Discrete Cosine and Sine Transforms

Britaňák, Vladimír

doi:10.1201/9781420037388.ch4

Cited by 42 publications

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“…A multi-dimensional transform (known as Fast Direct multi-dimensional DCT) can be carried out by using a composition of the one-dimensional magnification procedure along each dimension [9]. Equation (8) can then be immediately extended to 2D or 3D velocity models.…”

Section: Three-dimensional Space Reductionmentioning

confidence: 99%

“…Equation (8) can then be immediately extended to 2D or 3D velocity models. A detailed description of the extension of equation (8) to higher dimensions is given in [9]. In the case of three-dimensional data, given a 3D velocity model G of n = n x × n y × n z parameters, we ought to build a magnified 3D velocity model m of size N = N x × N y × N z n parameters.…”

Section: Three-dimensional Space Reductionmentioning

confidence: 99%

“…In our case, no information about the objective function will be used, thus our subspace technique can be applied regardless of the problem. Inspired by the methodology used in image compressing, we propose in this work a new procedure to construct this basis using a combination of sinusoidal and rectangular basis functions, more specifically a Discrete Cosine Transform (DCT) [9] and a step function transform. In fact, the step function used to magnify the vector parameter g ∈ R n , so that it fits the original space R N , usually leads to a pixelization effect.…”

Section: Search Space Reductionmentioning

confidence: 99%

“…As a smoothing procedure, we choose to work with a sinusoidal basis since it is one of the most popular subspace approaches for FWI to generate a smooth approximation vector using few coefficients [31]. We will smooth the pixelization effect in the magnified vector using a DCT [9].…”

Section: One-dimensional Space Reductionmentioning

confidence: 99%

“…The most used smoothing algorithms are based on Gaussian smoothing [1], bilateral filters [51], and sinusoidal based approaches [9]. As a smoothing procedure, we choose to work with a sinusoidal basis since it is one of the most popular subspace approaches for FWI to generate a smooth approximation vector using few coefficients [31].…”

Section: One-dimensional Space Reductionmentioning

confidence: 99%

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A parallel evolution strategy for an earth imaging problem in geophysics

et al. 2015

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In this paper we propose a new way to compute a rough approximation solution, to be later used as a warm starting point in a more refined optimization process, for a challenging global optimization problem related to Earth imaging in geophysics. The warm start consists of a velocity model that approximately solves a full-waveform inverse problem at low frequency. Our motivation arises from the availability of massively parallel computing platforms and the natural parallelization of evolution strategies as global optimization methods for continuous variables.Our first contribution consists of developing a new and efficient parametrization of the velocity models to significantly reduce the dimension of the original optimization space. Our second contribution is to adapt a class of evolution strategies to the specificity of the physical problem at hands where the objective function evaluation is known to be the most expensive computational part. A third contribution is the development of a parallel evolution strategy solver, taking advantage of a recently proposed modification of these class of evolutionary methods that ensures convergence and promotes better performance under moderate budgets.The numerical results presented demonstrate the effectiveness of the algorithm on a realistic 3D full-waveform inverse problem in geophysics. The developed numerical approach allows us to successfully solve an acoustic full-waveform inversion problem at low frequencies on a reasonable number of cores of a distributed memory computer.

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Section: Three-dimensional Space Reductionmentioning

confidence: 99%

Section: Three-dimensional Space Reductionmentioning

confidence: 99%

Section: Search Space Reductionmentioning

confidence: 99%

Section: One-dimensional Space Reductionmentioning

confidence: 99%

Section: One-dimensional Space Reductionmentioning

confidence: 99%

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A parallel evolution strategy for an earth imaging problem in geophysics

et al. 2015

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An Integer Approximation Method for Discrete Sinusoidal Transforms

Cintra

2011

Circuits Syst Signal Process

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Approximate methods have been considered as a means to the evaluation of discrete transforms. In this work, we propose and analyze a class of integer transforms for the discrete Fourier, Hartley, and cosine transforms (DFT, DHT, and DCT), based on simple dyadic rational approximation methods. The introduced method is general, applicable to several blocklengths, whereas existing approaches are usually dedicated to specific transform sizes. The suggested approximate transforms enjoy low multiplicative complexity and the orthogonality property is achievable via matrix polar decomposition. We show that the obtained transforms are competitive with archived methods in literature. New 8-point square wave approximate transforms for the DFT, DHT, and DCT are also introduced as particular cases of the introduced methodology.

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Energy-Efficient 8-Point DCT Approximations: Theory and Hardware Architectures

Cintra

Bayer

Coutinho

et al. 2015

Circuits Syst Signal Process

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Due to its remarkable energy compaction properties, the discrete cosine transform (DCT) is employed in a multitude of compression standards, such as JPEG and H.265/HEVC. Several low-complexity integer approximations for the DCT have been proposed for both 1-D and 2-D signal analysis. The increasing demand for low-complexity, energy efficient methods require algorithms with even lower computational costs. In this paper, new 8-point DCT approximations with very low arithmetic complexity are presented. The new transforms are proposed based on pruning state-of-the-art DCT approximations. The proposed algorithms were assessed in terms of arithmetic complexity, energy retention capability, and image compression performance. In addition, a metric combining performance and computational complexity measures was proposed. Results showed good performance and extremely low computational complexity. Introduced algorithms were mapped into systolic-array digital architectures and physically realized as digital prototype circuits using FPGA technology and mapped to 45 nm CMOS technology. All hardware-related metrics showed low resource consumption of the proposed pruned approximate transforms. The best proposed transform according to the introduced metric presents a reduction in power consumption of 21-25%. KeywordsDCT approximation image compressionFPGA pruned transforms 1 IntroductionTransform-based methods are widely employed in digital signal processing applications [1]. In this context, the efficient computation of discrete transforms has constantly attracted community efforts and the proposition of fast algorithms [2]. In particular, the 8-point discrete cosine transform (DCT) has a proven record of scientific and industrial applications, as demonstrated by the multitude of image and video coding standards that adopt it, such as: JPEG [3], MPEG [4-6], H.261 [7,8], H.263 [5,9], H.264/AVC [10,11], and the recent high efficiency video coding (HEVC) [12,13]. The HEVC is capable of achieving high compression * Renato J. Cintra is with the Signal Processing ).performance at approximately half the bit rate required by its predecessor H.264/AVC with same image quality [13][14][15][16]. On the other hand, the HEVC requires a significantly higher computational complexity in terms of arithmetic operations [14-17], being 2-4 times more computationally costly than H.264/AVC [14,16].In this context, the efficient computation of the DCT is a venue for improving the performance of abovementioned codecs.Since its inception, several fast algorithms for the DCT have been proposed [18][19][20][21][22][23]. However, traditional algorithms aim at the computation of the exact DCT, which requires several multiplication operations. Additionally, several algorithms have achieved theoretical multiplicative complexity lower-bounds [21,24]. As a consequence, the progress in this area headed to approximate methods [25][26][27]. In some applications, a simple DCT approximation can provide meaningful results at low arithmetic complexity [28]. Thus, appro...

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Discrete Cosine and Sine Transforms

Cited by 42 publications

References 0 publications

A parallel evolution strategy for an earth imaging problem in geophysics

A parallel evolution strategy for an earth imaging problem in geophysics

An Integer Approximation Method for Discrete Sinusoidal Transforms

Energy-Efficient 8-Point DCT Approximations: Theory and Hardware Architectures

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