Symmetric convolution and the discrete sine and cosine transforms

Martucci, S.A.

doi:10.1109/78.295213

Cited by 317 publications

(211 citation statements)

References 25 publications

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“…We choose to use DCT-I and DST-I, and their corresponding inverses, IDCT-I and IDST-I, which can be efficiently implemented using Fast Fourier transforms (FFT) [see Martucci, 1994]. In this way it can be said that where…”

Section: Numerical Proceduresmentioning

confidence: 99%

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Insights from a pseudospectral approach to the Elder problem

Reeuwijk

Mathias

Simmons

et al. 2009

Water Resources Research

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[1] The aim of this paper is to clarify and circumvent the issue of multiple steady state solutions in the Elder problem. A pseudospectral method is used to avoid numerical error associated with spatial discretization. The pseudospectral method is verified by comparison to an analytical solution at Rayleigh number, Ra = 0, and by reproducing the three stable steady state solutions that are known to exist at Ra = 400. A bifurcation diagram for 0 < Ra < 400, which is free of discretization error, confirms that multiple steady states are indeed an intrinsic characteristic of the Elder problem. The existence of multiple steady states makes the Ra = 400 Elder problem less suitable for benchmarking numerical models. To avoid the multiple steady states, we propose a benchmark at Ra = 60. The results for this Low Rayleigh Number Elder Problem are presented and compared to simulations with the commercial groundwater modeling package FEFLOW. Correspondence between the pseudospectral model and FEFLOW is excellent.

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Section: Numerical Proceduresmentioning

confidence: 99%

“…It should be noted that the DCT-I and DST-I methods correspond to cosine and sine transforms up to a prefactor [Martucci, 1994]. These prefactors should be applied before there is full correspondence.…”

Section: Numerical Proceduresmentioning

confidence: 99%

Insights from a pseudospectral approach to the Elder problem

Reeuwijk

Mathias

Simmons

et al. 2009

Water Resources Research

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“…Note that the transform block sizes for and are both lcm where lcm denotes the least common multiple, and hence the uniform TDF results can be applied. As a result, the pipelining multiplication matrices become (10) By defining lcm (11) the various parameters of the system are given only if lcm by (1)- (7). The equivalent system model is shown in Fig.…”

Section: B Ntdfmentioning

confidence: 99%

“…With respect to transform-domain filtering (TDF), a number of techniques have been applied to provide convolution-multiplication properties to the DCT [9]- [11]. Such "DCT filtering" approaches successfully reduce the number of multiplications and additions, but also possess some limitations and imperfections.…”

Section: Introductionmentioning

confidence: 99%

2-D transform-domain resolution translation

Lee

Eleftheriadis

2000

IEEE Trans. Circuits Syst. Video Technol.

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Abstract-The extensive use of discrete transforms in image and video coding suggests the investigation on filtering before downsampling (FBDS) and filtering after upsampling (FAUS) methods directly acting on the transform domain. In this paper, we describe the "transform-domain resolution translation" technique that gives flexibility to resize windows of each video conferencing session for server compositing without explicit decompression, spatial domain processing, and compression. We generalize transform-domain filtering (TDF) to include nonuniform and multirate cases to implement the transform-domain resolution translator. The former is defined as a TDF problem in which the original transform domain is of different size from the target one, while the latter considers the implementation of sampling rate conversion in the transform domain. The implementation architecture is based on a pipeline that involves matrix-vector product blocks and vector addition, but is not limited to particular hardware. Such techniques are particularly useful for fast algorithms for processing compressed images and video where transform coding is extensively used (e.g., in JPEG, H.261, MPEG-1, MPEG-2, and H.263). Index Terms-Multirate transform-domain filtering (MTFD), nonuniform transform-domain filtering (NTDF), transform-domain filtering (TDF), transform-domain resolution translation (TDRT).

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“…Martucci [2] investigated the relationships between symmetric convolution and the various Discrete Trigonometric Transforms. He demonstrated that DTTs could be used for an efficient implementation of the linear filtering of finite signals with reflective boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Efficient and Consistent Recursive Filtering of Images with Reflective Extension

Appleton

Talbot

2003

Scale Space Methods in Computer Vision

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Abstract. Recursive filters are commonly used in scale space construction for their efficiency and simple implementation. However these filters have an initialisation problem which either produces unusable results near the image boundaries or requires costly approximate solutions such as extending the boundary manually. In this paper, we describe a method for the recursive filtering of reflectively extended images for filters with symmetric denominator. We begin with an analysis of reflective extensions and their effect on nonrecursive filtering operators. Based on the non-recursive case, we derive a formulation of recursive filtering on reflective domains as a linear but time-varying implicit operator. We then give an efficient method for decomposing and solving the linear implicit system. This decomposition needs to be performed only once for each dimension of the image. This yields a filtering which is both stable and consistent with the ideal infinite extension. The filter is efficient, requiring the same order of computation as the standard recursive filtering. We give experimental evidence to verify these claims.

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Symmetric convolution and the discrete sine and cosine transforms

Cited by 317 publications

References 25 publications

Insights from a pseudospectral approach to the Elder problem

Insights from a pseudospectral approach to the Elder problem

2-D transform-domain resolution translation

Efficient and Consistent Recursive Filtering of Images with Reflective Extension

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