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

Mawardi Bahri,Ryuichi Ashino

doi:10.1155/2019/1062979

Abstract

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

Highlights

Introduction 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]
In [18,19,20], Kou et al proposed the quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform (QFT) in the LCTdomain. is generalization is obtained by replacing the Fourier kernel with the right-sided QFT kernel in the LCT definition
We study the convolution theorems associated with the QLCT, which can be useful in digital signal and image processing

Summary

Preliminaries

We have the following important result on the QFT of a convolution of two quaternion functions. Quaternion Linear Canonical Transform e linear canonical transform (LCT) is a linear integral transformation with three free parameters which has widely been used in various fields such as spectral analysis, image processing, and optical system analysis [35, 36] Several famous transforms such as the Fourier transform, the fractional Fourier transform, and the other transformations are special cases. We have applied the quaternion conjugation rule pq q p for p, q ∈ H, proving the theorem. Due to the noncommutativity of the kernel of the QLCT, we only have a left linearity property with quaternion constants; that is, LHA1,A2 􏼈αf + βg􏼉(v) αLHA1,A2 􏼈f􏼉(v) + βLHA1,A2 􏼈g􏼉(v). F(z) 􏽒R2 KA−11 (v1, z1)LHA1,A2 􏼈f􏼉(v)KA−21 (v2, z2)dv 􏽒Rvks KAs(zs, vs)KAs(ys, vs)dvs bks (− μ)− kδ(k)(zs − ys), k ∈ N ∪ {0}

Convolution Associated with QLCT

Correlation Theorem for QLCT

Uncertainty Principle for QLCT