Topic
Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: In this article, the Fourier coefficients are used to extract basic shape features from fiber cross-sections and shape complexity and other geometric attributes such as area, roundness, and ellipticity.
Abstract: . Quantitative characterization of fiber cross sections has attracted considerable in terlst, since cross-sectional size and shape have an important impact on the physical and mechanical properties of fibers, as well as the performance of end-use products. We present one application of automated measurement using image processing tech niques that extract basic shape features from fiber cross sections. Cross-sectional shapes are characterized with the aid of geometric and Fourier descriptors. Geometric de scriptors measure attributes such as area, roundness, and ellipticity. Fourier descriptors are derived from the Fourier series for the cumulative angular function of the cross- sectional boundary and are used to characterize shape complexity and other geometric attributes. Shape reconstruction based on Fourier coefficients is also discussed. We present the results of shape analysis for a wide variety of fiber types.
70 citations
TL;DR: In this paper, it was shown that the radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients, and the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analysis of solutions to parabolic and non-parabolic partial differential equations.
70 citations
TL;DR: This paper shows that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic, and demonstrates that they are particularly well suited for this problem.
Abstract: An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.
In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308---318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.
70 citations
TL;DR: In this article, an image processing algorithm, based on Fourier expansion, was developed to characterize the shape of apples, which reduced the dimensionality of the edge points of an image to a set of 24 Fourier coefficients.
70 citations
TL;DR: In this paper, a unified numerical analysis model is presented to solve the free vibration of composite laminated doubly-curved shells and panels of revolution with general elastic restraints by using the Fourier-Ritz method.
70 citations