# Quaternion Linear Canonical Wavelet Transform and The Corresponding
Uncertainty Inequalities

Aajaz A. Teali  Department of Mathematics, University of Kashmir, South
Campus, Anantnag-192101 Jammu and Kashmir, India<EMAIL_ADDRESS>
###### Abstract.

The linear canonical wavelet transform has been shown to be a valuable and
powerful time-frequency analyzing tool for optics and signal processing. In
this article, we propose a novel transform called quaternion linear canonical
wavelet transform which is designed to represent two dimensional quaternion-
valued signals at different scales, locations and orientations. The proposed
transform not only inherits the features of quaternion wavelet transform but
also has the capability of signal representation in quaternion linear
canonical domain. We investigate the fundamental properties of quaternion
linear canonical wavelet transform including Parseval’s formula, energy
conservation, inversion formula, and characterization of its range using the
machinery of quaternion linear canonical transform and its convolution. We
conclude our investigation by deriving an analogue of the classical
Heisenberg-Pauli-Weyl uncertainty inequality and the associated logarithmic
and local versions for the quaternion linear canonical wavelet transform.

###### Key words and phrases:

Quaternion algebra; quaternion-valued functions; quaternion wavelet transform,
uncertainty principles; quaternion Fourier transform; quaternion linear
canonical transform

##  1\. Introduction

Wavelet transforms serve as an important and powerful analyzing tool for time-
frequency analysis and have been applied in a number of fields including
signal processing, image processing, sampling theory, differential and
integral equations, quantum mechanics and medicine. However, the signal
analysis capability of the wavelet transform is limited in the time-frequency
plane as each wavelet component is actually a differently scaled bandpass
filter in the frequency domain and hence, it does not serve as an efficient
tool for processing the higher dimensional signal whose energy is not well
concentrated in the frequency domain. One of the examples of such signal is
chirp like signals [2]. Therefore, in order to obtain joint signal
representations in both time and frequency domains, the linear canonical
wavelet transform (LCWT) has been introduced in the context of time-frequency
analysis. The LCWT inherits the excellent mathematical properties of wavelet
transform and linear canonical transform along with some fascinating
properties of its own. In recent years, this transform has been paid a
considerable amount of attention, resulting in many applications in the areas
of optics, quantum mechanics, pattern recognition and signal processing [3, 4,
5, 6, 7].

On the other hand, considerable attention has been paid for the representation
of signals in quaternion domains as quaternion algebra is the closest in its
mathematical properties to the familiar system of the real and complex
numbers. The quaternion algebra offers a simple and profound representation of
signals wherein several components are to be controlled simultaneously. The
development of integral transforms for quaternion valued signals has found
numerous applications in 3D computer graphics, aerospace engineering,
artificial intelligence and colour image processing. The extension of
classical wavelet transform to quaternion algebra has been introduced in [8]
and [9], they also demonstrated their various properties. In [13], Traversoni
proposed a discrete quaternion wavelet transform, the application of which can
be found in [10, 11] and [12].

In the recent years, some authors have generalized the linear canonical
transform to quaternion-valued signals, known as the quaternionic linear
canonical transform (QLCT). The QLCT was firstly studied in [14] including
prolate spheroidal wave signals and uncertainty principles [15]. Some useful
properties of the QLCT such as linearity, reconstruction formula, continuity,
boundedness, positivity inversion formula and the uncertainty principle were
established in [16, 19, 20, 24]. An application of the QLCT to study of
generalized swept-frequency filters was introduced in [18]. Because of the
non-commutative property of multiplication of quaternions, there are mainly
three various types of 2D quaternion linear canonical transform (QLCTs): two-
sided QLCTs, left-sided QLCTs and right-sided QLCTs (refer to [15]). Based on
the (two-sided) QLCT [14], the quaternion windowed linear canonical transform
of 2D quaternionic signals has been introduced by Wen-Biao Gao and Bing-Zhao
Li in [31]. It can reveal the local QLCT-frequency contents and enjoys high
concentrations and eliminates the cross term, but it has limitation of having
fixed window localization.

The objective of this paper is to eliminate the redundancy of the quaternion
wavelet transform(QWT) and quaternion linear canonical transform(QLCT) by
introducing the quaternion linear canonical wavelet transform(QLCWT) i.e. the
quaternion version of linear canonical wavelet transform. QLCWT generalizes
the definition of the QWT in time-QLCD-frequency plane by using the modified
quaternion linear canonical convolution. The proposed QLCWT not only inherits
the features of quaternion wavelet transform but also has the capability of
signal representation in quaternion domain. Besides, it has explicit physical
interpretation and low complexity. It is hoped that this transform might be
useful in three dimensional color field processing, space color video
processing, crystallography, aerospace engineering, oil exploration and for
the solution of many types of quaternionic differential equations.

The article is organized as follows: We begin in Section 2 by presenting the
notation, quaternion algebra and wavelet theory needed to understand and place
our results in context. In Section 3, we introduce the concept of quaternion
linear canonical wavelet transform and obtain the expected properties of the
extended linear canonical wavelet transform including Parseval’s formula,
energy conservation, inversion formula, and characterization of its range. The
well known Heisenberg-Pauli-Weyl inequality, logarithmic and local uncertainty
principle are generalized in the quaternion Linear canonical domains in
Section 4.

##  2\. Preliminaries

###  2.1.  Quaternion Algebra

The theory of quaternions was initiated by the Irish mathematician Sir W.R.
Hamilton in 1843 and is denoted by $\mathbb{H}$ in his honour. The quaternion
algebra provides an extension of the complex number system to an associative
non-commutative four-dimensional algebra. The quaternion algebra $\mathbb{H}$
over $\mathbb{R}$ is given by

$\displaystyle\mathbb{H}=\Big{\\{}f=a_{0}+i\,a_{1}+j\,a_{2}+k\,a_{3}\,:\,a_{0},a_{1},a_{2},a_{3}\in\mathbb{R}\Big{\\}},$

where $i,j,k$ denote the three imaginary units, obeying the Hamilton’s
multiplication rules

$\displaystyle
ij=k=-ji,~{}jk=i=-kj\,,\,ki=j=-ik\,,\,{\text{and}}\,\,{i}^{2}={j}^{2}={k}^{2}={ijk}=-1.$

For quaternions ${f}_{1}=a_{0}+i\,a_{1}+j\,a_{2}+k\,a_{3}$ and
${f}_{2}=b_{0}+i\,b_{1}+j\,b_{2}+k\,b_{3}$, the addition is defined
componentwise and the multiplication is defined as

$\displaystyle{f}_{1}{f}_{2}$
$\displaystyle=(a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3})+i\,(a_{1}b_{0}+a_{0}b_{1}+a_{2}b_{3}-a_{3}b_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+j(a_{0}b_{2}+a_{2}b_{0}+a_{3}b_{1}-a_{1}b_{3})+k\,(a_{0}b_{3}+a_{3}b_{0}+a_{1}b_{2}-a_{2}b_{1}).$

The conjugate and norm of a quaternion $f=a_{0}+i\,a_{1}+j\,a_{2}+k\,a_{3},$
are given by $\overline{f}=a_{0}-i\,a_{1}-j\,a_{2}-k\,a_{3}$ and
${\|f\|}_{\mathbb{H}}=\sqrt{{a_{0}}^{2}+{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}$,
respectively. We also note that an arbitrary quaternion $h$ can be represented
by two complex numbers as
$f=(a_{0}+i\,a_{1})+j\,(a_{2}-i\,a_{3})=f_{1}+j\,f_{2}$, where
$f_{1},f_{2}\in\mathbb{C}$, and hence,
$\overline{f}=\overline{f_{1}}-j\,f_{2}$, with $\overline{f_{1}}$ denoting the
complex conjugate of $f_{1}$. Moreover, the inner product of any two
quaternions $f=f_{1}+\,jf_{2},$ and $g=g_{1}+j\,g_{2}$ in ${\mathbb{H}}$ is
defined by

$\displaystyle{\big{\langle}f,\,g\big{\rangle}}_{\mathbb{H}}$
$\displaystyle=f\overline{g}=(f_{1}{\overline{g}_{1}}+{\overline{f}_{2}}g_{2})+j(f_{2}{\overline{g}_{1}}-{\overline{f}_{1}}g_{2}).$

By virtue of the complex domain representation, a quaternion-valued function
$f:\mathbb{R}^{2}\to\mathbb{H}$ can be decomposed as $f(x)=f_{1}+j\,f_{2}$,
where $f_{1},f_{2}$ are both complex valued functions.

Let us denote $L^{2}(\mathbb{R}^{2},\mathbb{H})$, the space of all quaternion
valued functions $f$ satisfying

$\displaystyle{\big{\|}f\big{\|}}_{2}=\left\\{\int_{\mathbb{R}^{2}}\Big{(}|f_{1}({\bf
x})|^{2}+|f_{2}({\bf x})|^{2}\Big{)}\,d{\bf x}\right\\}^{1/2}<\infty.$

The norm on $L^{2}(\mathbb{R}^{2},\mathbb{H})$ is obtained from the inner
product of the quaternion valued functions $f=f_{1}+j\,f_{2},$ and
$g=g_{1}+j\,g_{2}$ as

$\displaystyle{\big{\langle}f,\,g\big{\rangle}}_{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}{\big{\langle}f,\,g\big{\rangle}}_{\mathbb{H}}\,d{\bf
x}$ $\displaystyle=\int_{\mathbb{R}^{2}}\bigg{\\{}\Big{(}f_{1}({\bf
x})\,\overline{g_{1}}({\bf x})+\overline{f_{2}}({\bf x})\,g_{2}({\bf
x})\Big{)}+j\Big{(}f_{2}({\bf x})\,\overline{g_{1}}({\bf
x})-\overline{f_{1}}({\bf x})\,g_{2}({\bf x})\Big{)}\bigg{\\}}\,d{\bf x}.$

An easy computation shows that $L^{2}(\mathbb{R}^{2},\mathbb{H})$ equipped
with above defined inner product is a Hilbert space.

Definition 2.1. For any quaternion valued function $f\in
L^{1}(\mathbb{R}^{2},\mathbb{H})\cap L^{2}(\mathbb{R}^{2},\mathbb{H})$, the
two-sided quaternion Fourier transform (QFT) s denoted by ${\mathcal{F}_{q}}$
and is given by

$\displaystyle{\mathcal{F}_{q}}\big{[}f({\bf x})\big{]}({\bf w})=\hat{f}({\bf
w})=\int_{\mathbb{R}^{2}}e^{-ix_{1}w_{1}}f({\bf x})\,e^{-jx_{2}w_{2}}\,d{\bf
x},$ (2.1)

where ${\bf x}=(x_{1},x_{2}),\,{\bf w}=(w_{1},w_{2})$ and the quaternion
exponential $e^{-ix_{1}w_{1}}$ and $e^{-jx_{2}w_{2}}$ are the quaternion
Fourier kernels. The corresponding inversion formula is given by

$\displaystyle f({\bf x})$
$\displaystyle=\dfrac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{ix_{1}w_{1}}f({\bf
x})\,e^{jx_{2}w_{2}}\,d{\bf w}.$ (2.2)

###  2.2.  Quaternion Linear Canonical Transform

Due to non-commutativity of quaternion multiplication, there are three types
of the quaternion offset linear canonical transform, the left-sided QLCT, the
right-sided QLCT, and two-sided QLCT. In this paper, we will extend the theory
of two-sided QLCT introduced by [14] as follows:

Definition 2.2.  Let $A_{s}=\left[\begin{array}[]{cc}a_{s}&b_{s}\\\
c_{s}&d_{s}\\\ \end{array}\right]$, be a matrix parameter such that
$a_{s},b_{s},c_{s},d_{s}\in\mathbb{R}$ and $a_{s}d_{s}-b_{s}c_{s}=1,$ for
$s=1,2.$ The two-sided quaternion linear canonical transform of any quaternion
valued function $f\in L^{2}(\mathbb{R}^{2},\mathbb{H})$, is given by

$\displaystyle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f({\bf
x})\big{]}({\bf
w})=\left\\{\begin{array}[]{cc}\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
x})\,K_{A_{2}}^{j}(x_{2},w_{2})d{\bf x},&b_{1}b_{2}\neq 0\\\
\sqrt{d_{1}d_{2}}e^{\frac{ic_{1}d_{1}}{2}w_{1}^{2}}f(d_{1}w_{1},d_{2}w_{2})e^{\frac{jc_{2}d_{2}}{2}w_{2}^{2}},&b_{1}b_{2}=0\\\
\end{array}\right.$

where ${\bf x}=(x_{1},x_{2}),\,{\bf w}=(w_{1},w_{2})$ and for $b_{1}b_{2}\neq
0$ ,the quaternion kernels $K_{A_{1}}^{i}(x_{1},w_{1})$ and
$K_{A_{2}}^{j}(x_{2},w_{2})$ are respectively given by

$\displaystyle K_{A_{1}}^{i}(x_{1},w_{1})=\dfrac{1}{\sqrt{2\pi
b_{1}}}\exp\left\\{\frac{i}{2b_{1}}\big{[}a_{1}x_{1}^{2}-2x_{1}w_{1}+d_{1}w_{1}^{2}-\frac{\pi
b_{1}}{2}\big{]}\right\\}$

and

$\displaystyle K_{A_{2}}^{j}(x_{2},w_{2})=\dfrac{1}{\sqrt{2\pi
b_{2}}}\exp\left\\{\frac{j}{2b_{2}}\big{[}a_{2}x_{2}^{2}-2x_{2}w_{2}+d_{2}w_{2}^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\},$

The corresponding inversion formula for two-sided QOLCT is given by

$\displaystyle f({\bf x})$
$\displaystyle=\int_{\mathbb{R}^{2}}\overline{K_{A_{1}}^{i}(x_{1},w_{1})}\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\,\overline{K_{A_{2}}^{j}(x_{2},w_{2})}\,d{\bf w}.$ (2.3)

Two quaternion functions $f,\,g\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ are
related to their two-sided QOLCT via the Parseval formula, given as

$\displaystyle\Big{\langle}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}[f],\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}[g]\Big{\rangle}_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=\big{\langle}f,g\big{\rangle}_{\mathbb{H}}.\,$
(2.4)

In particular, $\Big{\|}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f({\bf
x})\big{]}({\bf
w})\Big{\|}_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=\left\|f\right\|_{L^{2}(\mathbb{R}^{2},\mathbb{H})}.$

Definition 2.3. The set

$\mathcal{G}=\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO}(2)=\left\\{(a,{\bf
y},r_{\theta});\,a\in\mathbb{R}^{+},\,{\bf
y}\in\mathbb{R}^{2},\,r_{\theta}\in\text{SO}(2)\right\\}$

endowed with the operation

$\displaystyle\big{(}a,{\bf y},r_{\theta}\big{)}\odot\big{(}a^{\prime},{\bf
y}^{\prime},r_{\theta}^{\prime}\big{)}=\big{(}aa^{\prime},{\bf
y}+ar_{\theta}{\bf y}^{\prime},r_{\theta+\theta^{\prime}}\big{)},$

forms a group called a similitude group on $\mathbb{R}^{2}$ associated with
wavelets, where SO(2) is the special orthogonal group of rotation in
$\mathbb{R}^{2}$. The left Haar measures on $\mathcal{G}$ is given by
$d\eta={da\,d{\bf y}\,d\theta}/a^{3}$ . For more see [21]

##  3\. Quaternion Linear Canonical Wavelet Transform

In this section, we shall first introduce a new convolution structure for two
dimensional linear canonical transform and obtain the corresponding
convolution theorem. Then, we shall characterize the admissibility condition
in terms of two-sided QLCT and define the quaternion linear canonical wavelet
transform in terms of admissible canonical wavelets based on convolution
operator in LCT domain and investigate its fundamental properties.

Analogues to Wei et al [22], the generalized convolution theorem for two
dimensional linear canonical transform, based on generalized translation is
derived as follows.

The ${\bf y}$-generalized translation of signal $\psi({\bf x})$ is denoted by
$\psi({\bf x}\Theta{\bf y})$ is given by [27]

$\displaystyle\psi({\bf x}\Theta{\bf y})=\int\rho({\bf w})\Psi({\bf w})K({\bf
x},{\bf w})K^{*}({\bf y},{\bf w})d{\bf w},\,$ (3.1)

where $\Theta$ is argument of function, $\psi({\bf x}\Theta{\bf y})$ is
generalized delay operator for the generalized translation, $\rho$ is weight
function, $\Psi({\bf w})$ is transformed function $\psi({\bf x})$, and $K({\bf
x},{\bf w})$ is kernel of transformation. The corresponding generalized
translation in two sided QLC-domain is given by

$\displaystyle\psi_{A_{s}}({\bf x}\Theta{\bf y})$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{A_{1}}^{-i}(y_{1},w_{1})K_{A_{1}}^{i}(x_{1},w_{1})\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})K_{A_{2}}^{j}(x_{2},w_{2})K_{A_{1}}^{-j}(y_{2},w_{2})d{\bf
w}$
$\displaystyle=\dfrac{1}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{2}}\exp\left\\{\frac{i}{2b_{1}}\big{[}a_{1}(x_{1}^{2}-y_{1}^{2})-2(x_{1}-y_{1})w_{1}\big{]}\right\\}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}a_{2}(x_{2}^{2}-y_{2}^{2})-2(x_{2}-y_{2})w_{2}\big{]}\right\\}d{\bf
w}$
$\displaystyle=e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\left\\{\dfrac{1}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{2}}e^{\frac{i(y_{1}-x_{1})w_{1}}{b_{1}}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})e^{\frac{j(y_{2}-x_{2})w_{2}}{b_{2}}}d{\bf
w}\right\\}e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}.\,$ (3.2)

We now introduce a new convolution structure for two dimensional linear
canonical transform and obtain the corresponding convolution theorem.

Definition 3.1. For any two signals $f,\,\psi\in
L^{2}(\mathbb{R}^{2},\,\mathbb{H}),$ we define the generalized convolution
operator $\otimes^{A_{s}}$ by

$\displaystyle f\otimes^{A_{s}}\psi({\bf x})=\int_{\mathbb{R}^{2}}f({\bf
y})\psi_{A_{s}}({\bf x}\Theta{\bf y})d{\bf y},\,$ (3.3)

where $\psi_{A_{s}}({\bf x}\Theta{\bf y})$ is given by (3.2), and the
corresponding convolution theorem is

Theorem 3.2. If $f\otimes^{A_{s}}\psi({\bf x})$ is defined as (3.3) and
$\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\big{[}\psi({\bf x})\big{]}\in
L^{2}(\mathbb{R}^{2},\,\mathbb{R})$ denotes the two dimensional linear
canonical transform of a quaternion valued signal $\psi.$ Then we have

$\displaystyle\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\big{[}f\otimes^{A_{s}}\psi({\bf
x})\big{]}({\bf w})=\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\big{[}f({\bf
x})\big{]}({\bf w})\cdot\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\big{[}\psi({\bf
x})\big{]}({\bf w}).\,$ (3.4)

Proof. By definition of QLCT, and well known Fubini Theorem, we have

$\displaystyle\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\big{[}f\otimes^{A_{s}}\psi({\bf
x})\big{]}({\bf w})$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f\otimes^{A_{s}}\psi({\bf
x})\,K_{A_{2}}^{j}(x_{2},w_{2})d{\bf x}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,\int_{\mathbb{R}^{2}}f({\bf
y})\psi({\bf x}\Theta{\bf y})d{\bf y}\,K_{A_{2}}^{j}(x_{2},w_{2})d{\bf x}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
y})\,\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}$
$\displaystyle\qquad\times\left\\{\dfrac{1}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{2}}e^{\frac{i(y_{1}-x_{1})w_{1}}{b_{1}}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})e^{\frac{j(y_{2}-x_{2})w_{2}}{b_{2}}}d{\bf
w}\right\\}e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\,\,\,K_{A_{2}}^{j}(x_{2},w_{2})d{\bf
x}d{\bf y}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
y})\,\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}$
$\displaystyle\qquad\times\left\\{\dfrac{1}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}}e^{\frac{i(y_{1}-x_{1})w_{1}}{b_{1}}}dw_{1}\,\int_{\mathbb{R}}e^{\frac{j(y_{2}-x_{2})w_{2}}{b_{2}}}dw_{2}\right\\}e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\,\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})\,K_{A_{2}}^{j}(x_{2},w_{2})\,d{\bf x}d{\bf y}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
y})\,\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}$
$\displaystyle\qquad\times\left\\{\dfrac{1}{b_{1}b_{2}}b_{1}\delta(y_{1}-x_{1})\cdot
b_{2}\delta(x_{2}-y_{2})\right\\}e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\,\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})\,K_{A_{2}}^{j}(x_{2},w_{2})\,d{\bf x}d{\bf y}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
y})\,\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\delta(y_{1}-x_{1})\cdot\delta(x_{2}-y_{2})e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w})\,K_{A_{2}}^{j}(x_{2},w_{2})\,d{\bf x}d{\bf y}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
y})\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf x})\big{]}({\bf
w})\,K_{A_{2}}^{j}(x_{2},w_{2})\,d{\bf x}$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}K_{A_{1}}^{i}(x_{1},w_{1})\,f({\bf
y})K_{A_{2}}^{j}(x_{2},w_{2})\,d{\bf
x}\cdot\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}\psi({\bf x})\big{]}({\bf
w})\,$ $\displaystyle\qquad=\mathcal{L}_{A_{1},A_{2}}\big{[}f({\bf
x})\big{]}({\bf w})\cdot\mathcal{L}_{A_{1},A_{2}}\big{[}\psi({\bf
x})\big{]}({\bf w}).$

This completes the proof.

Keeping in view the generalized translation in equation (3.2), we have the
following definition of quaternion linear canonical wavelet family of $\psi\in
L^{2}(\mathbb{R}^{2},\,\mathbb{H})$.

Definition 3.3. For a quaternion signal $\psi\in
L^{2}(\mathbb{R}^{2},\,\mathbb{H}),$ we define quaternion linear canonical
wavelets as

$\displaystyle U_{a,{\bf y},\theta}\Psi({\bf x})=\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf
x})=a^{-1}e^{\frac{-ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf y})\big{)}\,e^{\frac{-ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})},\,$ (3.5)

where $U_{a,{\bf y},\theta}:L^{2}(\mathbb{R}^{2},\,\mathbb{H})\rightarrow
L^{2}(\mathcal{G},\,\mathbb{H})$ is a unitary operator,
$a\in\mathbb{R}^{+},{\bf y}\in\mathbb{R}^{2},$ and $r_{-\theta}\in$ SO(2),
with $r_{-\theta}{\bf
x}=(x_{1}\cos\theta+x_{2}\sin\theta,\,-x_{1}\sin\theta+x_{2}\cos\theta),\,0\leq\theta\leq
2\pi$. The order of the terms in (3.5) is fixed because of the non-
commutativity of quaternions.

We now prove a lemma which offers us the two-sided quaternion LCT of
$\Psi_{a,{\bf y},\theta}^{\mathbb{H}}({\bf x})$, defined in (3.5), in terms of
quaternion Fourier transform, which will be fruitful for investigating certain
fundamental properties.

Lemma 3.4. The two-sided quaternion LCT of $\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})$ , defined in (3.5) is given by

$\displaystyle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf w})$
$\displaystyle=\dfrac{a}{2\pi\sqrt{b_{1}b_{2}}}\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w_{1}^{2}+y_{1}^{2}a_{1}-2y_{1}w_{1}-\frac{\pi
b_{1}}{2}\big{]}\right\\}\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf b}}\right)$
$\displaystyle\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}.\,$ (3.6)

Proof. We have

$\displaystyle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf w})$
$\displaystyle\qquad=\dfrac{1}{2\pi
a\sqrt{b_{1}b_{2}}}\int_{\mathbb{R}^{2}}\exp\left\\{\frac{i}{2b_{1}}\big{[}a_{1}x_{1}^{2}-2x_{1}w_{1}+d_{1}w_{1}^{2}-\frac{\pi
b_{1}}{2}\big{]}\right\\}$ $\displaystyle\qquad\qquad\times
e^{\frac{-ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf
y})\big{)}\,e^{\frac{-ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\exp\left\\{\frac{j}{2b_{2}}\big{[}a_{2}x_{2}^{2}-2x_{2}w_{2}+d_{2}w_{2}^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}d{\bf x}$ $\displaystyle\qquad=\dfrac{1}{2\pi
a\sqrt{b_{1}b_{2}}}\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w_{1}^{2}-\frac{\pi
b_{1}+a_{1}y_{1}^{2}}{2}\big{]}\right\\}\int_{\mathbb{R}^{2}}e^{\frac{-ix_{1}w_{1}}{b_{1}}}\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf y})\big{)}\,e^{\frac{-jx_{2}w_{2}}{b_{2}}}\,d{\bf x}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}a_{2}y_{2}^{2}+d_{2}w_{2}^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$ $\displaystyle\qquad=\dfrac{1}{2\pi
a\sqrt{b_{1}b_{2}}}\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w_{1}^{2}-\frac{\pi
b_{1}+a_{1}y_{1}^{2}}{2}\big{]}\right\\}\int_{\mathbb{R}^{2}}e^{\frac{-i(az_{1}+y_{1})w_{1}}{b_{1}}}\psi\big{(}r_{-\theta}z\big{)}\,e^{\frac{-j(az_{2}+y_{2})w_{2}}{b_{2}}}\,a^{2}dz$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}a_{2}y_{2}^{2}+d_{2}w_{2}^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\qquad=\dfrac{a}{2\pi\sqrt{b_{1}b_{2}}}\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w_{1}^{2}-\frac{\pi
b_{1}+a_{1}y_{1}^{2}-2y_{1}w_{1}}{2}\big{]}\right\\}\int_{\mathbb{R}^{2}}e^{\frac{-i(az_{1}w_{1}}{b_{1}}}\psi\big{(}r_{-\theta}z\big{)}\,e^{\frac{-jaz_{2}w_{2}}{b_{2}}}\,dz$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}-2y_{2}w_{2}+a_{2}y_{2}^{2}+d_{2}w_{2}^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\qquad=\dfrac{a}{2\pi\sqrt{b_{1}b_{2}}}\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w_{1}^{2}-\frac{\pi
b_{1}+a_{1}y_{1}^{2}-2y_{1}w_{1}}{2}\big{]}\right\\}\,\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf b}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}-2y_{2}w_{2}+a_{2}y_{2}^{2}+d_{2}w_{2}^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}.$

Remark: From above lemma, we note the following relation

$\displaystyle\frac{2\pi\sqrt{b_{1}b_{2}}}{a}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf b}{\bf w})$
$\displaystyle\qquad=\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}(b_{1}w_{1})^{2}-\frac{\pi
b_{1}+a_{1}y_{1}^{2}-2y_{1}b_{1}w_{1}}{2}\big{]}\right\\}\mathcal{F}_{q}[\psi]\big{(}r_{-\theta}a{\bf
w}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}-2y_{2}b_{2}w_{2}+a_{2}y_{2}^{2}+d_{2}(b_{2}w_{2})^{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\,$ (3.7)

In a sequel, we have the following admissibility condition in terms of two-
sided quaternion linear canonical transform.

Definition 3.5(Admissible Canonical Quaternion Wavelet). A quaternion-valued
wavelet $\psi\in L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)$ is admissible,
if

$\displaystyle
C_{\psi}=\int_{\mathbb{R}^{+}}\int_{\text{SO}(2)}\left|\mathcal{F}_{q}[\psi]\big{(}r_{-\theta}a{\bf
w}\big{)}\right|^{2}\dfrac{dad\theta}{a}=\int_{\mathbb{R}^{+}\times\text{SO(2)}}\left|\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf
w})\right|^{2}\,\dfrac{dad\theta}{a^{3}}\,$ (3.8)

is invertible real constant for a.e. ${\bf w}\in\mathbb{R}^{2}$.

We are now ready to define the quaternion linear canonical wavelet transform
of two dimensional quaternion valued signals.

Definition 3.6. The quaternion linear canonical wavelet transform based on
2D-LCT convolution, of a quaternion-valued function $f\in
L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)$ with respect to an admissible
wavelet $\psi\in L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)$ is defined by

$\displaystyle\mathcal{W}_{\psi}^{\mathbb{H}}[f](a,{\bf y},\theta)$
$\displaystyle=f\otimes^{A_{s}}\psi\big{(}{\bf x}\big{)}$
$\displaystyle=a^{-1}\int_{\mathbb{R}^{2}}f({\bf
x})\,e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\,\overline{\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf y})\big{)}}\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\,d{\bf x}$
$\displaystyle=\left\langle f,\,\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}$
(3.9)

where $\Psi_{a,{\bf y},\theta}^{\mathbb{H}}$ is given by (3.5).

It is worth to note that the proposed quaternion linear canonical wavelet
transform (3.9) boils down to existing quaternion wavelet transform as well as
gives birth to some new quaternion wavelet transform which are not yet
reported in the open literature:

  * •

For the matrices $A_{s}=\left[\begin{array}[]{cc}0&1\\\ -1&0\\\
\end{array}\right]$ for $s=1,2$, the quaternion linear canonical wavelet
transform (3.9) boils down to the quaternion wavelet transform given by [11]

$\displaystyle\mathcal{W}_{\psi}^{\mathbb{H}}[f](a,{\bf
y},\theta)=a^{-1}\int_{\mathbb{R}^{2}}f({\bf
x})\overline{\psi\big{(}r_{-\theta}a^{-1}({\bf x}-{\bf y})\big{)}}\,d{\bf x}.$
(3.10)

  * •

For the matrices $A_{s}=\left[\begin{array}[]{cc}1&b_{s}\\\ 0&1\\\
\end{array}\right],\,b_{s}\neq 0$, for $s=1,2$, we can obtain a new quaternion
wavelet transform namely the quaternion Fresnel-Canonical wavelet transform
given by

$\displaystyle\mathcal{W}_{\psi}^{\mathbb{H}}[f](a,{\bf
y},\theta)=a^{-1}\int_{\mathbb{R}^{2}}f({\bf
x})\,e^{\frac{j}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\,\overline{\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf y})\big{)}}\,e^{\frac{i}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\,d{\bf x}.$
(3.11)

  * •

For the matrices $A_{s}=\left[\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\\\ \end{array}\right]$ for $s=1,2$, the quaternion
linear canonical wavelet transform (3.9) reduces to the quaternion fractional
wavelet transform given by

$\displaystyle\qquad\quad\mathcal{W}_{\psi}^{\mathbb{H}}[f](a,{\bf
y},\theta)=a^{-1}\int_{\mathbb{R}^{2}}f({\bf
x})\,e^{\frac{j\cot\alpha}{2}(x_{2}^{2}-y_{2}^{2})}\,\overline{\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf y})\big{)}}\,e^{\frac{i\cot\alpha}{2}(x_{1}^{2}-y_{1}^{2})}\,d{\bf
x}.$ (3.12)

For the demonstration of the quaternion linear canonical wavelet transform
(3.9), we shall present an illustrative example.

Example 3.7. Consider the two dimensional difference-of-Gaussian wavelets

$\displaystyle\psi({\bf
x})=\lambda^{-2}e^{\frac{-1}{2\lambda^{2}}(x_{1}^{2}+x_{2}^{2})}\,-\,e^{\frac{-1}{2}(x_{1}^{2}+x_{2}^{2})},\,0<\lambda<1.$
(3.13)

For $\theta=0,$ i.e. $r_{-\theta}=1,$ we have the corresponding quaternion
linear canonical wavelet family of $\psi$ as

$\displaystyle\Psi_{a,{\bf y},\theta}^{\mathbb{H}}({\bf x})$
$\displaystyle=a^{-1}\,e^{\frac{-ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\psi\big{(}a^{-1}({\bf
x}-{\bf y})\big{)}\,e^{\frac{-ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}$
$\displaystyle=a^{-1}\,e^{\frac{-ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\left\\{\lambda^{-2}e^{\frac{-\big{(}(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}\big{)}}{2a^{2}\lambda^{2}}}\,-\,e^{\frac{-\big{(}(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}\big{)}}{2a^{2}}}\right\\}e^{\frac{-ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}$

Then the quaternion linear canonical wavelet transform of the signal

$\displaystyle f({\bf
x})=e^{-\alpha_{1}x_{1}-\alpha_{2}x_{2}},\,\alpha_{1},\alpha_{2}\in\mathbb{R}$

with respect to above DOG wavelet $\psi$ (as defined in (3.13)) is given by

$\displaystyle\mathcal{W}_{\psi}^{\mathbb{H}}[f](a,{\bf y},\theta)$
$\displaystyle\qquad=\int_{\mathbb{R}^{2}}f({\bf x})\,\overline{\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})}\,d{\bf x}$
$\displaystyle\qquad=a^{-1}\int_{\mathbb{R}^{2}}e^{-\alpha_{1}x_{1}-\alpha_{2}x_{2}}\,e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\overline{\left\\{\lambda^{-2}e^{\frac{-\big{(}(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}\big{)}}{2a^{2}\lambda^{2}}}\,-\,e^{\frac{-\big{(}(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}\big{)}}{2a^{2}}}\right\\}}\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\,d{\bf
x}$
$\displaystyle\qquad=a^{-1}e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\int_{\mathbb{R}^{2}}e^{-\alpha_{1}x_{1}-\alpha_{2}x_{2}}\,e^{\frac{ja_{2}x_{2}^{2}}{2b_{2}}}\left\\{\lambda^{-2}e^{\frac{-\big{(}x_{1}^{2}+y_{1}^{2}-2x_{1}y_{1}+x_{2}^{2}+y_{2}^{2}-2x_{2}y_{2}\big{)}}{2a^{2}\lambda^{2}}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\left.-\,e^{\frac{-\big{(}x_{1}^{2}+y_{1}^{2}-2x_{1}y_{1})+(x_{2}^{2}+y_{2}^{2}-2x_{2}y_{2}\big{)}}{2a^{2}}}\right\\}\,e^{\frac{ia_{1}x_{1}^{2}}{2b_{1}}}\,d{\bf
x}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad=(\lambda^{2}a)^{-1}e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\int_{\mathbb{R}^{2}}e^{-\alpha_{2}x_{2}}\,e^{\frac{ja_{2}x_{2}^{2}}{2b_{2}}}e^{\frac{-\big{(}x_{1}^{2}+y_{1}^{2}-2x_{1}y_{1}+x_{2}^{2}+y_{2}^{2}-2x_{2}y_{2}\big{)}}{2a^{2}\lambda^{2}}}\,e^{-\alpha_{1}x_{1}}\,e^{\frac{ia_{1}x_{1}^{2}}{2b_{1}}}\,d{\bf
x}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad\qquad-a^{-1}e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\int_{\mathbb{R}^{2}}e^{-\alpha_{2}x_{2}}\,e^{\frac{ja_{2}x_{2}^{2}}{2b_{2}}}\,e^{\frac{-\big{(}x_{1}^{2}+y_{1}^{2}-2x_{1}y_{1}+x_{2}^{2}+y_{2}^{2}-2x_{2}y_{2}\big{)}}{2a^{2}}}\,e^{-\alpha_{1}x_{1}}\,e^{\frac{ia_{1}x_{1}^{2}}{2b_{1}}}\,d{\bf
x}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad=(\lambda^{2}a)^{-1}e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\,e^{-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}\lambda^{2}}}\int_{\mathbb{R}^{2}}e^{-\alpha_{2}x_{2}}\,e^{\frac{ja_{2}x_{2}^{2}}{2b_{2}}}e^{\frac{-\big{(}x_{1}^{2}+\\-2x_{1}y_{1}+x_{2}^{2}+-2x_{2}y_{2}\big{)}}{2a^{2}\lambda^{2}}}\,e^{-\alpha_{1}x_{1}}\,e^{\frac{ia_{1}x_{1}^{2}}{2b_{1}}}\,d{\bf
x}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad\qquad-a^{-1}e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\,e^{-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}}}\int_{\mathbb{R}^{2}}e^{-\alpha_{2}x_{2}}\,e^{\frac{ja_{2}x_{2}^{2}}{2b_{2}}}\,e^{\frac{-\big{(}x_{1}^{2}+y_{1}^{2}-2x_{1}y_{1}+x_{2}^{2}+y_{2}^{2}-2x_{2}y_{2}\big{)}}{2a^{2}}}\,e^{-\alpha_{1}x_{1}}\,e^{\frac{ia_{1}x_{1}^{2}}{2b_{1}}}\,d{\bf
x}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad=\dfrac{1}{\lambda^{2}a}e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\,e^{-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}\lambda^{2}}}\int_{\mathbb{R}}\exp\left\\{-x_{2}^{2}\left(\frac{1}{2a^{2}\lambda^{2}}-\frac{ja_{2}}{2b_{2}}\right)+x_{2}\left(-\alpha_{2}+\frac{y_{2}}{a^{2}\lambda^{2}}\right)\right\\}dx_{2}$
$\displaystyle\qquad\qquad\qquad\times\int_{\mathbb{R}}\exp\left\\{-x_{1}^{2}\left(\frac{1}{2a^{2}\lambda^{2}}-\frac{ia_{1}}{2b_{1}}\right)+x_{1}\left(-\alpha_{1}+\frac{y_{1}}{a^{2}\lambda^{2}}\right)\right\\}dx_{1}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad\qquad-\dfrac{1}{a}\,e^{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}}\,e^{-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}}}\int_{\mathbb{R}}\exp\left\\{-x_{2}^{2}\left(\frac{1}{2a^{2}}-\frac{ja_{2}}{2b_{2}}\right)+x_{2}\left(-\alpha_{2}+\frac{y_{2}}{a^{2}}\right)\right\\}dx_{2}$
$\displaystyle\qquad\qquad\qquad\times\int_{\mathbb{R}}\exp\left\\{-x_{1}^{2}\left(\frac{1}{2a^{2}}-\frac{ia_{1}}{2b_{1}}\right)+x_{1}\left(-\alpha_{1}+\frac{y_{1}}{a^{2}}\right)\right\\}dx_{1}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad=\dfrac{1}{\lambda^{2}a}\exp\left\\{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}\lambda^{2}}\right\\}\sqrt{\dfrac{2\pi
a^{2}\lambda^{2}b_{2}}{b_{2}-ja_{2}a^{2}\lambda^{2}}}\exp\left\\{\frac{(y_{2}-a^{2}\alpha_{2}\lambda^{2})^{2}}{a^{4}\lambda^{4}}\times\frac{2a^{2}\lambda^{2}b_{2}}{4(b_{2}-ja_{2}a^{2}\lambda^{2})}\right\\}$
$\displaystyle\qquad\qquad\qquad\times\sqrt{\dfrac{2\pi
a^{2}\lambda^{2}b_{1}}{b_{1}-ia_{1}a^{2}\lambda^{2}}}\exp\left\\{\frac{(y_{1}-a^{2}\alpha_{1}\lambda^{2})^{2}}{a^{4}\lambda^{4}}\times\frac{2a^{2}\lambda^{2}b_{1}}{4(b_{1}-ia_{1}a^{2}\lambda^{2})}\right\\}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad\qquad-\dfrac{1}{a}\exp\left\\{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}}\right\\}\sqrt{\dfrac{2\pi
a^{2}b_{2}}{b_{2}-ja_{2}a^{2}}}\exp\left\\{\frac{(y_{2}-a^{2}\alpha_{2})^{2}}{a^{4}}\times\frac{2a^{2}b_{2}}{4(b_{2}-ja_{2}a^{2})}\right\\}$
$\displaystyle\qquad\qquad\qquad\times\sqrt{\dfrac{2\pi
a^{2}b_{1}}{b_{1}-ia_{1}a^{2}}}\exp\left\\{\frac{(y_{1}-a^{2}\alpha_{1})^{2}}{a^{4}}\times\frac{2a^{2}b_{1}}{4(b_{1}-ia_{1}a^{2})}\right\\}\,e^{\frac{-ia_{1}y_{1}^{2}}{2b_{1}}}$
$\displaystyle\qquad=\dfrac{2\pi
a\sqrt{b_{1}b_{2}}}{\sqrt{b_{2}-ja_{2}a^{2}\lambda^{2}}}\exp\left\\{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}\lambda^{2}}+\frac{b_{2}(y_{2}-a^{2}\alpha_{2}\lambda^{2})^{2}}{a^{2}\lambda^{2}(b_{2}-ja_{2}a^{2}\lambda^{2})}\right\\}\sqrt{\dfrac{1}{b_{1}-ia_{1}a^{2}\lambda^{2}}}$
$\displaystyle\qquad\qquad\qquad\times\exp\left\\{\frac{b_{1}(y_{1}-a^{2}\alpha_{1}\lambda^{2})^{2}}{a^{2}\lambda^{2}(b_{1}-ia_{1}a^{2}\lambda^{2})}-\frac{ia_{1}y_{1}^{2}}{2b_{1}}\right\\}-\dfrac{2\pi
a\sqrt{b_{1}b_{2}}}{\sqrt{b_{2}-ja_{2}a^{2}}}$
$\displaystyle\qquad\qquad\times\exp\left\\{\frac{-ja_{2}y_{2}^{2}}{2b_{2}}-\frac{y_{1}^{2}+y_{2}^{2}}{2a^{2}}+\frac{b_{2}(y_{2}-a^{2}\alpha_{2})^{2}}{a^{2}(b_{2}-ja_{2}a^{2})}\right\\}\sqrt{\dfrac{1}{b_{1}-ia_{1}a^{2}}}\exp\left\\{\frac{b_{1}(y_{1}-a^{2}\alpha_{1})^{2}}{a^{2}(b_{1}-ia_{1}a^{2})}-\frac{ia_{1}y_{1}^{2}}{2b_{1}}\right\\}$

In the following theorem, we assemble some of the basic properties of the
proposed quaternion linear canonical wavelet transform (3.9).

Theorem 3.8. Let $\psi,\phi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be two
admissible wavelets, then for every $f,g\in L^{2}(\mathbb{R}^{2},\mathbb{H})$
linear canonical wavelet transform (3.9) satisfies the following properties:

(i) ${\text{\it
Linearity:}}\qquad{\mathcal{W}}_{\psi}^{\mathbb{H}}\Big{[}\alpha_{1}f+\alpha_{2}g\Big{]}(a,{\bf
y},\theta)=\alpha_{1}{\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)+\alpha_{2}{\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta),$

$\quad\text{where }\alpha_{1},\alpha_{2}\text{are quaternion constants in
}{\mathbb{H}}$

(ii) ${\text{\it Anti-
linearity:}}\qquad{\mathcal{W}}_{\alpha_{1}\psi+\alpha_{2}\phi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)={\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\overline{\alpha_{1}}+{\mathcal{W}}_{\phi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta)\overline{\alpha_{2}},$

(iii) ${\text{\it Translation:}}~{}~{}\text{For any constant {\bf
k}}\in\mathbb{R}^{2},$

${\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f({\bf x}-{\bf k})\big{]}(a,{\bf
y},\theta)=e^{\frac{ia_{1}k_{1}^{2}}{b_{1}}}\,{\mathcal{W}}_{\psi}^{\mathbb{H}}\Big{[}e^{\frac{ia_{1}k_{1}x_{1}}{b_{1}}}\,f({\bf
x})\,e^{\frac{ja_{2}k_{2}x_{2}}{b_{2}}}\Big{]}(a,{\bf y}-{\bf
k},\theta)\,e^{\frac{ja_{2}k_{2}^{2}}{b_{2}}}$

(iv) ${\text{\it Scaling:}}\,\,\text{For non-zero constant }\lambda,$

${\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f(\lambda\,{\bf x})\big{]}(a,{\bf
y},\theta)=\dfrac{1}{\lambda}{\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f(\lambda\,{\bf
x})\big{]}(a\lambda,{\bf y}\lambda,\theta),$

where matrices parameter in R.H.S is
$A^{\prime}_{s}=[a_{s}^{\prime},b_{s}^{\prime},c_{s},d_{s}]$ is related to
matrices in L.H.S i.e. $A_{s}=[a_{s},b_{s},c_{s},d_{s}]$ by a relation
$\dfrac{a_{s}^{\prime}}{b_{s}^{\prime}}=\dfrac{a_{s}}{b_{s}\lambda^{2}}$, for
$s=1,2.$

(v) ${\text{\it Parity:}}\quad{\mathcal{W}}_{P\psi}^{\mathbb{H}}\big{[}Pf({\bf
x})\big{]}(a,{\bf y},\theta)={\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f({\bf
x})\big{]}(a,-{\bf y},\theta),\,\text{where}\,Pf({\bf x})=f(-{\bf x}).$

(vi) ${\text{\it Dilation in
$\psi$:}}\quad{\mathcal{W}}_{D_{c}\psi}^{\mathbb{H}}\big{[}f({\bf
x})\big{]}(a,{\bf y},\theta)={\mathcal{W}}_{\psi}^{\mathbb{H}}\big{[}f({\bf
x})\big{]}(ac,{\bf y},\theta),\,\text{where}\,D_{c}\psi({\bf
x})=\frac{1}{c}\psi\left(\frac{{\bf x}}{c}\right).$

Proof. For brevity proof of these properties is omitted.

In our next theorem, we will show that the quaternion linear canonical wavelet
transform sets up an isometry from $L^{2}(\mathbb{R}^{2},\mathbb{H})$ to
$L^{2}(\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)},\mathbb{H})$.

Theorem 3.9(Parseval’s Formula). Suppose that $\psi\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$ be an admissible wavelets, then for every
$f,g\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$ we have

$\displaystyle\left\langle\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]},\,\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}\right\rangle_{L^{2}(\mathcal{G},\mathbb{H})}\,=\,C_{\psi}\langle
f,\,g\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}\,$ (3.13)

where $\mathcal{G}\,=\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}$ is
similitude group and $C_{\psi}$ is admissiblity given by (3.8).

Proof. Invoking Parseval formula (2.4) for the two-sided QLCT and implementing
Lemma 3.4, we have

$\displaystyle\left\langle\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]},\,\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}\right\rangle_{L^{2}(\mathcal{G},\mathbb{H})}$
$\displaystyle\quad=\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\overline{\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta)}\,\dfrac{dad{\bf y}d\theta}{a^{3}}$
$\displaystyle\quad=\int_{\mathcal{G}}\left\langle f,\,\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\overline{\left\langle
g,\,\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}\right\rangle}_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\dfrac{dad{\bf
y}d\theta}{a^{3}}$
$\displaystyle\quad=\int_{\mathcal{G}}\left\langle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]},\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf
x})\Big{]}\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\overline{\left\langle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]},\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf
x})\Big{]}\right\rangle}_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\dfrac{dad{\bf
y}d\theta}{a^{3}}$
$\displaystyle\quad=\int_{\mathcal{G}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}}({\bf w})d{\bf
w}\,\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf
w}^{\prime})\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}d{\bf w}^{\prime}\dfrac{dad{\bf y}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathcal{G}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})$
$\displaystyle\quad\times\overline{\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w_{1}^{2}+y_{1}^{2}a_{1}-2y_{1}w_{1}-\frac{\pi
b_{1}}{2}\big{]}\right\\}\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}}$
$\displaystyle\quad\quad\times\exp\left\\{\frac{i}{2b_{1}}\big{[}d_{1}w^{\prime^{2}}_{1}+y_{1}^{2}a_{1}-2y_{1}w^{\prime}_{1}-\frac{\pi
b_{1}}{2}\big{]}\right\\}\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf b}}\right)$
$\displaystyle\quad\quad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})},d{\bf w}d{\bf w}^{\prime}\,\dfrac{dad{\bf y}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathcal{G}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\quad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf
b}}\right)}\exp\left\\{\frac{-i}{2b_{1}}\big{[}d_{1}w_{1}^{2}+y_{1}^{2}a_{1}-2y_{1}w_{1}-\frac{\pi
b_{1}}{2}-d_{1}w^{\prime^{2}}_{1}-y_{1}^{2}a_{1}+2y_{1}w^{\prime}_{1}+\frac{\pi
b_{1}}{2}\big{]}\right\\}$
$\displaystyle\quad\quad\times\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,d{\bf w}d{\bf w}^{\prime}\,\dfrac{dad{\bf y}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathcal{G}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\quad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf
b}}\right)}\exp\left\\{\frac{-i}{2b_{1}}\big{[}d_{1}(w_{1}^{2}-w^{\prime^{2}}_{1})-2y_{1}(w_{1}-w^{\prime}_{1})\big{]}\right\\}$
$\displaystyle\quad\quad\times\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,d{\bf w}d{\bf w}^{\prime}\,\dfrac{dad{\bf y}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\quad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf
b}}\right)}\exp\left\\{\frac{-id_{1}}{2b_{1}}(w_{1}^{2}-w^{\prime^{2}}_{1})\right\\}\int_{\mathbb{R}}\exp\left\\{\frac{iy_{1}}{b_{1}}(w_{1}-w^{\prime}_{1})\right\\}\,dy_{1}$
$\displaystyle\quad\quad\times\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\,\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}d{\bf w}d{\bf w}^{\prime}\,\dfrac{dady_{2}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\qquad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf
b}}\right)}\exp\left\\{\frac{-id_{1}}{2b_{1}}(w_{1}^{2}-w^{\prime^{2}}_{1})\right\\}\delta(w_{1}-w^{\prime}_{1})\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf b}}\right)$
$\displaystyle\quad\qquad\times\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dady_{2}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\qquad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf b}}\right)}\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\qquad\times\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dady_{2}d\theta}{a^{3}}$

$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\overline{\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}}$
$\displaystyle\quad\qquad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf b}}\right)}\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}$
$\displaystyle\quad\qquad\times\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dady_{2}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\overline{\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}}$
$\displaystyle\quad\qquad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf
b}}\right)}\,\,\overline{\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)}}\overline{\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}w^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}w^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}}$
$\displaystyle\quad\qquad\times\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dady_{2}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf w}}{{\bf
b}}\right)\exp\left\\{\frac{j}{2b_{2}}\big{[}d_{2}w_{2}^{2}+y_{2}^{2}a_{2}-2y_{2}w_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}}$
$\displaystyle\quad\quad\times\,\,\overline{\exp\left\\{\frac{-j}{2b_{2}}\big{[}d_{2}{\bf
w}^{\prime^{2}}_{2}+y_{2}^{2}a_{2}-2y_{2}{\bf w}^{\prime}_{2}-\frac{\pi
b_{2}}{2}\big{]}\right\\}\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf
b}}\right)}}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dady_{2}d\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})$
$\displaystyle\quad\qquad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf b}}\right)\exp\left\\{\frac{jd_{2}}{2b_{2}}(w_{2}^{2}-{\bf
w}^{\prime^{2}}_{2})\right\\}\int_{\mathbb{R}}\exp\left\\{\frac{-jy_{2}}{b_{2}}(w_{2}-{\bf
w}^{\prime}_{2})\right\\}dy_{2}\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf b}}\right)}}$
$\displaystyle\quad\qquad\times\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dad\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})$
$\displaystyle\quad\qquad\times\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w}}{{\bf b}}\right)\exp\left\\{\frac{jd_{2}}{2b_{2}}(w_{2}^{2}-{\bf
w}^{\prime^{2}}_{2})\right\\}\,\delta({\bf
w}^{\prime}_{2}-w_{2})\overline{\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf
w^{\prime}}}{{\bf b}}\right)}}$
$\displaystyle\quad\qquad\times\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w}^{\prime})}\,dw^{\prime}_{2}dw_{1}dw_{2}\,\dfrac{dad\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\left|\mathcal{F}_{q}[\psi]\left(r_{-\theta}\,\dfrac{a{\bf w}}{{\bf
b}}\right)\right|^{2}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w})}\,\,dw_{1}dw_{2}\,\dfrac{dad\theta}{a^{3}}$
$\displaystyle\quad=\dfrac{a^{2}}{(2\pi)^{2}b_{1}b_{2}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\left|\frac{2\pi\sqrt{b_{1}b_{2}}}{a}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf
w})\right|^{2}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w})}\,dw_{1}dw_{2}\,\dfrac{dad\theta}{a^{3}}$
$\displaystyle\quad=\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\left|\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf
w})\right|^{2}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w})}\,\,d{\bf w}\,\dfrac{dad\theta}{a^{3}}$
$\displaystyle\quad=\int_{\mathbb{R}^{2}}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]}({\bf
w})\int_{\mathbb{R}^{+}\times\text{SO(2)}}\left|\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\Big{]}({\bf
w})\right|^{2}\,\dfrac{dad\theta}{a^{3}}\,\overline{\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}({\bf
w})}\,d{\bf w}$
$\displaystyle\quad=\,C_{\psi}\left\langle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}f\big{]},\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\big{[}g\big{]}\right\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}$
$\displaystyle\quad=\,C_{\psi}\langle
f,\,g\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}$

where $C_{\psi}$ is given by (3.8).

This completes the proof of the theorem.

Corollary 3.10(Energy Conservation). For $f=g$, we have the following
identity:

$\displaystyle\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}_{\mathbb{H}}^{2}\dfrac{da\,d{\bf
y}\,d\theta}{a^{3}}=C_{\psi}\,\big{\|}f\big{\|}^{2}_{L^{2}(\mathbb{R}^{2},\mathbb{H})}.$
(3.14)

Remark: We see that, except the factor $C_{\psi}$, the quaternion linear
canonical wavelet transform sets up an isometry from
$L^{2}(\mathbb{R}^{2},\mathbb{H})$ to
$L^{2}(\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)},\mathbb{H})$.

The next theorem guarantees the reconstruction of the input quaternion signal
from the corresponding quaternion linear canonical wavelet transform.

Theorem 3.11(Inversion Formula). Suppose that $\psi\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$ be an admissible wavelets, then any
quaternion signal $f\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ can be reconstructed
from the quaternion linear canonical wavelet transform
$\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf y},\theta)$ via the
following formula:

$\displaystyle f({\bf
x})=\dfrac{1}{C_{\Psi}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\Psi_{a,{\bf y},\theta}^{\mathbb{H}}({\bf x})\,\dfrac{da\,d{\bf
y}\,d\theta}{a^{3}},\,a.e.\,$ (3.15)

Proof. For arbitrary $g\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$ implication of
Theorem 3.9 yields

$\displaystyle C_{\psi}\langle
f,\,g\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}$
$\displaystyle=\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\overline{\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta)}\,\dfrac{dad{\bf y}d\theta}{a^{3}}$
$\displaystyle=\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\overline{\left\langle g,\,\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}\right\rangle}_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\dfrac{dad{\bf
y}d\theta}{a^{3}}$
$\displaystyle=\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\left\langle\Psi_{a,{\bf
y},\theta}^{\mathbb{H}},\,g\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\dfrac{dad{\bf
y}d\theta}{a^{3}}$
$\displaystyle=\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\int_{\mathbb{R}^{2}}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}},\overline{g({\bf x})}\,d{\bf x}\,\dfrac{dad{\bf
y}d\theta}{a^{3}}$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\Psi_{a,{\bf y},\theta}^{\mathbb{H}}\,\dfrac{dad{\bf
y}d\theta}{a^{3}}\overline{g({\bf x})}\,d{\bf x}$
$\displaystyle=\left\langle\int_{\mathcal{G}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\Psi_{a,{\bf y},\theta}^{\mathbb{H}}\,\dfrac{dad{\bf
y}d\theta}{a^{3}},\,\,g\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}$

where we applied Fobini’s theorem in getting second last equality. Therefore,
we have

$\displaystyle C_{\Psi}\,f({\bf
x})=\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\Psi_{a,{\bf y},\theta}^{\mathbb{H}}({\bf x})\,\dfrac{da\,d{\bf
y}\,d\theta}{a^{3}},\,a.e.$

This completes the proof of theorem.

Theorem 3.12(Characterization of range). For an admissible wavelet $\psi\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$, the range of the quaternion linear
canonical wavelet transform $\mathcal{W}_{\psi}^{\mathbb{H}}$ is a reproducing
kernel in
$L^{2}\left(\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)},\,\mathbb{H}\right)$
where kernel is given by

$\displaystyle K_{\psi}(a,{\bf y},\theta;a^{\prime},{\bf
y}^{\prime},\theta^{\prime})=C_{\psi}^{-1}\left\langle\Psi_{a,{\bf
y},\theta}^{\mathbb{H}},\,\,\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,$
(3.16)

Moreover,

$\displaystyle\left|K_{\psi}(a,{\bf y},\theta;a^{\prime},{\bf
y}^{\prime},\theta^{\prime})\right|\,\leq\,C_{\psi}^{-1}\big{\|}\psi\big{\|}_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)},\,\text{if}\,C_{\psi}>0.\,$
(3.17)

Proof. Invoking the inversion formula (3.15) in the definition of quaternion
linear canonical wavelet transform (3.8),we have

$\displaystyle\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a^{\prime},{\bf
y}^{\prime},\theta^{\prime})$ $\displaystyle=\int_{\mathbb{R}^{2}}f({\bf
x})\overline{\,\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}({\bf x})}\,d{\bf x}$
$\displaystyle=\int_{\mathbb{R}^{2}}\dfrac{1}{C_{\psi}}\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\Psi_{a,{\bf y},\theta}^{\mathbb{H}}({\bf x})\,\dfrac{da\,d{\bf
y}\,d\theta}{a^{3}}\overline{\,\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}({\bf x})}\,d{\bf x}$
$\displaystyle=\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\int_{\mathbb{R}^{2}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\dfrac{1}{C_{\psi}}\,\Psi_{a,{\bf y},\theta}^{\mathbb{H}}({\bf
x})\overline{\,\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}({\bf x})}\,d{\bf
x}\,\dfrac{da\,d{\bf y}\,d\theta}{a^{3}}$
$\displaystyle=\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\dfrac{1}{C_{\psi}}\,\int_{\mathbb{R}^{2}}\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\overline{\,\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}({\bf x})}\,d{\bf
x}\,\dfrac{da\,d{\bf y}\,d\theta}{a^{3}}$
$\displaystyle=\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)C_{\psi}^{-1}\left\langle\Psi_{a,{\bf
y},\theta}^{\mathbb{H}},\,\,\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}\right\rangle_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)}\,\dfrac{da\,d{\bf
y}\,d\theta}{a^{3}}$
$\displaystyle=\int_{\mathbb{R}^{+}\times\mathbb{R}^{2}\times\text{SO(2)}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)K_{\psi}(a,{\bf y},\theta;a^{\prime},{\bf
y}^{\prime},\theta^{\prime})\,\dfrac{da\,d{\bf y}\,d\theta}{a^{3}}$

Which completes the proof of first assertion.

Also, from the Definition 3.3, we have

$\displaystyle\left|K_{\psi}(a,{\bf y},\theta;a^{\prime},{\bf
y}^{\prime},\theta^{\prime})\right|\,$
$\displaystyle\leq\,\dfrac{1}{\left|C_{\psi}\right|}\int_{\mathbb{R}^{2}}\left|\Psi_{a,{\bf
y},\theta}^{\mathbb{H}}({\bf x})\,\overline{\Psi_{a^{\prime},{\bf
y}^{\prime},\theta^{\prime}}^{\mathbb{H}}({\bf x})}\right|\,d{\bf x}$
$\displaystyle=\,\dfrac{1}{\left|aa^{\prime}C_{\psi}\right|}\int_{\mathbb{R}^{2}}\left|\psi\left(\frac{{\bf
x}-{\bf
y}}{a}\right)\right|_{\mathbb{H}}\,\,\left|\overline{\psi\left(\frac{{\bf
x}-{\bf y}^{\prime}}{a^{\prime}}\right)}\right|_{\mathbb{H}}\,d{\bf x}$
$\displaystyle=\,\dfrac{1}{\left|aa^{\prime}C_{\psi}\right|}\int_{\mathbb{R}^{2}}\left|\psi\left(z-\frac{{\bf
y}}{a}\right)\right|_{\mathbb{H}}\,\left|\psi\left(z-\frac{{\bf
y}^{\prime}}{a^{\prime}}\right)\right|_{\mathbb{H}}\,aa^{\prime}dz$
$\displaystyle=\,\dfrac{1}{\left|C_{\psi}\right|}\int_{\mathbb{R}^{2}}\left|\psi\left(z\right)\right|^{2}_{\mathbb{H}}\,dz$
$\displaystyle=C_{\psi}^{-1}\,\big{\|}\psi\big{\|}_{L^{2}\left(\mathbb{R}^{2},\,\mathbb{H}\right)},\,\text{if}\,C_{\psi}>0.$

This completes the proof of theorem.

##  4\. Uncertainty Principles for Quaternion Linear Canonical Wavelet
Transform

Heisenberg’s uncertainty principle in harmonic analysis is of central
importance in time-frequency analysis as is it provides a lower bound for
optimal simultaneous resolution in the time and frequency domains (see [29]).
This principle has been extended to different time-frequency transforms and
several other versions of the uncertainty principle have been investigated
from time to time. For instance, Beckner [30] obtained a logarithmic version
of the uncertainty principle by using a sharp form of Pitt’s inequality and
showed that this version yields the classical Heisenberg’s inequality by
virtue of Jensen’s inequality. In this Section, we shall establish an analogue
of the well-known Heisenberg’s uncertainty inequality and the corresponding
logarithmic version for the quaternion linear canonical wavelet transform as
defined by (3.9). We first prove the following lemma.

Lemma 4.1. Let $\psi\in L^{2}(\mathbb{R}^{2},\,\mathbb{H})$ be an admissible
quaternion wavelet, then for every $f\in L^{2}(\mathbb{R}^{2},\,\mathbb{H}),$
we have

$\displaystyle\int_{\mathcal{G}}\,\Big{|}w_{k}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}\Big{]}({\bf
w})\Big{|}_{\mathbb{H}}^{2}d\eta\,=\,C_{\psi}\left\|w_{k}\mathcal{F}_{q}[f]({\bf
w})\right\|^{2}_{L^{2}(\mathbb{R}^{2},\,\mathbb{H})}\,$ (4.1)

Proof. Invoking Parseval formula (3.13) of quaternion linear canonical wavelet
transform, we have

$\displaystyle C_{\psi}\langle
f,\,g\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\,\overline{\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta)}\,\dfrac{dad{\bf y}d\theta}{a^{3}}$

Implementing Plancheral theorem of QFT(two-sided) on L.H.S and Plancheral
theorem of QLCT(two-sided) to the ${\bf y}$-integral on R.H.S of above
equation, we obtain

$\displaystyle
C_{\psi}\left\langle\mathcal{F}_{q}[f],\,\mathcal{F}_{q}[g]\right\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\left\langle\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}\Big{]},\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}\Big{]}\right\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}\,\dfrac{dad\theta}{a^{3}}$

Multiplying $\left|w_{k}\right|$ on both sides both sides of above equation,
we get

$\displaystyle C_{\psi}\left\langle
w_{k}\,\mathcal{F}_{q}[f],\,w_{k}\,\mathcal{F}_{q}[g]\right\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\left\langle
w_{k}\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}\Big{]},\,w_{k}\,\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}\Big{]}\right\rangle_{L^{2}(\mathbb{R}^{2},\mathbb{H})}\,\dfrac{dad\theta}{a^{3}}$

Finally, for $f=g$, above equation yields

$\displaystyle
C_{\psi}\int_{\mathbb{R}^{2}}\left|w_{k}\,\mathcal{F}_{q}[f]({\bf
w})\right|_{\mathbb{H}}^{2}d{\bf
w}=\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\,\Big{|}w_{k}\mathcal{L}^{\mathbb{H}}_{A_{1},A_{2}}\Big{[}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}\Big{]}({\bf
w})\Big{|}_{\mathbb{H}}^{2}\dfrac{dad{\bf w}d\theta}{a^{3}}\,$

This completes the proof of Lemma 4.1.

We are now ready to establish the Heisenberg-type inequalities for the
proposed quaternion linear canonical wavelet transform as defined by (3.9).

Theorem 4.2. Let $\psi\in L^{2}(\mathbb{R}^{2},\,\mathbb{H})$ be an admissible
quaternion wavelet, the quaternion linear canonical wavelet transform
$\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf y},\theta)$ given by
(3.9) satisfies the following uncertainty inequality:

$\displaystyle\int_{\mathbb{G}}\left|y_{k}^{2}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|_{\mathbb{H}}^{2}d{\bf
y}\cdot\int_{\mathbb{R}^{2}}\Big{|}w_{k}\,\mathcal{F}_{q}\big{[}f\big{]}({\bf
w})\Big{|}_{\mathbb{H}}^{2}d{\bf
w}\geq\dfrac{b_{k}^{2}}{4}\sqrt{C_{\psi}}\big{\|}f\big{\|}^{4}_{L^{2}(\mathbb{R}^{2},\,\mathbb{H})}$
(4.2)

Proof. Invoking the Heisenberg’s inequality for the quaternion linear
canonical transform [15], we can write

$\displaystyle\left\\{\int_{\mathbb{R}^{2}}{y_{k}}^{2}\big{|}f({\bf
y})\big{|}_{\mathbb{H}}^{2}d{\bf
y}\right\\}^{1/2}\left\\{\int_{\mathbb{R}^{2}}w_{k}^{2}\Big{|}\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\left[f\right]({\bf
w})\Big{|}_{\mathbb{H}}^{2}d{\bf
w}\right\\}^{1/2}\geq\dfrac{b_{k}}{2}\int_{\mathbb{R}^{2}}\big{|}f({\bf
x})\big{|}_{\mathbb{H}}^{2}d{\bf x}.$

Integrating both sides of the above inequality with respect to measure
$\frac{da\,d\theta}{a^{3}}$, we have

$\displaystyle\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\Bigg{[}\left\\{\int_{\mathbb{R}^{2}}{y_{k}}^{2}\big{|}f({\bf
y})\big{|}_{\mathbb{H}}^{2}d{\bf
y}\right\\}^{1/2}\left\\{\int_{\mathbb{R}^{2}}w_{k}^{2}\Big{|}\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\left[f\right]({\bf
w})\Big{|}_{\mathbb{H}}^{2}d{\bf
w}\right\\}^{1/2}\Bigg{]}\dfrac{da\,d\theta}{a^{3}}\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\dfrac{b_{k}}{2}\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\big{|}f({\bf
x})\big{|}_{\mathbb{H}}^{2}\dfrac{da\,d{\bf x}\,d\theta}{a^{3}}$

Now by implementing Cauchy-Schwartz inequality, we obtain

$\displaystyle\left\\{\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}{y_{k}}^{2}\big{|}f({\bf
y})\big{|}_{\mathbb{H}}^{2}d{\bf y}\dfrac{da\,d\theta}{a^{3}}\right\\}^{1/2}$
$\displaystyle\left\\{\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}w_{k}^{2}\Big{|}\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\left[f\right]({\bf
w})\Big{|}_{\mathbb{H}}^{2}d{\bf w}\dfrac{da\,d\theta}{a^{3}}\right\\}^{1/2}$
$\displaystyle\qquad\qquad\geq\dfrac{b_{k}}{2}\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\big{|}f({\bf
x})\big{|}_{\mathbb{H}}^{2}\dfrac{da\,d{\bf x}\,d\theta}{a^{3}}$

Considering $\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf y},\theta)$
as a function of ${\bf y}$ and replacing $f$ by
$\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf y},\theta)$ in above
equation to obtain

$\displaystyle\left\\{\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}{y_{k}}^{2}\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|_{\mathbb{H}}^{2}d{\bf
y}\dfrac{da\,d\theta}{a^{3}}\right\\}^{1/2}$
$\displaystyle\qquad\times\left\\{\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}w_{k}^{2}\Big{|}\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\left[\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right]({\bf w})\Big{|}_{\mathbb{H}}^{2}d{\bf
w}\dfrac{da\,d\theta}{a^{3}}\right\\}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\dfrac{b_{k}}{2}\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\big{|}_{\mathbb{H}}^{2}\dfrac{da\,d{\bf y}\,d\theta}{a^{3}}$

By virtue of Lemma 4.1, we obtain

$\displaystyle\left\\{\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}{y_{k}}^{2}\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|_{\mathbb{H}}^{2}d{\bf
y}\dfrac{da\,d\theta}{a^{3}}\right\\}^{1/2}\left\\{C_{\psi}\int_{\mathbb{R}^{2}}\left|w_{k}\,\mathcal{F}_{q}[f]({\bf
w})\right|_{\mathbb{H}}^{2}d{\bf w}\right\\}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\dfrac{b_{k}}{2}\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\big{|}_{\mathbb{H}}^{2}\dfrac{da\,d{\bf y}\,d\theta}{a^{3}}$

Furthermore, on implementing Corollary 3.10 on R.H.S of the above inequality
we have

$\displaystyle\left\\{\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}{y_{k}}^{2}\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|_{\mathbb{H}}^{2}d{\bf
y}\dfrac{da\,d\theta}{a^{3}}\right\\}^{1/2}\left\\{C_{\psi}\int_{\mathbb{R}^{2}}\left|w_{k}\,\mathcal{F}_{q}[f]({\bf
w})\right|_{\mathbb{H}}^{2}d{\bf w}\right\\}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\dfrac{b_{k}}{2}C_{\psi}\,\big{\|}f\big{\|}^{2}_{L^{2}(\mathbb{R}^{2},\mathbb{H})}$

Dividing both sides by $\sqrt{C_{\psi}}$, we obtain the desired result.

Before presenting our next result, we have the following definition of space
of rapidly decreasing smooth quaternion functions(see [16]).

Definition 4.3. For a multi-index
$\alpha=(\alpha_{1},\alpha_{2})\in\mathbb{R}^{+}\times\mathbb{R}^{+},$ the
Schwartz space in $L^{2}(\mathbb{R}^{2},\mathbb{H})$ is defined as

$\displaystyle\mathcal{S}(\mathbb{R}^{2},\mathbb{H})=\left\\{f\in\mathbb{C}^{\infty}(\mathbb{R}^{2},\mathbb{H});\sup_{t\in\mathbb{R}^{2}}\big{(}1+|t|^{k}\big{)}\Big{|}\dfrac{\partial^{\alpha_{1}+\alpha_{2}}[f(t)]}{\partial^{\alpha_{1}}_{t_{1}}\partial^{\alpha_{2}}_{t_{2}}}\Big{|}\,<\infty\right\\},$

where $\mathbb{C}^{\infty}(\mathbb{R}^{2},\mathbb{H})$ is the set of smooth
functions from $\mathbb{R}^{2}$ to $\mathbb{H}$.

We now establish the logarithmic uncertainty principle for the quaternion
linear canonical wavelet transform
$\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}$ as defined by (3.9).

Theorem 4.4.For an admissible quaternion wavelet
$\psi\in\mathcal{S}(\mathbb{R}^{2},\mathbb{H})$ and a signal
$f\in\mathcal{S}(\mathbb{R}^{2},\mathbb{H})$, the quaternion linear canonical
wavelet transform $\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}$ satisfies
the following logarithmic estimate of the uncertainty inequality:

$\displaystyle\int_{\mathcal{G}}\ln|{\bf
y}|\,\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta)\Big{|}_{\mathbb{H}}^{2}d\eta+C_{\psi}\int_{\mathbb{R}^{2}}\ln|{\bf
w}|\,\left|\mathcal{F}_{q}\big{[}f\big{]}({\bf
w})\right|_{\mathbb{H}}^{2}d{\bf
w}\,\geq\,C_{\psi}\big{(}\mathcal{D}+ln|b|\big{)}\int_{\mathbb{R}^{2}}\Big{|}f({\bf
x})\Big{|}^{2}_{\mathbb{H}}d{\bf x},\,$ (4.3)

where
$\mathcal{D}=\left(\dfrac{\Gamma^{\prime}(1/2)}{\Gamma(1/2)}-\ln\pi\right)$
and $\Gamma$ is a Gamma function.

Proof. For the quaternion-valued function $f\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$, the time and frequency spreads satisfy the
inequality [23]

$\displaystyle\int_{\mathbb{R}^{2}}\ln|{\bf y}|\,\Big{|}f({\bf
y})\Big{|}_{\mathbb{H}}^{2}d{\bf y}+\int_{\mathbb{R}^{2}}\ln|{\bf
w}|\,{\Big{|}\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\big{[}f\big{]}({\bf
w})\Big{|}_{\mathbb{H}}^{2}}\,d{\bf
w}\geq\,\big{(}\mathcal{D}+ln|b|\big{)}\int_{\mathbb{R}^{2}}\Big{|}f({\bf
y})\Big{|}_{\mathbb{H}}^{2}\,d{\bf y}$

Replacing $f({\bf y})$ by
$\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf y},\theta)$ in the
above inequality, we obtain

$\displaystyle\int_{\mathbb{R}^{2}}\ln|{\bf
y}|\,\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|^{2}_{\mathbb{H}}d{\bf y}+\int_{\mathbb{R}^{2}}\ln|{\bf
w}|\,\left|\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\left[\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}\right]({\bf
w})\right|_{\mathbb{H}}^{2}\,d{\bf w}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\,\big{(}\mathcal{D}+ln|b|\big{)}\int_{\mathbb{R}^{2}}\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|_{\mathbb{H}}^{2}\,d{\bf y}$

Integrating above equation with respect to measure $\frac{dad\theta}{a^{3}}$,
and then applying the Fubini theorem, we obtain

$\displaystyle\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\ln|{\bf
y}|\,\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|^{2}_{\mathbb{H}}d{\bf
y}\dfrac{dad\theta}{a^{3}}+\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\ln|{\bf
w}|\,\left|\mathcal{L}_{A_{1},A_{2}}^{\mathbb{H}}\left[\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}\right]({\bf
w})\right|_{\mathbb{H}}^{2}\,d{\bf w}\,\dfrac{dad\theta}{a^{3}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\,\big{(}\mathcal{D}+ln|b|\big{)}\int_{\mathbb{R}^{+}}\int_{\text{SO(2)}}\int_{\mathbb{R}^{2}}\left|\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\right|_{\mathbb{H}}^{2}\,d{\bf y}\,\dfrac{dad\theta}{a^{3}}$

Applying Lemma 4.1 for $w_{k}^{2}=\ln|w_{k}|$ on L.H.S and Corollary 3.10 on
R.H.S, we obtain the desired result

$\displaystyle\int_{\mathcal{G}}\ln|{\bf
y}|\,\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}g\big{]}(a,{\bf
y},\theta)\Big{|}_{\mathbb{H}}^{2}d\eta+C_{\psi}\int_{\mathbb{R}^{2}}\ln|{\bf
w}|\,\left|\mathcal{F}_{q}\big{[}f\big{]}({\bf
w})\right|_{\mathbb{H}}^{2}d{\bf
w}\,\geq\,C_{\psi}\big{(}\mathcal{D}+ln|b|\big{)}\int_{\mathbb{R}^{2}}\Big{|}f({\bf
x})\Big{|}^{2}_{\mathbb{H}}d{\bf x},\,$

This completes the proof of Theorem 4.4.

In the following, we establish a local type uncertainty principle for
quaternion linear canonical wavelet transform
$\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}$ as defined by (3.9).

Theorem 4.5. Given an admissible quaternion wavelet $\psi\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$ and a signal $f\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$, with
$\|\psi\|^{2}_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=1=\|f\|^{2}_{L^{2}(\mathbb{R}^{2},\mathbb{H})},$
such that for a measurable set
$E\subset\mathbb{R}^{2}\times\mathbb{R}^{2},\,\epsilon\geq 0,$ and

$\displaystyle\int\int_{E}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}_{\mathbb{H}}^{2}d{\bf y}d{\bf x}\,\geq\,1-\epsilon.$ (4.4)

We have $a(1-\epsilon)\leq\,\mu(E),$ where $\mu(E)$ is Lebesgue measure of
$E$.

Proof. From Definition 3.4, we have

$\displaystyle\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}_{\mathbb{H}}$
$\displaystyle=\Big{|}a^{-1}\int_{\mathbb{R}^{2}}f({\bf
x})\,e^{\frac{ja_{2}}{2b_{2}}(x_{2}^{2}-y_{2}^{2})}\,\overline{\psi\big{(}r_{-\theta}a^{-1}({\bf
x}-{\bf y})\big{)}}\,e^{\frac{ia_{1}}{2b_{1}}(x_{1}^{2}-y_{1}^{2})}\,d{\bf
x}\Big{|}_{\mathbb{H}}$
$\displaystyle\leq\,\dfrac{1}{a}\int_{\mathbb{R}^{2}}|f({\bf
x})|\,\big{|}\overline{\psi\big{(}r_{-\theta}a^{-1}({\bf x}-{\bf
y})\big{)}}\big{|}\,d{\bf x}.$

Taking Sup-norm on L.H.S and implementing well known Holders inequality on
R.H.S, we obtain

$\displaystyle\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}_{L^{\infty}(\mathbb{R}^{2},\,\mathbb{H})}\leq\,\dfrac{1}{a}\,\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2},\,\mathbb{H})}\big{\|}\psi\big{\|}_{L^{2}(\mathbb{R}^{2},\,\mathbb{H})}\,$
(4.5)

Plugging inequality (4.5) in (4.4), we get

$\displaystyle 1-\epsilon\,$
$\displaystyle\leq\,\int\int_{E}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}_{\mathbb{H}}^{2}d{\bf y}d{\bf x}$
$\displaystyle\leq\,\mu(E)\cdot\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}_{L^{\infty}(\mathbb{R}^{2},\,\mathbb{H})}$
$\displaystyle\leq\,\dfrac{\mu(E)}{a}\,\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2},\,\mathbb{H})}\big{\|}\psi\big{\|}_{L^{2}(\mathbb{R}^{2},\,\mathbb{H})}$
$\displaystyle\leq\,\dfrac{\mu(E)}{a}.$

This completes the proof of Theorem 4.5.

Theorem 4.6(Local uncertainty inequality). Let $E$ be a measurable subset of
$\mathbb{R}^{2}\times\mathbb{R}^{2}$, such that $0<\mu(E)<1,$ then for every
$f\in L(\mathbb{R}^{2},\mathbb{H})$ and an admissible quaternion wavelet
$\psi\in L(\mathbb{R}^{2},\mathbb{H})$ , we have

$\displaystyle\big{\|}f\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\big{\|}\psi\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\leq\dfrac{1}{\sqrt{1-\mu(E)}}\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}_{L^{2}(E^{c},\,\mathbb{H})}\,$ (4.6)

Moreover, for every $\alpha>0,$ there exist $C(\alpha)>0,$ such that

$\displaystyle\big{\|}f\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\big{\|}\psi\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\leq
C(\alpha)\,\Big{[}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\big{|}({\bf
y},{\bf
x})\big{|}^{2\alpha}\,\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}\,d{\bf y}d{\bf x}\Big{]}^{1/2}.\,$ (4.7)

Proof. By invoking the Corollary 3.10 and Theorem 4.5, we have

$\displaystyle\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}^{2}_{L^{2}(\mathbb{R}^{2}\times\mathbb{R}^{2},\,\mathbb{H})}$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf x}$
$\displaystyle=\int\int_{E}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf
x}+\int\int_{E^{c}}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf x}$
$\displaystyle\leq\,\mu(E)\big{\|}f\big{\|}^{2}_{L^{2}(\mathbb{R},\,\mathbb{H})}\big{\|}\psi\big{\|}^{2}_{L^{2}(\mathbb{R},\,\mathbb{H})}+\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}^{2}_{L^{2}(E^{c},\,\mathbb{H})}$

Equivalently,

$\displaystyle(1-\mu(E))\big{\|}f\big{\|}^{2}_{L^{2}(\mathbb{R},\,\mathbb{H})}\big{\|}\psi\big{\|}^{2}_{L^{2}(\mathbb{R},\,\mathbb{H})}\leq\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}^{2}_{L^{2}(E^{c},\,\mathbb{H})}$

Taking square root on both sides and then dividing both sides by
$\sqrt{1-\mu(E)},$ we get

$\displaystyle\big{\|}f\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\big{\|}\psi\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\leq\dfrac{1}{\sqrt{1-\mu(E)}}\Big{\|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{\|}_{L^{2}(E^{c},\,\mathbb{H})}$

which proves our first assertion.

Now, we fix ${\lambda}_{0}\in(0,1]$ small enough such that
$\mu\big{(}B_{{\lambda}_{0}}\big{)}<1,$ where $B_{{\lambda}_{0}}=\left\\{({\bf
y},{\bf x})\in\mathbb{R}^{2}\times\mathbb{R}^{2};\,\big{|}({\bf y},{\bf
x})\big{|}<\,{\lambda}_{0}\right\\},$ the ball of radius ${\lambda}_{0}$
centered at origin, we have from inequality (4.6),

$\displaystyle\big{\|}f\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}\big{\|}\psi\big{\|}_{L^{2}(\mathbb{R},\,\mathbb{H})}$
$\displaystyle\leq\,\dfrac{1}{\sqrt{1-\mu(B_{{\lambda}_{0}})}}\left[\int\int_{\big{|}({\bf
y},{\bf
x})\big{|}>\,{\lambda}_{0}}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf x}\right]^{1/2}$
$\displaystyle\leq\,\dfrac{1}{{\lambda}^{\alpha}_{0}\sqrt{1-\mu(B_{{\lambda}_{0}})}}\left[\int\int_{\big{|}({\bf
y},{\bf
x})\big{|}>\,{\lambda}_{0}}{\lambda}^{2\alpha}_{0}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf x}\right]^{1/2}$
$\displaystyle\leq\,\dfrac{1}{{\lambda}^{\alpha}_{0}\sqrt{1-\mu(B_{{\lambda}_{0}})}}\left[\int\int_{\big{|}({\bf
y},{\bf x})\big{|}>\,{\lambda}_{0}}\,\big{|}({\bf y},{\bf
x})\big{|}^{2\alpha}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf x}\right]^{1/2}$
$\displaystyle\leq\,C(\alpha)\left[\int\int_{\big{|}({\bf y},{\bf
x})\big{|}>\,{\lambda}_{0}}\,\big{|}({\bf y},{\bf
x})\big{|}^{2\alpha}\Big{|}\mathcal{W}_{\psi}^{\mathbb{H}}\big{[}f\big{]}(a,{\bf
y},\theta)\Big{|}^{2}_{\mathbb{H}}d{\bf y}d{\bf x}\right]^{1/2}$

where
$C(\alpha)=\dfrac{1}{{\lambda}^{\alpha}_{0}\sqrt{1-\mu(B_{{\lambda}_{0}})}}.$
This completes the proof of Theorem 4.6.

## References

  * [1]
  * [2] L. Debnath and F. A. Shah, Wavelet Transforms and Their Applications, Birkhäuser, New York, (2015).
  * [3] Wei, D. and Y. Li, Generalized wavelet transform based on the convolution operator in the linear canonical transform domain, Optik-Int. J. Light Electron Optics. 125(16) (2014) 4491-4496.
  * [4] Wang, J., Wang, Y., Wang, W. et al. Discrete linear canonical wavelet transform and its applications. EURASIP J. Adv. Signal Process. 2018, 29 (2018). https://doi.org/10.1186/s13634-018-0550-z
  * [5] Yong Guo and Bing-Zhao Li, The linear canonical wavelet transform on some function spaces. International Journal of Wavelets, Multiresolution and Information Processing, Vol. 16, No. 01, 1850010 (2018). https://doi.org/10.1142/S0219691318500108
  * [6] Prasad, A., Ansari, Z.A. Approximation of linear canonical wavelet transform on the generalized Sobolev spaces. J. Pseudo-Differ. Oper. Appl. 10, 855–881 (2019). https://doi.org/10.1007/s11868-019-00275-5
  * [7] Yong Guo, Li-Dong Yang, Bing-Zhao Li, Multiresolution Analysis for Linear Canonical Wavelet Transform, IAENG International Journal of Computer Science, 46:2, (2019)
  * [8] J.X. He, Continous wavelet transform on the space $L^{2}(\mathbb{R},\mathbb{H},\,dx)$, Appl. Math. Lett. 17(1) (2001) 111-121.
  * [9] J. Zhao, L. Peng, Quaternion-valued admissible wavelets associated with the 2-dimensional Euclidean group with dilations, J. Nat. Geom. 20 (1) (2001) 21–32.
  * [10] E. Bayro-Corrochano, The theory and use of the quaternion wavelet transform, J. Math. Imag. Vision 24 (1) (2006) 19–35.
  * [11] B. Mawardi, R. Ashino and R. Vaillancourt, Two-dimensional quaternion wavelet transform, Applied Mathematics and Computation, 218 (2011), 10–21.
  * [12] P. Fletcher and S. J. Sangwine, The development of the quaternion wavelet transform, Signal Processing, vol. 136, pp. 2–15, 2017.
  * [13] L. Traversoni, Imaging analysis using quaternion wavelet, in geometric algebra with applications, in: E.B. Corrochano, G. Sobczyk (Eds.), Science and Engineering, Birkhäuser, Boston, 2001.
  * [14] K. I. Kou, J. Morais, Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis, Mathematical Methods in the Applied Sciences 36 (9) (2013), pp. 1028-1041. doi:10.1002/mma.2657.
  * [15] K.I. Kou, J. Ou and J. Morais, On certainty principle for quaternion linear canonical transform, Abst. App. Anal. 2013(1), 24-121
  * [16] K.I. Kou, and J. Morais, Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem. Appl. Math. Comput. 247(15), 675–688 (2014)
  * [17] Y. Yang, K.I. Kou, Uncertainty principles for hypercomplex signals in the linear canonical transform domains, Signal Process. 95 (2014) 67–75.
  * [18] Bahri, M.: Quaternion linear canonical transform application. Glob. J. Pure Appl. Math. 11(1), 19–24 (2015)
  * [19] Mawardi, B., Ryuichi, A.A.: Simplified proof of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. 2016, (Article ID 5874930). https://doi.org/10.1155/2016/5874930
  * [20] Zhang, Y.N., Li, B.Z.: Novel uncertainty principles for two-sided quaternion linear canonical transform. Adv. Appl. Clifford Algebras 28(1), 15 (2018)
  * [21] J.P. Antoine, R. Murenzi, Two-dimensional directional wavelet and the scale-angle representation, Signal Process. 52 (3) (1996) 259–281.
  * [22] D.Y. Wei, Q.W. Ran and Y.M. Li, Multichannel sampling and reconstruction of bandlimited signals in the linear canonical transform domain. IET Signal Processing 5(8):717-727 (2011)
  * [23] M. Bahri , Resnawati, S. Musdalifah, A Version of Uncertainty Principle for Quaternion Linear Canonical Transform, Abstract and Applied Analysis, vol. 2018, Article ID 8732457, 7 pages, 2018.
  * [24] B. Hu, Y. Zhou, L. D. Li, and J. Y. Zhang, “Polar linear canonical transform in quaternion domain,” Journal of Information Hiding and Multimedia Signal Processing, vol. 6, no. 6, pp. 1185–1193, 2015.
  * [25] M. Bahri and A. Ryuichi, Logarithmic uncertainty principle for quaternionic linear canonical tranform, 2016 International Conference on Wavelet Aalysis and Pattern Recognition(ICWAPR) 140-145.
  * [26] J.F. Price, Inequalities and local uncertainty principles, Journel of Mathematical Physics 24,1711(1983).
  * [27] Marks II, R.J.: ‘Advanced topics in Shannon sampling and interpolation theory’ (Springer-Verlag, 1993).
  * [28] F. A. Shah, A. A. Teali, and A. Y. Tantary, Windowed special affine Fourier transform, J. Pseudo-Differ. Oper. Appl., (2020) https://doi.org/10.1007/s11868-019-00319-w
  * [29] G.B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3 (1997) 207-238.
  * [30] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995) 1897-1905.
  * [31] Wen-Biao Gao and Bing-Zhao Li, Quaternion Windowed Linear Canonical Transform of Two-Dimensional Signals. Adv. Appl. Clifford Algebras (2020) 30:16, https://doi.org/10.1007/s00006-020-1042-4
  * [32]