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

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

Clifford Multivector Toolbox (for MATLAB)

matlab ® is a numerical computing environment oriented towards manipulation of matrices and vectors (in the linear algebra sense, that is arrays of numbers). Until now, there was no comprehensive toolbox (software library) for matlab to compute with Clifford algebras and matrices of multivectors. We present in the paper an account of such a toolbox, which has been developed since 2013, and released publically for the first time in 2015. The paper describes the major design decisions made in implementing the toolbox, gives implementation details, and demonstrates some of its capabilities, up to and including the LU decomposition of a matrix of Clifford multivectors

Image Enhancement by Elliptic Discrete Fourier Transforms

This paper describes a method of enhancement of grayscale and color image in the frequency domain by the pair of two elliptic discrete Fourier transforms (EDFT). Unlike the traditional discrete Fourier transform (DFT), the EDFT is parameterized and the parameter defines ellipses (not circles) around which the input data are rotated. Methods of the traditional DFT are widely used in image enhancement, and the transform rotates data of images around the circles. The presented method of image enhancement proposes processing images on different set of ellipses for the direct and inverse transforms. Our preliminary experimental examples show effectiveness of the proposed method. The Illustrative examples of image enhancement are given

Noise Level Estimation for Digital Images Using Local Statistics and Its Applications to Noise Removal

In this paper, an automatic estimation of additive white Gaussian noise technique is proposed. This technique is built according to the local statistics of Gaussian noise. In the field of digital signal processing, estimation of the noise is considered as pivotal process that many signal processing tasks relies on. The main aim of this paper is to design a patch-based estimation technique in order to estimate the noise level in natural images and use it in blind image removal technique. The estimation processes is utilized selected patches which is most contaminated sub-pixels in the tested images sing principal component analysis (PCA). The performance of the suggested noise level estimation technique is shown its superior to state of the art noise estimation and noise removal algorithms, the proposed algorithm produces the best performance in most cases compared with the investigated techniques in terms of PSNR, IQI and the visual perception