A discrete analogue of the Weierstrass transform

Norman, Edward

doi:10.1090/s0002-9939-1960-0117512-5

Cited by 6 publications

(3 citation statements)

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“…Please note that this ground truth is also subject to discretisation artefacts and may not exactly reflect the desired solution. A more complicated alternative to this implementation has been proposed in [12], but this variant also suffers from similar truncation problems. Finally, let us note that we chose the Boat test image as a good representative for many real-world examples, since it contains many different frequencies and both homogeneous and textured regions.…”

Section: Qualitative Gainmentioning

confidence: 99%

Theoretical Foundations of Gaussian Convolution by Extended Box Filtering

Gwosdek

Grewenig

Bruhn

et al. 2012

Lecture Notes in Computer Science

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Abstract. Gaussian convolution is of fundamental importance in linear scale-space theory and in numerous applications. We introduce iterated extended box filtering as an efficient and highly accurate way to compute Gaussian convolution. Extended box filtering approximates a continuous box filter of arbitrary non-integer standard deviation. It provides a much better approximation to Gaussian convolution than conventional iterated box filtering. Moreover, it retains the efficiency benefits of iterated box filtering where the runtime is a linear function of the image size and does not depend on the standard deviation of the Gaussian. In a detailed mathematical analysis, we establish the fundamental properties of our approach and deduce its error bounds. An experimental evaluation shows the advantages of our method over classical implementations of Gaussian convolution in the spatial and the Fourier domain.

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Section: Qualitative Gainmentioning

confidence: 99%

Theoretical Foundations of Gaussian Convolution by Extended Box Filtering

Gwosdek

Grewenig

Bruhn

et al. 2012

Lecture Notes in Computer Science

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“…Impressive results on improved efficiency by means of recursive filtering can be found in [14] and [15], and the close relation between recursive filters and linear scalespace approaches has been clarified in [32]. Semidiscrete or fully discrete analogs of linear diffusion filtering have been proposed in [4], [26], [34] and [38].…”

Section: Efficient and Reliable Schemes For Nonlinear Diffusion Filtementioning

confidence: 99%

“…First we have to show that (34) is invertible. This is easily seen, because is strictly diagonally dominant (35) It is well known from linear algebra that strictly diagonally dominant matrices are invertible, see e.g., [35, p. 226].…”

Section: B Semi-implicit Schemementioning

confidence: 99%

Efficient and reliable schemes for nonlinear diffusion filtering

Weickert

Romeny²,

Viergever

1998

IEEE Trans. on Image Process.

1,033

718

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Laurent, Fourier, and Chebyshev-Padé Tables

Gragg¹

1977

Pade and Rational Approximation

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A discrete analogue of the Weierstrass transform

Cited by 6 publications

References 1 publication

Theoretical Foundations of Gaussian Convolution by Extended Box Filtering

Theoretical Foundations of Gaussian Convolution by Extended Box Filtering

Efficient and reliable schemes for nonlinear diffusion filtering

Laurent, Fourier, and Chebyshev-Padé Tables

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