37 research outputs found
The Study of Properties of n-D Analytic Signals and Their Spectra in Complex and Hypercomplex Domains
In the paper, two various representations of a n-dimensional (n-D) real signal u(x1,x2,…,xn) are investigated. The first one is the n-D complex analytic signal with a single-orthant spectrum defined by Hahn in 1992 as the extension of the 1-D Gabor’s analytic signal. It is compared with two hypercomplex approaches: the known n-D Clifford analytic signal and the Cayley-Dickson analytic signal defined by the Author in 2009. The signal-domain and frequency-domain definitions of these signals are presented and compared in 2-D and 3-D. Some new relations between the spectra in 2-D and 3-D hypercomplex domains are presented. The paper is illustrated with the example of a 2-D separable Cauchy pulse
Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula
We show that the discrete complex, and numerous hypercomplex, Fourier
transforms defined and used so far by a number of researchers can be unified
into a single framework based on a matrix exponential version of Euler's
formula , and a matrix root of -1
isomorphic to the imaginary root . The transforms thus defined can be
computed using standard matrix multiplications and additions with no
hypercomplex code, the complex or hypercomplex algebra being represented by the
form of the matrix root of -1, so that the matrix multiplications are
equivalent to multiplications in the appropriate algebra. We present examples
from the complex, quaternion and biquaternion algebras, and from Clifford
algebras Cl1,1 and Cl2,0. The significance of this result is both in the
theoretical unification, and also in the scope it affords for insight into the
structure of the various transforms, since the formulation is such a simple
generalization of the classic complex case. It also shows that hypercomplex
discrete Fourier transforms may be computed using standard matrix arithmetic
packages without the need for a hypercomplex library, which is of importance in
providing a reference implementation for verifying implementations based on
hypercomplex code.Comment: The paper has been revised since the second version to make some of
the reasons for the paper clearer, to include reviews of prior hypercomplex
transforms, and to clarify some points in the conclusion
Color Image Watermarking using JND Sampling Technique
This paper presents a color image watermarking scheme using Just Noticeable Difference (JND) Sampling Technique in spatial domain. The nonlinear JND Sampling technique is based on physiological capabilities and limitations of human vision. The quantization levels have been computed using the technique for each of the basic colors R, G and B respectively for sampling color images. A watermark is scaled to half JND image and is added to the JND sampled image at known spatial position. For transmission of the image over a channel, the watermarked image has been represented using Reduced Biquaternion (RB) numbers. The original image and the watermark are retrieved using the proposed algorithms. The detection and retrieval techniques presented in this paper have been quantitatively benchmarked with a few contemporary algorithms using MSE and PSNR. The proposed algorithms outperform most of them. Keywords: Color image watermarking, JND sampling, Reduced Biquaternion, Retrieva
Algebraic technique for mixed least squares and total least squares problem in the reduced biquaternion algebra
This paper presents the reduced biquaternion mixed least squares and total
least squares (RBMTLS) method for solving an overdetermined system in the reduced biquaternion algebra. The RBMTLS method is suitable when
matrix and a few columns of matrix contain errors. By examining real
representations of reduced biquaternion matrices, we investigate the conditions
for the existence and uniqueness of the real RBMTLS solution and derive an
explicit expression for the real RBMTLS solution. The proposed technique covers
two special cases: the reduced biquaternion total least squares (RBTLS) method
and the reduced biquaternion least squares (RBLS) method. Furthermore, the
developed method is also used to find the best approximate solution to over a complex field. Lastly, a numerical example is presented to
support our findings.Comment: 19 pages, 3 figure
L-structure least squares solutions of reduced biquaternion matrix equations with applications
This paper presents a framework for computing the structure-constrained least
squares solutions to the generalized reduced biquaternion matrix equations
(RBMEs). The investigation focuses on three different matrix equations: a
linear matrix equation with multiple unknown L-structures, a linear matrix
equation with one unknown L-structure, and the general coupled linear matrix
equations with one unknown L-structure. Our approach leverages the complex
representation of reduced biquaternion matrices. To showcase the versatility of
the developed framework, we utilize it to find structure-constrained solutions
for complex and real matrix equations, broadening its applicability to various
inverse problems. Specifically, we explore its utility in addressing partially
described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our
study concludes with numerical examples.Comment: 30 page
A Novel Representation for Two-dimensional Image Structures
This paper presents a novel approach towards two-dimensional (2D) image structures modeling. To obtain more degrees of freedom, a 2D image signal is embedded into a certain geometric algebra. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on it, a local representation for the intrinsically two-dimensional (i2D) structure is derived as the monogenic curvature signal. From it, independent features of local amplitude, phase and orientation are simultaneously extracted. Besides, a monogenic curvature scale-space can be built by applying a Poisson kernel to the monogenic curvature signal. Compared with the other related work, the remarkable advantage of our approach lies in the rotationally invariant phase evaluation of 2D structures in a multi-scale framework, which delivers access to phase-based processing in many computer vision tasks
Commutative Quaternion Matrices
In this study, we introduce the concept of commutative quaternions and
commutative quaternion matrices. Firstly, we give some properties of
commutative quaternions and their Hamilton matrices. After that we investigate
commutative quaternion matrices using properties of complex matrices. Then we
define the complex adjoint matrix of commutative quaternion matrices and give
some of their properties
Filtering and Tracking with Trinion-Valued Adaptive Algorithms
A new model for three-dimensional processes based on the trinion algebra is introduced for the first time. Compared
with the pure quaternion model, the trinion model is more compact and computationally more efficient, while having similar or
comparable performance in terms of adaptive linear filtering. Moreover, the trinion model can effectively represent the general
relationship of state evolution in Kalman filtering, where the pure quaternion model fails. Simulations on real-world wind
recordings and synthetic data sets are provided to demonstrate the potentials of this new modeling method