The processing of periodically sampled multidimensional signals

Mersereau, R.M.; Speake, T.

doi:10.1109/tassp.1983.1164018

Cited by 98 publications

(26 citation statements)

References 5 publications

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“…where the latter equality represents the sampling induced aliasing of the spectrum b f on the reciprocal (or dual) lattice generated by the matrix (2π/h)L −T [23]. We denote the reciprocal lattice as L…”

Section: Preliminariesmentioning

confidence: 99%

“…From this, by using the Strang-Fix criteria (7) for ψ and ϕ, it can be deduced that, if the approximation order of ψ is at least n, then the resultant filter Di satisfies (23) and provides an n-th order approximation of ∂ l i f when used in conjunction with ϕ i . Furthermore, this result holds true irrespective of the second-stage shift.…”

Section: Assessment Of the Two-stage Op Frameworkmentioning

confidence: 99%

“…In order for a directional derivative filter Di to be asymptotically optimal, it must satisfy the optimality criterion (23). This is tantamount to requiring that the Taylor-series expansion of the frequency response b Di(ω) match that of the ideal analysis function jli T ω b ϕ ⋆ (ω) upto order n + 1 where n is the desired approximation order.…”

Section: A Strategy For Designing Practical Filtersmentioning

confidence: 99%

“…, then the combined filter di satisfies (28) as well as the optimality criterion (23). The directional dependence due to the derivative and the shift are completely reflected in the response of the derivative filter ri making the filter p directionally independent.…”

Section: A Strategy For Designing Practical Filtersmentioning

confidence: 99%

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Gradient Estimation Revitalized

Alim

Möller

Condat

2010

IEEE Trans. Visual. Comput. Graphics

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Fig. 1. DVR images of the carp dataset. Left: Original Cartesian (256 3 ) dataset reconstructed using prefiltered tricubic B-splines and shaded using the analytic gradient. Middle: Downsampled Cartesian (128 3 ) dataset reconstructed using prefiltered tricubic B-splines and shaded by interpolating the gradient obtained through a fourth-order central differencing scheme (pFIR). Right: The middle dataset shaded using our proposed fourth-order shifted gradient estimation scheme (pFIR-s). Notice how the details in the bones and skull are much better preserved as compared to the central-differencing scheme even though the underlying scalar interpolation is the same. This gain in quality comes at no additional cost.Abstract-We investigate the use of a Fourier-domain derivative error kernel to quantify the error incurred while estimating the gradient of a function from scalar point samples on a regular lattice. We use the error kernel to show that gradient reconstruction quality is significantly enhanced merely by shifting the reconstruction kernel to the centers of the principal lattice directions. Additionally, we exploit the algebraic similarities between the scalar and derivative error kernels to design asymptotically optimal gradient estimation filters that can be factored into an infinite impulse response interpolation prefilter and a finite impulse response directional derivative filter. This leads to a significant performance gain both in terms of accuracy and computational efficiency. The interpolation prefilter provides an accurate scalar approximation and can be re-used to cheaply compute directional derivatives on-the-fly without the need to store gradients. We demonstrate the impact of our filters in the context of volume rendering of scalar data sampled on the Cartesian and Body-Centered Cubic lattices. Our results rival those obtained from other competitive gradient estimation methods while incurring no additional computational or storage overhead.

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Section: Preliminariesmentioning

confidence: 99%

Section: Assessment Of the Two-stage Op Frameworkmentioning

confidence: 99%

Section: A Strategy For Designing Practical Filtersmentioning

confidence: 99%

Section: A Strategy For Designing Practical Filtersmentioning

confidence: 99%

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Gradient Estimation Revitalized

Alim

Möller

Condat

2010

IEEE Trans. Visual. Comput. Graphics

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“…We nd that it is hard to construct orthogonal scaling functions Decomposed images with highpass lters Q (1) , Q (2) , Q…”

Section: Which Implies That (25) Holds If and Only If V (ω)mentioning

confidence: 99%

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Jiang

2008

IEEE Trans. Signal Process.

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Recently hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs fewer sampling points; it has better consistent connectivity; it has higher symmetry; its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal lter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal lter banks which are available in the literature have only 3-fold symmetry.In this paper we investigate the construction of orthogonal and PR FIR hexagonal lter banks with 6-fold symmetry. We obtain block structures of 7-size re nement (7-channel 2-D) orthogonal and PR FIR hexagonal lter banks with 6-fold rotational symmetry.√ 7-re nement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal lter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal lter banks with pseudo 6-fold axial symmetry.Index TermsHexagonal lattice, hexagonal image, lter bank with 6-fold symmetry, orthogonal hexagonal lter bank, biorthogonal hexagonal lter bank, √ 7-re nement wavelet, 7-size re nement multiresolution decomposition/reconstruction.

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Interpolation filters for 2-D hexagonally sampled signals

Fettweis

1997

Int. J. Circ. Theor. Appl.

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The processing of periodically sampled multidimensional signals

Cited by 98 publications

References 5 publications

Gradient Estimation Revitalized

Gradient Estimation Revitalized

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Interpolation filters for 2-D hexagonally sampled signals

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