The Gibbs Phenomenon for Piecewise-Linear Approximation

Foster, James; Richards, F. B.

doi:10.1080/00029890.1991.11995703

Cited by 47 publications

(27 citation statements)

References 1 publication

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“…The Gibbs phenomenon associated with spherical harmonics were first observed by Herman Weyl in 1968 [18]. The history and the overview of Gibbs phenomenon can be found in several literature [17, 27]. …”

Section: Discussionmentioning

confidence: 99%

Cosine series representation of 3D curves and its application to white matter fiber bundles in diffusion tensor imaging

Chung

Adluru

Lee

et al. 2010

Statistics and Its Interface

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We present a novel cosine series representation for encoding fiber bundles consisting of multiple 3D curves. The coordinates of curves are parameterized as coefficients of cosine series expansion. We address the issue of registration, averaging and statistical inference on curves in a unified Hilbert space framework. Unlike traditional splines, the proposed method does not have internal knots and explicitly represents curves as a linear combination of cosine basis. This simplicity in the representation enables us to design statistical models, register curves and perform subsequent analysis in a more unified statistical framework than splines. The proposed representation is applied in characterizing abnormal shape of white matter fiber tracts passing through the splenium of the corpus callosum in autistic subjects. For an arbitrary tract, a 19 degree expansion is usually found to be sufficient to reconstruct the tract with 60 parameters.

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Section: Discussionmentioning

confidence: 99%

Cosine series representation of 3D curves and its application to white matter fiber bundles in diffusion tensor imaging

Chung

Adluru

Lee

et al. 2010

Statistics and Its Interface

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show abstract

“…In the short period since 1992, more references have been found for the Gibbs Phenomenon in Fourier-Bessel series expansion [23][24][25][26][27], in Fourier series in higher dimensions [25,27]' in spline expansions [28][29][30][31][32], in the interpolation of the discrete Fourier transform (DFT) [33], and in how it depends on the way we measure it as an error (in Lp-approximation, for example) [34]. This is in addition to the recent active research for the Gibbs phenomenon in most of the continuous wavelets integral representations [35,36,37]' and the discrete wavelets series expansions [32,[38][39][40]30].…”

Section: The Gibbs-wilbraham Phenomenonmentioning

confidence: 99%

“…This is a credit to Wilbraham that is long overdue! More recent such referencing are those of Korner [73], Troutman and Bautista [74], Walter [40]' Foster and Richards [28] and in [22]. Before Gibbs explained the presence of overshoots and undershoots in 1898, it is known that Wilbraham's discovery of this phenomenon was mentioned in the 535 pages of Burchardt [67], which covered almost everything about the history of the trigonometric series until 1850.…”

Section: (2101)mentioning

confidence: 99%

“…Indeed, at about that time, there were a number of research activities that were also directed at the study of a Gibbs-like phenomenon in expansions other than the typical Fourier series and integrals. This includes the approximation of functions with jump discontinuities by splines [28][29][30][31]. In 1991, Foster and Richards' [28] started that for splines of order 2 (or degree 1), or what is usually described as piecewise-linear approximation.…”

Section: A Brief Overviewmentioning

confidence: 99%

“…This includes the approximation of functions with jump discontinuities by splines [28][29][30][31]. In 1991, Foster and Richards' [28] started that for splines of order 2 (or degree 1), or what is usually described as piecewise-linear approximation. They showed the existence of a Gibbslike phenomenon with maximum overshoot of 13.4% of the size of the jump discontinuity, which is much higher than the typical 8.95% of the Fourier series and integrals.…”

Section: A Brief Overviewmentioning

confidence: 99%

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The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations

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Low Reynolds number isotope transient kinetic modeling in isothermal differential tubular catalytic reactors

Otarod

Supkowski

2017

AIChE Journal

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A novel method is presented for modeling steady state isotope transient kinetics of heterogeneous catalytic reactions when the flow regime is laminar and conversion is differential. It is based on a factorization theorem which is deduced from the observation that transport functions fluctuate radially in porous beds. Factorization separates the radial from axial and temporal coordinates of the flow rate and concentration functions. It is shown that in transient tracing with a differential conversion, the radial components of the transport functions in the material conservation equations can be integrated into constant parameters to be determined from experimental data. The method is in particular useful as the knowledge of the radial profile of velocity and other transport functions and parameters are not prerequisites for data correlation. The methodology was successfully applied to the adsorption of carbon monoxide in Boudouard reaction on an alumina supported palladium catalyst. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1317–1329, 2018

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The Gibbs Phenomenon for Piecewise-Linear Approximation

Cited by 47 publications

References 1 publication

Cosine series representation of 3D curves and its application to white matter fiber bundles in diffusion tensor imaging

Cosine series representation of 3D curves and its application to white matter fiber bundles in diffusion tensor imaging

The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations

Low Reynolds number isotope transient kinetic modeling in isothermal differential tubular catalytic reactors

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