Volume 10, Issue 4 p. 541-552
Free Access

ariaDNE: A robustly implemented algorithm for Dirichlet energy of the normal

Shan Shan

Corresponding Author

Shan Shan

Department of Mathematics, Duke University, Durham, North Carolina

Shan Shan
Email: [email protected]Search for more papers by this author
Shahar Z. Kovalsky

Shahar Z. Kovalsky

Department of Mathematics, Duke University, Durham, North Carolina

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Julie M. Winchester

Julie M. Winchester

Department of Evolutionary Anthropology, Duke University, Durham, North Carolina

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Doug M. Boyer

Doug M. Boyer

Department of Evolutionary Anthropology, Duke University, Durham, North Carolina

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Ingrid Daubechies

Ingrid Daubechies

Department of Mathematics, Duke University, Durham, North Carolina

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First published: 11 January 2019
Citations: 19


  1. Shape characterizers are metrics that quantify aspects of the overall geometry of a three-dimensional (3D) digital surface. When computed for biological objects, the values of a shape characterizer are largely independent of homology interpretations and often contain a strong ecological and functional signal. Thus, shape characterizers are useful for understanding evolutionary processes. Dirichlet normal energy (DNE) is a widely used shape characterizer in morphological studies.
  2. Recent studies found that DNE is sensitive to various procedures for preparing 3D mesh from raw scan data, raising concerns regarding comparability and objectivity when utilizing DNE in morphological research. We provide a robustly implemented algorithm for computing the Dirichlet energy of the normal (ariaDNE) on 3D meshes.
  3. We show through simulation that the effects of preparation-related mesh surface attributes, such as triangle count, mesh representation, noise, smoothing and boundary triangles, are much more limited on ariaDNE than DNE. Furthermore, ariaDNE retains the potential of DNE for biological studies, illustrated by its effectiveness in differentiating species by dietary preferences.
  4. Use of ariaDNE can dramatically enhance the assessment of the ecological aspects of morphological variation by its stability under different 3D model acquisition methods and preparation procedure. Towards this goal, we provide scripts for computing ariaDNE and ariaDNE values for specimens used in previously published DNE analyses.


Developing methods to model and understand tempo and mode of macroevolution is an important goal for evolutionary biology (e.g. Eastman, Alfaro, Joyce, Hipp, & Harmon, 2011; Harmon et al., 2010; Ingram, & Mahler, 2013; Revell, 2012). Equally important are methods for effectively representing phenotypic differences between species (Adams & Otárola-Castillo, 2013; Pampush, Spradley et al., 2016; Winchester, 2016) without which many evolutionary modelling questions would be moot (Slater & Friscia, 2018). The goal of rapidly and objectively quantifying morphological phenotypes benefits greatly from the advent of easily accessible and widely available three-dimensional (3D) digital models of anatomical structures. The unprecedented accessibility of 3D data is a direct result of technology improvements and cost reductions for generating them (Copes, Lucas, Thostenson, Hoekstra, & Boyer, 2016), as well as proliferation and population of archives for sharing them (Boyer, Gunnell, Kaufman, & McGeary, 2016).

Better shape quantification is timely because analysing wrong or poorly justified traits may lead to misimpressions about which processes meaningfully describe a clade’s evolution (Slater & Friscia, 2018). Instead, justifications for which morphological traits to study and how to quantify those traits should be based on clade-specific hypotheses that propose both an evolutionary mode and an ecological explanation. In other words, demonstrating that one or more traits follow a particular evolutionary model does not go very far towards understanding the evolutionary processes at play in a clade, especially if there is no hypothesis relating variation in those traits to ecological variation. For instance, although Harmon et al. (2010) showed that the ‘adaptive radiation’ (Osborn, 1902) or ‘early burst’ (EB) model of evolution was rarely supported among dozens of clades tested, their study did not specify why the particular morphological traits they looked at should follow the EB model. Showing that different traits can have different evolutionary patterns in the same clade, Meloro and Raia (2010) found that tooth size and carnassial angle variables followed very different evolutionary patterns within Carnivora. Carnassial angle, arguably the more directly functional variable, followed an adaptive radiation model, while m1 size followed a simpler Brownian motion model. As another example of the importance of trait function, Cantalapiedra, Prado, Fernández and Alberdi (2017) chose to quantify relative tooth crown height (hypsodonty) in order to understand drivers of disparity and diversity in equids. Hypsodonty has seemingly obvious adaptive significance for grazing in many clades, even beyond horses. Moreover, hypsodonty has been formally demonstrated by Eronen, Evans, et al. (2010) and Eronen, Polly, et al. (2010) to be an ecometric (Eronen, Evans, et al., 2010; Eronen, Polly, et al., 2010; Polly & Head, 2015) for grassland use in equids.

Shape characterizers that quantify aspects of the overall geometry of a surface are a promising class of metrics (Evans, 2013). They are distinguished from ‘shape descriptors’, primarily including geometric morphometric quantifications of shape (Adams & Otárola-Castillo, 2013). Examples of shape characterizers include relief index (RFI; for example, M’kirera & Ungar, 2003), orientation patch count (OPC; for example, Evans & Janis, 2014; Evans, Wilson, Fortelius, & Jernvall, 2007; Melstrom, 2017) and Dirichlet normal energy (DNE; for example, Bunn et al., 2011; Pampush, Spradley et al., 2016; Winchester, 2016). RFI measures the relative height and sharpness of an object; OPC measures the complexity or rugosity of a surface; DNE measures the bending energy of a surface. When different teeth exhibit substantially different values of a shape characterizer, they also look different and have easily conceivable functional and ecological differences. For instance, a tooth with higher relief often has sharper blades or cusps that pierce food items more effectively than a tooth with lower relief. As another example, DNE differences in mammalian baculum shape potentially correspond to difference in genital form and function (Gardiner, Behnsen & Brassey, 2018).

Compared to popular shape characterizers like RFI and OPC, DNE has several advantages. Although DNE is landmark-free and independent of the surface’s initial position, orientation and scale, RFI and OPC rely on the orientation of the tooth relative to an arbitrarily defined occlusal plane. Thus, DNE is less susceptible to observer-induced error/noise. In addition, OPC relies on the orientation of the tooth with regard to rotation around the central vertical axis. Furthermore, direct comparisons show that DNE has a stronger dietary signal for teeth than RFI and OPC (Winchester et al., 2014). This greater success in dietary separation is likely due to its more effective isolation of information on the ‘sharpness’ of surface features. In contrast, RFI only measures the relative cusp and/or crown height which does not describe sharpness; OPC is less sensitive to changes in blade orientation due to its binning protocol (Boyer, Evans & Jernvall, 2010).

Mathematically, DNE computes a discrete approximation to the Dirichlet Energy of the normal, a continuous surface attribute, coming from differential geometry. This quantity is defined as the integral, over the surface, of change in the normal direction, indicating at each point of the surface, how much the surface bends. In practical applications, a continuous surface is represented as a triangular mesh, that is, a collection of points or nodes and triangles. (We note that the nomenclature is not standardized across all scientific fields; in computer science these would be called vertices and triangular faces respectively; see, for example, Botsch, Kobbelt, Pauly, Alliez & Lévy, 2010). To compute DNE on such a discrete mesh, normal directions must be estimated for each point/triangle. The sum of the change of normal directions over the points/triangles is then used to approximate the Dirichlet energy of the normal for the continuous surface that the mesh represents. However, the DNE algorithm published in MorphoTester (Winchester, 2016) and in the R package ‘molaR’ (Pampush, Spradley et al., 2016) is sensitive to varying mesh preparation protocols and requires special treatment for boundary triangles, which are triangles that have one side/node that fall on the boundary of the mesh (Pampush, Spradley et al., 2016, Spradley, Pampush, Morse & Kay, 2017), raising concerns regarding the comparability and reproducibility when utilizing DNE for morphological research.

Recent attempts to address this issue have developed protocols for standardizing the mesh preparation process (Spradley et al., 2017). Unlike previous work, we provide a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE), that is insensitive to a greater range of mesh preparation protocols. Figure 1 shows DNE and ariaDNE values on an example tooth and various meshes representing the same surface: (from left to right) 2k triangles, 20k triangles, a different mesh representation, 0.001 noise (by adding a random number normally distributed with mean 0 and standard deviation 0.001 to the points), 0.002 noise (similar to 0.001 noise but with standard deviation set to be 0.002) and smoothing. The red surface shading indicates the value of curvature as measured by each approach; it is uniformized across each row by the row’s highest local curvature value. To demonstrate this insensitivity empirically, we test the stability of our algorithm on tooth models with differing triangle count, mesh representation (i.e. a different set of points/nodes or triangles representing the same continuous surface) and simulated noise. We also test the effects of smoothing and boundary triangles as in Spradley et al. (2017). We furthermore assess the dietary differentiation power of ariaDNE.

Details are in the caption following the image
Comparing effects of triangle count, mesh representation, noise and smoothing on a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) (top) and dirichlet normal energy (DNE) (bottom). (a) shows the distribution of curvature as measured by each method overlaid in shades of red on a grey 3D rendering of the surface. Normalized ariaDNE and DNE values (by the values for the typical tooth) are shown above each surface and summarized in the bar plots (b), demonstrating the robustness of ariaDNE vs. DNE


2.1 Methodological analysis of ariaDNE

Bunn et al. (2011) noted the relevance of the differential geometry concept of Dirichlet energy of the normal for morphology and provided an algorithm called DNE calculating an approximation to this quantity on discrete surface meshes by summing the local energy over all triangles. The local energy on a triangle is defined by the total change in the normals; this provides a local estimate for the curvature of the surface. However, this change in normals is sensitive to how a continuous surface is discretized. That is, a different triangle count, mesh representation, or contamination by noise or small artefacts can all lead to significantly different numerical values.

To address this sensitivity problem, we leverage the observation that the local energy can be also expressed by the curvature at the query point on the surface (Willmore, 1965); another simple method for estimating curvature on discrete surfaces is by principal component analysis (PCA). The procedure is outlined as follows. For each query point, find all its neighbouring points within a fixed radius; the value of this radius is set as a parameter for the method (Yang, Lai, Hu & Pottmann, 2006). Then, apply PCA to the coordinates of those points; the plane spanned by the first two principal components typically approximates the tangent plane to the surface at the query point, with the third principal component approximating the normal direction. The corresponding smallest principal component score urn:x-wiley:2041210X:media:mee313148:mee313148-math-0001 indicates the deviation from the fitted plane, that is, the curvature.

There are two issues with this PCA method: (a) The third principal component does not always approximate the normal direction. Therefore, the smallest principal component score may not accurately reflect curviness as we discussed above, that is, the deviation of the surface from the tangent plane. Figure 2 (top) shows an erroneous normal approximation for a cusp, where the normals should be perpendicular to the surface but using standard PCA gives skewed estimation. (b) Standard PCA becomes numerically unstable (due to ill-conditioning) when the number of nearby neighbours is low. This implies that when the triangle count is low, there may not be enough points for PCA.

Details are in the caption following the image
Improved normal estimation with our modified method in a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE). Top: traditional principal component analysis (PCA) method gives skewed normal estimates on a pointed cusp, leading to erroneous curvature approximation. Bottom: our modification gives better normal approximation, and therefore improves curvature approximation

To resolve the first issue, we modify the algorithm to choose at each query point the principal component closest to its normal, and set the curvature at that point to be the score of the chosen principal component. Figure 2 (bottom) illustrates that this modification produces estimates more consistent with surface normals, thereby providing a better local estimate of the tangent plane, and in turn curvature. In practice, normals at a point are obtained by taking a weighted average of normals of adjacent triangles, easily computed on discrete meshes.

To resolve the second issue, we propose a modification to the traditional PCA method. Selecting neighbours within a fixed radius could result, near some point, in a small-sized neighbourhood where few or even no points would be selected; instead, we apply a ‘weighted PCA’, with weights decaying according to the distance away from the query point, retaining the rest of procedure. There are many ways to define the weight function. Indeed, using an indicator function that outputs one for points within the chosen radius and zero elsewhere recovers the tradition PCA method. For ariaDNE, we set the weight function to be the widely used Gaussian kernel urn:x-wiley:2041210X:media:mee313148:mee313148-math-0002.

The Gaussian kernel captures local geometric information on the surface. The parameter ε indicates the size of local influence. Figure 3 illustrates effects of different ε on the weight function: the larger ε, the more points on the mesh have significant weight values, resulting in larger principal component scores for those points. In consequence, when ε increases, ariaDNE becomes larger. In practice, we suggest using ε ranging from 0.04 to 0.1. If ε is too small, ariaDNE will be highly sensitive to trivial features that are most likely to be noise (similar to traditional DNE); if ε is too large, the approximation will simply become non-local. Choosing an appropriate value of ε depends on the application in hand.

Details are in the caption following the image
Effect of increasing the ε parameter (bandwidth) on the weight function (top; red indicates highest weight) and curvature computed by a robustly implemented lgorithm for Dirichlet energy of the normal (ariaDNE) for molar teeth. Choices for ε are 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, with surface shading similar to Figure 1. When ε is small, both Dirichlet normal energy (DNE) and ariaDNE capture fine-scale features on the tooth. When ε is larger, ariaDNE captures larger scale features

In summary, (a) we apply a weighted PCA, localized around each query point by means of the Gaussian kernel function; (b) we find the principal component that is closest to its normal and set the curvature to be its principal score; (c) we integrate this curvature estimate along the surface to obtain ariaDNE. See Appendix A in the Supporting Information for detailed procedure.

2.2 Study samples

Understanding the correlation between surface geometry and metrics like DNE or ariaDNE helps understand whether these metrics are relevant to questions concerning morphology, ecology and evolution. The meaningfulness and success of a metric have to be measured against relevant samples and the research questions.

Here, we use a sample of new world monkeys’ (platyrrhine) second mandibular molars downloaded from Morphosource (Winchester et al., 2014). The sample has substantial taxonomic breadth (seven extant platyrrhine primate genera) and depth (10 individuals per genus), consisting of meshes (117,623–665,001 points, 234,358–1,334,141 triangles) from Alouatta, Ateles, Brachyteles, Callicebus, Chiropotes, Pithecia and Saimiri. Platyrrhine dentitions have been essential for questions about dental variation and dietary preference (Allen, Cooke, Gonzales & Kay, 2015; Anthony & Kay, 1993; Dennis, Ungar, Teaford & Glander, 2004; Ledogar, Winchester, St, Clair & Boyer, 2013; Pampush, Spradley et al., 2016; Pampush, Winchester, et al., 2016; Winchester et al., 2014). Questions have included how dietarily diverse platyrrhines should be considered based on available behavioural data, whether and how dental morphology is reflective of diet differences, and how important tooth wear, individual variation and scale of geometric features are when considering tooth differences between species. In the following sections, we tested the stability of ariaDNE by perturbing attributes like triangle count and mesh representation. We also tested the effects of noise, smoothing and boundary triangles on ariaDNE. Furthermore, we assessed its power in differentiating the seven platyrrhine primate species according to dietary habits.

2.3 Sensitivity test

2.3.1 Triangle count

To evaluate the sensitivity of ariaDNE under varying triangle count, each tooth was downsampled to produce simplified surfaces with 20k, 16k, 12k, 8k, 4k and 2k triangles. We computed ariaDNE values (ε = 0.04,0.06,0.08,0.1) using the MATLAB function ‘ariaDNE’ provided in Section 23. For comparison, we also computed DNE values using the function ‘DNE’ (Section 23), a MATLAB port of the R function ‘DNE’ from ‘molaR’. Default parameters were used for DNE, with outlier percentile at 0.1 and boundary triangles excluded.

2.3.2 Mesh representation

A continuous surface can be represented by different discrete meshes; even with the same triangle count, they can differ by altering the position of points and their adjacency relations (i.e. triangles). We would like ariaDNE to be roughly the same for all meshes that represent the same continuous surface. To evaluate the sensitivity of ariaDNE under varying mesh representation, we tested on a surface generated by a mathematical function as well as real tooth samples. First, we tested it on the surface S defined by z = 0.3 sin (2x) sin (2y) where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 (Figure 4). To generate a mesh, we randomly picked 2,000 sets of (x,y) coordinates uniformly distributed on 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and calculated their accompanying z-values using the equation above. Each set of (x,y,z) coordinates represented a node/point in the mesh, and the triangles are obtained by applying Delaunay Triangulation to these points. We generated 100 meshes by repeating these steps and computed their DNE and ariaDNE values as in Section 4. We remark here that meshes generated by this procedure do not necessarily have evenly distributed points; some areas of the mesh can have smaller triangles than others.

Details are in the caption following the image
Effect of varying mesh representation on a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) and dirichlet normal energy (DNE) values computed for a synthetic surface (top) and a tooth from Ateles (bottom). Left panel: examples of different mesh representations. Right panel: scatter plots and box plots of ariaDNE (ε = 0.08) and DNE values computed for N meshes representing the synthetic surface (top, N = 100) and the tooth surface (bottom, N = 70)

Real tooth samples are already given as meshes; we generated new mesh representations for each tooth sample by computing pairwise surface correspondences. Specifically, points and triangles from one surface were taken to the other surface in the samples by correspondence maps computed using the methods in (Boyer et al., 2011), between all pairs of surfaces in the sample. These correspondences resulted in 70 different mesh representations for each tooth in the sample. We computed their DNE and ariaDNE (ε = 0.04,0.06,0.08,0.1) as in Section 4.

2.3.3 Simulated noise

We tested the stability of ariaDNE when simulated noise was added to the surface defined as in Section 5 and real tooth samples. First, given a mesh representing the same surface S as in Section 5, a noisy mesh was obtained by adding a random variable uniformly distributed on [−0.001,0.001] to the x,y,z coordinates of each node/point on the mesh (Figure S1). We then generated 100 noisy versions of the given mesh by repeating the previous steps.

For real tooth data, we generated a noisy mesh by adding a random variable uniformly distributed on [−0.003,0.003] to the x,y,z coordinates for each node/point in the mesh (Figure S1). The noise level was chosen arbitrarily; we added more noise to the tooth samples to increase diversity of the test cases. We obtained 100 noisy meshes per tooth, and computed their DNE and ariaDNE (ε = 0.04,0.06,0.08,0.1) values as in Section 4.

2.3.4 Smoothing

Smoothing is commonly used to eliminate noise produced during mesh preparation. Spradley et al. (2017) tested the effects of various smoothing operators and smoothing amounts on DNE with surface meshes of hemispheres and primate molars. They suggested that aggressive smoothing procedures like Laplacian smoothing and implicit fairing should be avoided. To evaluate the performance of ariaDNE under different smoothing algorithms, we randomly picked seven tooth models from our sample (one from each taxa) and generated their smooth surfaces by applying 100 iterations of the Avizo smoothing module, three iterations of the Meshlab function HC Laplacian Smoothing or three iterations of the implicit fairing method using MorphoTester. Then, we computed their DNE and ariaDNE (ε = 0.08) as in Section 4.

We further evaluated the effects of varying amounts of Avizo smoothing on ariaDNE; we iteratively applied the Avizo smoothing module to a single molar tooth from Ateles. The smoothing function was performed in intervals of 20 on the raw surface mesh, evenly spaced from 20 to 200 to generate 10 new surface meshes. Default value for lambda was kept (lambda = 0.6). We computed their DNE and ariaDNE (ε = 0.08) as in Section 4.

2.3.5 Boundary triangles

Triangles with one side/node that are on the boundary of the mesh have a large impact on DNE, calling for special treatment (Spradley et al., 2017). We assess how such boundary triangles affect ariaDNE on two molar teeth, one of Ateles where crown side walls are relatively bulged outwardly, and one from Brachyteles where crown side walls are relatively unbulged (Figure 6). For each tooth, we found its boundary triangles and computed their local energy using ariaDNE and DNE (‘BoundaryDiscard’ =‘none,’ that is, no boundary triangles will be removed).

2.4 Tests on species differentiation

Previous studies have revealed systematic variation among species with different dietary habits in the values of DNE and other topographic metrics, such as RFI. To test the differentiation power for species and their dietary preferences, we compared RFI, DNE and ariaDNE on the 70 mandibular second molars in our platyrrhine sample. Under the diet-classification scheme from Winchester et al. (2014), Alouatta and Brachyteles are folivorous, Ateles and Callicebus are frugivorous, Chiropotes and Pithecia are hard-object feeding and Saimiri is insectivorous. For each tooth, we computed RFI, DNE and ariaDNE (ε = 0.02, 0.04, 0.06, 0.08, 0.1, 0.12) as in Section 4. We then used ANOVA and multiple comparison tests to assess the differentiation power of these different metrics for dietary preferences.


3.1 Sensitivity tests

In numerical analysis, an algorithm is stable if perturbing inputs do not significantly affect outputs. To enable comparison, the change in the outputs can be quantified by coefficient of variation, which is the result of dividing the standard deviation by the mean. Tables S1–S4 provide coefficients of variation of DNE and ariaDNE values of the perturbed meshes in each collection per tooth model. For each tooth and each perturbed collection, the coefficient of variation of ariaDNE is less than that of DNE, meaning ariaDNE is relatively more stable than DNE under varying triangle count, mesh representation and noise. Table 1 summarizes results, indicating means of coefficients of variation from Tables S1–S4. Figure 5 illustrates effects of increasing triangle count on ariaDNE (ε = 0.10) and DNE values computed for seven arbitrarily chosen teeth (one per genus). The ariaDNE values for each tooth (maximum percent change: 3.42%) remain relatively unchanged compared to DNE (maximum percent change: 384%). Figure 4 illustrates ariaDNE is relatively more stable when the mesh representation is changed. In the scatter plots, ariaDNE and DNE values are normalized to have a mean one in each case; in the box plots, the values are normalized to have a median one in each case. Similar result holds for adding noise to the surface. See Figure S1.

Details are in the caption following the image
Effect of increasing triangle count on a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) (left) and Dirichlet normal energy (DNE) (right) values computed for seven teeth from Alouatta, Ateles, Brachyteles, Callicebus, Saimiri, Chiropotes, Pithecia. The ariaDNE values for each tooth remain relatively unchanged, compared to the DNE values, under varying triangle count
Table 1. Robustness of a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) under various mesh attributes perturbation. For each tooth in the 70 platyrrhine sample, we generated three collections of perturbed meshes by varying triangle count, mesh representation or adding simulated noise. We computed the coefficient of variation of their dirichlet normal energy (DNE) and ariaDNE values in each collection for each tooth (see Tables S1–S4). The numbers in the table are obtained by taking the mean across all 70 tooth samples
Method Triangle count Remeshing Noise
ariaDNE ε = 0.04 0.0213 0.0824 0.0055
ε = 0.06 0.0114 0.0429 0.0044
ε = 0.08 0.0117 0.0304 0.0039
ε = 0.10 0.0117 0.0293 0.0038
DNE 0.420 2.3075 0.0169

Table 2 shows percent change in ariaDNE and DNE values subject to different smoothing algorithms. After 100 iterations of Avizo smoothing, ariaDNE increased 2% of its original value, whereas DNE dropped to 46%. After three iterations of HC Laplacian smoothing and implicit fairing, ariaDNE dropped to approximately 90% of the original value, whereas DNE dropped to approximately 40%. The larger drop in values using Laplacian smoothing and implicit fairing is consistent with the discussion by Spradley et al. (2017). However, for all smoothing algorithms, the variation in ariaDNE is significantly lower than for DNE. This suggests that ariaDNE is relatively stable under varying smoothing algorithms. For Avizo smoothing, Figure S2 shows that the degree of overall change in ariaDNE from unsmoothed surfaces to smoothed surfaces is much less than the overall change in DNE. This suggests that ariaDNE is relatively more stable under varying Avizo smoothing iterations.

Table 2. Effect of different smoothing algorithms on Dirichlet normal energy (DNE) and a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) (ε = 0.08). The numbers in the table are DNE and ariaDNE values divided by values of raw surfaces indicating the percent change. The table also contains coefficients of variation (COV) of DNE and ariaDNE computed on the three smooth surfaces in each taxa. The table demonstrates: (a) The effect of smoothing is limited on ariaDNE vs. DNE; (b) ariaDNE is relatively more stable under varying smoothing algorithms
Raw Avizo Laplacian Fairing COV Raw Avizo Laplacian Fairing COV
Alouatta 1 0.38 0.35 0.48 0.120 1 1.00 0.90 0.90 0.049
Ateles 1 0.57 0.46 0.57 0.144 1 1.05 0.93 0.93 0.056
Brachyteles 1 0.54 0.45 0.57 0.127 1 1.06 0.96 0.95 0.053
Callicebus 1 0.46 0.39 0.52 0.122 1 0.99 0.90 0.91 0.074
Chiropotes 1 0.51 0.37 0.48 0.112 1 1.02 0.95 0.94 0.06
Pithecia 1 0.43 0.29 0.41 0.165 1 1.02 0.94 0.93 0.058
Saimiri 1 0.33 0.44 0.56 0.169 1 1.00 0.90 0.92 0.056

Figure 6 shows that the local energy of the boundary triangles computed with ariaDNE are among the smallest, whereas those computed with DNE have a few larger ones, which affect the DNE value for the whole surface. This histogram suggests that the effects of boundary triangles on ariaDNE are limited, and therefore no special treatment for them is needed. This represents another improvement for ariaDNE compared to DNE.

Details are in the caption following the image
The boundary triangles have less impact on a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) than Dirichlet normal energy (DNE). The left panel shows an Ateles molar (top), with a curvy side wall and a Brachyteles molar (bottom), with a straight side wall. The right panel shows histograms of local energy values of the boundary triangles, computed by ariaDNE and DNE. To enable comparison, the values are normalized by the mean of those of all triangles

3.2 Species differentiation power

For each shape characterizer (RFI, DNE and ariaDNE), ANOVA rejects the hypothesis with P < 0.05 that all dietary groups have the same mean, which indicates that some dietary differentiation was detected. To further determine which group means are different, we used multiple comparison tests and the results are summarized in Table 3. RFI separated folivore from frugivore and hard-object feeding; DNE in addition separated hard-object feeding from insectivore. As ε increases, ariaDNE further separated frugivore from hard-object feeding and insectivore. No metrics separated folivore and insectivore. However, similarity in their ariaDNE values are not surprising. Insect and leaf tissues tend to be high in structural carbohydrates, which sharpened dental blades are capable of shearing, and therefore high ariaDNE values. What is more important here is the separation from teeth that have low cusps and wide basins, as these are used for crushing motions to efficiently break down soft (i.e. fruit) and hard objects. For ε = 0.08, 0.1 and 0.12, the box plots of ariaDNE (Figure 7) converge on a pattern in which folivorous Alouatta, Brachyteles and insectivorous Saimiri have higher values, reflecting sharper cusps, whereas frugivorous Ateles and Callicebus have lower values and hard-object feeding Chiropotes and Pithecia have the lowest values, reflecting low unsharp cusps. The separation was not as clear for RFI and DNE.

Details are in the caption following the image
Box plots of RFI, Dirichlet normal energy (DNE), and a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) with ε = 0.02, 0.04, 0.06, 0.08, 0.1, 0.12 for Alouatta(Al), Ateles(At), Brachyteles(B), Callicebus(C), Chiropotes(Ch), Pithecia(P), Saimiri(S). Edges of the box indicate the 25th and 75th percentiles, and the outliers are plotted individually using the ‘+’ symbol. Colour indicates dietary preference: green represents folivore, purple represents frugivore, red represents hard-object feeding and yellow represents insectivore
Table 3. Multiple comparison tests on RFI, Dirichlet normal energy (DNE) and a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) (ε = 0.02, 0.04, 0.06, 0.08, 0.10, 0.12) values of folivore (Fo), frugivore (Fr), hard-object feeding (H) and insectivore (I). The numbers in the table are p values for the pairwise hypothesis test that the corresponding mean difference is not equal to 0. For ε = 0.08, 0.10, 0.12, ariaDNE differentiated folivore, frugivore and hard-object feeding. None of the metrics differentiated insectivore from folivore
ε = 0.02 0.04 0.06 0.08 0.10 0.12
Fo-Fr 0.0001 0.0224 0.0512 0.8362 0.0207 0.0002 0.0000 0.0000
Fo-H 0.0000 0.0000 0.0010 0.0003 0.0000 0.0000 0.0000 0.0000
Fo-I 0.2414 0.8463 0.8986 0.7564 0.6544 0.5584 0.5893 0.7376
Fr-H 0.9372 0.2295 0.5425 0.0049 0.0000 0.0000 0.0000 0.0000
Fr-I 0.1888 0.3910 0.4738 0.3461 0.0034 0.0000 0.0000 0.0000
H-I 0.0689 0.0125 0.0627 0.0002 0.0000 0.0000 0.0000 0.0000

For ε = 0.02, ariaDNE shows a pattern similar to DNE. This suggests that when ε is small, both methods capture fine and/or local features on tooth models, and as ε becomes larger, ariaDNE starts capturing larger scale features, ignoring smaller scale features. Figure 3 demonstrates the feature scale of DNE and ariaDNE with various ε values. The pattern is particularly interesting in Callicebus, Chiropotes and Pithecia, which evince less pointed cusps, but which exhibit more fine details on the basin (such as enamel crenulations for the pitheciines). In these teeth, ariaDNE values are high when ε is small, but drop with larger ε. The pattern is more pronounced in the teeth of Pithecia because their high-energy features—the enamel crenulations—are even smaller than those of Callicebus and so are erased more completely by using a high ε value.

So is it good or bad to erase small-scale features? We do not believe there is an objective answer or a universal optimum for ε. Berthaume and Schroer (2017) emphasized the importance of small-scale features in their analyses of dental topography of extant apes, which also exhibit crenulated enamel similar to Pitheciines. Additionally, erasing small-scale features makes the mandibular second molars of Pithecia more similar to those of Aye Ayes (Daubentonia). Previous studies have argued that the two species are analogous from an ecological point of view (Winchester et al., 2014). On the other hand, small-scale features could reflect an important functional ability of Pithecia not available to Daubentonia (Ledogar et al., 2013). Considering other research questions, these small-scale features align Pithecia with Callicebus, which may be evidence of a close phylogenetic relationship between them—one that was debated prior to availability of genetic data, based on a dearth of obvious unique anatomical similarities.


4.1 Bandwidth and multi-scale quantifications

Even with a less sensitive implementation, ariaDNE still requires choices on the parameter ε. In Section 2, we discussed the origin and interpretation of ε. We showed how ε affects values of ariaDNE and the power to differentiate primates with differing dietary preferences in Section 13. To summarize: (a) for a given ε, values of ariaDNE remained relatively unchanged compared to DNE, when the input mesh is perturbed (Figure 1). This suggests that ε is independent of mesh attributes like triangle count, mesh representation, noise level, smoothness, etc. (b) The parameter ε indicates the size of local influence: the larger ε, the more points on the mesh are considered important to quantify the local energy of the query point, and therefore larger ariaDNE values. This means ε determines the scale of features to be included in geometric quantification. Small ε will make surfaces with finer features have higher ariaDNE values, and large ε will make surfaces with large-scale features have higher ariaDNE values.

Parameter tuning was often achieved through optimization based on a priori goals, yet a single choice of parameter may not satisfy all goals. For example, the parameter that maximizes the differentiation between species in different diet groups may be different from that which minimizes the effect of wear or optimizes the differentiation between species irrespective of diet. The requirement of choosing a uniform scale applies to quantitative methods generally, and perhaps this is their biggest weakness compared to qualitative analyses of more traditional comparative morphology, where multiple scales of perception were naturally integrated into prose describing observed similarities and differences. However, the freedom to check patterns under different parameters also presents potential for more informative comparisons, as seen in (Figure 7). Future work should aim to characterize samples using values computed across a range of ε values.

4.2 Wider applicability of ariaDNE

Many other applications of ariaDNE beyond functional questions of teeth are possible (Figure 8). For instance, in bivalves, burrowing benthic forms should benefit from shells with greater rugosity (higher ariaDNE) to help them stay embedded in the sea floor, whereas more planktonic forms should benefit from smoother, more hydrodynamic shells (lower ariaDNE). It might also be helpful to use ariaDNE on distal phalanges (bones supporting the nail/claw) as claws suited for climbing are narrower and sharper (higher ariaDNE) while those suited for burrowing (or grasping) will be broader and blunter (lower ariaDNE). In addition to studying shape complexity across species, ariaDNE might also be used for complexity in a shape over developmental time. Salvador-Martínez and Salazar-Ciudad (2017), for example, used DNE to study embryo shape development and compared it with disparity in gene expression.

Details are in the caption following the image
a robustly implemented algorithm for Dirichlet energy of the normal (ariaDNE) values for surfaces representing astragulus (ankle bone) of Oryctolagus (saltatorial) and Hemicentetes (ambulatory), molars of Mammut (folivorous) and Mammuthus (grazing) and shells of Lirobittium rugatum and Tornatellaria adelinae. Surface shading indicates curvature computed by our algorithm; ariaDNE values are above each surface

Comparing the distribution of ariaDNE values over surfaces will likely provide even more insight into ecologically meaningful shape variation. For example, two surfaces with the same total ariaDNE may have very different distributions: one may have greater spatial variance in ariaDNE, with high ariaDNE features more clustered in one case than another. In all, ariaDNE opens doors to defining other interesting shapes metrics that could potentially assist our understanding in morphology, evolution and ecology.

4.3 ariaDNE for previously published DNE analysis

The insensitivity of ariaDNE under varying mesh preparation protocols makes it more widely usable than DNE for comparing and combining results from studies with varying samples or mesh preparation protocols. The computed ariaDNE values for previously published DNE studies (Berthaume, Delezene, & Kupczik, 2018; Berthaume & Schroer, 2017; Bunn et al., 2011; López-Torres, Selig, Prufrock, Lin, & Silcox, 2018; Pampush, Spradley et al., 2016; Pampush, Winchester, et al., 2016; Pampush et al., 2018; Prufrock, Boyer, & Silcox, 2016; Winchester et al., 2014) are included in the supplementary materials and available to download as csv files from https://sshanshans.github.io/articles/ariadne.html. We will continue to update our website as we obtain access to more data samples.

4.4 Conclusion

We provided a robust algorithm for computing the Dirichlet energy of the normal by utilizing weighted PCA. Compared to DNE, ariaDNE is stable under a greater range of mesh preparation protocols. Specifically, analyses indicated that the effects of differing triangle count, mesh representation, noise, smoothing and boundary triangles are much more limited on ariaDNE than DNE. Furthermore, ariaDNE retains the potential of DNE for biological studies, illustrated by it effectively differentiating platyrrhine primate species according to dietary preferences. While increasing the ε parameter of the method can erase small-scale features and significantly affect how ariaDNE characterizes structures with small-scale features compared to those with larger features (as it did with Chiropotes and Pithecia primates in our sample), we think this property can be leveraged to provide more informative comparisons. Future work should aim to characterize samples using values computed across a range of ε values. In this type of analysis, parameters could be optimized according to model selection criteria. Finally, as with other topographic metrics, ariaDNE is likely most informative when deployed in combination with other shape metrics to achieve the goal of more accurately inferring morphological shape attributes.


D.M.B. and J.M.W. were supported in this work by NSF BCS 1552848 and DBI 1661386. I.D., S.Z.K. and S.S. were supported in this work by Math + X Investigators Award 4000837.


This research was led by S.S. All authors contributed significantly to the manuscript and give final approval for publication.


Sample locations

The platyrrhine sample was published by Winchester et al. (2014), and is available on Morphosource, a project-based data archive for 3D morphological data: https://www.morphosource.org/Detail/ProjectDetail/Show/project_id/89.

Matlab scripts

Matlab scripts are available from the GitHub repository: https://github.com/sshanshans/ariaDNE_code and are archived with Zenodo (Shan, 2018): https://doi.org/10.5281/zenodo.1465949.