A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

Abstract

The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

“…It can be observed that for 1 ≤ p ≤ 2 we may change L 2 -norm to L p -norm on left-hand side of (26) and obtain the next result. Theorem 2.…”

“…Remark 2. The non-commutativity of the QFT kernel implies that Theorem 2 is slightly different to the right-sided QFT (compare to Theorem 12 of [26]). Below, Theorem 3 is not valid for the right-sided QFT and left-sided QFT.…”

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

“…Some important properties such as Parseval's theorem, reconstruction formula, uncertainty principles, and asymptotic behaviour are discussed. Like the uncertainty principle for the QFT [3], they also showed that only a two-dimensional Gaussian signal minimizes the uncertainty. However, the convolution theorem is an important result of the QLCT which does not hold using this construction because of the noncommutative property of the right-sided quaternion Fourier kernel.…”

“…e quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using the quaternion algebra. e QFT has been shown to relate to the other quaternion signal analysis tools, such as quaternion wavelet transform [1][2][3], fractional quaternion Fourier transform [4,5], quaternionic windowed Fourier transform [6][7][8][9], and quaternion Wigner transform [10]. Because of the noncommutative property of quaternion multiplication, we obtain at least three different kinds of two-dimensional QFTs as follows (see [11][12][13][14][15]):…”

Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

“…Before proving the relationship between the QAF-LCT and the continuous quaternion wavelet transform (CQWT), we first introduce the definition of the CQWT (see [15][16][17]). …”