Fast Discrete Cosine Transform Algorithms Research Articles

High Precision DCT CORDIC Architectures for Maximum PSNR

This paper proposes two optimal Cordic Loeffler based DCT (Discrete Cosine Transform algorithm) architectures: a fast and low Power DCT architecture and a high PSNR DCT architecture. The rotation parameters of CORDIC angles required for these architectures have been calculated using a MATLAB script. This script allows the variation of the angle’s precision from 10-1 to 10-4. The experimental results show that the fast and low Power DCT architecture correponds to the precision 10-1. Its complexity is even lower than the BinDCT which is a reference in terms of low complexity and its power has been enhanced in comparison with the conventional Cordic Loeffler DCT by 12 mW. The experimental results also show that the high PSNR DCT architecture corresponds to the precision 10-3 for which the PSNR has been improved by 6.55 dB in comparison with the conventional Cordic Loeffler DCT. Then, the hardware implementation and the generated RTL of some required Cordics are presented.

An Efficient Image Watermarking Method Based on Fast Discrete Cosine Transform Algorithm

This paper proposes an image watermarking method based on the fast discrete cosine transform (DCT) algorithm for implementation in digital signal processor. A digital watermark can be effectively embedded and efficiently extracted without the host image. The keys in watermarking process include four frequency coefficients in DCT, two random permutation vectors, and a quantization matrix for normalizing the watermark and the host image. The fast DCT algorithm has been shown to reduce the complexity of two-dimensional image transformation so that embedding/decoding an image watermark can be completed in real time within 0.33 seconds. The quality of both watermarked image and extracted (retrieved) watermark remains excellent. It is shown that the watermarking method is efficient in and robust to data cropping, transmission loss, and compression/decompression.

A Fast DCT Algorithm for Watermarking in Digital Signal Processor

Discrete cosine transform (DCT) has been an international standard in Joint Photographic Experts Group (JPEG) format to reduce the blocking effect in digital image compression. This paper proposes a fast discrete cosine transform (FDCT) algorithm that utilizes the energy compactness and matrix sparseness properties in frequency domain to achieve higher computation performance. For a JPEG image of8×8block size in spatial domain, the algorithm decomposes the two-dimensional (2D) DCT into one pair of one-dimensional (1D) DCTs with transform computation in only 24 multiplications. The 2D spatial data is a linear combination of the base image obtained by the outer product of the column and row vectors of cosine functions so that inverse DCT is as efficient. Implementation of the FDCT algorithm shows that embedding a watermark image of 32 × 32 block pixel size in a 256 × 256 digital image can be completed in only 0.24 seconds and the extraction of watermark by inverse transform is within 0.21 seconds. The proposed FDCT algorithm is shown more efficient than many previous works in computation.

Fast DCT algorithms for EEG data compression in embedded systems

Electroencephalography (EEG) is widely used in clinical diagnosis, monitoring and Brain - Computer Interface systems. Usually EEG signals are recorded with several electrodes and transmitted through a communication channel for further processing. In order to decrease communication bandwidth and transmission time in portable or low cost devices, data compression is required. In this paper we consider the use of fast Discrete Cosine Transform (DCT) algorithms for lossy EEG data compression. Using this approach, the signal is partitioned into a set of 8 samples and each set is DCT-transformed. The least-significant transform coefficients are removed before transmission and are filled with zeros before an inverse transform. We conclude that this method can be used in real-time embedded systems, where low computational complexity and high speed is required.

An algebraic method for synthesizing fast algorithms of discrete cosine transform of arbitrary size

An algebraic method for synthesizing fast algorithms of the discrete cosine transform (DCT) of arbitrary size is proposed. The method is based on the polynomial algebra \(\mathbb{F}{{[x]} \mathord{\left/ {\vphantom {{[x]} {p(x)}}} \right. \kern-\nulldelimiterspace} {p(x)}}\) associated with the DCT. The fast DCT algorithm comes as a result of the step-by-step decomposition of this algebra. In turn, the decomposition requires step-by-step factorization of the polynomial p(x). This problem is solved using Galois’s theory, which allows finding all the subfields of the splitting field of the polynomial p(x) where p(x) can be factorized.

Novel DCT-Based Real-Valued Discrete Gabor Transform and Its Fast Algorithms

The oversampled Gabor transform is more effective than the critically sampled one in many applications. The biorthogonality relationship between the analysis window and the synthesis window of the Gabor transform represents the completeness condition. However, the traditional discrete cosine transform (DCT)-based real-valued discrete Gabor transform (RGDT) is available only in the critically sampled case and its biorthogonality relationship for the transform has not been unveiled. To bridge these important gaps, this paper proposes a novel DCT-based RDGT, which can be applied in both the critically sampled case and the oversampled case, and their biorthogonality relationships can be derived. The proposed DCT-based RDGT involves only real operations and can utilize fast DCT algorithms for computation, which facilitates computation and implementation by hardware or software as compared to that of the traditional complex-valued discrete Gabor transform. This paper also develops block time-recursive algorithms for the efficient and fast computation of the RDGT and its inverse transform. Unified parallel lattice structures for the implementation of these algorithms are presented. Computational complexity analysis and comparisons have shown that the proposed algorithms provide a more efficient and faster approach for discrete Gabor transforms as compared to those of the existing discrete Gabor transform algorithms. In addition, an application in the noise reduction of the nuclear magnetic resonance free induction decay signals is presented to show the efficiency of the proposed RDGT for time-frequency analysis.

Fast Algorithm Designs for Low-Complexity 4 x 4 Discrete Cosine Transform

In the letter, the fast one-dimensional (1-D) and two-dimensional (2-D) algorithms for realizing low-complexity 4 × 4 discrete cosine transform (DCT) for H.264 applications are developed. Through applying matrix utilizations with Kronecker product and direct sum, the efficient fast 2-D 4 × 4 DCT algorithm can be developed from the proposed fast 1-D 4 × 4 DCT algorithm by matrix decompositions. The fast 1-D and 2-D low-complexity 4 × 4 DCT algorithms requires fewer multiplications and additions than other fast DCT algorithms. Owing to regular modularity, the proposed fast algorithms can achieve real-time H.264 video signal processing with VLSI implementation.

Fast algorithms for multidimensional DCT-to-DCT computation between a block and its associated subblocks

In this paper, we first propose an efficient algorithm for computing one-dimensional (1-D) discrete cosine transform (DCT) for a signal block, given its two adjacent subblocks in the DCT domain and then introduce several algorithms for the fast computation of multidimensional (m-D) DCT with size N/sub 1//spl times/N/sub 2//spl times/.../spl times/N/sub m/ given 2/sup m/ subblocks of DCT coefficients with size N/sub 1//2/spl times/N/sub 2//2/spl times/.../spl times/N/sub m//2, where N/sub i/(i=1,2,...,m) are powers of 2. Obviously, the row-column method, which employs the most efficient algorithms along each dimension, reduces the computational complexity considerably, compared with the traditional method, which employs only the one-dimensional (1-D) fast DCT and inverse DCT (IDCT) algorithms. However, when m/spl ges/2, the traditional method, which employs the most efficient multidimensional DCT/IDCT algorithms, has lower computational complexity than the row-column method. Besides, we propose a direct method by dividing the data into 2/sup m/ parts for independent fast computation, in which only two steps of r-dimensional (r=1,2,...,m) IDCT and additional multiplications and additions are required. If all the dimensional sizes are the same, the number of multiplications required for the direct method is only (2/sup m/-1)/m2/sup m-1/ times of that required for the row-column method, and if N/spl ges/2/sup 2m-1/, the computational efficiency of the direct method is surely superior to that of the traditional method, which employs the most efficient multidimensional DCT/IDCT algorithms.

Modeling DCT coefficients for fast video encoding

Digital video coding standards such as H.263 and MPEG are becoming more and more important for multimedia applications. Due to the huge amount of computations required, there are significant efforts to speed up the processing of video encoders. Previously, the efforts were mainly focused on the fast motion-estimation algorithm. However, as the motion-estimation algorithm becomes optimized, to speed up the video encoders further we also need to optimize other functions such as the discrete cosine transform (DCT) and inverse DCT (IDCT). In this paper, we propose a theoretical model for DCT coefficients. Based on the model, we develop an adaptive algorithm to reduce the computations of DCT, IDCT, quantization, and inverse quantization. We also present a fast DCT algorithm to speed up the calculations of DCT further when the quantization step size is large. We show, by simulations, that significant improvement in the processing speed can be achieved with negligible video-quality degradation, We also implement the algorithm in a real-time PC-based platform to show that it is effective and practical.

Fast recursive algorithms for short-time discrete cosine transform

Local adaptive signal processing can be carried out using the short-time discrete cosine transform (DCT). Two fast recursive algorithms for computing the short-time DCT are presented. The algorithms are based on a recursive relationship between three subsequent local DCT spectra. The computational complexity of the algorithms is compared with that of fast DCT algorithms.

A new fast DCT algorithm and its systolic VLSI implementation

The authors present a new fast algorithm along with its systolic array implementation for computing the N-point discrete cosine transform (DCT), where N is a power of two. The architecture requires log/sub 2/N multipliers and can evaluate one complete N-point DCT (i.e., N transform samples) every N clock cycles. Due to the features of regularity and modularity, it is well suited to VLSI implementation. As compared to existing systolic DCT designs with the same throughput performance, the proposed one involves much less hardware complexity.

Analysis of pruning in fast cosine transform

The discrete cosine transform (DCT) is now the established technique for image and video coding. In most image coding set schemes, a large number of high-frequency transform coefficients can be neglected and need not be calculated. Therefore, significant savings in computation can be obtained by applying pruning in the fast algorithms. We analyze pruning in Chan and Ho's (1990) fast DCT algorithm and evaluate the saving of operations.

A 0.8 μ 100-MHz 2-D DCT core processor

The discrete cosine transform (DCT) has been commonly adopted in many transformation applications such as image, video, and facsimile. A VLSI architecture and implementation of a high speed 2-dimensional DCT core processor with 0.8 /spl mu/ technology is presented. This architecture applies a fast DCT algorithm and multiplier-accumulator based on the distributed algorithm, which has contributed to reduce the hardware requirement and to achieve high speed operation. The transpose memory inserted between each dimension of DCT is partitioned in order to reduce further hardware overhead. Furthermore, this 2-dimensional DCT scheme satisfies the accuracy specification of CCITT recommendation MPEG.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A fault detecting DCT architecture

The discrete cosine transform (DCT) has been successfully applied to speech and image data compression. With the demands of high speed and high reliability, it becomes important for engineers to design the implementation network with a fault tolerant capability. In this paper, we first present a fault detecting architecture for DCT implementation based on properties on the fast DCT algorithm and on the technique of recomputing by an alternate path. It is shown that the scheme presented has favourable fault coverage and lower hardware overheads.

A 100-MHz 2-D discrete cosine transform core processor

A 100-MHz two-dimensional discrete cosine transform (DCT) core processor applicable to the real-time processing of HDTV signals is described. An excellent architecture utilizing a fast DCT algorithm and multiplier accumulators based on distributed arithmetic have contributed to reducing the hardware amount and to enhancing the speed performance. A layout scheme with a column-interleaved memory and a new ROM circuit are introduced for the efficient implementation of memory-based signal processing circuits. Furthermore, mean values of errors generated in the core were minimized to enhance the computational accuracy with the word-length constraints. Consequently, it features the fastest operating speed and the smallest area with sufficient accuracy to satisfy the specifications in CCITT recommendation H.261. The core integrates about 102 K transistors and occupies 21 mm/sup 2/ using 0.8- mu m double-metal CMOS technology. >

DCT algorithms for VLSI parallel implementations

Two algorithms are presented for computing the discrete cosine transform (DCT) on existing VLSI structures. First, it is shown that the N-point DCT can be implemented on the existing systolic architecture for the N-point discrete Fourier transform (DFT) by introducing some modifications. Second, a new prime factor DCT algorithm is presented for the class of DCTs of length N=N/sub 1/*N/sub 2/, where N/sub 1/ and N/sub 2/ are relatively prime and odd numbers. It is shown that the proposed algorithm can be implemented on the already existing VLSI structures for prime factor DFT. The number of multipliers required is comparable to that required for the other fast DCT algorithms. It is shown that the discrete sine transform (DST) can be computed by the same structure. >

A Fast Computational Algorithm for the Discrete Cosine Transform

A Fast Discrete Cosine Transform algorithm has been developed which provides a factor of six improvement in computational complexity when compared to conventional Discrete Cosine Transform algorithms using the Fast Fourier Transform. The algorithm is derived in the form of matrices and illustrated by a signal-flow graph, which may be readily translated to hardware or software implementations.