The Quaternion Fourier transform (QFT) is one of the key tools in studying color image processing. Indeed, a deep understanding of the QFT has created the color images to be transformed as whole, rather than as color separated component. In addition, understanding the QFT paves the way for understanding other integral transform, such as the Quaternion Fractional Fourier transform (QFRFT), Quaternion linear canonical transform (QLCT) and Quaternion Wigner-Ville distribution. The aim of this paper is twofold: first to provide some of the theoretical background regarding the Quaternion bound variation function. We then apply it to derive the Quaternion Fourier and linear canonical inversion formulas. Secondly, to provide some in tuition for how the Quaternion Fourier and linear canonical inversion theorems work on the absolutely integrable function space.
In this paper we generalize the continuous quaternion windowed Fourier transform called the multivariate two sided continuous quaternion windowed Fourier transform. Using the two sided quaternion Fourier transform we derive several important properties.
The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Euclidean Fourier transform. In this paper, we establish Miyachi's theorem for this transform.
In this paper, we define a new transform called the Gabor quaternionic Fourier transform (GQFT), which generalizes the classical windowed Fourier transform to quaternion valued-signals, we give several important properties such as the Plancherel formula and inversion formula. Finally, we establish the Heisenberg uncertainty principles for the GQFT.
By the important applications of Gabor transform in time-frequency analysis and signal analysis, in this paper, we consider the Gabor quaternion Fourier transform (GQFT), and we prove of it a version Benedicks-type uncertainty principle for GQFT and some local concentration uncertainty principles.
The main objective of the present paper is to establish a new uncertainty principle (UP) for the two-sided quaternion Fourier transform (QFT). This result is an extension of a result of Benedicks, Amrein and Berthier, which states that a nonzero function in $L^1\left({\mathbb{R}}^2, {\mathbb{H}}\right)$ and its two-sided QFT cannot both have support of finite measure.
The quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space…
In this paper we propose a novel transform called continuous quaternionic stockwell transform. We express the admissibility condition in term of the (two-sided) quaternion Fourier transform . We show that its fundamental properties, such as Plancherel, Parseval and inversion formula, can be established whenever the continuous quaternion stockwell satisfy a particular admissibility condition. We present several examples of the continuous quaternion stockwell transform. We apply the continuous quaternion stockwell transform properties and the two sided quaternion Fourier transform to establish a…
The purpose of this article is to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform (QFT). We study some fundamental properties of this extension such as scaling, translation, rotation, Parseval's identity, inversion theorem, and a reproducing kernel, then we derive the associated Heisenberg-Pauli-Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP and Hardy's UP to the CQWT domain.
In this paper, we generalize the continuous quaternion shearlet transform on $\mathbb{R}^{2}$ to $\mathbb{R}^{2d}$, called the multivariate two sided continuous quaternion shearlet transform. Using the two sided quaternion Fourier transform, we derive several important properties such as (reconstruction formula, reproducing kernel, plancherel's formula, etc.). We present several example of the multivariate two sided continuous quaternion shearlet transform. We apply the multivariate two sided continuous quaternion shearlet transform properties and the two sided quaternion Fourier transform to …
In this paper, the uncertainty principle of discrete signals associated with Quaternion Fourier transform is investigated. It suggests how sparsity helps in the recovery of missing frequency.
Gabor transform is one of the performed tools for time-frequency signal analysis. The principal aim of this paper is to generalize the Gabor Fourier transform to the quaternion linear canonical transform. Actually, this transform gives us more flexibility to studied nonstationary and local signals associated with the quaternion linear canonical transform. Some useful properties are derived, such as Plancherel and inversion formulas. And we prove some uncertainty principles: those including Heisenberg's, Lieb's and logarithmic inequalities. We finish by analogs of concentration and Benedick's t…
The Wigner-Ville distribution (WVD) and quaternion offset linear canonical transform (QOLCT) are a useful tools in signal analysis and image processing. The purpose of this paper is to define the Wigner-Ville distribution associated with quaternionic offset linear canonical transform (WVD-QOLCT). Actually, this transform combines both the results and flexibility of the two transform WVD and QOLCT. We derive some important properties of this transform such as inversion and Plancherel formulas, we establish a version of Heisenberg inequality, Lieb's theorem and we give the Poisson summation form…
The quaternionic offset linear canonical transform (QOLCT) can be thought as a generalization of the quaternionic linear canonical transform (QLCT). In this paper we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact we prove the Plancherel formula, and some properties related to the QOLCT, then we generalize some different uncertainty principles (UPs), including Heisenberg-Weyls UP, Hardys UP, Beurlings UP, and logarithmic UP to the QOLCT domain in a broader sense
In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation $df(t)-iωf(t)=g(t)$ for a regulated function $f$, where $ω$ is a complex number with positive imaginary part. This …
If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(πx)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{α<β}|\int_α^βf(x)\,dx|$ is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by $dF$ where $F$ is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is $f\in L^1({\mathbb R})$ such that $\lim_{S\to\in…
The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration…
In this paper, we prove both of the Titchmarsh theorems associated with Two-Sided Quaternionic Fourier Transform and we conclude about Short-Time Two-Sided Quaternionic Fourier Transform.
We consider non oscillatory functions and prove an everywhere Fourier Inversion Theorem for functions of very moderate decrease. The proofs rely on some ideas in nonstandard analysis.
Discrete analogs of the classical Fourier-Jacobi transform are introduced and investigated. It involves series and integrals with respect to parameters of the Gauss hypergeometric function ${}_2F_1(a+in/2,a-in/2;\ c; -x^2 ), \ x >0, n \in \mathbb{N}, a,c > 0, i $ is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established.
We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurem…
