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Needs Improvement Love it! Wilson, H. Tagare, F. Dougherty, eds. SPIE, vol. New Anatomist — Modern Morphometrics 41 Burnaby, T. Carroll, J. Cole, T. Dryden, I. Feder, J. Galton, F. Gilbert, J. Goodall, C. Statistical Society, Series B, — Gould, S. Gower, J. Hanihara, T. Howells, W. Harvard University, Cambridge, MA. Huxley, J. Reprinted , Dover Publications, New York. Johnson, R. Kendall, D. Kent, J. Slice Klingenberg C. Klingenberg, C. Klingenberg C.

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Kolar, J. Thomas Publishers Ltd. Krzanowski, W. Kuhl, F. Lele, S. III, , A new test for shape differences when variancecovariance matrices are unequal, J. Lele S. Long, C. Mahalanobis, P. Mandelbrot, B. Freeman and Company, New York. Modern Morphometrics 43 Manly, B. Marcus, L. Rohlf and F. Bookstein, eds. Mardia, K. Moreton, I. Monteiro, L. Morant, G. Mosier, C. Mosimann, J.

Palmer, A. Palmqvist, P. Pearson, K. Peitgen, H. Rao, C. Slice Rao, C. Reyment, R. Naylor and D. Richtsmeier, J. Robins, G. Rohlf, F. Version 1. Siegel, A. Modern Morphometrics 45 Siegel, A. Series 2, Department of Statistics, Princeton University. Slice, D. Small, C. Smith, D. Sneath, P. Stoyan, D. Bookstein M y argument here springs principally out of the fact that the morphometrics of named location data is essentially complete. Over a busy decade of development there has emerged a complete multivariate methodology for biometric sizeand-shape analysis of named location data, whatever the geometric scheme, that helps to further a wide variety of studies concerned with mean differences, variation and ordination, or covariances of form with its causes or effects.

For instance, intersections of curves must be named landmarks, and also sharp corners or centers of curvature of sharply curved arcs of otherwise smooth curves; likewise, intersections of curves with surfaces. Intersections of surfaces must be named curves, and also ridges along which smooth surfaces are particularly sharply curved. Fred L. Bookstein the review by Adams, Rohlf, and Slice , or the chapters by Slice and by Mitteroecker et al. And so it is not too early to begin speculating on the next toolkit even as our research and teaching communities assimilate this one.

Quantitative morphology has long dealt with information beyond the coordinates of named points, curves, or surface structures, and many of these other types of information deserve morphometric methods of their own. Sometimes the underweighted information is at relatively large scale, such as a growth-gradient; sometimes it is at relatively small scale, such as spacing of nearby structures; and sometimes it is multilocal, such as pertains to bilateral symmetry or to patterns of spacing down an axis.

Under each heading I sketch the limitations of the current methodological toolkit, limitations often cryptic or rarely noted elsewhere, and then show a range of real examples that suggest possible new or newly applied formalisms for data alignment, variance and covariance, linear models, and graphic displays. Whether or not the map is linear, the integral can be thought of as a summary of the extent to which the resulting deformed grid cells are not all the same—their variation as little graphical objects on their own.

Morphometrics shares these useful functions with several other domains of applied mathematics, including mathematical geology and environmetrics. In these other communities, the part with zero bending is called drift. Now any quadratic trend-surface map cf. Sneath, is considered bending-free inasmuch as all of the third derivatives of a quadratic polynomial are identically zero. Bookstein Bookstein, , sec. In view of the cognitive neuropsychology of human vision, a new discipline will be required for interpreting the alternative grids.

There is thus a plausible alternative to the method of creases Bookstein, as well. The onedimensional equivalent for this new construal of drift a quadratic trend, that is, a map with constant second derivative is the analogous search for extrema of the second derivative of an interpolation. Detailed lore of this new version of the old thin-plate spline will be published elsewhere. Notice that for two-dimensional pictorial data, the quadratic trend subspace has six more dimensions than the linear trend subspace, a total of eight instead of two.

Here I would like to present just one example of how a familiar data set becomes reinterpreted by this substitution, in which the new spline is applied to the classic Vilmann data set of landmark octagons from the midline neural skull After Landmarks 53 of the rodent. Figure 1 shows the analyses of deformations from the age-7 mean to the mean shapes at ages of 14, 21, 30, 40, 60, and 90 days. In the top row is the ordinary linear-drift thin-plate spline; in the second row, the new version with quadratic drift term. Thin-plate spline representations for the Vilmann rat midsagittal neurocranial data set, I: analysis of shape change from the age-7 average.

Second row: the suggested alternate quadratic-trend thin-plate spline TPSQ for the same comparisons. Third row: quadratic trend components of the same comparisons. All trend deformations combine vertical compression with a relative shortening of the upper margin of the calva. Fourth row: residuals local bending from the quadratic trend. All show a bending along the top of the calva. Grid sectors outside the form should not be interpreted. Upper row: quadratic-trend thin-plate splines. Second row: Quadratic components. Third row: residuals from trend. All deformations are exaggerated threefold.

As is customary [Rohlf and Bookstein, ], the trend term is estimated by least-squares in Procrustes distance, not the corresponding term of the exactly interpolating spline. This points to a new feature, the bending of the upper margin of the calva, that did not emerge as a feature from analyses using the ordinary thin-plate spline. Each transformation is exaggerated threefold. From the second row, we see that the quadratic trend, a combination of relative shortening of the upper margin and relative compression of height, characterizes each of the six growth intervals separately.

Issues of appropriate spacing of landmarks in connection with estimates of a quadratic trend term are different from those pertinent to a linear trend. After Landmarks 55 Residuals from the quadratic trend, bottom row, show a continual infusion of the bending of the upper margin already seen in Figure 1. The successive shape changes are now modest enough in magnitude that none of the interpolations fold in the vicinity of the data, even at threefold extrapolation. The comparison of Figures 1 and 2 is instructive.

For instance, we see quite clearly how, as with the ordinary linear-drift thin-plate spline, the new quadratic-drift grids are not put forward as models for actual tissue changes, but instead as guideposts for the extraction of biometric features. As the quadratic drift shows third row , there is no such vertical gradient in the landmark data. How does this rolling-up arise? Notice the sharp discrepancy across the form between the strong bending of the landmarks of the upper margin and the nearly invariant geometry of those along the lower margin.

When there is no bending cost to a quadratic term, as with this particular spline, the exact interpolation represents the ordinate as if it lay on a parabolic cylinder of axis nearly aligned with the line along which landmark locations are hardly changing the cranial base. The paraboloid can be as bent as it needs to be around that line; hence the line-singularity of the spline map there.

Figures 1 and 2 are wholly consistent in this respect. Reparametrizing—in effect, regridding—in this way often leads to greater constancy of apparent morphogenetic factors. For an extended discussion in more mathematical language, see Miller et al. The limitation to twelve of the twenty rats is purely for reasons of legibility. Bookstein rat 9 drift rat 9 local rat 10 drift rat 10 local rat 11 drift rat 11 local rat 12 drift rat 12 local rat 13 drift rat 13 local rat 14 drift rat 14 local rat 15 drift rat 15 local rat 16 drift rat 16 local rat 17 drift rat 17 local rat 18 drift rat 18 local rat 19 drift rat 19 local rat 20 drift rat 20 local Figure 3.

Quadratic-trend thin-plate spline deformations for twelve individual rats, 14 to 40 days of age. Columns 1, 3, 5: global quadratic trend. Columns 2, 4, 6: residual bending. Notice the strong resemblance of the individual trend deformations, and the variable bending of the top of the calva in the residual grids.

Each deformation is extrapolated twofold for legibility. The doubling is merely for legibility. Ninety percentage of the total Procrustes sum of squares for quadratic trend is carried by one single dimension, the combination of vertical compression and trapezoid formation we have already noted. The remaining variation of the global trend is spherical. The local bending part of these transformations is characterized by After Landmarks 57 one single relative warp for which the sample extreme scores belong to animal 11 and animal 20; evidently this component concerns the degree of bending of the upper calvarial margin.

The crease noted in Bookstein for each of these animals actually arises by virtue of the combination of the large-scale trapezoidal process with this bending factor; it is always very near the bottom of that bend. Notice, also, that the extent of this bending along the upper margin is correlated to the appearance of the singularity here, the rolling-up of the residual grid along the lower margin already discussed in connection with Figure 1.

For data in two dimensions, it is estimated using eight degrees of freedom, considerably fewer than the full tangent-space dimensionality for this example of Nothing up our sleeve here: We have moved from full shape space to an a-priori subspace, just as in the consideration of the more usual linear uniform term itself. One might expect, however, that in other applications, such as tumor growth, these local terms might have a larger role to play.

This display was originally developed to suit the 58 Fred L. As the scope of the new morphometrics was extended, its graphics did not keep pace: the conventional diagram style has not hitherto been extended to exploit the special properties of semilandmarks on curves or surfaces. When the form under comparison is characterized by extended features, it is an impoverishment to report its comparisons by a pictorial grammar making no reference to those features.

In this section I sketch three extensions of the current toolkit that might aid the task of biological understanding in particular applications. While the arguments of the three-dimensional thin-plate spline formulas are constant on mutually perpendicular planes, it is not necessary to subordinate the report of a transformation grid to any such algebraic straitjacket.

The Edgewarp program package Bookstein and Green, includes a display mode that intersects an arbitrary three-dimensional thin-plate spline incorporating either version, linear or quadratic, of the trend formalism by an arbitrary sequence of query planes. The data for this example, courtesy of Philipp Gunz and Philipp Mitteroecker of the University of Vienna, are part of a larger study of sexual dimorphism and allometry in the anthropoids that is still in preparation.

The example involves landmarks and semilandmarks for the comparison of an adult male chimpanzee skull to the skull of a human two-year-old for the semilandmark scheme, see Mitteroecker et al. Figure 4 traces the parietal crest of the chimpanzee skull outer surface black line and shows one of a series of section planes constructed precisely normal to this ridge curve.

From the continuum of deformed squares, Figure 5 shows a selection of eight in three different perpendicular views. After Landmarks 59 Figure 4. Surface of a chimpanzee skull, with the parietal ridge traced and one normal plane indicated. There are semilandmark points on this surface of 15, triangles.

Bilateral Asymmetry While the geometric-statistical structure of asymmetry analysis in Procrustes space is thoroughly understood Mardia et al. The methods apply to any form made up out of some landmarks or curves that are unpaired—that lie along a putative midplane—and also landmarks or curves that are paired left and right.

The set of precisely symmetrical forms of this sort, regardless of the counts of paired and unpaired features, constitutes a mathematical hyperplane, and the mirroring is represented by a vector perpendicular to that hyperplane, for which Procrustes coordinate shifts sum to zero over the paired landmarks Fred L. Bookstein 0. Eight sections normal to the parietal curve of Figure 4 from the thin-plate spline that warps the semilandmarks of the chimp skull onto the homologously placed landmarks for the skull of a human two-year-old. In the limit of small deformations, the thin-plate splines for the mirroring deformation, left side vs right side, are inverses.

In the conventional thin-plate spline, the trend for a mirroring deformation like this is limited to a simple shear along the manifold of unpaired structures. It is thereby of hardly any help in the description of asymmetry since only rarely is asymmetry characterized by such a shear. The spline introduced in the preceding section, with a quadratic trend term, is much better suited to this application.

Figure 6 compares the two as applied to a previously published Figure 6. The spline on 32 landmarks 9 bilateral pairs and 14 unpaired is sectioned in a conventional facial plane. A midline shift centered on Orbitale is clear in both presentations. L, R: left, right. Lo: lateral orbital point. Nas: Nasion. Sla: Sella. Mo: Medial orbital point. Po: Porions. Or: Orbitale. Rhn: Rhinion. Pi: Pyriform aperture lateral extrema. Bsn: Basion. Go: Gonion. Sdl: Supradentale. Lie: Lower incisal edge at midline. Uie: Upper incisal edge at midline.

Idl: Infradentale. Bpt: B point.

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    After Landmarks 61 62 Fred L. Bookstein threee-dimensional data set, the grand average of all the three-dimensionalized images from the entire Broadbent-Bolton normative male craniofacial sample see Dean et al. The principal feature of the grid is a lateral shift centered on the semilandmark Orbitale. In the center panel is the corresponding grid i. The graphics of the midline shift is the same, but now there is a new and more obvious visual feature also, one corresponding very well to the familiar size dominance of one hemifacial complex over the other.

    The global nature of this asymmetry is even clearer in Figure 7 left , which shows the quadratic thin-plate spline using a sectioning plane twice as large in every direction. The size contrast is global; the shift of the midline per se, merely local. Large-scale and small-scale aspects of asymmetry. Left: The same quadratic spline, sectioned at twice the scale. Right: The same after three bilaterally paired landmarks in the orbital region are deleted. The local asymmetry shift of the midline has now vanished, leaving only the familiar isotropic left—right size difference.

    After Landmarks 63 key in Figure 6. The midline shift is now obliterated, leaving only the global quadratic trend for mean left—right asymmetry in every direction of this frontal presentation. Other Coordinate Systems with Their Own Symmetries The same Edgewarp display mode that underlay the dynamic grids of Figure 5 can also be used to show the actual content of solid biomedical images. Edgewarp has built-in utilities that access Eve directly over the Internet, and its manual Bookstein and Green, explains how to construct dynamic displays of arbitrary sections of Eve and compare them to analogous sections of your own CT or MRI image or any other three-dimensional anatomical resource to which you have access.

    For publication these continuously moving displays must be reduced, like the grids in Figure 5, to a discrete series of static diagrams in a common coordinate system. The other planes shown are a selection from a full circle of normal sections of the callosum centered precisely on its midcurve which is not a plane structure. For an explanation of this construction, along with an example of its medical importance, see Bookstein et al. These two examples have in common the liberation of the report of anatomy, or of its comparisons, from any dependence on the coordinate systems 64 Fred L.

    Bookstein characterizing the imaging physics by which the raw data were gathered. When morphometric data were limited to landmark points, these issues did not arise, as landmarks do not specify any particular frame of reference, and all the Procrustes methods for landmark analysis are coordinate-free.

    But the biology of the extensions of the landmark methods to curves and surfaces is not coordinate-free—these structures represent the real processes by which the coordinates have been literally coordinated. The combination of the thin-plate spline with Procrustes methods for analysis of this data cf. Mitteroecker et al. This was the method of edgels edge information at landmarks of Bookstein and Green, An edgel is made from a landmark by adding additional coordinates—augmenting the actual Cartesian coordinates of location by additional information regarding derivatives of the deformation presumably based on knowledge coming from edges or textures of the original data.

    The mathematics of the edgel will not be reviewed here—it is set out in full in the paper—but the general colonization of three-dimesional data by the old two-dimesional toolkit suggests that now might be a good time to resurrect that possibility as well. That study After Landmarks 65 was of a total of 24 anthropoids represented by 15 landmark points and 21 semilandmarks derived from simulated midsagittal sections of actual CT images of 16 Homo sapiens, six fossil Homo, and two chimpanzees.

    The interesting part of the data analysis involved representation of these two borders of the sectioned frontal bone by a pair of semilandmark structures. The corresponding grid Figure 10, top panel clearly indicates the concentration of the shape difference at the frontal sinus. Figure Analysis of an archel in Edgewarp. Top: Comparison of average form of the midsagittal frontal bone and vicinity, 16 Homo sapiens vs six fossils, using 26 landmarks and semilandmarks as published in Bookstein et al.

    Bottom: A nearly identical grid representing the thickness of the frontal bone by derivative information edgels instead of the locations of the second curve of semilandmarks. Bookstein The idea of the edgel is that the effect of a grid on pairs of nearby landmarks, or landmarks paired with nearby semilandmarks, can be modeled as an explicit datum pertaining to the derivative of the mapping. Below in Figure 10 is the edgel representation of the same deformation scheme. Now there is only one curve of semilandmarks, rather than two, and the separation between inner and outer tables of the frontal bone is represented instead by ratios of change of length and, to a limited extent, direction of the little vectors drawn.

    In reality, each of these vectors is simply double the separation of the corresponding pair of semilandmarks from the upper panel, projected in the direction normal to the inner curve. Evidently this alternative formalism results in very nearly the same description of deformation as the extended two-curve semilandmark representation above. Combinations of these two formalisms, semilandmarks and edgels, make possible a great increase in the descriptive power of morphometrics for actual structural questions at multiple scales of geometric observation simultaneously. The original edgel formalism can specify any number of derivatives at a landmark or, as here, at a semilandmark , and can specify direction and magnitude of directional derivatives separately.

    Sexual dimorphism of the senile human corpus callosum, exaggerated threefold. Left: In three dimensions, the kernel of the linear-trend thin-plate spline insulates the space surrounding an arch or sheet from reorientations of that structure. Right: The quadratic-trend spline propagates such changes into the surrounding grid. Data are from Davatzikos et al. There is an interesting interaction between the archel and the algebra of the thin-plate spline as I have been exploring it in this chapter. The kernel function r of the ordinary three-dimensional thin-plate spline is nondifferentiable at landmarks or semilandmarks.

    As a consequence, transformations applying the old linear-drift TPS to sheets of semilandmarks in three dimensions do not constrain the grid to bridge the deformation on one side of the pair of sheets smoothly to the deformation on the other side of the pair. The result cf. Figure 11, left is to insulate changes within arches from interaction with the structures around them—not a helpful property.

    The alternative thin-plate spline introduced in this chapter, with kernel r 3 for data from landmarks or semilandmarks in three dimensions, does not have this problem Figure 11, right. In allowing changes within a sheet of tissue to extrapolate into the tissue normal to the sheet for some distance, it presents an alternative report of local features that may be a better guide to actual morphogenetic processes in several applications.

    Bookstein origin in more classical applications of signal-processing to emphasize a different formalism in which the image contents are represented as functions of Cartesian coordinates rather than coordinate values per se. They may have been more prophetic than they intended, in that it is typical of revolutions in the quantitative sciences cf. Kuhn, that parts of a data tradition are left behind in the rush to establish new formalisms that are more powerful in particular applications. That is certainly the case for the new geometric morphometrics, which is overdue to go back and pick up pictorial information, textures, and all the rest of the information that was available in earlier data representations such as photographs beyond the landmarks and semilandmarks reviewed here.

    The trick is to expand the variation of image contents in the vicinity of a mean image in a Taylor series for shifts of arguments taken at the locations of actual landmarks; then at least for small ranges of image variation the least-squares functional analysis of the Taylor series expansions approximates the geometric morphometrics of the same landmark locations. That no simple worked examples of this next round of techniques are available for inclusion in this chapter owes solely to the obduracy of certain standing study sections at NIH.

    In other words, the combinations of semilandmark schemes with derivative information need not be discretized. Instead, composite data resources could be constructed that in effect balance this After Landmarks 69 formalism of deformation grid against the present formalism of generalized landmarks points, curves, surfaces itself. Bookstein Bookstein, F.

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    Macleod, ed. Zaidel and M. Iacoboni, eds. Bookstein, J. Duncan, N. Lange, and D. Wilson, eds. SPIE Proceedings, volume , pp. Dean, D. Kuhn, T. Woolf, ed. Miller, M. After Landmarks 71 Rohlf, F. Wahba, G. Bookstein oday there is a fully developed statistical toolkit for data that come as coordinates of named point locations or landmarks. Because all the statistical methods require these landmarks to be homologous among the specimens under investigation it is challenging to include information about the curves and surfaces in-between the landmarks in the analysis.

    The problem is that these correspond biologically as extended structures rather than lists of distinct points. This chapter is devoted to the method of semilandmarks Bookstein, , which allows these homologous curves and surfaces to be studied with the existing statistical toolkit. Information from the interior of homogeneous tissue blocks is not accessible by these methods. Here we explicitly extend the algebra to curves and surfaces in three dimensions and give practical advice on how to collect and interpret this kind of data. The notion of homology invoked in this assumption is not the classic biological notion of that name, which entails similarity of structure, physiology, or development owing to common descent Ax, ; Cain, ; Mayr, , ; Remane, In this classic diction, only explicit entities of selection or development can be considered homologous.

    Since points per se are not likely to be explicit targets of selection, this criterion is too strict—it would rule out almost any use of point coordinates in the course of evo-devo research. Hence for some 30 years morphometrics has used a distinct but related notion of homology, traceable perhaps to an article by Jardine , that centers on variation in the relationships among locations of structures across samples. The landmarks and semilandmarks that serve as data for the methods of this chapter both arise as careful spatial samples of this underlying mapping function.

    In three dimensions, however, the number of truly homologous point locations is Semilandmarks in Three Dimensions 75 often very limited. On the skull, true landmarks are typically located on bony processes, at the intersections of sutures, or at foramina Richtsmeier et al. But many curving structures lack punctate landmarks of this sort, and on others candidate points cannot be declared with any assurance to correspond across realistic ranges of variation. The method of semilandmarks begins with structures that are known to correspond as parts the classic biological notion of homology , and then represents them by geometric curves or surfaces that, in turn, generate reasonable mapping functions.

    In this way the biological notion of homology has most of its power and sweep restored, as the notion of point-landmark has proved too stringent for effective biometrics in most three-dimensional anthropological applications. Extensions of the thin-plate spline to include curvature information can be found in Bookstein and Green , see also: Bookstein this volume and Little and Mardia A statistical analysis separates the total geometric signal into one part from the true landmark points, together with the residual.

    Andresen et al. Ridge lines, characterized by a minimax property of directional surface curvature, are extracted and matched in order to establish object correspondence. The semilandmarks are mapped into Procrustes space and analyzed using principal coordinates. Each of these approaches is ad-hoc or algebraically inconsistent in one or another important way. There are some Procrustes steps, some Euclidean projection steps, some unwarping steps, and some operations of equal spacing, under the control of no particular governing equation.

    It would be preferable to have an approach that is matrix-driven at all its steps, so that in studies of modest variation, such as characterizes most quantitative evo-devo work in vertebrate zoology, the variation and covariation of all parameters, whether interpreted, modelled, or discarded as nuisance or noise, can be treated together. The combination of these two steps results in an essentially unique set of shape coordinates for the semilandmarks describing most realistic assemblages of landmarks, curves, and surfaces on three-dimensional forms. To justify a method more complicated than equally spaced points on curves or even triangulations of surfaces, it is necessary to show what goes wrong with those temptingly simple alternatives.

    Figure 1a shows a rectangle with one landmark in the lower left corner along with 27 other points spaced equally around the outline. Figures 1b and 1c show a slightly different rectangle with two different sets of semilandmarks. The left thin-plate spline grid in Figure 2 shows a remarkably suggestive pattern of gradients and twists. The comparisons we publish, and the statistics that support them, need to apply Semilandmarks in Three Dimensions 77 Figure 1. Figure 2. Thin-plate splines corresponding to Figure 1. The only way we can think of to achieve this invariance is to produce the spacing as a by-product of the statistical analysis itself.

    Figure 3 shows a similar problem for outline structures that bend at large scale. When the points are distributed on the bent form under the criterion of equidistancy 3b , their positions relative to the corners do not correspond to the points in 3a. A better solution is presented in 3c. Figure 4 shows that the TPS grid from 3a to 3c is much smoother and thus, in this application, less misleading than the one from 3a to 3b.

    While the two generic examples of elongating or bending rectangles might have been resolved in part by placing true landmarks at the corners and Type III landmarks at the midpoints of the sides, in many applications Nature is less generous with sharp corners or other shape features that could serve as landmarks. This is the case for the midline of the corpus callosum, the structure that connects the two hemispheres of the brain.

    Figure 5 shows a dataset composed of corpus callosum outlines taken from midsagittal sections of MRI scans representing 78 Philipp Gunz et al. Figure 3. Figure 4. Splines corresponding to Figure 3.

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    Figure 5. Midsagittal section of an MRI scan and some corpus callosum outlines. Semilandmarks in Three Dimensions 79 normal variation of adults and children. These curves elongate and bend but have only one landmark rostrum. Figure 6 shows deformation grids between the average consensus form and a form with equidistant points compared with the same form captured by semilandmarks. When the consensus is compared to the specimen with the equidistant points, the thin-plate spline deformation grid shows strong local shape differences.

    Show all. From the reviews: "The chapters in Modern Morphometrics in Physical Anthropology are from a symposium at the meeting of the American Association of Physical Anthropologists.

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