Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. The hypercomplex representation may be interpreted as an ordered pair of complex signals or as a quaternion signal. We discuss its derivation and properties and show that its quaternion Fourier transform is one-sided. It is shown how to derive from the hypercomplex representation a complex envelope and a phase. A classical result in the case of real signals is that an amplitude modulated signal may be analysed into its envelope and carrier using the analytic signal provided that the modulating signal has frequency content not overlapping with that of the carrier. We show that this idea extends to the complex case, provided that the complex signal modulates an orthonormal complex exponential. Orthonormal complex modulation can be represented mathematically by a polar representation of quaternions previously derived by the authors. As in the classical case, there is a restriction of non-overlapping frequency content between the modulating complex signal and the orthonormal complex exponential. We show that, under these conditions, modulation in the time domain is equivalent to a frequency shift in the quaternion Fourier domain. Examples are presented to demonstrate these concepts

Color Image Analysis by Quaternion-Type Moments

International audienceIn this paper, by using the quaternion algebra, the conventional complex-type moments (CTMs) for gray-scale images are generalized to color images as quaternion-type moments (QTMs) in a holistic manner. We first provide a general formula of QTMs from which we derive a set of quaternion-valued QTM invariants (QTMIs) to image rotation, scale and translation transformations by eliminating the influence of transformation parameters. An efficient computation algorithm is also proposed so as to reduce computational complexity. The performance of the proposed QTMs and QTMIs are evaluated considering several application frameworks ranging from color image reconstruction, face recognition to image registration. We show they achieve better performance than CTMs and CTM invariants (CTMIs). We also discuss the choice of the unit pure quaternion influence with the help of experiments. appears to be an optimal choice

Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images

In the present work hypercomplex spectral methods of the processing and analysis of images are introduced. The thesis is divided into three main chapters. First the quaternionic Fourier transform (QFT) for 2D signals is presented and its main properties are investigated. The QFT is closely related to the 2D Fourier transform and to the 2D Hartley transform. Similarities and differences of these three transforms are investigated with special emphasis on the symmetry properties. The Clifford Fourier transform is presented as nD generalization of the QFT. Secondly the concept of the phase of a signal is considered. We distinguish the global, the local and the instantaneous phase of a signal. It is shown how these 1D concepts can be extended to 2D using the QFT. In order to extend the concept of global phase we introduce the notion of the quaternionic analytic signal of a real signal. Defining quaternionic Gabor filters leads to the definition of the local quaternionic phase. The relation between signal structure and local signal phase, which is well-known in 1D, is extended to 2D using the quaternionic phase. In the third part two application of the theory are presented. For the image processing tasks of disparity estimation and texture segmentation there exist approaches which are based on the (complex) local phase. These methods are extended to the use of the quaternionic phase. In either case the properties of the complex approaches are preserved while new features are added by using the quaternionic phase

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

Image Description using Radial Associated Laguerre Moments

This study proposes a new set of moment functions for describing gray-level and color images based on the associated Laguerre polynomials, which are orthogonal over the whole right-half plane. Moreover, the mathematical frameworks of radial associated Laguerre moments (RALMs) and associated rotation invariants are introduced. The proposed radial Laguerre invariants retain the basic form of disc-based moments, such as Zernike moments (ZMs), pseudo-Zernike moments (PZMs), Fourier-Mellin moments (OFMMs), and so on. Therefore, the rotation invariants of RALMs can be easily obtained. In addition, the study extends the proposed moments and invariants defined in a gray-level image to a color image using the algebra of quaternion to avoid losing some significant color information. Finally, the paper verifies the feature description capacities of the proposed moment function in terms of image reconstruction and invariant pattern recognition accuracy. Experimental results confirmed that the associated Laguerre moments (ALMs) perform better than orthogonal OFMMs in both noise-free and noisy conditions