Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T08:47:19.783Z Has data issue: false hasContentIssue false
  • English
  • Français

Perimeter-based Fourier analysis: a new morphometric method applied to the trilobite cranidium

Published online by Cambridge University Press:  19 May 2016

Mike Foote
Mike Foote*
Affiliation:
Committee on Evolutionary Biology, The University of Chicago, Chicago, Illinois 60637
Get access Rights & Permissions [Opens in a new window]

Abstract

A new Fourier method is presented to quantify shapes too complex to be described by conventional polar Fourier analysis. The length along a closed curve serves as the independent variable. The centroid of the curve is determined and for each point on the curve two different dependent variables are defined, based on: 1) the angle defined by the starting point, the centroid, and the point on the curve; and 2) the radial distance from the centroid to the point on the curve. The method is used to describe the trilobite cranidium, and 12 harmonic coefficients are found to summarize 99 percent of the shape information contained in the cranidial outline. In an application to trilobite evolution during the Cambrian and Ordovician, it is found that higher taxa of trilobites become progressively more distinct morphologically. This result is in agreement with previous qualitative observations, and is attributable to an increase in morphologic dispersion among higher taxa, but not to a decrease in morphologic dispersion within higher taxa.

Type
Research Article
Information
Journal of Paleontology , Volume 63 , Issue 6 , November 1989 , pp. 880 - 885
Copyright
Copyright © The Paleontological Society 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anstey, R. L., and Delmet, D. A. 1973. Fourier analysis of zooecial shapes in fossil tubular bryozoans. Geological Society of America Bulletin, 84:1753–1764.2.0.CO;2>CrossRefGoogle Scholar
Benson, R. H., Chapman, R. E., and Siegel, A. F. 1982. On the measurement of morphology and its change. Paleobiology, 8:328–339.CrossRefGoogle Scholar
Bookstein, F. L. 1977. The study of shape transformation after D'Arcy Thompson. Mathematical Biosciences, 34:177–219.Google Scholar
Bookstein, F. L., Strauss, R. E., Humphries, J. M., Chernoff, B., Elder, R. L., and Smith, G. R. 1982. A comment on the uses of Fourier methods in systematics. Systematic Zoology, 31:85–92.Google Scholar
Bookstein, F. L., Chernoff, B., Elder, R. L., Humphries, J. M., Smith, G. R., and Strauss, R. E. 1985. Morphometrics in Evolutionary Biology. The Academy of Natural Sciences of Philadelphia, Special Publication 15, 277 p.Google Scholar
Canfield, D. J., and Anstey, R. L. 1981. Harmonic analysis of cephalopod suture patterns. Mathematical Geology, 13:23–35.Google Scholar
Cherry, L. M., Case, S. M., Kunkel, J. G., Wyles, J. S., and Wilson, A. C. 1982. Body shape metrics and organismal evolution. Evolution, 36:914–933.CrossRefGoogle ScholarPubMed
Clark, M. W. 1981. Quantitative shape analysis: a review. Mathematical Geology, 13:303–320.CrossRefGoogle Scholar
Davis, J. C. 1986. Statistics and Data Analysis in Geology, 2nd ed. John Wiley and Sons, New York, 646 p.Google Scholar
Ehrlich, R., Pharr, R. B. Jr., and Healy-Williams, N. 1983. Comments on the validity of Fourier descriptors in systematics: a reply to Bookstein et al. Systematic Zoology, 32:202–206.Google Scholar
Ehrlich, R., and Weinberg, R. 1970. An exact method for characterization of grain shape. Journal of Sedimentary Petrology, 40:205–212.Google Scholar
Ferson, S., Rohlf, F. J., and Koehn, R. K. 1985. Measuring shape variation of two-dimensional outlines. Systematic Zoology, 34:59–68.Google Scholar
Foote, M. 1988. Changing patterns of morphospace occupation among Cambrian and Ordovician trilobites. Geological Society of America, Abstracts with Programs, 20:A46.Google Scholar
Fortey, R. A., and Chatterton, B. D. E. 1988. Classification of the trilobite suborder Asaphina. Palaeontology, 31:165–222.Google Scholar
Fortey, R. A., and Owens, R. M. 1975. Proetida—a new order of trilobites. Fossils and Strata, 4:227–239.Google Scholar
Fox, W. T. 1987. Harmonic analysis of periodic extinctions. Paleobiology, 13:257–271.CrossRefGoogle Scholar
Gevirtz, J. L. 1976. Fourier analysis of bivalve outlines: implications on evolution and autecology. Mathematical Geology, 8:151–163.CrossRefGoogle Scholar
Harrington, H. J. 1959. Classification, p. O145–O170. In Moore, R. C. (ed.), Treatise on Invertebrate Paleontology, Pt. O, Arthropoda 1. Geological Society of America and University of Kansas Press, Lawrence.Google Scholar
Kaesler, R. L., and Waters, J. A. 1972. Fourier analysis of the ostracode margin. Geological Society of America Bulletin, 83:1169–1178.Google Scholar
Kuhl, F. P., and Giardina, C. R. 1982. Elliptic Fourier features of a closed contour. Computer Graphics and Image Processing, 18:236–258.Google Scholar
Lane, P. D., and Thomas, A. T. 1983. A review of the trilobite suborder Scutelluina. Special Papers in Palaeontology, 30:141–160.Google Scholar
Lochman-Balk, C., and Wilson, J. L. 1958. Cambrian biostratigraphy in North America. Journal of Paleontology, 32:312–350.Google Scholar
Lohmann, G. P. 1983. Eigenshape analysis of microfossils: a general morphometric procedure for describing changes in shape. Mathematical Geology, 15:659–672.Google Scholar
Palmer, A. R. 1958. Morphology and ontogeny of a Lower Cambrian ptychoparioid trilobite from Nevada. Journal of Paleontology, 32:154–170.Google Scholar
Rasetti, F. 1954. Phylogeny of the Cambrian trilobite family Catillicephalidae and the ontogeny of Welleraspis . Journal of Paleontology, 28:599–612.Google Scholar
Rasetti, F. 1961. Dresbachian and Franconian trilobites of the Conococheague and Frederick limestones of the central Appalachians. Journal of Paleontology, 35:104–124.Google Scholar
Rohlf, F. J., and Sokal, R. R. 1981. Statistical Tables, 2nd ed. W. H. Freeman and Company, San Francisco, 219 p.Google Scholar
Ross, R. J. Jr., et al. 1982. The Ordovician System in The United States. International Union of Geological Sciences, Publication No. 12, 73 p.Google Scholar
Shaw, A. B. 1957. Quantitative trilobite studies II. Measurement of the dorsal shell of non-agnostidean trilobites. Journal of Paleontology, 31:193–207.Google Scholar
Sneath, P. H. A. 1967. Trend-surface analysis of transformation grids. Journal of Zoology, 151:65–122.Google Scholar
Sokal, R. R., and Rohlf, F. J. 1981. Biometry, 2nd ed. W. H. Freeman and Company, San Francisco, 859 p.Google Scholar
Temple, J. T., and Tripp, R. P. 1979. An investigation of the Encrinurinae (Trilobita) by numerical taxonomic methods. Transactions of the Royal Society of Edinburgh, 70:223–250.Google Scholar
Thompson, D'A. W. 1961 [1917, 1942]. On Growth and Form, abridged ed. Cambridge University Press, Cambridge, 346 p.Google Scholar
Waters, J. A. 1977. Quantification of shape by use of Fourier analysis: the Mississippian blastoid genus Pentremites . Paleobiology, 3:288–299.Google Scholar
Whittington, H. B. 1966. Phylogeny and distribution of Ordovician trilobites. Journal of Paleontology, 40:696–737.Google Scholar
Younker, J. L., and Ehrlich, R. 1977. Fourier biometrics: harmonic amplitudes as multivariate shape descriptors. Systematic Zoology, 26:336–342.Google Scholar
Zahn, C. T., and Roskies, R. Z. 1972. Fourier descriptors for plane closed curves. IEEE Transactions on Computers, C-21:269–281.Google Scholar