An Integrated Approach for Landmark-Based Resistant Shape Analysis in 3D

The study of shape changes in morphology has seen a significant renovation in the last 20 years, particularly as a consequence of the development of geometric morphometric methods based on Cartesian coordinates of points. In order to extract information about shape differences when Cartesian coordinates are used, it is necessary to establish a common reference frame or system for all specimens to be compared. Therefore, a central issue in coordinate-based methods is which criterion should be used to align these configurations of points, since shape differences highly depend on those alignments. This is usually accomplished by aligning the configurations in a way that the sum of squared distances between coordinates of homologous points (landmarks) is minimized: the least-squares superimposition method. However, it is widely recognized that this method has some limitations when shape differences are not homogeneous across landmarks. Here we present an integrated approach for the resistant shape comparison of 3D landmark sets. It includes a new ordinary resistant Procrustes superimposition and its corresponding generalized resistant Procrustes version. In addition, they are combined with existing resistant multivariate statistical techniques for depicting the results. We demonstrate, by using both simulated and real datasets, that resistant Procrustes better detects and measures localized shape variation whenever present in up to half but one of the landmarks. The resistant Procrustes results are highly concordant with a priori biological information, and might dramatically improve the quality of inferences on patterns of shape variation.     

Morphometric analysis of Cartesian coordinates of the human skull

A method for locating the three dimensional coordinates of cranial landmarks with respect to the Frankfort, midsagittal, and coronal planes is presented. Sliding calipers were used to obtain the distances from left and right porion and apex to each landmark, except for a few points where spreading calipers are required. In the present example, 35 landmarks (for a total of 105 measurements) were located for each of 35 Peruvian precolumbian skulls. These distances were entered into a program (SKULL) which calculates the Cartesian coordinates of each landmark. The XYZ coordinates of each landmark contain all the information necessary for calculation of the distances between any two landmarks, and these distances may also be obtained as output from program SKULL, if desired (595 distances if all 35 landmarks are used). Reliability of the location of coordinates was determined by comparing computed distances among selected landmarks from program SKULL with traditional anthropometric measurements. Satisfactory agreements were found. Direct multivariate analysis of the coordinates of the landmarks produced insights not available in traditional multivariate analysis of conventional anthropometric measurements.     

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.     

Bias and error in estimates of mean shape in geometric morphometrics

Sampling experiments were performed to investigate mean square error and bias in estimates of mean shape produced by different geometric morphometric methods. The experiments use the isotropic error model, which assumes equal and independent variation at each landmark. The case of three landmarks in the plane (i.e., triangles) was emphasized because it could be investigated systematically and the results displayed on the printed page. The amount of error in the estimates was displayed as RMSE surfaces over the space of all possible configurations of three landmarks. Patterns of bias were shown as vector fields over this same space. Experiments were also performed using particular combinations of four or more landmarks in both two and three dimensions.It was found that the generalized Procrustes analysis method produced estimates with the least error and no pattern of bias. Averages of Bookstein shape coordinates performed well if the longest edge was used as the baseline. The method of moments (Stoyan, 1990, Model. Biomet. J. 32, 843) used in EDMA (Lele, 1993, Math. Geol. 25, 573) exhibits larger errors. When variation is not small, it also shows a pattern of bias for isosceles triangles with one side much shorter than the other two and for triangles whose vertices are approximately collinear causing them to resemble their own reflections. Similar problems were found for the log-distance method of Rao and Suryawanshi (1996, Proc. Nat. Acad. Sci. 95, 4121). These results and their implications for the application of different geometric morphometric methods are discussed.     

Phylogenetic morphometrics (I): the use of landmark data in a phylogenetic framework

A method for the direct use of aligned landmark data (2D or 3D coordinates of comparable points) in phylogenetic analysis is described. The approach is based on finding, for each of the landmark points, the ancestral positions that minimize the distance between the ancestor/descendant points along the tree. Doing so amounts to maximizing the degree to which similar positions of the landmarks in different taxa can be accounted for by common ancestry, i.e. parsimony. This method requires no transformation of the aligned data or the results: the data themselves are the x, y, z coordinates of the landmarks, and the output of mapping a character onto a given tree is the x, y, z coordinates for the hypothetical ancestors. In the special case of collinear points, the results are identical to those of optimization of (continuous) additive characters.     

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.     

APPLICATION OF THIN‐PLATE SPLINE TRANSFORMATIONS TO FINITE ELEMENT MODELS,OR, HOW TO TURN A BOG TURTLE INTO A SPOTTED TURTLE TO ANALYZE BOTH

Finite element (FE) models are popular tools that allow biologists to analyze the biomechanical behavior of complex anatomical structures. However, the expense and time required to create models from specimens has prevented comparative studies from involving large numbers of species. A new method is presented for transforming existing FE models using geometric morphometric methods. Homologous landmark coordinates are digitized on the FE model and on a target specimen into which the FE model is being transformed. These coordinates are used to create a thin‐plate spline function and coefficients, which are then applied to every node in the FE model. This function smoothly interpolates the location of points between landmarks, transforming the geometry of the original model to match the target. This new FE model is then used as input in FE analyses. This procedure is demonstrated with turtle shells: a Glyptemys muhlenbergii model is transformed into Clemmys guttata and Actinemys marmorata models. Models are loaded and the resulting stresses are compared. The validity of the models is tested by crushing actual turtle shells in a materials testing machine and comparing those results to predictions from FE models. General guidelines, cautions, and possibilities for this procedure are also presented.     

几何形态测量法在生物形态学研究中的应用

形态分析是生物系统学及其多样性研究中很重要的一部分。随着统计学的发展,我们可以对非常复杂的数据进行分析,这在客观上导致了多变量形态测量的出现。在20世纪80年代,在数据收集和分析上产生了重要突破——标点和标点相对位置的几何信息的匹配,从而可以将多变量分析的标点集叠加到生物原始图上,它不仅仅是生成散点图,而是试图客观反映生物的形态性状。这项研究被称为几何形态测量法(geometricmorphometrics),Rohlf和Marcus(1993)称其为形态测量方法上的一次革命。文章简要介绍了该方法。     

A geometric morphometric study into the sexual dimorphism of the human scapula

Sex determination is vital when attempting to establish identity from skeletal remains. Two approaches to sex determination exists: morphological and metrical. The aim of this paper was to use geometric morphometrics to study the shape of the scapula and its sexual dimorphism.The sample comprised 45 adult black male and 45 adult black female scapulae of known sex. The scapulae were photographed and 21 homologous landmarks were plotted to use for geometric morphometric analysis with the ‘tps’ series of programs, as well as the IMP package. Consensus thin-plate splines and vector plots for males and females were compared. The CVA and TwoGroup analyses indicated that significant differences exist between males and females. The lateral and medial borders of females are straighter while the supraspinous fossa is more convexly curved than that of males. More than 91% of the females and 95% of the males were correctly assigned. Hotelling's T²-test yielded a significant p-value of 0.00039. In addition, 100 equidistant landmarks representing the curve only were also assigned. These, however, yielded considerably poorer results. It is concluded that it is better to use homologous landmarks rather than curve data only, as it is most probable that the shape of the outline relative to the fixed homologous points on the scapula is sexually dimorphic.     

Technical note: Quantification of neurocranial shape variation using the shortest paths connecting pairs of anatomical landmarks

Three‐dimensional geometric morphometric techniques have been widely used in quantitative comparisons of craniofacial morphology in humans and nonhuman primates. However, few anatomical landmarks can actually be defined on the neurocranium. In this study, an alternative method is proposed for defining semi‐landmarks on neurocranial surfaces for use in detailed analysis of cranial shape. Specifically, midsagittal, nuchal, and temporal lines were approximated using Bezier curves and equally spaced points along each of the curves were defined as semi‐landmarks. The shortest paths connecting pairs of anatomical landmarks as well as semi‐landmarks were then calculated in order to represent the surface morphology between landmarks using equally spaced points along the paths. To evaluate the efficacy of this method, the previously outlined technique was used in morphological analysis of sexual dimorphism in modern Japanese crania. The study sample comprised 22 specimens that were used to generate 110 anatomical semi‐landmarks, which were used in geometric morphometric analysis. Although variations due to sexual dimorphism in human crania are very small, differences could be identified using the proposed landmark placement, which demonstrated the efficacy of the proposed method. Am J Phys Anthropol 151:658–666, 2013. © 2013 Wiley Periodicals, Inc.     

Technical note: Different techniques,different results—a comparison of photogrammetric and caliper-derived measurements

The primary goal of our study was to compare photogrammetric measurements with caliper-derived measurements. We also looked at the difference between caliper-derived measurements that were taken with and without the landmarks marked. Thirteen facial measurements were repeated ten times on two adult subjects as follows: 1) Calipers were used to take the measurements before the landmarks were marked on each subject's face; 2) the landmarks were then marked with a black pencil, and the calipers were used to take the measurements again; and 3) images were taken of each subject with the markings left on the face, and the measurements were extracted from these images. Compared with the caliper-derived data taken with the landmarks marked, the photogrammetric means and standard deviations were typically larger, leading us to conclude that there was a systematic difference between the data. The generally greater variation in the photogrammetric measurements was ascribed to poor conditions, such as shadows, oblique markings, and unmarked landmarks. When the data gathered by caliper with and without the landmarks marked were compared, a systematic difference was suggested by the number of statistically significant t-test probabilities. Marking the landmarks reduced the standard deviations in some measurements by controlling two sources of variation: differing pressure on the skin and slippage of the calipers. Anthropologists, medical geneticists, and others who use measurements for diagnostic or classificatory purposes should be aware that data gathered by different techniques may yield different results. Am J Phys Anthropol 106:547–552, 1998. © 1998 Wiley-Liss, Inc.     

A method for reconstructing three-dimensional positions of swarming mosquitoes

We developed a new field method for reconstructing the three-dimensional positions of swarming mosquitoes. This method overcame certain inherent difficulties accompanied by conventional stereoscopic methods and is applicable to three-dimensional measurements of other insect species. Firstly, we constructed a probabilistic model for stereoscopy; if mosquitoes and six reference points with known coordinates were photographed simultaneously from two or more perspectives, then from the positions of images of mosquitoes and the reference points on the photographs, 1) the position of each camera with respect to the reference points is estimated; 2) stereo images which correspond to an identical real mosquito are matched; and 3) the spatial positions of the mosquitoes are determined. We automated the processes 1), 2) and 3), developing computer programs based on our model. We then constructed a portable system for three-dimensional measurements of swarming mosquitoes in the field. Initial data that illustrate the application of our method to studying mosquito swarming were presented.     

Computerized restoration of nonhomogeneous deformation of a fossil cranium based on bilateral symmetry

We developed a computerized method of correcting plastic deformation of a fossil skull, based on bilateral symmetry with respect to the midsagittal plane, and applied this method to reconstruction of a fossilized Proconsul heseloni cranium (KNM-RU-7290A). A three-dimensional (3D) model of the fossil was generated using consecutive cross-sectional images retrieved from computed tomography. 3D coordinates of anatomical landmarks that should be located on the midsagittal plane and pairs of landmarks that should be symmetrical with respect to this plane were acquired. These landmarks were then repositioned so that geometrical constraints were satisfied, while translated distances of landmarks were minimized. We adopted a thin-plate spline function to mathematically describe the 3D nonlinear volumetric transformation between acquired and repositioned landmarks. Using this function, the entire fossil shape was transformed, and the effect of reversing the deformation could be visualized. The results indicated that the proposed method was effective in eliminating nonhomogeneous deformation of the fossil skull. The antemortem appearance of the skull cannot be completely restored by this method alone, due to methodological limitations. However, the presented method has a role as an adjunct in complementing conventional restoration techniques on account of its objective nature.     

Covariation between forelimb muscle anatomy and bone shape in an Australian scratch-digging marsupial: Comparison of morphometric methods

The close association between muscle and bone is broadly intuitive; however, details of the covariation between the two has not been comprehensively studied. Without quantitative understanding of how muscle anatomy influences bone shape, it is difficult to draw conclusions of the significance of many morphological traits of the skeleton. In this study, we investigated these relationships in the Quenda (Isoodon fusciventer), a scratch-digging marsupial. We quantified the relationships between forelimb muscle anatomy and bone shape for animals representing a range of body masses (124–1,952 g) using two-block partial least square analyses. Muscle anatomy was quantified as muscle mass and physiological cross-sectional area (PCSA), and we used two morphometric methods to characterize bone shape: seven indices of linear bone proportions, and landmarks analysis. Bone shape was significantly correlated with body mass, reflecting allometric bone growth. Of the seven bone indices, only shoulder moment index (SMI) and ulna robustness index (URI) showed a significant covariation with muscle anatomy. Stronger relationships between muscle anatomy and forelimb bone shape were found using the landmark coordinates: muscle mass and PCSA were correlated with the geometric shape of the scapula, humerus, and third metacarpal, but to a lesser extent with shape of the ulna. Overall, our data show that landmark coordinates are more sensitive than bone indices to capturing shape changes evident throughout ontogeny, and is therefore a more appropriate method to investigate covariation with forelimb muscle anatomy. Single-species studies investigating ontogeny require refined methods to accurately develop understanding of the important relationships between muscle force generation and bone shape remodeling. Landmark analyses provide such a method.     

林窗内光照强度的测量方法

林窗内光强存在复杂的时空变化,对植物更新和生长有着重要影响,因此,林窗光照强度的快速测量方法是生态学家十分关注的问题.目前,测量林窗光强的方法可分为3类:(1)直接测量法采用光量子探头等仪器直接测量光强,但测量林窗光强异质性时十分费时费力.(2)模型估测法通过几何计算可快速估测林窗任意位置光强,但模型估测法将林窗简化为圆柱体或椭圆柱体,并忽略了许多林窗光强的影响因素,这极大影响了它的测量精度.(3)相片法采用半球面影像等相片间接计算相片拍摄点的光强,但测量林窗光强异质性时需要在林窗内拍摄大量相片;相片法具有较高精度,可区分直射光和散射光,其中,基于半球面影像的林窗光指数(gap light index)精度最高,使用广泛;基于几何计算的林窗光指数不仅具有较高精度,且可以快速测量林窗任意位置光强,该方法适用于林窗光强水平分布格局、垂直结构以及光组成成分(直射光和散射光)特征研究.     

The Neanderthal "chignon": variation, integration, and homology

The occipital bun ("chignon") is cited widely as a Neanderthal derived trait. It encompasses the posterior projection/convexity of the occipital squama and is associated with lambdoid flattening on the parietal. A 'hemibun' in some Upper Paleolithic Europeans is thought by some authors to indicate interbreeding between Neanderthals and early modern Europeans. However, 'bunning' is difficult to measure, and the term has been applied to a range of morphological patterns. Furthermore, its usefulness in phylogenetic reconstruction and its homologous status across modern and fossil humans have been disputed. We present a geometric morphometric study that quantitatively evaluates the chignon, assesses its usefulness in separating Neanderthals from modern humans, and its degree of similarity to Upper Paleolithic 'hemibuns.' We measured the three-dimensional coordinates of closely spaced points along the midsagittal plane from bregma to inion and of anatomical landmarks in a large series of recent human crania and several Middle and Late Pleistocene European and African fossils. These coordinate data were processed using the techniques of geometric morphometrics and analyzed with relative warps, canonical variates, and singular warps. Our results show no separation between Neanderthals and modern humans, including early modern Europeans, when the shape of the occipital plane midsagittal-profile is considered alone. On the other hand, Neanderthals are well separated from both recent and fossil modern humans when information about the occipital's relative position and relative size are also included. Furthermore, the occurrence of a highly convex and posteriorly projecting midline occipital profile (interpreted as the occipital bun) is highly correlated (>0.8) with a flat parietal midsagittal profile and with antero-superiorly positioned temporal bones across both our recent and our fossil human samples. We conclude that the shape of the occipital profile alone should not be considered an independent trait, as it is very tightly integrated with braincase shape. Our analysis does not support differences in integration of the posterior midsagittal profile and the cranial base in Pleistocene and recent humans.     

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.     

Morphological disparity in theropod jaws: comparing discrete characters and geometric morphometrics

Disparity, the diversity of form and function of organisms, can be assessed from cladistic or phenetic characters, and from discrete characters or continuous characters such as landmarks, outlines, or ratios. But do these different methods of assessing disparity provide comparable results? Here we provide evidence that all metrics correlate significantly with each other and capture similar patterns of morphological variation. We compare three methods of capturing morphological disparity (discrete characters, geometric morphometric outlines and geometric morphometric landmarks) in coelurosaurian dinosaurs. We standardize our study by focusing all our metrics on the mandible, so avoiding the risk of confounding disparity methods with anatomical coverage of the taxa. The correlation is strongest between the two geometric morphometric methods, and weaker between the morphometric methods and the discrete characters. By using phylogenetic simulations of discrete character and geometric morphometric data sets, we show that the strength of these correlations is significantly greater than expected from the evolution of random data under Brownian motion. All disparity metrics confirm that Maniraptoriformes had the highest disparity of all coelurosaurians, and omnivores and herbivores had higher disparity than carnivores.     

Three-dimensional surface reconstruction software system for IBM personal computers

Surface and volumetric three-dimensional imaging methods have found application in fields as diverse as diagnostic medical imaging and paleontological research. The acquisition, modeling, classification, and computer graphics rendering of discrete image volumes will be introduced. Applications in diagnosis (craniofacial, orthopedic, cardiovascular, and others) as well as reconstruction methods for generic serial sections will be described. C language software for three-dimensional reconstruction which operates on an IBM PC/AT clone is described.