (SD=0.109)	2.569 (SD=1.135)	0.543 (SD=0.056)	0.478 (SD=0.054)

Figure 4:

To generate the RMSE scores for each landmark method, each specimen is warped using a TPS transform to the group template using the original landmarks and the semi-landmarks generated by that method. In the example in this figure, the specimen (b) was warped to the Gorilla gorilla template mesh (a) using the patch-TPS method. The maximum mesh distance between the template and the transformed specimen was calculated and is displayed at each point on the template in (c). The RMSE for the facial region defined by the extent of the facial landmarks shown in (d) for this specimen was 0.871.

The patch-based semi-landmarks were placed using 20 grid patches on each template, for a total of 880 semi-landmarks in the facial region. This method provided an improvement over the TPS warping using the manual landmarks alone for the Gorilla gorilla dataset. For the Pan troglodytes and Pongo pygmaeus datasets, the variation in the datasets caused the patch placement to fail in regions resulting in no improvement over the manual landmarks. The patch-based semi-landmark method using the same semi-landmark model templates transferred to other specimens using a TPS-based projection resulted in an improvement over the manual landmarks in each dataset. An example showing how the patch-TPS method adds robustness in the presence of variation in the dataset is shown in Figure 5. The pseudo-landmark sampling method was used with 1523 pseudo-landmarks distributed over the Gorilla gorilla template surface, 1520 pseudo-landmarks on the Pan troglodytes template, and 1557 pseudo-landmarks on the Pongo pygmaeus template. The pseudo-landmark numbers were dependent on the template geometry due to the spatial filtering step. Landmarks on the teeth were removed before analysis as the teeth are highly variable and missing in some specimens. The pseudo-landmark method resulted in an improvement over the manual landmarks alone in each dataset.

Figure 5:

This example demonstrates where the patch-TPS placement method outperforms the patch method. The patch method alone succeeds on the template in (a) but when applied to the individual sample in (b) noise such as holes in the surface on the right side of the skull cause the projection vectors from the semi-landmark patches to the surface to be estimated incorrectly. The semi-landmark patches applied using the patch-TPS method in (c) provide a significant improvement as individual variability is removed by first warping the subject to the template and then projecting the landmarks to the surface using the surface normals from the template which are smoother than the surface normals from an individual. The RMSE for this individual using the patch semi-landmarks method is 3.176 compared to 0.593 for the patch-TPS landmarks.

Evaluating the difference between the species in morphospace

As a second criteria to evaluate the effectiveness of the semi-landmarking methods, we used the ability of the patch and patch-TPS methods to differentiate the species of apes in the dataset in morphospace after applying generalized Procrustes analysis (GPA) followed by principal component analysis (PCA) to the combined dataset. The scaling step of the GPA is skipped (i.e., partial Procrustes superimposition) to preserve the physical scale of data. In this experiment, it is not expected that the inclusion of dense shape information will result in greater discrimination between groups, especially as interspecies variations are expected to be gross shape differences that can be captured by the manual landmarks. However, the inclusion of a large number of semi-landmark points, which may introduce noise if there are errors in the point correspondences, has the potential to obscure the original signal. The goal of this analysis is to demonstrate that the species separation is maintained, despite the addition of dense shape information.

The spherical surface sampling method is omitted from this analysis, as it produces a variable number of landmark points for each template image, due to the final spatial filtering step, hence it is not possible to conduct a joint superimposition. This illustrates one of the primary drawbacks of the spherical surface sampling method in doing multi-species analysis where no common template is available. In multispecies studies, where the morphological disparity across the species are low this could potentially be addressed by generating a multi-species template.

The projection of the subjects into the first two principal components of Procrustes shape space for each method are shown in Figure 6. The manual landmarks (Figure 6a) and the patch-TPS semilandmarks (Figure 6c) were able to recover three distinct species clusters. However, the patch method produced poor separation between groups (Figure 6b). This is consistent with the high warping error results for the Pan troglodytes and Pongo pygmaeus species using the patch method seen in Table 1. Due to differences in morphology in the dataset, the application of the triangular patches that succeeded on the template failed to be placed correctly in several subjects, as demonstrated in Figure 5. The failure of a single patch results in large distortions in the semi-landmark configurations and subjects with these large distortions were outliers in morphospace.

Figure 6:

To demonstrate the capability of the semi-landmarking methods to discriminate between species in morphospace, the subjects are projected onto the first two principal components of Procrustes shape space. Gorilla gorilla are shown in black, Pan troglodytes in red, and Pongo pygmaeus in green. The manual landmarks (a) and the patch-TPS semi-landmarks (c) were able to discriminate between the three groups, while the patch semi-landmarks (b) produced poor separation.

Discussion

Although semi-landmarking of surfaces promises to yield detailed shape information for samples that are difficult to analyze using manual landmarks, its use has been limited in part by the challenges posed by acquiring large numbers of regularly sampled points in three dimensions (Gonzalez et al., 2016). In several recent studies, surface semi-landmarks are placed individually on a template, using the same procedure as placing landmarks points (Bardua et al., 2019). The semi-landmarks are transferred semi-automatically to each subject in a dataset from the template (Schlager, 2017). When the number of semi-landmark points is high, manually placing each semi-landmark point can be time consuming and prone to user variation. For a specimen with spherical topology (i.e. closed surface without holes) surfaces, mesh simplification can be used to randomly subsample the surface (Aristide et al., 2016). Checkpoint is a widely used commercial program for 3D shape analysis that provides the ability to sample semi-landmark patches but requires purchase of a license. Methods such as Auto3DGM (Boyer et al., 2015) that produce a very dense set of pseudo-landmarks have been proven to be highly interesting, but can provide drawbacks in terms of computation time, complexity, and the variation allowed between specimens. The goal of the methods implemented in this work were to provide flexible, fast, and freely available methods to generate dense surface samples. The semi-landmarks produced by these methods can be further refined by manually editing the point placement on a template for transfer across specimen or sliding landmark procedures (Gunz et al., 2005; Gunz & Mitteroecker, 2013).

When the data quality is high (e.g. mesh quality, specimen completeness) and the specimens have similar morphology, the three methods implemented here have similar performance, as is demonstrated in the Gorilla gorilla dataset. In such a case, each method can provide a slight improvement in the estimate of shape when compared to the manual landmarks only and provide spatial information about regions between landmark points that may be difficult to capture with manual landmarks alone. The image quality, region of interest, and amount of morphological variation expected in the dataset should be considered when selecting a semi-landmarking method as each has different drawbacks and benefits.

The patch semi-landmarking method provides shape information between manual landmark points such that each semi-landmark point has a geometric interpretation relative to the original landmark points. No population average or template mesh is required to generate the initial grid and it can be applied to new subjects without the comparison to the original dataset. The primary drawback is the sensitivity of the patch placement to variation in the dataset. The samples digitized may have actual damage or reabsorption pits that create holes in the mesh surface. In addition, the process of converting the scan data to a 3D model can introduce errors. Typically, the mesh is generated by selecting a single threshold for a dataset to define the isosurface for each sample. Variations in imaging quality and acquisition or ambiguous surface topology can result in holes in the mesh or the incorporation of background noise into the mesh surface. These artifacts can cause errors in the estimation of the surface normal vectors that are used to project the semi-landmarks from the template to an individual specimen. Errors in these projection vectors cause the semi-landmarks to be projected to non-homologous regions, resulting in distortion in the landmark configuration. Increasing the smoothing of the normal vectors in the first step of this algorithm can help to reduce this source of noise although it adds to the processing time. The option to adjust this smoothing parameter is provided in the SlicerMorph extension module. Patch placement failures can also occur when the specimens have large morphological differences from the template used to construct the grid of patches used for a dataset. Expert knowledge is incorporated in the construction of the initial grid to optimize coverage in the region of interest and there is an assumption that the region selected is free from holes, noise, and has a geometry that can be approximated by a smooth triangular patch. Successful placement of a grid on a template or representative individual typically indicates that the grid can be applied automatically across the group. However, if the morphological differences are large, placement on an initial specimen or template may not guarantee that a patch can be placed in this region across specimen, as the normal vector used for projection may not be a reasonable approximation of the orientation of the normal vector on the surface of the target surface. The distortion introduced to the landmark configuration when even a small number of semi-landmarks are projected to the wrong surface can cause large differences in warping error, as shown in Table 1, and a loss of discriminating power in morphospace, as demonstrated in Figure 6. Additionally, the semi-landmarks generated by this method can only be placed in regions defined by the manual landmarks, so their extent is limited to regions covered by the original landmark set.

The patch-TPS semi-landmark method improves the robustness of the patch placement process over a group of samples. The initial TPS transform of each specimen to the template removes large-scale size and shape differences from the mean before placement. The template meshes typically have smoother surfaces due to averaging and better data quality than the individual meshes, especially when a synthetic population average is used (Maga et al., 2017). Using the smoother surface normal vectors for projection and estimating the surface normal at each point on the surface instead of from the triangular patch vertices reduces errors in the estimation of projection vectors due to noise in the subject meshes and limitations imposed by using a placement patch lying in a single plane. The robustness to noise and variation in the dataset provided by this method can result in improved performance, as is seen with the Pan troglodytes and Pongo pygmaeus datasets. The joint GPA analysis demonstrated that including semi-landmark points still allows for separation between groups (Figure 6) despite the addition of information about finer scale changes in facial morphology.

It is critical to note that the homology of the semi-landmarks produced by this method are dependent on the template used to transfer the semi-landmarks. When adding a new specimen to a group, the semi-landmarks should be transferred using the same template image. Conducting GPA and PCA across groups with semi-landmarks placed by projection from different group templates as in the validation demonstrated in this study can introduce a source of possible distortion. While this distortion is limited as each patch placement is constrained by landmark points known to be homologous across the species, it could be mitigated further by using a synthetic hybrid template. Similar to the patch method, the patch-TPS method is limited to the extent of the manual landmarks. Cases where the manual landmarks are limited in the region of interest or the underlying geometry cannot be estimated by a triangular patch, can pose significant challenges to the application of this method.

The pseudo-landmark sampling method outperforms both semi-landmarking methods for each species when evaluated using the average template estimation metric. In addition, it provides dense coverage over the entire image, not limited to the region bounded by the facial landmarks. This demonstrates the impact of the improvements to surface coverage and sampling regularity provided by pseudo-landmarks when the requirement to preserve known biological homology is relaxed. The pseudo-landmark method is especially suited to cases where shape information is needed in a region of interest where manual landmark coverage is very sparse or unavailable. The user selects the initial sampling rate, which influences the final semi-landmark number, although this cannot be determined in advance due to the final spatial filtering step. When the parameters have been set to generate a reasonable sampling of a template image surface it can provide a pseudo-landmark template that can be transferred to the remaining specimen using TPS-based projection with high accuracy. One drawback of this method is that it requires all samples, including the one used as a template, to be identical in terms of preserved anatomy. If the regions do not correspond, the semi-landmark points may end up mapped to regions with no underlying homology. To improve the point placement, features which should be excluded from morphological analysis can be segmented and removed before analysis from all samples. Alternatively, the template landmark set can be edited to remove landmarks from these locations before transferring them to the group. A second drawback presented by this method is that the number of semi-landmark points cannot be selected and is dependent on the template model geometry. For this study, while the method performs well at estimating the mean population template, it could not be evaluated by joint GPA and PCA analysis due the different number of semi-landmark points and the lack of homology between the semi-landmarks chosen on the template models from different groups. Where the patch and patch-TPS placement methods produce semi-landmark points with a geometric interpretation dependent on the manual landmark points, the points identified by the spherical sampling method represent a random surface sampling. As with the patch-TPS method, this could be addressed by defining a common template for transferring the semi-landmark points across groups.