Intelligent OFDM telecommunication system. Part 2. Examples of complex and quaternion many-parameter transforms

In this paper, we propose unified mathematical forms of many-parametric complex and quaternion Fourier transforms for novel Intelligent OFDM-telecommunication systems (OFDM-TCS). Each many-parametric transform (MPT) depends on many free angle parameters. When parameters are changed in some way, the type and form of transform are changed as well. For example, MPT may be the Fourier transform for one set of parameters, wavelet transform for other parameters and other transforms for other values of parameters. The new Intelligent-OFDM-TCS uses inverse MPT for modulation at the transmitter and direct MPT for demodulation at the receiver. © 2019 IOP Publishing Ltd. All rights reserved

The development of the quaternion wavelet transform

The purpose of this article is to review what has been written on what other authors have called quaternion wavelet transforms (QWTs): there is no consensus about what these should look like and what their properties should be. We briefly explain what real continuous and discrete wavelet transforms and multiresolution analysis are and why complex wavelet transforms were introduced; we then go on to detail published approaches to QWTs and to analyse them. We conclude with our own analysis of what it is that should define a QWT as being truly quaternionic and why all but a few of the “QWTs” we have described do not fit our definition

Holistic Processing of Colour Images Using Novel Quaternion-Valued Wavelets on the Plane

We investigate the applicability of quaternion-valued wavelets on the plane to holistic colour image processing. We present a methodology for decomposing and reconstructing colour images using quaternionic wavelet filters associated to recently developed quaternion-valued wavelets on the plane. We consider compression, enhancement, segmentation, and denoising techniques to demonstrate quaternion-valued wavelets as a promising tool for holistic colour image processing

Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

Robust hashing for image authentication using quaternion discrete Fourier transform and log-polar transform

International audienceIn this work, a novel robust image hashing scheme for image authentication is proposed based on the combination of the quaternion discrete Fourier transform (QDFT) with the log-polar transform. QDFT offers a sound way to jointly deal with the three channels of color images. The key features of the present method rely on (i) the computation of a secondary image using a log-polar transform; and (ii) the extraction from this image of low frequency QDFT coefficients' magnitude. The final image hash is generated according to the correlation of these magnitude coefficients and is scrambled by a secret key to enhance the system security. Experiments were conducted in order to analyze and identify the most appropriate parameter values of the proposed method and also to compare its performance to some reference methods in terms of receiver operating characteristics curves. The results show that the proposed scheme offers a good sensitivity to image content alterations and is robust to the common content-preserving operations, and especially to large angle rotation operations

Vector extension of monogenic wavelets for geometric representation of color images

14 pagesInternational audienceMonogenic wavelets offer a geometric representation of grayscale images through an AM/FM model allowing invariance of coefficients to translations and rotations. The underlying concept of local phase includes a fine contour analysis into a coherent unified framework. Starting from a link with structure tensors, we propose a non-trivial extension of the monogenic framework to vector-valued signals to carry out a non marginal color monogenic wavelet transform. We also give a practical study of this new wavelet transform in the contexts of sparse representations and invariant analysis, which helps to understand the physical interpretation of coefficients and validates the interest of our theoretical construction

The Applications of Discrete Wavelet Transform in Image Processing: A Review

This paper reviews the newly published works on applying waves to image processing depending on the analysis of multiple solutions. the wavelet transformation reviewed in detail including wavelet function, integrated wavelet transformation, discrete wavelet transformation, rapid wavelet transformation, DWT properties, and DWT advantages. After reviewing the basics of wavelet transformation theory, various applications of wavelet are reviewed and multi-solution analysis, including image compression, image reduction, image optimization, and image watermark. In addition, we present the concept and theory of quadruple waves for the future progress of wavelet transform applications and quadruple solubility applications. The aim of this paper is to provide a wide-ranging review of the survey found able on wavelet-based image processing applications approaches. It will be beneficial for scholars to execute effective image processing applications approaches

Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)

We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general \textit{multivector square roots} \in \cl{p,q} \textit{of} -1 are used to split multivector signals, and to construct the left and right CFT kernel factors. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1 .Comment: 19 pages, 1 figur