Some approaches for location of centroids of quartz grain outlines to increase homology between Fourier amplitude spectra

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.     

FUZZY QMODEL—A new approach for linear unmixing

Many data sets can be viewed as a collection of samples representing mixtures of a relatively small number of end members. When end members are present in the sample set, the algorithm QMODEL by Klovan and Miesch can efficiently determine proportionate contributions. EXTENDED QMODEL by Full, Ehrlich, and Klovan was designed to deduce the composition of realistic end members when the end members are not represented by samples. However, in the presence of high levels of random variation or outliers not belonging to the system of interest, EXTENDED QMODEL may not be reliable inasmuch as it is largely dependent on extreme values for definition of an initial mixing polyhedron. FUZZY QMODEL utilizes the fuzzy c-means algorithm of Bezdek to provide an alternative initial mixing polyhedron. This algorithm utilizes the collective property of all the data rather than outliers and so can produce suitable solutions in the presence of noisy or “messy” data points.     

付里叶形状分析方法及其在地质学中的应用

作为1种有效的定量分析方法。付里叶形状分析已被应用到地质学诸多分支学科的研究中，如沉积学、岩石学、微体古生物学等。文中简要介绍付里叶形状定量分析的基本原理和方法，特别是闭合式付里叶分析方法及其在地质学中的应用。应用实例之一是南极沉积物源的解释，即怎样区分冰期海洋沉积物和冰蓣物；之二是研究反映生态环境变化的浮游有孔虫G．truncatulionoides的形状。其付里叶形状定量分析结果与δ^18O同位素数据相吻合。     

Book review

Mathematical Geosciences -     

Fundamental problems associated with “eigenshape analysis” and similar “factor” analysis procedures

Recent advances in image analysis techniques have allowed rapid generation of peripheral points of two-dimensional objects. Such points, defining overall shape of an object, have been analyzed using several techniques including geometric shape analysis, Fourier analysis, and, more recently, by “eigenshape” analysis. The latter technique purports to represent a general, optimal, and universally best technique to analyze shapes of fossils. Such a technique was devised allegedly due to failure of exisiting techniques. We seriously question the validity of the eigenshape technique and discuss limitations of such an approach. Objections presented in this paper are applicable to a variety of other multivariable data reduction techniques as well.