Principles and methods of geometric morphometrics

The basic concepts, notions and methods of geometric morphometrics (GM) are considered. This approach implies multivariate analysis of landmark coordinates located following certain rules on the surface of a morphological object. The aim of GM is to reveal differences between morphological objects by their shapes as such, the "size factor" being excluded. The GM is based on the concept of Kendall's space (KS) defined as a hypersphere with points distributed on its surface. These points are the shapes defined as aligned landmark configurations. KS is a non-Euclidian space, its metrics called Procrustes is defined by landmark configuration of a reference shape relative to which other shapes are aligned and compared. The differences among shapes are measured as Procrustes distances between respective points. For the linear methods of multivariate statistics to be applied to comparison of shapes, the respective points are projected onto the tangent plane (tangent space), the tangent point being defined by the reference. There are two principal methods of shape comparisons in GM: the Procrustes superimposition (a version of the least squares analysis) and thin-plate spline analysis. In the first case, Procrustes residuals are the outcome shape variables which remain after isometric alignment of the shapes being compared. Their summation over all landmarks yields Procrustes distances among these shapes. The Procrustes distances can be used in multivariate analyses just as the Euclidian distances. In the second case, the shapes are fitted to the references by stretching/compressing and shearing until complete identity of their landmark configurations. Eigenvectors of resulting bending energy matrix are defined as new shape variables, principal warps which yield another shape space with the origin defined by the reference. Projections of the shapes being compared onto principal warps yield partial warps, and their covariance matrix decomposition into eigenvectors yields relative warps which are similar to principal components (in particular, they are mutually orthogonal). Both partial and relative warps can be used in many multivariate statistic analyses as quantitative shape variables. Results of thin-plate spline analysis can be represented graphically by transformation grid which displays type, amount and localization of the shape differences. Basis rules of sample composition and landmark positioning to be used in GM are considered. At present, rigid (with minimal degrees of freedom) 2D morphological objects are most suitable for GM applications. It is important to recognize three type of real landmarks, and additionally semi-landmarks and "virtual" landmarks. Some procedures of thin-plate spline analysis are considered exemplified by some study cases, as well as applications of some standard multivariate methods to GM results. They make it possible to evaluate correlation between different shapes, as well as between a shape and some non-shape variables (linear measurements etc); to evaluate the differences among organisms by shape of a morphological structure; to identify landmarks which most accounted for both correlation and differences between the shapes. An annotated list of most popular softwares for GM is provided.     

Analysis of quantitative trait locus effects on the size and shape of mandibular molars in mice

While >50 genes have been found to influence the development of teeth in mice, we still know very little about the genetic basis for the adaptive characteristics of teeth, such as size and shape. We applied interval mapping procedures to Procrustes size and shape data obtained from 10 morphological landmarks on the mandibular molar row of the F(2) progeny from a cross between the LG/J and SM/J strains of mice. This revealed many more QTL for molar shape (18) than for molar centroid size (3), although levels of dominance effects were comparable among QTL for size and shape. Comparisons of patterns of Procrustes additive and dominance shape effects and ordination of QTL effects by principal components analysis suggested that the effects of the shape QTL were dispersed among the three molars and thus that none of these molars represents a genetically distinct developmental structure. The results of an analysis of co-occurrence of QTL for molar shape, mandible shape, and cranial dimensions in these mice suggested that many of the QTL for molar shape may be the same as those affecting these other sets of characters, although in some cases this could be due to effects of closely linked genes.     

Comparison of dental measurement systems for taxonomic assignment of first molars

Morphometrics of the molar crown is based traditionally on diameter measurements but is nowadays more often based on 2D image analysis of crown outlines. An alternative approach involves measurements at the level of the cervical line. We compare the information content of the two options in a three-dimensional (3D) digital sample of lower and upper first molars (M(1) and M(1) ) of modern human and Neanderthal teeth. The cervical outline for each tooth was created by digitizing the cervical line and then sectioning the tooth with a best fit plane. The crown outline was projected onto this same plane. The curves were analyzed by direct extraction of diameters, diagonals, and area and also by principal component analysis either of the residuals obtained by regressing out these measurements from the radii (shape information) or directly by the radii (size and shape information). For M(1) , the crown and cervical outline radii allow us to discriminate between Neanderthals and modern humans with 90% and 95% accuracy, respectively. Fairly good discrimination between the groups (80-82.5%) was also obtained using cervical measurements. With respect to M(1) , general overlap of the two groups was obtained by both crown and cervical measurements; however, the two taxa were differentiable by crown outline residuals (90-97%). Accordingly, while crown diameters or crown radii should be used for taxonomic analysis of unworn or slightly worn M(1) s, the crown outline, after regressing out size information, could be promising for taxonomic assignment of lower M1s.     

Dental morphology and odontometry of the Tibetan immigrants

Dental features such as size, shape, cusp number and groove pattern, deflecting wrinkle, protostylid, form and size of dental arch and occlusion pattern, as well as crown diameters of maxillary and mandibular dentition of the Tibetan immigrants in India are described. Sex differences in various measurements are also noted. Overall reduction in size, presence of hypodontia of the third molar, and absence of Carabelli's trait are a distinct progressive/evolutionary trend in the Tibetan dentition, while presence of a shovel-shaped lingual surface of central and lateral incisors is a retrogressive/primitive condition. In some of the dental features Tibetans resemble Caucasoid and modern populations, in certain others they resemble other Mongoloid populations, and in still other features they resemble aboriginal populations and fossil hominids.     

Alternative dental measurements: proposals and relationships with other measurements

Most archaeological and fossil teeth are heavily worn, and this greatly limits the usefulness of tooth crown diameter measurements, as they are usually defined at the widest points of the crown. There are alternatives, particularly measurements at the cervix of the tooth, where the crown joints the root, and measurements along a diagonal axis in molars, that are much less affected by wear. These would allow a wider range of specimens to be included, e.g., in the study of dental reduction in Upper Palaeolithic and Mesolithic Homo sapiens. In addition, they would allow the little-worn teeth of children to be compared directly with well-worn teeth in adults. These alternatives, however, have been little used, and as yet there have not been any studies of the repeatability with which they can be measured, or of the extent to which they are related to the more usual crown diameters. The present study is based on a group of unworn teeth, where direct comparisons could be made between the alternative measurements, which are not much affected by wear, with the usual crown diameters, which are very much affected. In an interobserver-error study of this material, cervical and diagonal measurements could be recorded as reliably as the usual crown diameters. The buccolingual cervical measurement was strongly correlated with the normal bucclingual crown diameter in all teeth, whereas the mesiodistal cervical measurement was highly correlated with the normal mesiodistal crown diameter in incisors and canines, but less so in premolars and molars. The molar diagonal measurements showed high correlations with all other measurements. Crown areas (robustness index) calculated from the usual diameters were strongly correlated with crown areas calculated from cervical measurements, and crown areas calculated from molar diagonals were strongly correlated with both other areas. Despite the long usage of the more usual maximum crown diameters, the alternative dental measurements could be measured just as reliably, could record similar information about tooth crown size, and would be better measures for the worn dentitions seen in archaeological and fossil material.     

Cranial integration in Homo: singular warps analysis of the midsagittal plane in ontogeny and evolution

This study addresses some enduring issues of ontogenetic and evolutionary integration in the form of the hominid cranium. Our sample consists of 38 crania: 20 modern adult Homo sapiens, 14 sub-adult H. sapiens, and four archaic Homo. All specimens were CT-scanned except for two infant H. sapiens, who were imaged by MR instead. For each specimen 84 landmarks and semi-landmarks were located on the midsagittal plane and converted to Procrustes shape coordinates. Integration was quantified by the method of singular warps, a new geometric-statistical approach to visualizing correlations among regions. The two classic patterns of integration, evolutionary and ontogenetic, were jointly explored by comparing analyses of overlapping subsamples that span ranges of different hypothetical factors. Evolutionary integration is expressed in the subsample of 24 adult Homo, and ontogenetic integration in the subsample of 34 H. sapiens. In this data set, vault, cranial base, and face show striking and localized patterns of covariation over ontogeny, similar but not identical to the patterns seen over evolution. The principal differences between ontogeny and phylogeny pertain to the cranial base. There is also a component of cranial length to height ratio not reducible to either process. Our methodology allows a separation of these independent processes (and their impact on cranial shape) that conventional methods have not found.     

Relative warp analysis of cranial and upper molar shape in rock miceApodemus mystacinus sensu lato

We studied phenotypic relationships among 13 samples of two rock mice species:Apodemus mystacinus (Danford and Alston, 1877) from Anatolia (n = 38) andA. epimelas (Nehring, 1902) from the Balkans (n = 71). Cartesian coordinates of landmarks were collected on the skull and on the occlusal projection of the upper molars (18 landmarks). Centroid size (a measure of overall size) suggested that molars vary independently of overall skull size in both species. Discriminant function analysis on relative warp scores classified >80% of specimens into the correct species, with the best results obtained for the ventral aspect of the skull and for molars. Projection of the 1st discriminant function scores against centroid size provided good separation between the two species. Analysis of vector displacements associated with extremes of variation suggested considerable phenetic differences on the ventral side of the skull and in the molar shape of the two species. The great majority of shifts in landmarks were in a longitudinal direction and the rearrangements of molar cusps were more complex than was the case with the cranium. A bivariate plot of the posterior hard palate length against the incisive foramen length separatedA. mystacinus andA. epimelas well.     

Multivariate analysis of tooth crown morphology in Japanese-American F1 hybrids

Mesiodistal crown diameters of I1 through M1 and six non-metric crown traits in permanent dentition of Japanese-American F₁ hybrids were compared with those of the parental populations. The hybrids were born of Japanese females and American males, both Caucasians and American Blacks, after World War II and brought up at Elizabeth Saunders Home in Kanagawa Prefecture, Japan. The comparisons were undertaken by means of multivariate analysis methods such as principal component analysis, distance and similarity coefficients and multidimensional scaling. The F₁ hybrids generally occupy an intermediate position of the two parental populations, and this is particularly evident in size component of the crown measurements and in distance analysis of frequencies of the non-metric crown traits. The shape component of the crown measurements, however, not necessarily follows such a rule and suggests a more complicated gene control than in the case of the size component. Also, each measurement and frequency of non-metric trait of tooth crowns in F₁ hybrids is not uniformly at the middle position between their parental populations but this is true when they are analyzed as multivariables.     

现代各主要人群额骨3D几何形态的对比

额骨是连接面颅和脑颅的重要头骨组成部分,关于现代各个人种的额骨形态是否存在明显的差别,这些人种额骨的基本形态如何,变异范围以及与其他人群的相似与差异等问题都尚未完全厘清。而额骨的很多特征由于技术手段的限制很难进行测量和准确的描述比较。鉴于这些问题,本文将采用基于三维表面半标志点的几何形态测量方法,研究东亚现代人额骨的表面形态及其变异范围,并与欧洲,东南亚,美洲,非洲以及澳洲的现代人群的额骨形态进行对比,为对比不同人群的形态研究建立基础数据。结果显示,额骨形态的变异主要表现在:1)额骨鳞部的额结节和正中矢状脊共同向前隆起或回缩以及相对额骨宽度;2)眉弓的粗壮程度,额结节的侧向发育程度和正中矢状脊的发育情况。为了进一步揭示中国现代人与其他人群在额骨形态上的关系,本文还探讨了额骨大小在不同人群中的差异。结果显示,东亚现代人和欧洲现代人额骨中心大小值的中位数最大,澳洲现代人的最小。东亚现代人的额骨形态与澳洲,欧洲和非洲均有非常显著的差异。为了检验额骨的形态是否与遗传距离一致,作者还对额骨形态距离和遗传距离做了相关性分析。结果显示,不同人群的额骨形态与其遗传距离呈显著的相关性,说明本研究结果中不同人群额骨形态上的差异大小可以在一定程度上反映其遗传距离,并可能进一步反映人群历史。东亚现代人的额骨平均形态在与各个人群比较过程中表现出一致性特征,可能在一定程度上反映了东亚现代人群的进化过程是相对独立的。未来额骨的三维几何形态测量可通过扩大标本数量进一步探讨不同性别和不同演化阶段之间的差异。     

Photogrammetry: a useful tool for three‐dimensional morphometric analysis of small mammals

Recently, the improvement of methods for shape analysis has revolutionized the field of morphometrics. While three‐dimensional (3D) imaging technology is increasingly available, many studies of 3D structures still use two‐dimensional (2D) data, even when this may result in the loss of important information. This is particularly conspicuous in the study of small mammals, as devices precise enough for 3D digitization of small objects are the most expensive. Thus, the development of low‐cost methods aimed to recover 3D shape from small mammals would be of wide interest. Photogrammetry allows for obtaining 3D data with a lower cost than other 3D techniques, but it has not been previously applied to the study of small mammals. Accordingly, here we test the suitability of photogrammetric techniques to obtain 3D landmarks on mouse skulls as a model for small mammals. Shape and size of 3D models obtained with photogrammetric techniques were consistent among replicates, even when different sets of photographs were used. The linear measurements obtained from 3D models produced here were highly correlated with measurements obtained with callipers on actual crania, and differences among both sets of measures were smaller than those among individuals in most of the tested measures. These results show for the first time that photogrammetry is a precise technique for 3D shape analysis of small mammals. Photogrammetry also proved to be accurate for obtaining linear measurements between 3D landmarks; however, further studies are needed to demonstrate that this technique is also accurate to recreate 3D shapes.     

Evolution of shape in oligocene and mioceneNotocarinovalva (Ostracoda, Crustacea): A multivariate statistical study

Largely due to L. F. Bookstein, geometric morphometrics has been developed as an amalgamation of techniques drawn from mathematical statistics, non-Euclidean geometry and computer graphics, applied to labelled points (landmarks) and the biological images upon which they are registered. A tool of fundamental importance is the method of interpolation known as the thin-plate spline. Bookstein's sample-oriented procedure of relative warps, applied to data composed of coordinate pairs observed on eight landmarks on three samples of two species of fossil marine ostracods (bivalved microcrustacean),Notocarinovalva airella andN. yulecartensis, separated in time, is used to obtain weight matrices, which, when appropriately partitioned, constitute the familiar data matrices of multivariate statistical analysis. Standard multivariate statistical analysis of samples may be carried out in the tangent space to shape space at the Procrustes average shape. Linear discriminant function scores were used for assessing, approximately, the evolutionary relevance of shape change and change in size (based on standard distance measures) in the species from the upper and lower Oligocene and lower middle Miocene of Victoria, Australia. The findings indicate that a small, though consistent, change in non-uniform shape of the carapace has taken place. The analysis indicates that the evolution in shape of the carapace could have been caused by random genetic drift in moderately large populations, tempered by weak selection.     

Carabelli's trait and tooth size of human maxillary first molars

Carabelli's trait is a morphological feature that can occur on the protocone of human maxillary molars. This study tests the hypothesis that Carabelli's trait is correlated statistically with the dimensions of the crown's four principal cusps or whether, as a cingular feature, the trait truly accretes onto an otherwise unaffected crown. Computer-assisted image analysis was used to measure the 6 intercusp distances and 12 angular relationships among cusp tips on the permanent first molar of 300 young adult American whites. Carabelli's complex was scored using an 8-grade ordinal scheme. Crown size was quantified in three ways, namely as 1) maximum mesiodistal and buccolingual diameters, 2) the 6 intercusp distances, and 3) the 12 angular cusp arrangements. There was no sex difference in the morphological expression of Carabelli's trait in this sample. Overall crown size and intercusp distances were significantly and progressively larger in molars with larger Carabelli's trait expressions. There are graded size responses between crown size (mesiodistal and buccolingual diameters), sizes of the four principal cusps, and morphological stage of Carabelli's complex, though the statistical relationships are appreciably stronger in males than females. Carabelli's trait occurs preferentially in larger molars. In contrast, angular (shape) relationships among cusp tips are not discernibly affected by trait size in either sex. There is the situation, then, that Carabelli's trait is developmentally correlated with crown size, but with no apparent alteration of cusp arrangements, suggesting that the increases are isometric across the occlusal table. Why the association is much weaker in females remains speculative, but these data provide yet another line of evidence that, within a population, tooth size is associated in a positive fashion with crown complexity.     

Variation in scale shape among alternative sympatric phenotypes of Arctic charr Salvelinus alpinus from two lakes in Scotland

Landmark‐based geometric morphometric analysis was used to detect differences in scale shape between ecologically distinct phenotypes of Arctic charr Salvelinus alpinus coexisting in the same lake. Relative warp analysis and standard multivariate analyses of the partial warps, obtained after a Procrustes superimposition, showed that scale landmarks were efficient in discriminating among two closely related alternative phenotypes within each of the two lakes. In Loch Tay, S. alpinus exhibited a bimodal body size‐frequency distribution among sexually mature fish, whereas in Loch Awe, S. alpinus are unimodal in body size but segregated into two distinct spawning phenotypes. In both lakes, alternative phenotypes showed significant differences in foraging ecology, habitat use and life history. It is probable that differences in scale shape reflect differences in ecology of these forms.     

Inferring the Nature of Allometry from Geometric Data

The form of an organism is the combination of its size and its shape. For a sample of forms, biologists wish to characterize both mean form and the variation in form. For geometric data, where form is characterized as the spatial locations of homologous points, the first step in analysis superimposes the forms, which requires an assumption about what measure of size is appropriate. Geometric morphometrics adopts centroid size as the natural measure of size, and assumes that variation around the mean form is isometric with size. These assumptions limit the interpretation of the resulting estimates of mean and variance in form. We illustrate these problems using allometric variation in shape. We show that superimposition based on subsets of relatively isometric points can yield superior inferences about the overall pattern of variation. We propose and demonstrate two superimposition techniques based on this idea. In subset superimposition, landmarks are progressively discarded from the data used for superimposition if they result in significant decreases in the variation among the remaining landmarks. In outline superimposition, regularly distributed pseudolandmarks on the continuous outline of a form are used as the basis for superimposition of the landmarks contained within it. Simulations show that these techniques can result in dramatic improvements in the accuracy of estimated variance-covariance matrices among landmarks when our assumptions are roughly satisfied. The pattern of variation inferred by means of our superimposition techniques can be quite different from that recovered from full generalized Procrustes superimposition. The pattern of shape variation in the wings of drosophilid flies appears to meet these assumptions. Adoption of superimposition procedures that incorporate biological assumptions about the nature of size and of the variation in shape can dramatically improve the ability to infer the pattern of variation in geometric morphometric data.     

Reduction of maxillary molars in Homo sapiens sapiens: a different perspective.

Crown and cusp areas, and buccolingual and mesiodistal diameters of maxillary molars of complete upper tooth rows (30 males, 30 females) were analysed in order to quantify changes in size and shape from the first to the third molar. Uni- and multivariate analyses revealed the mesial cusps, in particular the protocone (mesiolingual cusp), to be more stable than the other cusps. Although there is a gradient in size from the first to third molar, shape changes were found to be marked. Overall, the findings are in keeping with the field theory and the hypotheses of environmental constraints on later developing teeth. However, not all of the results could be entirely explained by these concepts. Functional aspects seem to account for the relative stability of the protocone and the buccolingual crown diameter. It appears that this functional complex is relatively stable despite the overall reduction of tooth size, which is probably secondary to processes occurring in the jaws and the cranium. This finding may have implications for studies on tooth reduction between populations of different time periods.     

«Relative growth» from the dentino-enamel junction in primate maxillary molars

The dentino-enamel junction is critical throughout growth to mature crown configurations, being the interface between the papilla and the dental cap. Enamel deposition occurs relatively late and often causes changes from the pattern residing in the dentino-enamel junction. Primate teeth (mostly M¹) have been stripped of enamel after measurement and mapping of the original crown. Relative growth, a variant of static adult (allomorphic) allometries, is assessed by displacement of enamel basal crown component landmarks from dentine homologues relative to tooth size. The hypothesis that differential enamel growth reflects evolutionary history is supported by the positive allometry and shape differences in enamel versus dentine landmarks among phyletically enlarged and dentally-reduced primates.     

Allometric and non‐allometric consequences of inbreeding on Drosophila melanogaster wings

Inbreeding is expected to increase the variability in size and shape within populations. The distinct effects of inbreeding on size and shape suggest that they are governed by different developmental pathways. One unresolved question is whether the non‐allometric shape component is partially unconstrained developmentally and therefore whether shape is evolvable. In the present study, we utilized a mass outbred population of Drosophila melanogaster maintained at standard laboratory conditions. Eight lines with equivalent expected levels of inbreeding (F ≈ 0.67) were obtained by restricting the size of each population to two pairs for nine generations. Nine landmarks were measured on Drosophila wings of the inbreed lines and compared with those of the mass population. Wing landmarks comprise an excellent model system for studying evolution of size and shape. Landmark measurements were analyzed with a Procrustes generalized least squares procedure. To visualize global shape changes among samples, we reconstructed the mean shape and the shape changes related to both the allometric and non‐allometric components. An increased variability in the non‐allometric shape component was found with inbreeding. This indicated that shape was not entirely developmentally constrained, and therefore that shape appears to be evolvable. © 2011 The Linnean Society of London, Biological Journal of the Linnean Society, 2011, 102 , 626–634.