Application of sine transform in image processing

The sine transform has frequently been suggested as an alternative to the more commonly used Walsh-Hadamard and discrete cosine transforms for the purpose of image coding. It is demonstrated here that there are fundamental theoretical drawbacks to the transform which result in poor performance when it is employed in a practical coding situation.     

Block-diagonal structure of Walsh-Hadamard/discrete cosine transform

The A-matrix, the conversion matrix for Walsh-Hadamard/discrete cosine transform, is known for its efficient block-diagonal structure. This associates with the even/odd structure of the transform kernels. In the letter we present a direct matrix derivation by using the intrinsic properties of the discrete cosine transform and the Walsh-Hadamard transform.     

Transform methods for seismic data compression

The authors consider the development and evaluation of transform coding algorithms for the storage of seismic signals. Transform coding algorithms are developed using the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT). These are evaluated and compared to a linear predictive coding algorithm for data rates ranging from 150 to 550 bit/s. The results reveal that sinusoidal transforms are well-suited for robust, low-rate seismic signal representation. In particular, it is shown that a DCT coding scheme reproduces faithfully the seismic waveform at approximately one-third of the original rate     

On Interframe Transform Coding

This paper is concerned with interframe coding of monochrome pictures using 3-dimensional transforms. First, an algorithm which enables the vector processing of 3-dimensional arrays is derived. Next, this algorithm is used in an interframe transform coding experiment. Picture data (6 bits/pel) is processed in (4 times 4 times 4) blocks using the Walsh-Hadamard transform (WHT), and the discrete cosine transform (DCT). The corresponding reconstructed pictures (1 bit/pel) are included.     

Fast progressive reconstruction of images using the DCT

Fast progressive reconstruction (FPR) of images based on discrete Fourier transform (DFT) and Walsh-Hadamard transform (WHT) has been developed by Takikawa [3]. This technique is now extended to the discrete cosine transform (DCT). The quality of reconstructed images during the intermediate stages based on these transforms is analyzed. This comparison is both subjective and objective. The feasibility of the DCT in this FPR scheme is discussed.This paper is based on part of the research carried out by M. Miran toward his M.Sc. Thesis at the University of Texas at Arlington. It was also presented at the Fifth European Signal Processing Conference, Barcelona, September 18–21, 1990.     

Classification of Vectorcardiograms Using Walsh And Cosine Orthogonal Transforms

In this study, the usefulness of two-dimensional transforms in classifying human vectorcardiograms is investigated. The transforms used are the Walsh-Hadamard (WHT) and the discrete cosine (DCT). Experimental results included in the paper demonstrate that about 80-85 percent correct classification may be achieved.     

Automating the modeling and optimization of the performance ofsignal transforms

Fast implementations of discrete signal transforms, such as the discrete Fourier transform (DFT), the Walsh-Hadamard transform (WHT), and the discrete trigonometric transforms (DTTs), can be viewed as factorizations of their corresponding transformation matrices. A given signal transform can have many different factorizations, with each factorization represented by a unique but mathematically equivalent formula. When implemented in code, these formulas can have significantly different running times on the same processor, sometimes differing by an order of magnitude. Further, the optimal implementations on various processors are often different. Given this complexity, a crucial problem is automating the modeling and optimization of the performance of signal transform implementations. To enable computer modeling of signal processing performance, we have developed and analyzed more than 15 feature sets to describe formulas representing specific transforms. Using some of these features and a limited set of training data, we have successfully trained neural networks to learn to accurately predict performance of formulas with error rates less than 5%. In the direction of optimization, we have developed a new stochastic evolutionary algorithm known as STEER that finds fast implementations of a variety of signal transforms. STEER is able to optimize completely new transforms specified by a user. We present results that show that STEER can find discrete cosine transform formulas that are 10-20% faster than what a dynamic programming search finds     

Vectorcardiographic Data Compression via Walsh And Cosine Transforms

In this study the use of two-dimensional transforms for compressing human vectorcardiographic (VCG) data is investigated. The VCG signal is two-dimensional in nature, one dimension consisting of the spatial axes and the other consisting of the samples in time along a particular spatial axis. The discrete cosine transform (DCT) and the Walsh-Hadamard transform (WHT) were used. The variance criterion was employed for selecting components to be retained. The training set was formed from 225 VCG records from three different diagnostic classes. The DCT yielded compression ratios from 3:1 to 5:1, while compression ratios of around 2:1 were obtained with the WHT.     

Energy Packing Efficiency for the Generalized Discrete Transforms

The concept of energy packing efficiency (EPE) of the Walsh-Hadamard transform (WHT) first proposed by Kitajima [1], is extended to other discrete orthogonal transforms. The concept as a criterion for evaluating the transforms is discussed. It is shown that the EPE is invariant for the generalized transforms as long as the ordering is the same.     

Multichannel transforms for signal/image processing

This paper presents a novel approach to the Fourier analysis of multichannel time series. Orthogonal matrix functions are introduced and are used in the definition of multichannel Fourier series of continuous-time periodic multichannel functions. Orthogonal transforms are proposed for discrete-time multichannel signals as well. It is proven that the orthogonal matrix functions are related to unitary transforms (e.g., discrete Hartley transform (DHT), Walsh-Hadamard transform), which are used for single-channel signal transformations. The discrete-time one-dimensional multichannel transforms proposed in this paper are related to two-dimensional single-channel transforms, notably to the discrete Fourier transform (DFT) and to the DHT. Therefore, fast algorithms for their computation can be easily constructed. Simulations on the use of discrete multichannel transforms on color image compression have also been performed.     

A comparative review of real and complex Fourier-related transforms

Major continuous-time, discrete-time, and discrete Fourier-related transforms as well as Fourier-related series are discussed both with real and complex kernels. The complex Fourier transforms, Fourier series, cosine, sine, Hartley, Mellin, Laplace transforms, and z-transforms are covered on a comparative basis. Generalizations of the Fourier transform kernel lead to a number of novel transforms, in particular, special discrete cosine, discrete sine, and real discrete Fourier transforms, which have already found use in a number of applications. The fast algorithms for the real discrete Fourier transform provide a unified approach for the optimal fast computation of all discrete Fourier-related transforms. The short-time Fourier-related transforms are discussed for applications involving nonstationary signals. The one-dimensional transforms discussed are also extended to the two-dimensional transforms     

Methods for designing efficient parallel‐recursive filter structures for computing discrete transforms

Analytical and numerical approaches are presented for the design of first‐order and second‐order recursive digital filter banks for computing linear, discrete transforms. This subject has been studied extensively for the case of trigonometric transforms. The focus of this paper is on discrete polynomial transforms, and Clenshaw's recurrence formulae are used to design the second‐order filters. The efficiency of the implementation for a specific transform is dependent upon the characteristics of recurrence relations for the transform basis vectors. Efficient implementations are derived for the discrete cosine transform and the inverse discrete Legendre transform from analytical expressions for basis vector recurrence relations. A numerical procedure is presented to search for the existence and parameters of an efficient implementation when analytical expressions for the basis vector recurrence relations are unknown. This revised version was published online in June 2006 with corrections to the Cover Date.     

Parameterisation of slant-Haar transforms

A parameterisation of the slant-Haar transform is presented, which includes an existing version of the slant-Haar transform. An efficient algorithm for the slant-Haar transform is developed and its computational complexity is estimated. The parametric slant-Haar transforms are compared to the Karhunen-Loeve transform. The parametric slant-Haar is shown to perform better than the commonly used slant-Haar and slant-Hadamard transforms for the first-order Markov model and also performs better than the discrete cosine transform for images approximated by the generalised image correlation model     

A combined-transform coding (CTC) scheme for medical images

A combined-transform coding (CTC) scheme is proposed to reduce the blocking artifact of conventional block transform coding and hence to improve the subjective performance. The proposed CTC scheme is described and its information-theoretic properties are investigated. Computer simulation results for a class of chest X-ray images are presented. A comparison between the CTC scheme and the conventional discrete cosine transform (DCT) and discrete Walsh-Hadamard transform (DWHT) demonstrates the performance improvement of the proposed scheme. In addition, combined coding can also be used in noiseless coding, yielding a slight improvement in the compression performance if it is used properly.     

Behaviour of a generalised covariance model in picture coding

We propose a generalised exponential model for the covariance function of real image that is more realistic than the two conventional models, i.e. separable and isotropic models. The theoretical coding performance based on this model is shown to agree well with that based on the real covariance function in the discrete cosine transform (DCT) coding.     

Image and video processing using discrete fractional transforms

The mathematical transforms such as Fourier transform, wavelet transform and fractional Fourier transform have long been influential mathematical tools in information processing. These transforms process signal from time to frequency domain or in joint time–frequency domain. In this paper, with the aim to review a concise and self-reliant course, the discrete fractional transforms have been comprehensively and systematically treated from the signal processing point of view. Beginning from the definitions of fractional transforms, discrete fractional Fourier transforms, discrete fractional Cosine transforms and discrete fractional Hartley transforms, the paper discusses their applications in image and video compression and encryption. The significant features of discrete fractional transforms benefit from their extra degree of freedom that is provided by fractional orders. Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractional cosine transform is better in encryption of image and video. Mean square error and peak signal-to-noise ratio with optimum fractional order are considered quality check parameters in image and video.     

快速变换算法三十年的发展

傅里叶变换快速算法发展已三十年，本文综述了高散变换快速算法的发展，特别是近几年的发展，其中包括传统的基2、基4、基8、分裂基算法的发展以及多维离散傅里叶变换、多维离散余（正）弦变换、多维离散W变换（哈特莱变换）的快速算法．阐述各种算法是如何将多维变换转换为一维变换的计算，并讨论了在有理数域上计算上述各种变换所需最小实数乘法的次数。     

Fast algorithms for the computation of sliding discrete sinusoidal transforms

Fast algorithms for computing various discrete cosine transforms and discrete sine transforms in a sliding window are proposed. The algorithms are based on a recursive relationship between three subsequent local transform spectra. Efficient inverse algorithms for signal processing in a sliding window are also presented. The computational complexity of the algorithms is compared with that of known fast discrete sinusoidal transforms and running recursive algorithms.     

Group testing for image compression using alternative transforms

This paper extends the group testing for wavelets (IEEE Trans. Image Process. 11 (2002) 901) algorithm to code coefficients from the wavelet packet transform, the discrete cosine transform, and various lapped transforms. Group testing offers a noticeable improvement over zerotree coding techniques on these transforms; is inherently flexible; and can be adapted to different transforms with relative ease. The new algorithms are competitive with many recent state-of-the-art image coders that use the same transforms.     

CORDIC-Based Unified Architectures for Computation of DCT/IDCT/DST/IDST

In this paper, CORDIC (coordinate rotation digital computer)-based Cooley-Tukey fast Fourier transform (FFT)-like algorithms for power-of-two point discrete cosine transform/discrete sine transform/inverse discrete cosine transform/inverse discrete sine transform are proposed and their corresponding unified architectures are developed by fully reusing the unique two basic processing elements. The proposed algorithms have some distinguished advantages, such as FFT-like regular data flow, unique post-scaling factor, and arithmetic-sequence rotation angles. The developed unified architectures can compute four different transforms by simple routing the data flow according to the specific transform without feeding different transform coefficients or different transform kernels. The unfolding technique is used to overcome the problem of difficult to realize pipeline that occur in iterative CORDIC algorithms. Compared to existing unified architectures, the proposed architectures have a superior performance in terms of hardware complexity, control complexity, throughput, scalability, modularity, and pipelinability.