Abstract
The ability to test for similarities and differences among families of shapes by closed-form Fourier expansion is greatly enhanced by the concept of homology. Underlying this concept is the assumption that each term of a Fourier series, when compared to the same term in another series, represents the “same thing”. A method that ensures homology is one which minimizes the “centering error,” as reflected in the first harmonic term of the Fourier expansion. The problem is to chose a set of edge points derived from a much larger, but variable, number of edge points such that a valid homologous Fourier series can be calculated. Methods are reviewed and criteria given to define a “proper” solution. An algorithm is presented which takes advantage of the fact that minimization of the “error term” can be accomplished by minimizing the distance between the origin of the polar coordinate system in the calculation of the Fourier series and the shape centroid. The use of this algorithm has produced higher quality solutions for quartz grain provenance studies.
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Beddow, J. K. and Philip, G. C., 1975, One the use of a Fourier analysis technique for describing the shape of individual particles: Planseeberichte fur Pulvermetallurgie, v. 23, n. 1, p. 3–14.
Ehrlich, R., Brown, P., Yarus, J., and Przygocki, R., 1980, The origin of shape frequency distributions and the relationship between size and shape: Jour. Sed. Pet., v. 50, n. 2, p. 475–484.
Ehrlich, R. and Weinberg, B., 1970, An exact method for the characterization of grain shape: Jour. Sed. Pet., v. 40, n. 1, p. 205–212.
Fico, C., 1980, Automated particle shape analysis—Development of a microprocessor controlled image analysis system: unpublished masters thesis, Univ. of South Carolina.
Full, William E., Ehrlich, R., and Klovan, J. E. 1981, EXTENDED QMODEL—Objective definition of external end members in the analysis of mixtures: Jour. Math. Geol., v. 13, p. 331–344.
Klovan, J. E. and Miesch, A. T., 1976, EXTENDED CABFAC and QMODEL computer programs for Q-mode factor analysis of compositional data: Comput. Geosci., v. 1, p. 161–178.
Meloy, T. P., 1977, A hypothesis for morphological characterization of particle shape and physiochemical properties, Powder Tech., v. 16, n. 2, p. 223–253.
Purcell, Edwin J., 1965, Calculus with analytic geometry: Appleton-Century-Crofts, New York, N.Y.
Schwarcz, H. P. and Shane, K. C., 1969, Measurement of particle shape by Fourier analysis: Sedimentology, v. 13, n. 314, p. 213–232.
Zahn, C. T. and Roskies, R. Z., 1972, Fourier descriptors for plane closed curves: IEEE Trans. Comput., C-21, p. 269–281.
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Full, W.E., Ehrlich, R. Some approaches for location of centroids of quartz grain outlines to increase homology between Fourier amplitude spectra. Mathematical Geology 14, 43–55 (1982). https://doi.org/10.1007/BF01037446
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DOI: https://doi.org/10.1007/BF01037446