Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine transform (DCT), which is often considered to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of these two discrete transforms from the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are considered. (A) Unlike the continuous extension of the DFT, the continuous extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$ between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT converges to the derivative of $g(t)$. And (C), for CEDCT the principle of locality is valid. Finally, we use the continuous extension of 2-dimensional DCT to illustrate its potential for interpolation, as well as for the data compression of 2D images.Comment: submitted to JMP on April 3, 2003; still waiting for the referee's Repor

Fast multi-dimensional scattered data approximation with Neumann boundary conditions

An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2\pi i kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (\pi kx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem

An efficient psychovisual threshold technique in image compression

Nowadays, psychovisual model plays a critical role in an image compression system. The psychovisual threshold gives visual tolerance to the human visual system by reducing the amount of frequency image signals. The sensitivity of the human eye can be fully explored and exploited in the qualitative experiment by describing what has been seen or by image quality judgment. However, the result of the psychovisual threshold through qualitative experiment depends on the test condition of the human visual systems and through repetitive viewing sessions. In a modern image compression, there is a need to provide some flexibility to obtain quality levels of the image output based on user preferences. The concept of psychovisual threshold is designed to determine quality levels of the image output. The psychovisual threshold represents an optimal amount of frequency image signals in image compression. This research proposes the psychovisual threshold through a quantitative experiment that can automatically predict an optimal balance between image quality and compression rate in image compression. The contribution of its frequency image signals to the image reconstruction will be the primitive of psychovisual threshold in image compression. It is very challenging to develop a psychovisual threshold from the contribution of the frequency image signals for each frequency order. In this research, the psychovisual threshold prescribes the quantization values and bit allocation for image compression. The psychovisual threshold is the basic primitive prior to generating quantization tables in image compression. The psychovisual threshold allows a developer to design adaptively customized quantization values according to his or her target image quality. The psychovisual threshold is also elementary and primitive for generating a set of bit allocation for frequency image signals. A set of bit allocation based on psychovisual threshold assigns the amount of bits for frequency image signals. A set of bit allocation refers to the psychovisual threshold instead of the quantization process in image compression. This research investigates the basic understanding of the psychovisual threshold in image compression. The experimental results provide significant improvement in the image compression. The psychovisual threshold which is presented as quantization tables, customized quantization tables and as a set of bit allocation gives a significant improvement on both of the quality of the image reconstruction and the average bit length of Huffman code. This research shows that psychovisual threshold is practically the best measure for optimal frequency image signals on image compression

Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis

The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well-known and widely used mathematical formulations for modeling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose the novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal (FSAS) representations with following properties: (1) real part of eight FSAS representations is the original signal and imaginary part is the FCQT of the real part, (2) imaginary part of eight FSAS representations is the original signal and real part is the FSQT of the real part, (3) like the GAS, Fourier spectrum of the all FSAS representations has only positive frequencies, however unlike the GAS, the real and imaginary parts of the proposed FSAS representations are not orthogonal to each other. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set of small number of Fourier intrinsic band functions which are AM-FM components. This study also proposes a new formulation of the FDM using the discrete cosine transform (DCT) with the GAS and FSAS representations, and demonstrate its efficacy for improved time-frequency-energy representation and analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Survey of Hybrid Image Compression Techniques

A compression process is to reduce or compress the size of data while maintaining the quality of information contained therein. This paper presents a survey of research papers discussing improvement of various hybrid compression techniques during the last decade. A hybrid compression technique is a technique combining excellent properties of each group of methods as is performed in JPEG compression method. This technique combines lossy and lossless compression method to obtain a high-quality compression ratio while maintaining the quality of the reconstructed image. Lossy compression technique produces a relatively high compression ratio, whereas lossless compression brings about high-quality data reconstruction as the data can later be decompressed with the same results as before the compression. Discussions of the knowledge of and issues about the ongoing hybrid compression technique development indicate the possibility of conducting further researches to improve the performance of image compression method