Manual Morphometrics for Nonmorphometricians

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Interesting results have been obtained for ostracods and cephalopods, for example, Reyment and Kennedy for ammonites and Reyment and McKenzie for Ostracoda from southern Australia. Remarks on Spline-Based Methods There are two main openings available for charting differences in form by means of coordinates. One of these takes each form, superimposes it in relation to others, and then computes differences in terms of reference-point displacements relative to this registration. The second tack is concerned with describing differences in point configurations as deformations of a grid produced by mapping one form into the other and visualizing the shrinkings and stretchings that are generated by the procedure.

The analysis of the registrations of the coordinates may be done in several ways. One may register to a common baseline by translating, rotating and scaling so that most points fit well, or register by minimizing the sum of squared differences between the equivalent landmarks of forms. This is usually referred to as generalized Procrustean fitting. The result is scaled by division by the centroid size. In Physics, the idea of affine transformation is known as an homogeneous deformation Klein, , p. Decomposition of coordinate-based data by the thin-plate spline technique exhibits a close analogy with the well known decomposition by means of Fourier trigonometric series.

The constant term in such a series is a global parameter and the trigonometric coefficients are local parameters at successively smaller scales Dryden and Mardia, , p. Dryden and Mardia , p. For small 20 R. Reyment variations, registration methods and distance methods lead to identical conclusions about shape. Unfortunately, there is no general rule available to support this belief.

Possible Sources of Error Errors can and do arise in the multivariate analytical processing of data generated by geometric morphometric procedures. The intuitively attractive method of triangulating data, Bookstein Coordinates, whereby coordinates are registered on a common edge induces spurious correlations and consequently invalid covariance matrices Dryden and Mardia, , p.

This notwithstanding, the basic theorem that everything that can be achieved using shape-coordinates can also be realized by means of ratios of distances, as demonstrated in the original paper on shape coordinates Bookstein, The possibility of applying the theory of compositional data analysis Aitchison, to the problem has not yet been more than briefly considered. Another source of error lies with imperfections in the data.

This is more likely to be a problem in palaeontological work where deviations from multivariate normality are a cause of misleading results. PhD Thesis, unpublished seems to be the first worker to have given serious attention to practical aspects of consistency and repeatability in multivariate analysis, in particular canonical variates and discriminant functions, but also principal components. Supplementary Reading R. Benson : Benson, a specialist on fossil and extant ostracods, was concerned with features of shell morphology, explicitly the reticulate pattern which he believed was under the control of evolutionary and environmental factors.

This theory is not unchallengeable but it does contain an element of bio-geometric interest. He devised a method of graphical pattern analysis to be used for defining elements of the reticulum in a lavishly illustrated monograph. The data were not analysed statistically, but remained at the visual level. These publications treat the problem of achieving stable results in methods of multivariate analysis, primarily canonical variate analysis but also principal components, with emphasis on the stability of the elements of the vector components.

This is a question of more than trivial significance when the results obtained for, say, relative warps are to be expressed graphically or to be inserted into a standard extension of multivariate 2 Morphometrics: An Historical Essay 21 techniques. As far as I have been able to judge, this awareness has yet to make itself apparent in geometric morphometrics. Every student of morphometrics should consult that thesis now and then. The generalization of the field to encompass affine and non-affine changes in shape by Bookstein is likewise a classic.

Both papers mark the dawning of the new era for the study of shape and growth. A very important advance in the analysis of shape-data is the paper by Mardia and Dryden This work may be seen as marking the beginning of the use of statistical models for shape studies. Concluding Remarks The brief account of morphometrics presented in the foregoing pages can do no more than highlight a few of the most important features and events of what is a complicated subject caught up in a phase of expansive development.

The introduction of advanced geometrical thinking into statistics is not widespread and very few statistical textbooks take it into consideration. The level of mathematical theory involved in shape-theory is certainly beyond the reach and learning of many statisticians and certainly transcends the ability of geologists and biologists.

The truth of this statement can be assessed by consulting the volume by Kendall et al. The method of crease analysis examines the effect of expansions forwards and backwards in time in phylogenetic reconstructions Bookstein, The crease method also holds promise of being a means of developing the automated description of diagnostic contrasts between reference specimens of species, that is, image-based taxonomy.

I express my gratitude to Professor Fred L. Bookstein for generously reading and constructively commenting on the manuscript and to Professor Kanti Mardia for valuable advice. References Aitchison J The statistical analysis of compositional data, Chapman and Hall, London, xv, pp Barnard MM The secular variation of skull characters in four series of Egyptian skulls.

Annals of Eugenics London 7: 89 pp 22 R. Reyment Bartlett MS Multivariate statistics. Theoretical and mathematical biology, Chapter 8, Blaisdell Publishing Company, Newyork, — pp Benson, RH The Bradleya problem, with descriptions of two new psychospheric ostracode genera, Agrenocythere and Poseidonamicus Ostracoda, Crustacea.

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Biometrics — Blackith RE A synthesis of multivariate techniques to distinguish patterns of growth in grasshoppers. Biometrics 28—40 Blackith RE Morphometrics. Growth — Bookstein FL The measurement of biological shape and shape change. Lecture Notes in Biomathematics. Statistical Science 1: — Bookstein FL Principal warps: thin-plate splines and the decomposition of deformations. Cambridge University Press, Cambridge, xvii, pp Bookstein FL Landmark methods for forms without landmarks: localizing group differences in outline shape.

Medical Image Analysis 4: 93— Burnaby TP a Allometric growth of ammonite shells: a generalization of the logarithmic spiral. Nature — Burnaby TP b Growth-invariant discriminant functions and generalized distances. Biometrics 96— Campbell NA Robust procedures in multivariate analysis. II Robust canonical variate analysis. Biometrics 12—38 Gower JC A general coefficient of similarity and some of its properties.

Biometrics — Gower JC Growth-free canonical variates and generalized distances. Bulletin of the Geological Institutions of the University of Uppsala. Biometrics — Huxley JS Problems of relative growth. Methuen, London, xix, pp Jolicoeur P The degree of generality of robustness in Martes americana.

Band 2: Geometrie. Springer Verlag, Berlin, pp Lohmann GP Eigenshape analysis of microfossils: a general morphometric procedure for describing changes in shape. Mathematical Geology — Macleod N Phylogenetic signals in morphometric data. Morphology, shape and phylogeny; Systematics association special volume 64, Taylor and Francis, London, — pp Mahanolobis PC On the generalized distance in statistics. Proceedings of National Institute Science, India 2: 49—55 Mosimann JE Size allometry: size and shape variables with characterizations of the lognormal and generalized gamma distributions.

Journal of American. Biometrika — Pearce SC Some recent applications of multivariate analysis to data from fruit trees. Biometrika 1— Quenouille MH Associated measurements.

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  4. Biometrika — Reyment RA Multidimensional palaeobiology. In Marcus LF et al. Zeitschrift der deutschen geologischen Gesellschaft. University Press, Oxford, No. Contributions to morphometrics, Museo nacional de ciencias naturales, Madrid, — pp Rohlf FJ Geometric morphometrics and phylogeny. Biometrics, — Sprent P The mathematics of size and shape. Biometrics, 23—28 24 R. Les variants sexuels de Maia squinada.

    Biotypologie, 7: 73—96 Thompson DW [] On growth and form. Krieger Idea and Aims Curvature and asymmetry are ever present in biological datasets as sources of shape variation, potentially confounded with other measurements of shape. This chapter examines potential methods for minimizing the impact of curvature on leaf shape measurement. Specifically: can curvature be ignored, should a curvature reference be included in analyses, or is mathematical straightening required? Methods used to measure leaf shape are reviewed. Introduction Botanical morphological diversity starts with leaf shape.

    Leaves are present throughout the land plants, found in vast quantities in the fossil record, and easy to preserve relatively unchanged as pressed herbarium specimens. For at least three hundred years, botanists have found applications for qualitative descriptions of tip shape, base shape, overall shape, and innumerable other characters.

    The number of terms developed to qualitatively describe leaf morphology is astounding, and they vary depending on the group of plants being studied ferns, conifers, flowering plants. For the small set of fern species used in this study, there were six terms for tip shape, seven for base shape, and eleven for overall shape Fig. Despite their long history, qualitative descriptions have shortcomings that limit their usefulness. However, leaves are notoriously difficult to consistently score for traits that may vary continuously Wilf ; Wiemann et al.

    Terms apply to the lamina of simple leaves and to the blade for dissected leaves. All terms shown have been applied to the species included in this study. Definitions modified and figures redrawn from Lellinger amalgam of terminology can be used to describe subtle differences in shape, but the volume of available terms makes the descriptions hard to reproduce between investigators; it is seldom clear where the boundaries among categories are drawn.

    And, illustrating their lack of concern for our carefully crafted categories, individual characters may exhibit asymmetric variation spanning several states, e. Nevertheless, these descriptors have proven useful e. These methods are all focused on recording a shape and extracting information with some correspondence to common qualitative descriptors. An unanswered—and seemingly unasked—question is: can you accurately analyze leaf shape data without first removing curvature from the data?

    By curvature, what is meant here is deviation of the midvein—the axis of bilateral symmetry in leaves—from a straight line. This curvature can manifest as both localized and large-scale variation. Morphometric methods typically control for position, rotation, and size.

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    Curvature and asymmetry represent additional sources of variation, and they may blur the boundaries between character states e. This can be quite prevalent Krieger et al. Studies of leaf shape have not addressed leaf curvature, or have simply set a cutoff at which leaves are too curved to include in the analysis; e. Approaches which use only the outline most Fourier and eigenshape analyses cannot account for curvature; additional information is required, such as the location of the midvein relative to the outline.

    In the absence of such a reference, asymmetry cannot be discriminated from curvature. Most studies of leaf shape have used measures which do not incorporate the midvein, except at the base and apex e. West and Noble developed a method to control for curvature, though not for outline analysis per se, drawing a series of lines perpendicular to the midvein in order to properly determine leaf width at all points along the midvein. The limitations of this method are discussed below see Methods. The goal of this study was to assess the impact of curvature on the extraction of shape measures from leaf outline data.

    The method I selected to measure shape is eigenshape analysis. The hypotheses tested were: 1. A sufficiently large sample size, relative to the amount of curvature in the dataset, will allow the eigenanalysis to identify and isolate curvature as a separate source of shape variation. The inclusion of the midvein as a reference curve, appended to the outline data, will allow the eigenanalysis to isolate variation due to curvature.

    Mathematical removal of curvature is necessary to recover meaningful shape descriptors. In the first case, when sample size is sufficiently large, the eigenanalysis identifies curvature using the covariance structure of the data. This has not previously been 30 J. Krieger tested, nor is it clear what sample size is sufficient. Without the midline, it is not possible to distinguish between asymmetry and curvature see Methods , so H1 seems unlikely.

    Since curvature is defined by the midvein, H2 seems reasonable: if the midvein is included in the analysis, the eigenanalysis may detect curvature in patterns of covariation between the outline and midvein. However, with no straightforward method to quantify curvature, much less to partition that curvature into orthogonal measures representing different types of curvature-based variation as an eigenanalysis would be expected to do , it is necessary to mathematically remove curvature from the dataset, to assess the other two approaches. The ultimate goal of this study is to assist botanists in making informed decisions about how to measure shape in ways that are consistent with classic qualitative terminology.

    Many morphometric tools are primarily concerned with discrimination among groups. More desirable is the ability to extract meaningful characters; this allows one to ask questions about the location and extent of variation, and to identify the characters which are useful in discrimination of groups. Each method will be assessed with respect to its ability to describe shape using characters interpretable in the context of classic, qualitative descriptors. Linear Measurements One approach is to record length, distilling all of the interesting shape information into a single mark along a ruler.

    The morphospace defined by measuring the length of five leaves is one-dimensional Fig. It is not possible to measure leaves with negative length, although you might argue that there are leaves with zero length their absence identified by leaf scars. In this example, the empirical morphospace defined by the set of measurements only extends to 10— 20cm or so, depending on the type of leaf.

    Measuring both length and width creates a two-dimensional morphospace Fig. A single point in a morphospace is the complete description of shape for an object, for the measurements taken: a point along the axis in Fig. These descriptions become more interesting as you increase the dimensionality of the observations that is, as more measurements are recorded for each object.

    Arabidopsis phenotyping through geometric morphometrics

    Clearly, length and width capture a tiny fraction of leaf shape; compare the leaf silhouettes to the cross-hairs in Fig. In fact, Fig. The utility of the 2D space with respect to qualitative descriptors is very limited; for example, ovate and obovate leaves see Fig. A The one-dimensional morphospace arranges leaves in a line based on length of the leaf blade.

    B The two-dimensional morphospace. Slight changes to the methodology, such as defining the leaf base using the lowest fourth of arc length along the outline, versus along the midvein, can lead to very different shape measurements. For the two leaves shown at far right, base angle as measured using midvein length is unchanged, whereas using arc length shows the first leaf as having a broader base with interesting axes, axes encompassing shape variation that tells us something about biology.

    Several linear measurements—blade length, width at widest point, petiole length, and their ratios—are commonly used, and these can even be automatically extracted from images Bylesjo et al. Rarely, the location of widest part of the leaf is recorded Dale et al. However, linear measurements tend to be highly correlated e. Even apparently simple measurements such as these are not without their complications.

    For example, base angle is computed using lines drawn from the base to a normal drawn at one quarter the length of the midline. An alternative, which is arguably more representative of the shapes given in Fig. It would also be more sensitive to lobing. One solution has been to elaborate the linear measurements. Any combination of linear measurements can be subjected to a principal components analysis PCA , for example, PCA of length, width at widest point, petiole length, and location of the widest point Dancik and Barnes This extracts variance-optimized the first axis explains the most variance, each subsequent axis explains the same or less , orthogonal measures.

    While these measures may contain more shape information than simple length or width, they are difficult to relate to useful qualitative descriptors. Another option is to incorporate additional geometric information, without actually performing geometric morphometric analysis. Linear measurements can be thought of as recording the distances between pairs of points landmarks on the leaf. A network of trusses among these points may be used e. Euclidean Distance Matrix Analysis EDMA, Lele and Richtsmeier ; Lele, , a truss method that analyzes the complete set of distance measurements among all landmarks, has been shown to introduce structure to the data that interferes with statistical analyses, a significant disadvantage relative to geometric morphometric approaches Rohlf a, b; summarized in Monteiro et al.

    Additionally, linear measures—including trusses—discard much of the positional information present in the points used for measurement Zelditch et al. Investigators will on occasion explicitly state that their linear measures were unable to discriminate taxa which they felt had obvious morphological differences e. It is in these situations that geometric methods tend to excel.

    Relative Warps Analysis Geometric morphometric methods use the positional information of landmarks, curves, and surfaces. For two closely related species with very similar leaf morphologies, landmark methods may be used for discrimination, where they have been 3 Controlling for Curvature in the Quantification of Leaf Form 33 found to outperform both linear methods coupled with principal components analysis and elliptic Fourier analysis Jensen et al.

    Landmark methods have the potential to isolate characters Bookstein , though it appears that a priori selection of clusters of landmarks corresponding to characters may be preferable MacLeod b. For most leaves, landmarks can be placed only at the tip and base Jensen , though Jensen et al. Jensen ; Jensen et al. However, figures presented in their studies Fig.

    The two leaves shown in Fig. An alternate landmarking approach Fig. This may be entirely acceptable if the goal is only to discriminate taxa, without generating meaningful shape measures, or even appropriate, if the lobes were serially homologous, but the expectation is that the whole leaf is homologous Hageman ; Kaplan , This issue can become more pronounced: in the dissected-leaf members of Pleopeltis s.

    One way around this is to select certain pinna-positions as the relocatable landmark points. The leaves shown in a exhibit more variable morphology than suggested by this diagram. Sizes of the first four lobes defined by landmarks 1—7 and 12—15 appear comparable. Because of the change in number of lobes, the difference in shape from b is recorded either as an extension of the tip landmark 11 or expansion between the first four lobes and the rest of the leaf illustrated with gray boxes. Neither of these patterns seems to capture the essential differences in shape between the two leaves 34 J.

    Krieger and Barrington used the leaf apex and the tips and bases of a subset of pinnae for landmarks: pinna-pairs 1, 2, 3 and 6. This approach was able to identify four characters to discriminate two fern taxa. However, the degree of homology of these landmarks will break down with increased variability in number of pinnapairs between position 6 and the leaf tip. The power of landmark analysis of simple leaves is, obviously, limited by discarding the shape information between landmarks Jensen ; Jensen et al.

    Elliptic Fourier and EFA-PCA Their essentially two-dimensional nature means that much of the shape of leaves can captured in the outline; this excludes venation patterns, but captures both large and small scale e. One widely used quantitative approach to comparing leaf outlines shapes is elliptic Fourier analysis EFA; Kuhl and Giardina , which mathematically decomposes outlines into sinusoidal functions Fig.

    Objects are described as closed outlines of x,y coordinates, and the changes in x and y positions are separately decomposed as periodic functions. The original outline is shown next to reconstructions made with different numbers of Elliptic Fourier Analysis EFA harmonics used. The parametric equations for x and y are used to generate the reconstructed outlines, based on EFA coefficients a, b, c, and d, which are calculated from the original outline.

    The first leaf is reconstructed with the terms for the zeroth and first harmonics. The fourth leaf uses all the terms shown. As the number of harmonics increases, the reconstructed outline resembles the original more closely 3 Controlling for Curvature in the Quantification of Leaf Form 35 to 45 degree increments around a circle. However, this is not critical to the method; the chain codes are transformed into changes in x and y, and the raw x,y data can be used instead.

    Two of the improvements introduced to Fourier analysis by EFA were that points could be unequally spaced, and the outline need not be single-valued Rohlf and Archie ; objects where radii cross the outline in more than one place violate the single-value limitation, which can be an issue for structures with wavy outlines such as lobed leaves. Shape can be recreated from the resulting Fourier harmonics, with the number of harmonics determining the degree of fidelity to the original form Fig.

    These are typically transformed to remove rotation, size, and starting position of the outline. Adjusting the outlines for starting point is reasonable for the original aims of EFA: the automatic recognition of silhouettes. However, in a biological context, it is often possible to specify a relocatable starting point using definitions of homology e. For taxonomic discrimination by leaf shape, Fourier harmonics have been found to outperform linear measures of leaf shape White et al. EFA has been successful for discrimination in hazelnut cultivars Menesatti et al. It has been used to demonstrate continuous variation in leaf shape among species in Sternbergia Gage and Wilkin and Hoya Torres et al.

    Ollson et al. EFA has also been applied to evolutionary studies, quantifying additive genetic variance Black-Samuelsson et al. One of the earliest uses of Fourier analysis in botany, Kincaid and Schneider , is often incorrectly cited as an EFA study, but they actually use a different Fourier method Rohlf and Archie The goal in using a method like EFA is, typically, to develop a set of meaningful reduced-variance shape descriptors.

    While Fourier analysis is well-suited to the discrimination of groups, its ability to extract morphological characters to allow interpretation of those differences appears to be severely limited. Unlike a linear measurement such as leaf length, which directly captures localized information about developmental processes by indirectly recording the number and size of cells within a region of the leaf, each harmonic in the Fourier decomposition describes a type of diffuse change across the whole shape.

    Bookstein et al. The simplest reconstruction, using a 36 J. Krieger single harmonic, approximates the leaf as an ellipse. Each subsequent harmonic refines the overall shape, slowly resolving localized characters. Ehrlich et al. However, studies of leaf form do not seem to support the use of Fourier harmonics in this way. Jensen et al. Premoli interpreted only one component, which corresponded to the location of the widest part of the leaf. As the first harmonic to differentiate ovate shape from an ellipse, the second harmonic necessarily describes this type of variation Fig.

    This seems at best trivially related to ovate-obovate variation in leaf shape. Even if characters cannot be identified, variance reduction can be achieved by selecting a subset of Fourier harmonics for further analysis. Typically, a qualitative assessment is made of how many harmonics are needed for reconstructed objects to become sufficiently similar to the original shapes. In Fig. However, for specific characters, such as tip or base shape, truncation of the harmonic series eliminates information, and the effects, e.

    This approach is not typically used; instead, the PCA is preceded by a reduction in the number of Fourier components e. It may be that truncation of the Fourier harmonics is appropriate for quantification of leaf features of a periodic nature, such as lobes and teeth, but this has not been studied. There remain limitations to EFA as currently used. Shape must be described as a closed outline, because a periodic signal is required. However, this is not an inherent limitation for Fourier analysis.

    Windowed short-timer Fourier transform STFT methods applied to ammonoid suture morphology Allen look promising for analysis of leaf teeth along open segments of the leaf outline, but it is less clear whether STFT would work for large-scale features in open curves. It is also unclear whether there would be a benefit to segmenting curves using relocatable features e. This requires separately interpolating different segments of the outline, which is allowable in EFA, because point spacing can be variable.

    Whether this would improve the results of an analysis, as seen in extended eigenshape analysis, remains to be tested. Despite its origin at the same time as eigenshape analysis, elliptic Fourier analysis has benefited little from developments in the field of geometric morphometrics. Applications include systematics Ray ; Jensen et al.

    Eigenshape ES analysis features four steps: description of shape, alignment of specimens, decomposition into orthogonal functions, and modeling of these shape functions Fig. Shape is commonly described using x,y coordinate data for landmarks and curves representing relocatable features i. This step is common to all outline analyses. While seemingly trivial, this is often the most time consuming step, and image resolution may impact the results.

    Progress is being made both in automated identification of leaves in an image and the extraction of curve data Belongie and Malik ; Belongie et al. The degree to which pixelation at the leaf boundary introduces noise to the data may be minimized through smoothing or curve fitting, for example, with spline curves Chi et al. In standard eigenshape, the entire curve is interpolated to equally spaced points, either in terms of distance between points or arc length along the curve. These points form the basis for comparison across specimens.

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    Extended eigenshape analysis constrains the curves into segments between landmarks e. The landmark constraints on the curve are used to specify regions that are separately interpolated, so all specimens will have equal numbers of points on a given segment. Specimen alignment is a procedure that removes size, position and rotation from curve data.

    This is typically achieved in eigenshape analysis by conversion to phi functions Fig. Phi functions describe curves as functions of cumulative angular change Fig. The coordinate data are transformed to changes in angle required to follow the curve; this information is unchanged under translation, rotation, and scaling. Reconstructing the original shapes requires both the phi functions and the segment lengths or, at least the relative lengths 38 A J. Krieger shape description alignment decomposition x,y to phi conversion digitized shape outline data extraction modeling eigenvectors eigenvalues eigenscores model Singular value decomposition of covariance or correlation matrix shape models Proc.

    Shape description transforms an image or, an actual object into analyzable data, e. Alignment removes position, rotation, and scale from the objects, leaving shape information. The eigenanalysis occurs in the decomposition stage, generating a set of variance-optimized, orthogonal measurements. Modeling, which is possible when complete object geometries are analyzed, allows the visualization of the morphospace. The four steps could equally be applied to relative warps analysis. The phi function describes cumulative change, so it remains constant when the angle between individual line segments does not change.

    Phi functions for closed, clockwise curves begin at 0. Redrawn and recomputed from Fig. A single leaf is shown overlaid on the sample mean shape. Specimens are rescaled to unit centroid size, then rotated and rescaled to minimize the distances between object outline points and the corresponding points on the mean shape which is recomputed after each round of reorientation of segments, if they are size-standardized. The latter information is not typically included in the eigenanalysis. If the outlines were interpolate to equidistant points, the x,y data may be reconstructed from the phi functions; this may not be the case if interpolation is carried out using equal distances along the curves, though the latter approach is truer to the Zahn and Roskies formulation of phi functions.

    In the decomposition step, a correlation or covariance matrix of these phi functions is subjected to a singular value decomposition, and the resulting eigenvectors 3 Controlling for Curvature in the Quantification of Leaf Form 39 describe a morphospace. The final step is modeling, visualizing shape for points in the morphospace.

    This is a particularly useful feature of eigenshape analysis, allowing you to identify the nature of variation along individual axes. Models can, in fact, be created for any space that is built through the linear combination of variables Rohlf , such as the eigenanalysis of EFA-PCA, but they are rarely used outside of relative warps and eigenshape analysis.

    Shape variation along an eigenshape axis hence, the variation represented by that eigenshape can be visualized by varying the amplitude of the eigenshape for that axis and adding this to the consensus shape Lohmann ; MacLeod The consensus can be defined in various ways, typically as the sample mean shape computed during alignment. These models are used to determine the nature of shape variation along individual axes in the shape space, and interpretations of the modeled variation can be confirmed by examining the distribution of object shapes or their qualitative descriptions along individual eigenshape axes e.

    Coordinate-point eigenshape CPES , which is only beginning to be adopted, dispenses with the phi conversion and instead compares the coordinate data of specimens, in two or three dimensions MacLeod , , Like phi function based eigenshape analysis phi-ES , curves may be constrained with landmarks.

    Unlike phi-ES, alignment is achieved through Procrustes Generalized Least Squares GLS superposition Rohlf and Slice , making this technique mathematically equivalent to an unweighted relative warps analysis e. CPES can also include isolated landmarks along with curve data. Clearly, there is no way to describe an isolated point in a framework of cumulative angular change.

    It is trivial to extend the technique to three dimensions by simply adding another dimension to the calculations for landmarks and curves; it is more complicated for surfaces; see MacLeod ; the same can be done for phi-ES, with a bit more effort MacLeod Sampson et al. Interpolated outlines were aligned using Procrustes superimposition, but then the points were allowed to slide along the outline before another round of Procrustes alignment.

    This approach was later refined by Bookstein as sliding semilandmarks, and it suffered from predating the development of extended eigenshape analysis see the discussion of their study in MacLeod To the degree the Sampson et al. It is an open question, the degree to which sliding semilandmarks distort the raw coordinate data.

    Eigenshape analysis, as applied to leaf outlines, has been shown to outperform Fourier analysis for discrimination of taxa McLellan and Endler Shape variation can be localized to regions as small as a pair of curve segments or coordinate points, and this size limit is set by the resolution of the outlines.

    Therefore, eigenshape analysis has the capacity to extract very small, localized shape characters. It 40 J. Krieger also has the potential to isolate leaf shape characters e. Other Eigenanalysis-Based Methods Several recently developed methods for leaf analysis are confusingly similar to existing geometric morphometric methods.

    All of these methods share eigenanalysis, which means they can utilize powerful shape visualization tools, but they typically fall short in their approaches to shape description. These methods are described here in the hopes of limiting further development of ad hoc methods, and referencing the ever-growing suite of approaches to the geometric morphometrics framework.

    Langlade et al. A set of 19 points were placed on each leaf. Four were relocatable: the base and tip, and two points at the distal limit of the pedicel where the lamina constricts to the midvein. These landmarks describe the positions of homologous regions of the leaf; the same is not true of the next two landmarks, placed on the outline at the widest point of the leaf.

    The remaining semilandmarks were interpolated along the lamina four on each side and midvein five. It would have been preferable to interpolate the leaf outline using the first four relocatable landmarks to segment it into two curves along the lamina and a third along the midvein.

    Additionally, describing the lamina with so few points is unnecessary, because variance reduction is better achieved in the principal components analysis to follow, and this downsampling is likely to miss subtle patterns of shape variation. Leaves were rotated and translated to make the midvein horizontal and place the centroid at the origin. The alignment is not a Procrustes superimposition, nor is it exactly a use of Bookstein coordinates Bookstein , which would rotate, scale, and translate to place the ends of the baseline at —0.

    This begs the question of why they chose to ignore well-established methods. The LeafAnalyser program Weight et al. They describe an analysis of Antirrhinum leaf data standardized in this way. The alignment is not a Procrustes superimposition. Using the tip and centroid as a baseline means that it is closer to using Bookstein coordinates than the Langlade methods, but the baseline may suffer from instability due to the centroid, the position of which can vary depending on the interpolation of the original data. Additionally, aligning the leaves in this way transfers variance away from the baseline, hence the leaf tip, to other semilandmarks.

    As the number of points is decreased, this will decrease the ability of the eigenanalysis to recognize 3 Controlling for Curvature in the Quantification of Leaf Form 41 variation in tip shape. The aligned leaves were then mean-centered, but not rescaled, and subjected to a principle components analysis. They describe the first principal component as size, despite the allometry clearly visible in the shape models for that axis. The initial automated outline extraction provided by LeafAnalyser is very useful, but the analysis should instead use a Procrustes superimposition and, if scale is to be left in, specimens should be scaled to the square root of centroid size.

    As with Langlade et al. The Langlade et al. The leaf outlines were interpolated following an unreferenced extended eigenshape protocol, interpolating among the first four landmarks described above tip, base, and two points at the limit of the pedicel. Leaves were then aligned using Procrustes superposition. Therefore, they applied coordinate-point eigenshape analysis, without actually recognizing it as such. Other Descriptions of Leaf Form Architectural approaches specify parameters such as frequency and angle of branching in stem or vein networks.

    These approaches are focused primarily on theory, generating shapes with apparent similarity to real plants Niklas , a, ; Stein and Boyer , though Niklas has certainly approached model development critically e. Simulation for the sake of making leaf-like objects is taken farther still with L-systems Prusinkiewicz and Lindenmayer ; Hammel et al.

    Recent developments in this area have potential to illuminate the origins of leaves Stein and Boyer , but the statistical behavior of the resulting morphospaces is poorly known. The venation of simple leaves can be directly compared to the branching pattern of dissected leaves Plotze et al. This type of approach would be problematic for most herbarium specimens, where veins are not typically visible and the leaves cannot be cleared, eliminating a significant source of data. There is also the open question of how venation relates to outline shape.

    The classic approach of marking an immature leaf surface with a grid of points e. Fractal dimension is a single metric which has been used to quantify margin roughness McLellan and Endler and branching patterns Campbell The fractal description of margin roughness is highly correlated with dissection index the standardized ratio of perimeter to square root of area , which is much easier to calculate McLellan and Endler Neither measure is describing localized morphometric characters.

    Multiscale fractal dimension has been shown to generate 42 J. Krieger better discrimination than Fourier analysis and linear measures Plotze et al. Fractals do not describe shape in the same way as a qualitative descriptor or outline-based measure; rather, they look at patterns of self-similarity and generate a compact description of form Campbell , either as a single value of dimension or as a curve, in the case of the multiscale fractal dimension.

    In general, singleparameter measurements of shape are problematic. It is unlikely that shape can be distilled to a single, meaningful value, and it is often the case that similar values result from very different shapes. Like architectural approaches, fractals have been used primarily to generate models of fern leaves Campbell , and the rare efforts to tie these models to real shapes rely more on subjective comparison than empirical analysis of shape variation e.

    Neither architectural nor fractal methods seem appropriate to the decomposition of leaf shape into interpretable characters. Materials and Methods This study focuses on simple-leaves species in the fern genus Pleopeltis Humb. Leaves are the primary source of macroscopic morphological diversity in ferns.

    Over the course of evolutionary history, ferns have spanned the morphological variation represented by all vascular plants, and most of that variation is still present in extant lineages Boyce and Knoll Most fern species have pinnate-determinate leaves—the archetypal fern frond—but simpleleaved species are distributed throughout the leptosporangiate ferns. Recent molecular work by Haufler C. Haufler, , University of Kansas, personal communication and Schneider Schneider et al. Schneider, , personal communication and preliminary monographic work by Smith A.

    Smith, , University of California, Berkeley, personal communication has recircumscribed the group to include some members usually treated in Polypodium L. Taxa were selected to represent most of the widespread species from Mexico to northern South America. Lists of the species used in this study are given in Table 3. I included all eight simple-leaved species of Pleopeltis used by Hooper , though I have not split Pl. Overall, sample sizes are comparable to or slightly larger than other published eigenshape analyses. Taxa with several varieties were included Pl. Two taxa which may Table 3.

    The number of individuals is the number of herbarium sheets, each a separate accession, from which one or more leaves were sampled. Additional information is given in Krieger Hooper, Amer. Fern J. Tryon, Rhodora Moore, Index Fil. Lellinger, Proc. Smith, Candollea Moore, Ind. Wendt, Amer. Krieger not belong to Pleopeltis s. Smith, , personal communication. Pleopeltis is an evolutionarily young group, estimated as having radiated less than 10 million years ago H.

    Schneider, , personal communication , lessening the chance that extinction of lineages will result in overestimating the morphological disparity that exists between extant, putative sister taxa. The molecular phylogenies support at least two origins of simple leaves within clades of dissected leaves. The recent diversification of Pleopeltis s.

    Of the herbarium sheets that had usable leaves, typically half or more of the leaves were damaged or obscured, and roughly half of the leaves that remained were highly curved. Therefore, it was critical to find a means to include the curved leaves. Image Processing Individual leaves and scale bars were digitally isolated from herbarium sheet images and converted to silhouettes in Photoshop various versions, Adobe Systems, Inc. This allowed manipulation of settings in tpsDig ver 1. Three tpsDig format data sets were created: landmarks, lamina outline, and midvein outline Fig.

    The landmarks file included one tip landmark and four petiole landmarks. The other two were placed to best approximate petiole length for a separate study. Leaves were typically overlapping and often damaged. Three types of data were saved for each simple leaf: lamina outline, midvein outline, and landmarks. Four landmarks were placed along the petiole to measure length, including petiole-rhizome junction and petiole-lamina junction.

    A fifth landmark was placed at the leaf tip. Additional landmarks could be placed in the centers of the most basiscopic sori arrows in b number of petiole landmarks could be used if greater accuracy is desired. The lamina and midvein files were digitized as closed outlines of at most 6, points. Eigenshape Analysis All analyses were performed in Mathematica version 7. The three leaf files were imported and visualized. When initially digitized the ends of the midvein and lamina outlines were not necessarily at the tip or base petiole-lamina junction landmarks; the analysis first identified the sections of the outlines corresponding to the landmarks, then reordered the points to place the outline and midvein ends at the base landmark this allows for the input of messy data, making the method easier to apply.

    The outlines were split into two segments each for the variance reduction associated with extended eigenshape analysis of homologous segments. To ensure that just the lamina was being compared, lamina outlines were pruned at the intersection with the midvein. This was necessary as several of the species used in this analysis had asymmetric extension of the lamina at the tip and base.

    Krieger for a comparison. Both the phi and coordinate-point analyses used mean-centering, where the consensus shape was subtracted from all specimens. Mean-centering eliminated that axis, making the results more comparable to a relative warps analysis identical, for CPES. For coordinate-point eigenshape, the mean centered specimens were aligned using Procrustes GLS superposition. Otherwise, both analyses were identical, as illustrated in Fig.

    Analysis 1: Sample Size To test the effects of sample size, runs were performed on random subsamples, from ten specimens up to the total sample size. Eigenshape analysis was performed on the phi functions or aligned coordinate-point data to extract specimen scores for the subsampled datasets. Alignments varied with each sample in the CPES analyses. Mean centering was performed as in the complete analysis, using a mean shape computed from each subsample.

    To assess how quickly the analyses converged on the final morphospace, a correlation was computed between scores of objects in the subsampled space and their scores in the full analysis. The absolute value of the correlation was used, because the sign of individual eigenshape axes is irrelevant being mutually orthogonal, individual axes can be reversed with no effect on the analysis. Resampling was performed for the first five eigenshape axes.

    ES30 was used for comparison, because this axis explained a small amount of the variance and would be expected to be highly sample dependent, thus more variable for small sample sizes. In principal, the eigenshapes could be compared between the original analysis and subsamples, but it is more appropriate to compare specimen scores, which represent measurements on the actual objects. Several axes explained very similar amounts of variance e.

    Failing to do this would greatly underestimate the similarity between the subsampled and full morphospaces. Analysis 2: Appending the Midvein A second eigenanalysis leaf analysis 2; LA2 was performed using a combination of raw outline and smoothed midvein data. To compare analyses before LA1 and after LA2 addition of the midveins, shape models were generated for the first seven eigenshape axes.

    Each of the eigenshapes represents a type of shape measurement, and the score of a specimen on that axis is the value of the measurement. As above, the absolute value of the correlation was used. The highest value was noted, and values less than 0.

    Analysis 3: Straightening To assess the benefits of removing curvature, leaves were mathematically straightened. A number of options were examined for straightening the lamina outline. Straightening presents several geometric issues, including reducing error due to small-scale noise in the reference curve, managing point densities, preserving point order, and retaining features such as invaginations and cardioid bases and tips.

    In this study, the reference curve used for straightening was the midvein, which was present, easily visible, and extended all the way to the tip for all leaves. To reduce the effect of noise due to pixelation and error during tracing, the midveins were smoothed by a ten-point average. The midveins averaged 1, points long, so the ten-point average retained large-scale curvature and much of the small-scale curvature, as well.

    A five-point average did not adequately remove small-scale noise. West and Noble also performed midvein smoothing before extracting width measures, using a five-point linear regression.

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    The straightening approach used by West and Noble to collect linear measures is the most intuitive automated approach: at each point along the smoothed midvein, a line is drawn perpendicular to the tangent at that point. The width at each midvein point could then be used to reconstruct the leaf in a straightened form though West and Noble did not perform this step. The normal lines often cross one another Fig. Effectively, the order of outline points is ignored. The algorithm also truncates cardioid bases and tips, and it follows only half of the contour of an invagination.

    The West algorithm could be modified to retain all of the points where the normals cross the leaf margin, but this would amplify the issues with point ordering. The sample analyzed here included a small number of leaves with cardioid emarginate tips and many leaves with deep furrows along the margin, a potentially taxonomically informative feature that needed to remain in the analysis.

    The algorithm used in this study gave priority to the lamina outline, to ensure that all points along this curve were present in the straightened outline and retained in the correct order. The lamina was straightened using a series of vectors referenced to tangents along the smoothed midvein Fig. For each point along the lamina, a vector was calculated from the closest point on the midvein. The tangent to the midvein at this point was approximated as the vector between the midvein points before and after the current point; because of the ten-point smoothing small-scale noise had a minimal effect on the tangent vectors.

    The angle between this tangent and the x-axis was then used to rotate the midveinlamina vector since all points along the straightened midvein will have the x-axis as 48 J. The unstraightened segments portray a leaf that is flat on one side and curved on the other. This suggests an asymmetric leaf, when the actual shape is better captured after straightening using the midvein as a reference. Since this is the smoothed midvein, these two points are actually the average of the surrounding 12 points, so the variation in segment lengths here is an exaggeration.

    The midvein-lamina vector is translated to the corresponding new midvein point along the x-axis. The angle between this vector and the tangent is used to rotate the translated vector, giving the straightened lamina outline point their tangent. The midvein points were placed along the x-axis, spaced according to the Euclidean distances between points along the midvein. The rotated vectors were used to generate the outline points relative to the transformed midvein points. Mining Res. Morphometrics for Nonmorphometricians. Abdelhady, A. Elewa, Evolution of the Upper Cretaceous oysters: Traditional morphometrics approach.

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