The Least-Squares Property of the Lanczos Derivative

Burch, Nathanial; Fishback, Paul E.

doi:10.2307/30044192

Cited by 14 publications

(16 citation statements)

References 9 publications

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“…He called this differentiation by integration. His work got quite a lot of citations, see for instance [25], [60], [32], [54], [8], [9], [66]. The name Lanczos derivative, notated as f L (x), became common for a value obtained from the right-hand side of (3.7).…”

Section: Lanczos' 1956 Workmentioning

confidence: 99%

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Differentiation by integration using orthogonal polynomials, a survey

Diekema¹,

Koornwinder

2012

Journal of Approximation Theory

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This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing.Comment: 35 pages, 3 figures; minor corrections, subsection 3.11 added; accepted by J. Approx. Theor

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Section: Lanczos' 1956 Workmentioning

confidence: 99%

“…The name Lanczos derivative, notated as f L (x), became common for a value obtained from the right-hand side of (3.7). In [54] and [8] also (3.6) (the Legendre case for general n) was rediscovered.…”

Section: Lanczos' 1956 Workmentioning

confidence: 99%

Differentiation by integration using orthogonal polynomials, a survey

Diekema¹,

Koornwinder

2012

Journal of Approximation Theory

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“…The pioneering work done by Lanczos [55,56] designs lowpass responses by truncating the Fourier series coefficients and minimizing the energy of Gibbs oscillation in the least square sense. Different extensions of this work have been also reported in [9,19,32,66] using Lagrange and Legendre polynomials. A more generalized design in least square sense would be the SG filters [31,62,68,83,84] that have been applied in many engineering problems.…”

mentioning

confidence: 91%

“…• The filter should not contain any perturbing residues after the cutoff level. However in most cases they are usually represented as ripples (side-lobes) such as in least square designs [9,31,45,62,62,66,83,84] or slow decay responses in smoothing kernels such as derivative-of-Gaussian (DoG) filters [18,29,64]. Such residuals are shown in 1.2.…”

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confidence: 99%

Finite Differences in Forward and Inverse Imaging Problems: MaxPol Design

Hosseini¹,

Plataniotis²

2017

SIAM J. Imaging Sci.

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Abstract. A systematic and comprehensive framework for finite impulse response (FIR) lowpass/fullband derivative kernels is introduced in this paper. Closed form solutions of a number of derivative filters are obtained using the maximally flat technique to regulate the Fourier response of undetermined coefficients. The framework includes arbitrary parameter control methods that afford solutions for numerous differential orders, variable polynomial accuracy, centralized/staggered schemes, and arbitrary side-shift nodes for boundary formulation. Using the proposed framework four different derivative matrix operators are introduced and their numerical stability is analyzed by studying their eigenvalues distribution in the complex plane. Their utility is studied by considering two important image processing problems, namely gradient surface reconstruction and image stitching. Experimentation indicates that the new derivative matrices not only outperform commonly used method but provide useful insights to the numerical issues in these two applications.Key words. numerical differentiation, boundary condition, derivative matrix, fullband/lowpass FIR differentiation, maxflat technique, inverse Vandermonde, gradient surface recovery, image stitching 1. Introduction. Finite difference (FD) methods are used extensively in variety of image processing tasks. For example, they are used in digital approximation of order moments such as gradients, Hessian, and high-order tensor. They find numerous applications in either forward imaging problems to encode certain features such as edge and corner [11,44,60,61,64,69,78], or discretizing the differential operators used in inverse problems for image restoration [1,8,23,33,59,67,76,77]. Despite extensive applications, the problem of numerical differentiation suffers from several deficiencies such as the lack of estimation accuracy, sensitivity over perturbing artifacts, and improper formulation of boundary condition (BC). Such deficiencies can easily make the overall imaging applications highly complicated (ill-posed) and sometimes turn the whole exercise to be useless. Recent works highlight the need for a systematic and comprehensive framework of numerical differentiation to overcome these issues by reducing the complexities of error propagation in the system design and furthermore prevent computational burdens. Applications can be found in gradient surface reconstruction [30,33], edge detection [2,3,5,10], edge synthesis [22,35,81], feature extraction [16,17,30], and many more. This article concerns with the main question of what is the best numerical approach to approximate derivatives in an image processing related task? With this question in mind we present a comprehensive framework for differentiation applied in forward and inverse imaging problems. The "forward" here is referred to find a solution of direct approximation in a measurement system A(X) = Y where A is an operator and X is a given signal/image to find Y . While, the "inverse" is referred to inferring X from a simi...

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“…In (2) and (3) we employ two derivatives, however, also there is the orthogonal derivative (Washburn 2006, Groetsch 1998, Shen 1999, Hicks & Liebrok 2000, Burch et al 2005, Diekema & Koornwinder 2012 obtained by Lanczos (1956), Cioranescu (1938) and Haslam-Jones (1953), hence it is natural to ask if this ultimate derivative leads to relation (5). The answer is yes, to see the next section.…”

Section: Introductionmentioning

confidence: 99%