Biometrics, biomathematics and the morphometric synthesis. (English) Zbl 0855.92002

Summary: At the core of contemporary morphometrics – the quantitative study of biological shape variation – is a synthesis of two originally divergent methodological styles. One contributory tradition is the multivariate analysis of covariance matrices originally developed as biometrics and now dominant across a broad expanse of applied statistics. This approach, couched solely in the linear geometry of covariance structures, ignores biomathematical aspects of the original measurements. The other tributary emphasizes the direct visualization of changes in biological form. However, making objective the biological meaning of the features seen in those diagrams was always problematical; also, the representation of variation, as distinct from pairwise difference, proved infeasible.
To combine these two variants of biomathematical modeling into a valid praxis for quantitative studies of biological shape was a goal earnestly sought through most of this century. That goal was finally achieved in the 1980s when techniques from mathematical statistics, multivariate biometrics, non-Euclidean geometry and computer graphics were combined in a coherent new system of tools for the complete regionalized quantitative analysis of landmark points together with the biomedical images in which they are seen.
In this morphometric synthesis, correspondence of landmarks (biologically labeled geometric points, like “bridge of the nose”) across specimens is taken as a biomathematical primitive. The shapes of configurations of landmarks are defined as equivalence classes with respect to the Euclidean similarity group and then represented as single points in D. G. Kendall’s shape space [Bull. Lond. Math. Soc. 16, 81-121 (1984; Zbl 0579.62100)], a Riemannian manifold with Procrustes distance as metric. All conventional multivariate strategies carry over to the study of shape variation and covariation when shapes are interpreted in the tangent space to the shape manifold at an average shape. For biomathematical interpretation of such analyses, one needs a basis for the tangent space compatible with the reality of local biotheoretical processes and explanations at many different geometric scales, and one needs graphics for visualizing average shape differences and other statistical contrasts there. Both of these needs are managed by the thin-plate spline, a deformation function that has an unusually helpful linear algebra. The spline also links the biometrics of landmarks to deformation analysis of the images from which the landmarks originally arose.
This article reviews the history and principal tools of this synthesis in their biomathematical and biometrical context and demonstrates their usefulness in a study of focal neuroanatomical anomalies in schizophrenia.


92B05 General biology and biomathematics
62P10 Applications of statistics to biology and medical sciences; meta analysis


Zbl 0579.62100


Full Text: DOI


[1] Andreasen, N., S. Arndt, V. Swayze, T. Cizadlo, M. Flaum, D. O’Leary, J. Ehrhardt and W. Yuh. 1994. Thalamic abnormalities in schizophrenia visualized through magnetic resonance image averaging.Science 266, 294–298.
[2] Blackith, R. and R. Reyment. 1971.Multivariate Morphometrics. New York: Academic Press. · Zbl 0271.92001
[3] Bookstein, F. L. 1978.The Measurement of Biological Shape and Shape Change.Lecture Notes in Biomathematics, Vol. 24. New York: Springer. · Zbl 0376.92003
[4] Bookstein, F. L. 1982a. On the cephalometrics of skeletal change.Amer. J. Orthodontics 82, 177–198.
[5] Bookstein, F. L. 1982b. Foundations of morphometrics.Ann. Rev. Ecology and Systematics 13, 451–470.
[6] Bookstein, F. L. 1984a. A statistical method for biological shape comparisons.J. Theor. Biol. 107, 475–520.
[7] Bookstein, F. L. 1984b. Tensor biometrics for changes in cranial shape.Ann. Human Biol. 11, 413–437.
[8] Bookstein, F. L. 1986. Size and shape spaces for landmark data in two dimensions.Statis. Sci. 1, 181–242. · Zbl 0614.62144
[9] Bookstein, F. L. 1989a. Principal warps: thin-plate splines and the decomposition of deformations.IEEE Trans. Pattern Anal. Machine Intelligence 11, 567–585. · Zbl 0691.65002
[10] Bookstein, F. L. 1989b. ”Size and shape”: a comment on semantics.Systematic Zoology 38, 173–180.
[11] Bookstein, F. L. 1991.Morphometric Tools for Landmark Data. New York: Cambridge University Press. · Zbl 0770.92001
[12] Bookstein, F. L. 1994. Landmarks, edges, morphometrics, and the brain atlas problem. InFunctional Neuroimaging: Technical Foundations, R. Thatcher, M. Hallett, T. Zeffiro, E. John and M. Huerta (Eds), pp. 107–119. New York: Academic Press.
[13] Bookstein, F. L. 1995a. A standard formula for the uniform shape component in landmark data. InAdvances in Morphometrics: Proceedings of the 1993 NATO ASI on Morphometrics, L. F. Marcus et al. (Eds). New York: Plenum. To appear.
[14] Bookstein, F. L. 1995b. Combining the tools of geometric morphometrics. InAdvances in Morphometrics: Proceedings of the 1993 NATO ASI on Morphometrics, L. F. Marcus et al. (Eds). New York: Plenum. To appear.
[15] Bookstein, F. L. 1995c. Combining ”vertical” and ”horizontal” features from medical images.Computer Vision, Virtual Reality, and Robotics in Medicine.Lecture Notes in Computer Science, N. Ayache (Ed), Vol. 905, pp. 184–191. Berlin: Springer.
[16] Bookstein, F. L. 1995d. How to produce a landmark point: the statistical geometry of incompletely registered images. InVision Geometry IV. S. P. I. E. Proceedings, (R. A. Melter et al. (Eds), Vol. 2573. Bellingham, WA: SPIE, pp. 266–277.
[17] Bookstein, F. L. 1995e. Utopian skeletons in the biometric closet. Occasional Papers of the Institute for the Humanities, number 2, Institute for the Humanities, University of Michigan.
[18] Bookstein, F. L., B. Chernoff, R. Elder, J. Humphries, G. Smith and R. Strauss. 1985.Morphometrics in Evolutionary Biology. Philadelphia: Academy of Natural Sciences of Philadelphia.
[19] Bookstein, F. L. and W. D. K. Green. 1993. A feature space for edgels in images with landmarks.J. Math. Imaging and Vision 3, 231–261. · Zbl 0797.68177
[20] Bookstein, F. L. and W. D. K. Green. 1994a. Edgewarp: A flexible program package for biometric image warping in two dimensions. InVisualization in Biomedical Computing 1994.SPIE Proceedings. R. Robb (Ed), Vol. 2359. Bellingham, WA: SPIE, pp. 135–147.
[21] Bookstein, F. L. and W. D. K. Green. 1994b. Edgewarp: A program for biometric warping of medical images. Videotape, 26 minutes.
[22] Boyd, E. 1980.Origins of the Study of Human Growth. University of Oregon Health Sciences Center. Portland, OR.
[23] Burnaby, T. P. 1966. Growth-invariant discriminant functions and generalized distances.Biometrics 22, 96–110. · Zbl 0149.38403
[24] DeQuardo, J. R., F. L. Bookstein, W. D. K. Green, J. Brumberg and R. Tandon. 1995. Spatial relationships of neuroanatomic landmarks in schizophrenia.Psychiatry Research: Neuroimaging.
[25] Duchon, J. 1976. Interpolation des fonctions de deux variables suivant la principe de la flexion des plaques minces.RAIRO Anal. Numé. 10, 5–12.
[26] Duncan, O. D. 1984.Notes on Social Measurement: Historical and Critical. New York: Russell Sage Foundation.
[27] Dürer, A. 1528.Vier Bücher von Menschlicher Proportion. Dietikon-Zürich: Josef Stocker, 1969.
[28] Friston, K. J. 1994. Statistical parametric mapping. InFunctional Neuroimating: Technical Foundations R. Thatcher, M. Hallett, T. Zeffiro, E. John and M. Heurta (Eds), pp. 79–93. New York: Academic Press.
[29] Goodall, C. R. 1983. The statistical analysis of growth in two dimensions. Doctoral dissertation, Department of Statistics, Harvard University.
[30] Goodall, C. R. 1991. Procrustes methods in the statistical analysis of shape.J. Roy. Statist. Soc. Ser. B 53, 285–339. · Zbl 0800.62346
[31] Goodall, C. R. and K. V. Mardia. 1991. A geometric derivation of the shape density.Adv. in Appl. Probab. 23, 496–514. · Zbl 0736.60012
[32] Gower, J. C. 1971. A general coefficient of similarity and some of its properties.Biometrics 27, 857–874.
[33] Grenander, U. and M. Miller. 1994. Representations of knowledge in complex systems.J. Roy. Statist. Soc. Ser. B 56, 549–603. · Zbl 0814.62009
[34] Hopkins, J. W. 1966. Some considerations in multivariate allometry.Biometrics 22, 747–760.
[35] Hotelling, H. 1936. Relations between two sets of variables.Biometrika 28, 321–377. · Zbl 0015.40705
[36] Huxley, J. 1932.Principles of Relative Growth. London: Methuen.
[37] Jolicoeur, P. 1963. The multivariate generalization of the allometry equation.Biometrics 19, 497–499.
[38] Kendall, D. G. 1984. Shape-manifolds, Procrustean metrics, and complex projective spaces.Bull. London Math. Soc. 16, 81–121. · Zbl 0579.62100
[39] Kent, J. T. 1994. The complex Bingham distribution and shape analysis.J. Roy. Statist. Soc. Ser. B 56, 285–299. · Zbl 0806.62040
[40] Kent, J. T. and K. V. Mardia. 1994. The link between kriging and thin-plate splines. InProbability, Statistics, and Optimisation, F. P. Kelly (Ed), pp. 325–339. New York: Wiley. · Zbl 0861.41006
[41] Koenderink, J. 1990.Solid Shape. Cambridge, MA: M.I.T. Press.
[42] Kuhn, T. S. 1959. The function of measurement in modern physical science. InQuantification, H. Woolf (Ed), pp. 31–63. Indianapolis: Bobbs-Merrill.
[43] Latour, B. 1987.Science in Action. Cambridge, MA: Harvard University Press.
[44] Lewis, J. L. W. Lew and J. Zimmerman. 1980. A nonhomogeneous anthropometric scaling method based on finite element principles.J. Biomechanics 13, 815–824.
[45] Lohmann, G. P. 1983. Eigenshape analysis of microfossils: a general morphometric procedure for describing changes in shape.Math. Geology 15, 659–672.
[46] Marcus, L. F., E. Bello and A. García-Valdecasas (Eds). 1993.Contributions to Morphometrics. Madrid: Monografias, Museo Nacional de Ciencias Naturales, Consejo Superior de Investigaciones Cientificas.
[47] Marcus, L. F., M. Corti, A. Loy, G. Naylor and D. Slice (Eds). 1995.Advances in Morphometrics: Proceedings of the 1993 NATO ASI on Morphometrics. New York: Plenum. To appear.
[48] Mardia, K. V. 1995. Shape advances and future perspectives. InProceedings in Current Issues in Statistical Shape Analysis, K. V. Mardia and C. A. Gill (Eds), pp. 57–75. Leeds, U.K.: Leads University Press.
[49] Mardia, K. V. and I. Dryden. 1989. The statistical analysis of shape data.Biometrika 76, 271–281. · Zbl 0666.62056
[50] Mardia, K. V. and C. A. Gill (Eds). 1995.Proceedings in Current Issues in Statistical Shape Analysis. Leeds, U.K.: Leeds University Press.
[51] Meinguet, J. 1979. Multivariate interpolation at arbitrary points made simple.Z. Angewandte Math. Phys. 30, 292–304. · Zbl 0428.41008
[52] Mosimann, J. E. 1970. Size allometry: size and shape variables with characterizations of the log-normal and generalized gamma distributions.J. Amer. Statist. Assoc. 65, 930–945. · Zbl 0223.62025
[53] Netter, F. H. 1989.Atlas of Human Anatomy. Summit, NJ: Ciba-Geigy Corporation.
[54] Oxnard, C. E. 1973.Form and Pattern in Human Evolution. Chicago: University of Chicago Press.
[55] Oxnard, C. E. 1978. On biologist’s view of morphometrics.Ann. Rev. Ecology and Systematics 9, 219–241.
[56] Pearson, K. 1914–1930.The Life, Letters and Labours of Francis Galton (Three volumes bound as four). Cambridge, U.K.: University Press.
[57] Porteous, I. R. 1994.Geometric Differentiation for the Intelligence of Curves and Surfaces. Cambridge, U.K.: Cambridge University Press. · Zbl 0806.53001
[58] Reyment, R. A. 1991.Multivariate Palaeobiology. Oxford: Pergamon.
[59] Richards, O. W. and A. C. Kavanagh. 1943. The analysis of relative growth-gradients and changing form of growing organisms: illustrated by the tobacco leaf.American Naturalist 77, 385–399.
[60] Rohlf, F. J. 1993. Relative warp analysis and an example of its application to mosquito wings. InContributions to Morphometrics, L. F. Marcus et al. (Eds), pp. 131–159. Madrid: Monografias, Museo Nacional de Ciencias Naturales, Consejo Superior de Investigaciones Cientificas.
[61] Rohlf, F. J. and F. J. Bookstein (Eds). 1990.Proceedings of the Michigan Morphometrics Workshop. Ann Arbor, MI: University of Michigan Museums.
[62] Rohlf, F. J. and D. Slice, 1990. Extensions of the Procrustes method for the optimal superposition of landmarks.Systematic Zoology 39, 40–59.
[63] Sibson, B. 1978. Studies in the robustness of multidimensional scaling: Procrustes statistics.J. Roy. Statist. Soc. Ser. B. 40, 234–238. · Zbl 0389.62086
[64] Sneath, P. H. A. 1967. Trend-surface analysis of transformation grids.J. Zoology 151, 65–122.
[65] Sneath, P. H. A. and R. R. Sokal, 1963.Principles of Numerical Taxonomy. San Francisco: W. H. Freeman.
[66] Stigler, S. M. 1986.The History of Statistics: The Measurement of Uncertainty Before 1900. Cambridge, MA: Harvard University Press. · Zbl 0656.62005
[67] Thompson, D’A. W. 1917.On Growth and Form. London: Macmillan. · JFM 46.0375.03
[68] Timoshenko, S. and S. Woinowsky-Krieger. 1959.Theory of Plates and Shells, 2nd ed. New York: McGraw-Hill. · Zbl 0114.40801
[69] Wahba, G. 1990.Spline Models for Observational Data. Philadelphia: Society for Industrial and Applied Mathematics. · Zbl 0813.62001
[70] Wright, S. 1968.Evolution and the Genetics of Populations. Vol. 1: Genetic and Biometric Foundations. Chicago: University of Chicago Press.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.