# On Uncertainty Principle for Quaternionic Linear Canonical Transform

Authors: Kit Ian Kou; Jian-Yu Ou; Joao Morais
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.


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## Body

## 1. Introduction

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform (FT) cannot both be sharply localized. The uncertainty principle plays an important role in signal processing [1–11] and physics [12–21]. In quantum mechanics an uncertainty principle asserts that one cannot make certain of the position and velocity of an electron (or any particle) at the same time. That is, increasing the knowledge of the position decreases the knowledge of the velocity or momentum of an electron. In quaternionic analysis some papers combine the uncertainty relations and the quaternionic Fourier transform (QFT) [22–24].The QFT plays a vital role in the representation of (hypercomplex) signals. It transforms a real (or quaternionic) 2D signal into a quaternion-valued frequency domain signal. The four components of the QFT separate four cases of symmetry into real signals instead of only two as in the complex FT. In [25] the authors used the QFT to proceed color image analysis. The paper [26] implemented the QFT to design a color image digital watermarking scheme. The authors in [27] applied the QFT to image preprocessing and neural computing techniques for speech recognition. Recently, certain asymptotic properties of the QFT were analyzed and straightforward generalizations of classical Bochner-Minlos theorems to the framework of quaternionic analysis were derived [28, 29]. In this paper, we study the uncertainty principle for the QLCT and the generalization of the QFT to the Hamiltonian quaternionic algebra.The classical LCT being a generalization of the FT, was first proposed in the 1970s by Collins [30] and Moshinsky and Quesne [31]. It is an effective processing tool for chirp signal analysis, such as the parameter estimation, sampling progress for nonbandlimited signals with nonlinear Fourier atoms [32], and the LCT filtering [33–35]. The windowed LCT [36], with a local window function, can reveal the local LCT-frequency contents, and it enjoys high concentrations and eliminates cross terms. The analogue of the Poisson summation formula, sampling formulas, series expansions, Paley-Wiener theorem, and uncertainly relations is studied in [36, 37]. In view of numerous applications, one is particularly interested in higher-dimensional analogues to Euclidean space. The LCT was first extended to the Clifford analysis setting in [38]. It was used to study the generalized prolate spheroidal wave functions and their connection to energy concentration problems [38]. In the present work, we study the QLCT which transforms a quaternionic 2D signal into a quaternion-valued frequency domain signal. Some important properties of the QLCT are analyzed. An uncertainty principle for the QLCT is established. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. To the best of our knowledge, the study of a Heisenberg-type uncertainty principle for the QLCT has not been carried out yet. The results in this paper are new in the literature. The main motivation of the present study is to develop further general numerical methods for differential equations and to investigate localization theorems for summation of Fourier series in the quaternionic analysis setting. Further investigations and extensions of this topic will be reported in a forthcoming paper.The paper is organized as follows. Section2 gives a brief introduction to some general definitions and basic properties of quaternionic analysis. The LCT of 2D quaternionic signal is introduced and studied in Section 3. Some important properties such as Parseval’s and inversion theorems are obtained. In Section 4, we introduce and discuss the concept of QLCT and demonstrate some important properties that are necessary to prove the uncertainty principle for the QLCT. The classical Heisenberg uncertainty principle is generalized for the QLCT in Section 5. This principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. Some conclusions are drawn in Section 6.


## 2. Preliminaries

The quaternionic algebra was invented by Hamilton in 1843 and is denoted byℍ in his honor. It is an extension of the complex numbers to a 4D algebra. Every element of ℍ is a linear combination of a real scalar and three orthogonal imaginary units (denoted, resp., by i, j, and k) with real coefficients
(1)ℍ:={q=q0+iq1+jq2+kq3∣q0,q1,q2,q3∈ℝ},
where the elements i, j, and k obey Hamilton’s multiplication rules
(2)i2=j2=k2=-1;ij=-ji=k,jk=-kj=i,ki=-ik=j.
For every quaternionic number q=q0+q_,  q_=iq1+jq2+kq3, the scalar and nonscalar parts of q are defined as Sc(q):=q0 and NSc(q):=q_, respectively.Every quaternionq=q0+q_ has aquaternionic conjugateq-=q0-q_. This leads to a norm of q∈ℍ defined as
(3)|q|≔qq-=q02+q12+q22+q32.
Let |q| and θ (∈ℝ) be polar coordinates of the point (q0,q_)∈ℍ that corresponds to a nonzero quaternion q=q0+q_. q can be written in polar form as
(4)q=|q|(cosθ+e_sinθ),
where q0=|q|cosθ, |q_|=|q|sinθ, θ=arctan(|q_|/q0), and e_=q_/|q_|. If q≡0, the coordinate θ is undefined; so it is always understood that q≠0 whenever θ=argq is discussed.The symbolee_θ, or exp(e_θ), is defined by means of an infinite series (or Euler’s formula) as
(5)ee_θ:=∑n=0∞‍(e_θ)nn!=cosθ+e_sinθ,
where θ is to be measured in radians. It enables us to write the polar form (4) in exponential form more compactly as
(6)q=|q|ee_θ=|q|exp(q_|q_|arctan(|q_|q0)).
Quaternions can be used for three- or four-entry vector analyses. Recently, quaternions have also been used for color image analysis. For q=q0+iq1+jq2+kq3∈ℍ, we can use q1,q2, and q3 to represent, respectively, the R, G, and B values of a color image pixel and set q0=0.Forp=1 and 2, the quaternion modules Lp(ℝ2;ℍ) are defined as
(7)Lp(ℝ2;ℍ)≔{∫ℝ2‍f∣f:ℝ2⟶ℍ,∥f∥Lp(ℝ2;ℍ)222222222222≔∫ℝ2‍|f(x1,x2)|pdx1dx2<∞}.
For two quaternionic signals f,g∈L2(ℝ2;ℍ) the quaternionic space can be equipped with a Hermitian inner product,
(8)〈f,g〉L2(ℝ2;ℍ)≔∫ℝ2‍f(x1,x2)g(x1,x2)-dx1dx2,
whose associated norm is
(9)∥f∥L2(ℝ2;ℍ)≔(∫ℝ2‍|f(x1,x2)|2dx1dx2)1/2.
As a consequence of the inner product (9), we obtain the quaternionic Cauchy-Schwarz inequality(10)|Sc(〈f,g〉L2(ℝ2;ℍ))|≤|〈f,g〉L2(ℝ2;ℍ)|≤∥f∥L2(ℝ2;ℍ)∥g∥L2(ℝ2;ℍ)
for any f,g∈L2(ℝ2;ℍ).In [39, 40] a Clifford-valued generalized function theory is developed. In the following, we adopt the definition that T is called a tempered distribution, if T is a continuous linear functional from 𝕊:=𝕊(ℝ2) to ℍ, where 𝕊(ℝ2) is the Schwarz class of rapidly decreasing functions. The set of all tempered distributions is denoted by 𝕊′. If T∈𝕊′, we denote this value for a test function ϕ by writing
(11)T[ϕ]≔∫ℝT(x1,x2)ϕ(x1,x2)dx1dx2,
using square brackets. (In the literature one often sees the notation 〈T,ϕ〉, but we shall avoid this, since it does not completely share the properties of the inner product.)This is equivalent to the one defined in [39] using modules and enables us to define Fourier transforms on tempered distributions, by the formula
(12)T^[ϕ]=T[ϕ^],∀ϕ∈𝕊,
which is just to perform Fourier transform
(13)ϕ^(ω1,ω2)=∫ℝ2ϕ(x1,x2)ei(x1ω1+x2ω2)dx1dx2
on each of the components of the distribution. We will use the following results:
(14)1^(ω1,ω2)=(2π)2δ(ω1,ω2),(15)(i-|α|Dαδ)^(ω1,ω2)=ω1α1ω2α2,
where α=(α1,α2), |α|=α1+α2, Dα=(∂/∂x1)α1(∂/∂x2)α2, and δ is the usual Dirac delta function.In the following we introduce the LCT for 2D quaternionic signals.


## 3. LCTs of 2D Quaternionic Signals

The LCT was first introduced in the 70s and is a four-parameter class of linear integral transform, which includes among its many special cases the FT, the fractional Fourier transform (FRFT), the Fresnel transform, the Lorentz transform, and scaling operations. In a way, the LCT has more degrees of freedom and is more flexible than the FT and the FRFT, but with similar computation cost as the conventional FT [41]. Due to the mentioned advantages, it is natural to generalize the classical LCT to the quaternionic algebra.


### 3.1. Definition

Using the definition of the LCT [33, 42], we extend the LCT to the 2D quaternionic signals. Let us define the left-sided and right-sided LCTs of 2D quaternionic signals.Definition 1 (left-sided and right-sided LCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1, for i=1,2. The left-sided and right-sided LCTs of 2D quaternionic signals f∈L1(ℝ2;ℍ) are defined by(16)
L
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,respectively.Note that, forbi=0(i=1,2), the LCT of a signal is essentially a chirp multiplication and it is of no particular interest for our objective in this work. Hence, without loss of generality, we set bi≠0 in the following sections unless stated otherwise. Therefore
(17)Lli(f)(u1,x2)=∫ℝ‍KA1i(x1,u1)f(x1,x2)dx1,b1≠0,(18)Lrj(f)(x1,u2)=∫ℝ‍f(x1,x2)KA2j(x2,u2)dx2,b2≠0,
where the kernel functions
(19)KA1i(x1,u1)≔1i2πb1ei((a1/2b1)x12-(1/b1)x1u1+(d1/2b1)u12),(20)KA2j(x2,u2):=1j2πb2ei((a2/2b2)x22-(1/b2)x2u2+(d2/2b2)u22),
respectively.


### 3.2. Properties

The following proposition summarizes some important properties of the kernel functionsKA1i (and KA2j) of the left-sided (and right-sided) LCTs which will be useful to study the properties of LCTs, such as the Plancherel theorem.Proposition 2.
Let the kernel functionKA be defined by (19) or (20). Then (i)
KA(-x,ω)=KA(x,-ω);(ii)
KA(-x,-ω)=KA(x,ω);(iii)
KA(x,ω)-=KB(x,ω), where B=(a-b-cd);(iv)
∫ℝ‍KA1(x,ω)KA2(ω,y)dω=KA1A2(x,y), where A1A2 corresponds to matrix product.The proofs of properties (i) to (iii) follow from definitions (19) and (20). The proof of property (iv) can be found in [33, 35].Note that some properties of the LCT for 2D quaternionic signals follow from the one-dimensional case [35, 42].Proposition 3.
LetLAi(f)(i=1,2) be defined by Lli in (17) or Lrj in (18), respectively. If f,g∈L1⋂L2(ℝ2;ℍ), then the following properties hold.(i)
Additivity:(21)LA2(LA1)(f)=LA2A1(f),forLAi≔Lli,LA2(LA1)(f)=LA1A2(f),forLAi≔Lrj.(ii)
Reversibility:(22)LAi-1(LAi)(f)=f,forLAi≔Lli,LAi-1(LAi)(f)=f,forLAi≔Lrj.(iii)
Plancherel Theorem (right-sided LCT): Iff,g∈𝕊, then
(23)〈f,g〉L2(ℝ2;ℍ)=〈Lrj(f),Lrj(g)〉L2(ℝ2;ℍ).
In particular, withf=g, we get the Parseval theorem; that is,
(24)∥f∥L2(ℝ2;ℍ)2=∥Lrj(f)∥L2(ℝ2;ℍ)2.Proof.
By Fubini’s theorem, property (iv) of Proposition2 establishes the additivity property (i) of left-sided LCTs,
(25)LA2(LA1(f)(u1,x2))(u1,x2)=∫ℝ‍KA2i(u1,y1)LA1(f)(u1,x2)du1=∫ℝ‍(∫ℝ‍KA2i(u1,y1)KA1i(x1,u1)du1)f(x1,x2)dx1=LA2A1(f)(y1,x2).
The proof of the right-sided LCT LAi:=Lrj is similar.
Reversibility property (ii) is an immediate consequence of additivity property (i) once we observe thatA1=Ai and A2=Ai-1.
To verify property (iii), applying Fubini’s theorem, it suffices to see that(26)〈Lrj(f),Lrj(g)〉L2(ℝ2;ℍ)=∫ℝ2‍Lrj(f)(u1,x2)Lrj(g)(u1,x2)-du1dx2=∫ℝ4‍f(x1,x2)KA1j(x1,u1)2×KA1j(y1,u1)-g(y1,x2)-du1dx1dy1dx2=12πb1∫ℝ4‍f(x1,x2)ej(a1/2b1)(x12-y12)222222×e-j(1/b1)u1(x1-y1)g(y1,x2)-du1dx1dy1dx2=∫ℝ3‍f(x1,x2)ej(a1/2b1)(x12-y12)2×δ(y1-x1)g(y1,x2)-dx1dy1dx2=〈f,g〉L2(ℝ2;ℍ),
where we have used (14).Notice that the left-sided and right-sided LCTs of quaternionic signals are unitary operators onL2(ℝ2;ℍ). In signal analysis, it is interpreted in the sense that (right-sided) LCT of quaternionic signal preserves the energy of a signal.Remark 4.
Note that the Plancherel theorem is not valid for the two-sided or left-sided LCT of 2D quaternionic signal. For this reason, we study the right-sided LCT of 2D quaternionic signals in the following.It is worth noting that whenA1=A2=[01-10], the left-sided and right-sided LCTs of f reduce to the left-sided and right-sided FTs of f. That is,
(27)Lli(f)(u1,x2)=12πi∫ℝ‍e-ix1u1f(x1,x2)dx1Lli(f)(u1,x2)=12πiFli(f)(u1,x2),Lrj(f)(x1,u2)=12πj∫ℝ‍f(x1,x2)e-jx2u2dx2Lrj(f)(x1,u2)=12πjFrj(f)(x1,u2),
respectively. Here
(28)Fli(f)(u1,x2)≔∫ℝ‍e-ix1u1f(x1,x2)dx1,Frj(f)(x1,u2)≔∫ℝ‍f(x1,x2)e-jx2u2dx2
are the left-sided FT and right-sided FT of f, respectively.We now formulate the linear canonical integral representation of a 2D quaternionic signalf.Theorem 5 (linear canonical inversion theorem).
Suppose thatf∈L1(ℝ2;ℍ), that f is continuous except for a finite number of finite jumps in any finite interval, and that f(s,t)=(1/2)(f(s,t+)+f(s,t-)) for all t and s. Then
(29)f(s,t0)=limα→∞∫-αα‍Lri(f)(s,ω)KA-1i(ω,t0)dω
for every t0 and s where f has (generalized) left and right partial derivatives. In particular, if f is piecewise smooth (i.e., continuous and with a piecewise continuous derivative), then the formula holds for all t0 and uniformly in s.Proof.
Put(30)I(s,t0;α)≔∫-αα‍Lri(f)(s,ω)KA-1i(ω,t0)dω
and rewrite this expression by inserting the definition of Lri(f),
(31)I(s,t0;α)=∫-αα‍(∫ℝ‍f(s,t)KAi(t,ω)dt)KA-1i(ω,t0)dω=∫ℝ‍∫-αα‍f(s,t)KAi(t,ω)KA-1i(ω,t0)dωdt=∫ℝ‍f(s,t)[12πbei(a/2b)(t2-t02)∫-αα‍ei(1/b)(t0-t)ωdω]dt=14πe-i(a/2b)t02∫ℝ‍f(s,t)(ei(a/2b)t2sin((α/b)(t0-t))t0-t)dt=14πe-i(a/2b)t02∫ℝ‍f(s,t0-u)×(ei(a/2b)(t0-u)2sin((α/b)u)u)du.
Switching the order of integration is permitted, because the improper double integral is absolutely convergent over the strip (t,ω)∈ℝ×[-α,α], and in the last step we have put t0-t=u. Using the formula
(32)∫0∞‍ei(a/2b)(t0-u)2sin((α/b)u)udu=2πei(a/2b)t02,forα,b>0,
we can write(33)12πe-i(a/2b)t02∫0∞‍f(s,t0-u)22222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du-f(s,t0-)=12πe-i(a/2b)t02∫0∞‍(f(s,t0-u)-f(s,t0-))222222222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du.
Now let ϵ>0 be given. Since we have assumed that f∈L1(ℝ2;ℍ), there exists a number β such that
(34)12π∫β∞‍|f(s,t0-u)|du<ϵ.
Changing the variable, we find that
(35)∫β∞‍ei(a/2b)(t0-u)2sin((α/b)u)udu=∫αβ/b∞‍ei(a/2b)(t0-bx/α)2sinxxdx⟶0,asαb⟶∞.
The last integral in (33) can be split into three terms:
(36)12πe-i(a/2b)t02∫0β‍f(s,t0-u)-f(s,t0-)u22222222×(ei(a/2b)(t0-u)2sin((α/b)u))du+12πe-i(a/2b)t02∫β∞‍f(s,t0-u)2222222222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du-12πe-i(a/2b)t02f(s,t0-)×∫β∞‍(ei(a/2b)(t0-u)2sin((α/b)u)u)du=I1+I2-I3.
The term I3 tends to zero as b>0 and α→∞, because of (35). The term I2 can be estimated:
(37)|I2|=|12πe-i(a/2b)t02∫β∞‍f(s,t0-u)2222222222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du12πe-i(a/2b)t02∫β∞‍|≤12π∫β∞‍|f(s,t0-u)|du≤ϵ.
In the term I1 we have the function g(s,u)=(f(s,t0-u)-f(s,t0-))/u. This is continuous except for jumps in the interval ℝ×(0,β), and it has the finite limit g(s,0+)=(∂/∂t)fL(s,t0) as u↘0; this means that g is bounded uniformly in s and thus integrable on the interval. By the Riemann-Lebesgue lemma, we conclude that I1→0 as β→∞. All this together gives, since ϵ can be taken as small as we wish,
(38)12πe-i(a/2b)t02×∫0∞‍f(s,t0-u)2×(ei(a/2b)(t0-u)2sin((α/b)u)u)du⟶f(s,t0-),222222222222222222222222222222asαb⟶∞.
A parallel argument implies that the corresponding integral over (-∞,0) tends to f(s,t0+) uniformly in s. Taking the mean value of these two results, we have completed the proof of the theorem.Remark 6.
IfLri(f)∈L1(ℝ2;ℍ), then (29) can be written as the absolutely convergent integral
(39)f(s,t0)=∫ℝ‍Lri(f)(s,ω)KA-1i(ω,t0)dω.The following lemma gives the relationship between the left-(right-) sided LCTs and Left-(right-) sided FTs off.Lemma 7.
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1, for i=1,2. Let f∈L1(ℝ2;ℍ); then one has
(40)Lli(f)(u1,x2)=ei(d1/2b1)u12Fli(1i2πb1ei(a1/2b1)(·)2f(·,x2))Lli(f)(u1,x2)=×(u1b1,x2),(41)Lrj(f)(x1,u2)=(Frj(f(x1,·)1j2πb2ej(a2/2b2)(·)2)×(x1,u2b2))ej(d2/2b2)u22.Proof.
By the definition ofLli(f) in (17), a direct computation shows that
(42)Lli(f)(u1,x2)=∫ℝ‍1i2πb1ei((a1/2b1)x12-(1/b1)x1u1+(d1/2b1)u12)f(x1,x2)dx1=ei(d1/2b1)u12×(∫ℝ‍e-ix1(u1/b1)(1i2πb1ei(a1/2b1)x12f(x1,x2))dx1)=ei(d1/2b1)u12Fli(1i2πb1ei(a1/2b1)(·)2f(·,x2))(u1b1,x2).
Similarly, by the definition of Lrj(f) in (18), we obtain (41).The LCT can be further generalized into the offset linear canonical transform (offset LCT) [33, 43, 44]. It has two extra parameters which represent the space and frequency offsets. The basic theories of the LCT have been developed including uncertainty principles [20, 45], convolution theorem [42, 46], the Hilbert transform [11, 47], sampling theory [32, 42], and discretization [41, 48, 49], which enrich the theoretical system of the LCT. On the other hand, since the LCT has three free parameters, it is more flexible and has found many applications in radar system analysis, filter design, phase retrieval, pattern recognition, and many other areas [35, 42].


## 3.1. Definition

Using the definition of the LCT [33, 42], we extend the LCT to the 2D quaternionic signals. Let us define the left-sided and right-sided LCTs of 2D quaternionic signals.Definition 1 (left-sided and right-sided LCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1, for i=1,2. The left-sided and right-sided LCTs of 2D quaternionic signals f∈L1(ℝ2;ℍ) are defined by(16)
L
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f
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≔
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=
0
,respectively.Note that, forbi=0(i=1,2), the LCT of a signal is essentially a chirp multiplication and it is of no particular interest for our objective in this work. Hence, without loss of generality, we set bi≠0 in the following sections unless stated otherwise. Therefore
(17)Lli(f)(u1,x2)=∫ℝ‍KA1i(x1,u1)f(x1,x2)dx1,b1≠0,(18)Lrj(f)(x1,u2)=∫ℝ‍f(x1,x2)KA2j(x2,u2)dx2,b2≠0,
where the kernel functions
(19)KA1i(x1,u1)≔1i2πb1ei((a1/2b1)x12-(1/b1)x1u1+(d1/2b1)u12),(20)KA2j(x2,u2):=1j2πb2ei((a2/2b2)x22-(1/b2)x2u2+(d2/2b2)u22),
respectively.


## 3.2. Properties

The following proposition summarizes some important properties of the kernel functionsKA1i (and KA2j) of the left-sided (and right-sided) LCTs which will be useful to study the properties of LCTs, such as the Plancherel theorem.Proposition 2.
Let the kernel functionKA be defined by (19) or (20). Then (i)
KA(-x,ω)=KA(x,-ω);(ii)
KA(-x,-ω)=KA(x,ω);(iii)
KA(x,ω)-=KB(x,ω), where B=(a-b-cd);(iv)
∫ℝ‍KA1(x,ω)KA2(ω,y)dω=KA1A2(x,y), where A1A2 corresponds to matrix product.The proofs of properties (i) to (iii) follow from definitions (19) and (20). The proof of property (iv) can be found in [33, 35].Note that some properties of the LCT for 2D quaternionic signals follow from the one-dimensional case [35, 42].Proposition 3.
LetLAi(f)(i=1,2) be defined by Lli in (17) or Lrj in (18), respectively. If f,g∈L1⋂L2(ℝ2;ℍ), then the following properties hold.(i)
Additivity:(21)LA2(LA1)(f)=LA2A1(f),forLAi≔Lli,LA2(LA1)(f)=LA1A2(f),forLAi≔Lrj.(ii)
Reversibility:(22)LAi-1(LAi)(f)=f,forLAi≔Lli,LAi-1(LAi)(f)=f,forLAi≔Lrj.(iii)
Plancherel Theorem (right-sided LCT): Iff,g∈𝕊, then
(23)〈f,g〉L2(ℝ2;ℍ)=〈Lrj(f),Lrj(g)〉L2(ℝ2;ℍ).
In particular, withf=g, we get the Parseval theorem; that is,
(24)∥f∥L2(ℝ2;ℍ)2=∥Lrj(f)∥L2(ℝ2;ℍ)2.Proof.
By Fubini’s theorem, property (iv) of Proposition2 establishes the additivity property (i) of left-sided LCTs,
(25)LA2(LA1(f)(u1,x2))(u1,x2)=∫ℝ‍KA2i(u1,y1)LA1(f)(u1,x2)du1=∫ℝ‍(∫ℝ‍KA2i(u1,y1)KA1i(x1,u1)du1)f(x1,x2)dx1=LA2A1(f)(y1,x2).
The proof of the right-sided LCT LAi:=Lrj is similar.
Reversibility property (ii) is an immediate consequence of additivity property (i) once we observe thatA1=Ai and A2=Ai-1.
To verify property (iii), applying Fubini’s theorem, it suffices to see that(26)〈Lrj(f),Lrj(g)〉L2(ℝ2;ℍ)=∫ℝ2‍Lrj(f)(u1,x2)Lrj(g)(u1,x2)-du1dx2=∫ℝ4‍f(x1,x2)KA1j(x1,u1)2×KA1j(y1,u1)-g(y1,x2)-du1dx1dy1dx2=12πb1∫ℝ4‍f(x1,x2)ej(a1/2b1)(x12-y12)222222×e-j(1/b1)u1(x1-y1)g(y1,x2)-du1dx1dy1dx2=∫ℝ3‍f(x1,x2)ej(a1/2b1)(x12-y12)2×δ(y1-x1)g(y1,x2)-dx1dy1dx2=〈f,g〉L2(ℝ2;ℍ),
where we have used (14).Notice that the left-sided and right-sided LCTs of quaternionic signals are unitary operators onL2(ℝ2;ℍ). In signal analysis, it is interpreted in the sense that (right-sided) LCT of quaternionic signal preserves the energy of a signal.Remark 4.
Note that the Plancherel theorem is not valid for the two-sided or left-sided LCT of 2D quaternionic signal. For this reason, we study the right-sided LCT of 2D quaternionic signals in the following.It is worth noting that whenA1=A2=[01-10], the left-sided and right-sided LCTs of f reduce to the left-sided and right-sided FTs of f. That is,
(27)Lli(f)(u1,x2)=12πi∫ℝ‍e-ix1u1f(x1,x2)dx1Lli(f)(u1,x2)=12πiFli(f)(u1,x2),Lrj(f)(x1,u2)=12πj∫ℝ‍f(x1,x2)e-jx2u2dx2Lrj(f)(x1,u2)=12πjFrj(f)(x1,u2),
respectively. Here
(28)Fli(f)(u1,x2)≔∫ℝ‍e-ix1u1f(x1,x2)dx1,Frj(f)(x1,u2)≔∫ℝ‍f(x1,x2)e-jx2u2dx2
are the left-sided FT and right-sided FT of f, respectively.We now formulate the linear canonical integral representation of a 2D quaternionic signalf.Theorem 5 (linear canonical inversion theorem).
Suppose thatf∈L1(ℝ2;ℍ), that f is continuous except for a finite number of finite jumps in any finite interval, and that f(s,t)=(1/2)(f(s,t+)+f(s,t-)) for all t and s. Then
(29)f(s,t0)=limα→∞∫-αα‍Lri(f)(s,ω)KA-1i(ω,t0)dω
for every t0 and s where f has (generalized) left and right partial derivatives. In particular, if f is piecewise smooth (i.e., continuous and with a piecewise continuous derivative), then the formula holds for all t0 and uniformly in s.Proof.
Put(30)I(s,t0;α)≔∫-αα‍Lri(f)(s,ω)KA-1i(ω,t0)dω
and rewrite this expression by inserting the definition of Lri(f),
(31)I(s,t0;α)=∫-αα‍(∫ℝ‍f(s,t)KAi(t,ω)dt)KA-1i(ω,t0)dω=∫ℝ‍∫-αα‍f(s,t)KAi(t,ω)KA-1i(ω,t0)dωdt=∫ℝ‍f(s,t)[12πbei(a/2b)(t2-t02)∫-αα‍ei(1/b)(t0-t)ωdω]dt=14πe-i(a/2b)t02∫ℝ‍f(s,t)(ei(a/2b)t2sin((α/b)(t0-t))t0-t)dt=14πe-i(a/2b)t02∫ℝ‍f(s,t0-u)×(ei(a/2b)(t0-u)2sin((α/b)u)u)du.
Switching the order of integration is permitted, because the improper double integral is absolutely convergent over the strip (t,ω)∈ℝ×[-α,α], and in the last step we have put t0-t=u. Using the formula
(32)∫0∞‍ei(a/2b)(t0-u)2sin((α/b)u)udu=2πei(a/2b)t02,forα,b>0,
we can write(33)12πe-i(a/2b)t02∫0∞‍f(s,t0-u)22222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du-f(s,t0-)=12πe-i(a/2b)t02∫0∞‍(f(s,t0-u)-f(s,t0-))222222222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du.
Now let ϵ>0 be given. Since we have assumed that f∈L1(ℝ2;ℍ), there exists a number β such that
(34)12π∫β∞‍|f(s,t0-u)|du<ϵ.
Changing the variable, we find that
(35)∫β∞‍ei(a/2b)(t0-u)2sin((α/b)u)udu=∫αβ/b∞‍ei(a/2b)(t0-bx/α)2sinxxdx⟶0,asαb⟶∞.
The last integral in (33) can be split into three terms:
(36)12πe-i(a/2b)t02∫0β‍f(s,t0-u)-f(s,t0-)u22222222×(ei(a/2b)(t0-u)2sin((α/b)u))du+12πe-i(a/2b)t02∫β∞‍f(s,t0-u)2222222222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du-12πe-i(a/2b)t02f(s,t0-)×∫β∞‍(ei(a/2b)(t0-u)2sin((α/b)u)u)du=I1+I2-I3.
The term I3 tends to zero as b>0 and α→∞, because of (35). The term I2 can be estimated:
(37)|I2|=|12πe-i(a/2b)t02∫β∞‍f(s,t0-u)2222222222222×(ei(a/2b)(t0-u)2sin((α/b)u)u)du12πe-i(a/2b)t02∫β∞‍|≤12π∫β∞‍|f(s,t0-u)|du≤ϵ.
In the term I1 we have the function g(s,u)=(f(s,t0-u)-f(s,t0-))/u. This is continuous except for jumps in the interval ℝ×(0,β), and it has the finite limit g(s,0+)=(∂/∂t)fL(s,t0) as u↘0; this means that g is bounded uniformly in s and thus integrable on the interval. By the Riemann-Lebesgue lemma, we conclude that I1→0 as β→∞. All this together gives, since ϵ can be taken as small as we wish,
(38)12πe-i(a/2b)t02×∫0∞‍f(s,t0-u)2×(ei(a/2b)(t0-u)2sin((α/b)u)u)du⟶f(s,t0-),222222222222222222222222222222asαb⟶∞.
A parallel argument implies that the corresponding integral over (-∞,0) tends to f(s,t0+) uniformly in s. Taking the mean value of these two results, we have completed the proof of the theorem.Remark 6.
IfLri(f)∈L1(ℝ2;ℍ), then (29) can be written as the absolutely convergent integral
(39)f(s,t0)=∫ℝ‍Lri(f)(s,ω)KA-1i(ω,t0)dω.The following lemma gives the relationship between the left-(right-) sided LCTs and Left-(right-) sided FTs off.Lemma 7.
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1, for i=1,2. Let f∈L1(ℝ2;ℍ); then one has
(40)Lli(f)(u1,x2)=ei(d1/2b1)u12Fli(1i2πb1ei(a1/2b1)(·)2f(·,x2))Lli(f)(u1,x2)=×(u1b1,x2),(41)Lrj(f)(x1,u2)=(Frj(f(x1,·)1j2πb2ej(a2/2b2)(·)2)×(x1,u2b2))ej(d2/2b2)u22.Proof.
By the definition ofLli(f) in (17), a direct computation shows that
(42)Lli(f)(u1,x2)=∫ℝ‍1i2πb1ei((a1/2b1)x12-(1/b1)x1u1+(d1/2b1)u12)f(x1,x2)dx1=ei(d1/2b1)u12×(∫ℝ‍e-ix1(u1/b1)(1i2πb1ei(a1/2b1)x12f(x1,x2))dx1)=ei(d1/2b1)u12Fli(1i2πb1ei(a1/2b1)(·)2f(·,x2))(u1b1,x2).
Similarly, by the definition of Lrj(f) in (18), we obtain (41).The LCT can be further generalized into the offset linear canonical transform (offset LCT) [33, 43, 44]. It has two extra parameters which represent the space and frequency offsets. The basic theories of the LCT have been developed including uncertainty principles [20, 45], convolution theorem [42, 46], the Hilbert transform [11, 47], sampling theory [32, 42], and discretization [41, 48, 49], which enrich the theoretical system of the LCT. On the other hand, since the LCT has three free parameters, it is more flexible and has found many applications in radar system analysis, filter design, phase retrieval, pattern recognition, and many other areas [35, 42].


## 4. QLCTs of 2D Quaternionic Signals

### 4.1. Definition

This section leads to thequaternionic linear canonical transforms (QLCTs). Due to the noncommutative property of multiplication of quaternions, there are many different types of QLCTs: two-sided QLCTs, left-sided QLCTs, and right-sided QLCTs.Definition 8 (two-sided QLCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. The two-sided QLCTs of signals f∈L1(ℝ2;ℍ) are the functions ℒi,j(f):ℝ2→ℍ given by
(43)ℒi,j(f)(u1,u2)≔∫ℝ2‍KA1i(x1,u1)f(x1,x2)KA2j(x2,u2)dx1dx2,
where u1,u2=u1e1+u2e2, with KA1i(x1,u1) and KA2j(x2,u2) given by (19) and (20), respectively.Definition 9 (left-sided QLCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. The left-sided QLCTs of signals f∈L1(ℝ2;ℍ) are the functions ℒli,j(f)∈L(ℝ2;ℍ) given by
(44)ℒli,j(f)(u1,u2)≔∫ℝ2‍KA1i(x1,u1)KA2j(x2,u2)f(x1,x2)dx1dx2,
where the kernels KA1i and KA2j are given by (19) and (20), respectively.Due to the validity of the Plancherel theorem, we study the right-sided QLCTs of 2D quaternionic signals in this paper.Definition 10 (right-sided QLCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. The left-sided QLCTs of signals f∈L1(ℝ2;ℍ) are the functions ℒri,j(f)∈L1(ℝ2;ℍ) given by
(45)ℒri,j(f)(u1,u2)≔∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2,
where KA1i and KA2j are given by (19) and (20), respectively.It is significant to note that whenA1=A2=[01-10], the QLCT of f reduces to the QFT of f. We denote it by
(46)ℱri,j(f)(u1,u2)≔∫ℝ2‍f(x1,x2)e-ix1u1e-jx2u2dx1dx2.Remark 11.
In fact, the right-sided QLCTs defined above can be generalized as follows:(47)ℒre1,e2(f)(u1,u2)=∫ℝ2‍f(x1,x2)KA1e1(x1,u1)KA2e2(x2,u2)dx1dx2,
where e1=e1,ii+e1,jj+e1,kk and e2=e2,ii+e2,jj+e2,kk so that
(48)e1,i2+e1,j2+e1,k2=e2,i2+e2,j2+e2,k2=1222222222222(i.e.,e12=e22=-1)e1,ie2,i+e1,je2,j+e1,ke2,k=0.
Equation (45) is the special case of (47) in which e1=i and e2=j.Remark 12.
Forbi≠0(i=1,2) and f∈L1(ℝ2;ℝ), the (right-sided) QLCT of a 2D signal f∈L1(ℝ2;ℍ) in (45) has the closed-form representation:
(49)ℒri,j(f)(ω1,ω2)≔Φ0(ω1,ω2)+Φ1(ω1,ω2)+Φ2(ω1,ω2)+Φ3(ω1,ω2),
where we put the integrals
(50)Φ0(ω1,ω2)=∫ℝ2‍f(x1,x2)12πkb1b222×cos(a12b1x12-1b1x1ω1+d12b1ω12)22×cos(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2,Φ1(ω1,ω2)=∫ℝ2‍f(x1,x2)i2πjb1b222×sin(a12b1x12-1b1x1ω1+d12b1ω12)22×cos(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2,Φ2(ω1,ω2)=∫ℝ2‍f(x1,x2)j2πib1b222×cos(a12b1x12-1b1x1ω1+d12b1ω12)22×sin(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2,Φ3(ω1,ω2)=∫ℝ2‍f(x1,x2)k2πb1b222×sin(a12b1x12-1b1x1ω1+d12b1ω12)22×sin(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2.
These equations clearly show how the QLCTs separate real signals into four quaternion components, that is, the even-even, odd-even, even-odds and odd-odd components of f.Let us give an example to illustrate expression (45).Example 13.
Consider the quaternionic distribution signal, that is, the QLCT kernel of (45)
(51)f(x1,x2)=KA2-j(x2,u0)KA1-i(x1,v0).
It is easy to see that the QLCT off is a Dirac quaternionic function; that is,
(52)ℒri,j(f)(u1,u2)=(2π)2δ(u1,u2-t_),t_=t1e1+t2e2.


### 4.2. Properties

This subsection describes important properties of the QLCTs that will be used to establish the uncertainty principles for the QLCTs.We now establish a relation between the right-sided LCTs and the right-sided QLCTs of 2D quaternion-valued signals.Lemma 14.
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. For f∈L1(ℝ2;ℍ), one has
(53)ℒri,j(f)(u1,u2)Lrj(Lri‍(f))(u1,u2).Proof.
By using the defnition of right-sided QLCTs (45),
(54)ℒri,j(f)(u1,u2)=∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2=∫ℝLri‍(f)(u1,x2)KA2j(x2,u2)dx2=Lrj(Lri‍(f))(u1,u2).We then establish the Plancherel theorems, specific to the right-sided QLCTs.Theorem 15 (the Plancherel theorems of QLCTs).
Fori=1,2, let fi∈𝕊; the inner product (8) of two quaternionic module functions and their QLCTs is related by
(55)〈f1,f2〉L2(ℝ2;ℍ)=〈ℒri,j(f1),ℒri,j(f2)〉L2(ℝ2;ℍ).
In particular, with f1=f2=f, we get the Parseval identity; that is,
(56)∥f∥L2(ℝ2;ℍ)2=∥ℒri,j(f)∥L2(ℝ2;ℍ)2.Proof.
By the inner product (8) and definition of right-sided QLCTs (45), a straightforward computation and Fubini’s theorem show that
(57)〈ℒri,j(f1),ℒri,j(f2)〉=∫ℝ2‍(∫ℝ2‍f1(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2)×(∫ℝ2‍f2(y1,y2)KA1i(y1,u1)KA2j(y2,u2)dy1dy2-)du1du2=∫ℝ6‍f1(x1,x2)KA1i(x1,u1)2×(KA2j(x2,u2)KA2j(y2,u2)-)×f2(y1,y2)KA1i(y1,u1)-dx1dx2dy1dy2du1du2=∫ℝ6‍f1(x1,x2)KA1i(x1,u1)2×(12πbej(a2/2b2)(x22-y22)e-j(1/b2)u2(x2-y2))2×f2(y1,y2)KA1i(y1,u1)-du1dx1dx2dy1dy2du2=∫ℝ5‍f1(x1,x2)KA1i(x1,u1)2×(ej(a2/2b2)(x22-y22)δ(y2-x2))2×f2(y1,y2)KA1i(y1,u1)-dx1dx2dy1dy2du2=〈Lri(f1),Lri(f2)〉=〈f1,f2〉,
where we have used the Plancherel theorem of right-sided LCTs (24) and formula (14).Remark 16.
Note that the Plancherel theorem is not valid for the two-sided or left-sided QLCT of quaternionic signals. For this reason, we choose to apply the right-sided QLCT of 2D quaternionic signals in the present paper.Theorem15 shows that the total signal energy computed in the spatial domain equals the total signal energy in the quaternionic domain. The Parseval theorem allows the energy of a quaternion-valued signal to be considered on either the spatial domain or the quaternionic domain and the change of domains for convenience of computation.To proceed with, we prove the following derivative properties.Lemma 17.
Fori=1,2, let Ai=[aibicidi]∈ℝ2×2 be a matrix parameter,bi≠0, and aidi-bici=1.  If  f∈𝕊, then
(58)∫ℝ2‍ui2|ℒri,j(f)(u1,u2)|2du1du2=bi2∫ℝ2‍|∂∂xif(x1,x2)|2dx1dx2.Proof.
Fori=1, using (14), (15), and Fubini’s theorem, we have
(59)∫ℝ2‍u12|ℒri,j(f)(u1,u2)|2du1du2=∫ℝ2‍u12(∫ℝ2‍f(s1,s2)KA1i(s1,u1)KA2j(s2,u2)ds1ds2)×(∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2-)du1du2=∫ℝ6‍u12f(s1,s2)KA1i(s1,u1)(KA2j(s2,u2)KA2j(x2,u2)-)2×KA1i(x1,u1)-f(x1,x2)-du2ds1ds2dx1dx2du1=∫ℝ5‍u12f(s1,s2)KA1i(s1,u1)(ej(a2/2b2)(s22-x22)δ(x2-s2))×KA1i(x1,u1)-f(x1,x2)-ds1ds2dx1dx2du1=∫ℝ4‍f(s1,s2)(u12KA1i(s1,u1)KA1i(x1,u1)-)2×f(x1,x2)-du1ds1dx1dx2=-b12∫ℝ3‍f(s1,x2)222×(ei(a1/2b1)(s12-x12)∂2∂x12δ(x1-s1))f(x1,x2)-ds1dx1dx2=-b12∫ℝ2‍f(x1,x2)∂2∂x12f(x1,x2)-dx1dx2=b12∫ℝ2‍|∂∂x1f(x1,x2)|2dx1dx2.
To prove the case i=2, we argue in the same spirit as in the proof of the case i=1. Applying (14), (15), and Fubini’s theorem, we have
(60)∫ℝ2‍u22|ℒri,j(f)(u1,u2)|2du1du2=∫ℝ2‍u22(∫ℝ2‍f(s1,s2)KA1i(s1,u1)KA2j(s2,u2)ds1ds2)×(∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2-)du1du2=∫ℝ6‍f(s1,s2)KA1i(s1,u1)(u22KA2j(s2,u2)KA2j(x2,u2)-)2×f(x1,x2)KA1i(x1,u1)-du2ds1ds2dx1dx2du1=-b22∫ℝ5‍f(s1,s2)KA1i(s1,u1)×(ej(a2/2b2)(s22-x22)∂2∂x22δ(x2-s2))×f(x1,x2)KA1i(x1,u1)-ds1ds2dx1dx2du1=-b22∫ℝ5‍f(s1,x2)KA1i(s1,u1)×(∂2∂x22f(x1,x2))KA1i(x1,u1)-ds1dx1dx2du1=-b22∫ℝ4‍f(s1,x2)(KA1i(s1,u1)KA1i(x1,u1)-)×∂2∂x22f(x1,x2)-du1ds1dx1dx2=-b22∫ℝ3‍f(s1,x2)(ei(a1/2b1)(s12-x12)δ(x1-s1))×∂2∂x22f(x1,x2)-ds1dx1dx2=-b22∫ℝ2‍f(x1,x2)∂2∂x22f(x1,x2)-dx1dx2=b22∫ℝ2‍|∂∂x2f(x1,x2)|2dx1dx2.Some properties of the QLCT are summarized in Table1. Let f1 and f2∈𝕊, the constants α and β∈ℝ, Ai=[aibicidi]∈ℝ2×2,  bi≠0, and aidi-bici=1.Table 1
Properties of the QLCT.
Property
Function
QLCT
Real linearity
α
f
1
(
x
1
,
x
2
)
+
β
f
2
(
x
1
,
x
2
)
α
ℒ
r
i
,
j
(
f
)
(
u
1
,
u
2
)
+
β
ℒ
r
i
,
j
(
f
2
)
(
u
1
,
u
2
)
Formula
Plancherel
〈
f
1
,
f
2
〉
L
2
(
ℝ
2
;
ℍ
)
=
〈
ℒ
r
i
,
j
(
f
1
)
,
ℒ
r
i
,
j
(
f
2
)
〉
L
2
(
ℝ
2
;
ℍ
)
Parseval
∥
f
∥
L
2
(
ℝ
2
;
ℍ
)
=
∥
ℒ
r
i
,
j
(
f
)
∥
L
2
(
ℝ
2
;
ℍ
)
Derivatives
∫
ℝ
2
u
i
2
|
ℒ
r
i
,
j
(
f
)
(
u
1
,
u
2
)
|
2
d
u
1
d
u
2
=
b
i
2
∫
ℝ
2
|
∂
∂
x
i
f
(
x
1
,
x
2
)
|
2
d
x
1
d
x
2


## 4.1. Definition

This section leads to thequaternionic linear canonical transforms (QLCTs). Due to the noncommutative property of multiplication of quaternions, there are many different types of QLCTs: two-sided QLCTs, left-sided QLCTs, and right-sided QLCTs.Definition 8 (two-sided QLCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. The two-sided QLCTs of signals f∈L1(ℝ2;ℍ) are the functions ℒi,j(f):ℝ2→ℍ given by
(43)ℒi,j(f)(u1,u2)≔∫ℝ2‍KA1i(x1,u1)f(x1,x2)KA2j(x2,u2)dx1dx2,
where u1,u2=u1e1+u2e2, with KA1i(x1,u1) and KA2j(x2,u2) given by (19) and (20), respectively.Definition 9 (left-sided QLCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. The left-sided QLCTs of signals f∈L1(ℝ2;ℍ) are the functions ℒli,j(f)∈L(ℝ2;ℍ) given by
(44)ℒli,j(f)(u1,u2)≔∫ℝ2‍KA1i(x1,u1)KA2j(x2,u2)f(x1,x2)dx1dx2,
where the kernels KA1i and KA2j are given by (19) and (20), respectively.Due to the validity of the Plancherel theorem, we study the right-sided QLCTs of 2D quaternionic signals in this paper.Definition 10 (right-sided QLCTs).
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. The left-sided QLCTs of signals f∈L1(ℝ2;ℍ) are the functions ℒri,j(f)∈L1(ℝ2;ℍ) given by
(45)ℒri,j(f)(u1,u2)≔∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2,
where KA1i and KA2j are given by (19) and (20), respectively.It is significant to note that whenA1=A2=[01-10], the QLCT of f reduces to the QFT of f. We denote it by
(46)ℱri,j(f)(u1,u2)≔∫ℝ2‍f(x1,x2)e-ix1u1e-jx2u2dx1dx2.Remark 11.
In fact, the right-sided QLCTs defined above can be generalized as follows:(47)ℒre1,e2(f)(u1,u2)=∫ℝ2‍f(x1,x2)KA1e1(x1,u1)KA2e2(x2,u2)dx1dx2,
where e1=e1,ii+e1,jj+e1,kk and e2=e2,ii+e2,jj+e2,kk so that
(48)e1,i2+e1,j2+e1,k2=e2,i2+e2,j2+e2,k2=1222222222222(i.e.,e12=e22=-1)e1,ie2,i+e1,je2,j+e1,ke2,k=0.
Equation (45) is the special case of (47) in which e1=i and e2=j.Remark 12.
Forbi≠0(i=1,2) and f∈L1(ℝ2;ℝ), the (right-sided) QLCT of a 2D signal f∈L1(ℝ2;ℍ) in (45) has the closed-form representation:
(49)ℒri,j(f)(ω1,ω2)≔Φ0(ω1,ω2)+Φ1(ω1,ω2)+Φ2(ω1,ω2)+Φ3(ω1,ω2),
where we put the integrals
(50)Φ0(ω1,ω2)=∫ℝ2‍f(x1,x2)12πkb1b222×cos(a12b1x12-1b1x1ω1+d12b1ω12)22×cos(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2,Φ1(ω1,ω2)=∫ℝ2‍f(x1,x2)i2πjb1b222×sin(a12b1x12-1b1x1ω1+d12b1ω12)22×cos(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2,Φ2(ω1,ω2)=∫ℝ2‍f(x1,x2)j2πib1b222×cos(a12b1x12-1b1x1ω1+d12b1ω12)22×sin(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2,Φ3(ω1,ω2)=∫ℝ2‍f(x1,x2)k2πb1b222×sin(a12b1x12-1b1x1ω1+d12b1ω12)22×sin(a22b2x22-1b2x2ω2+d22b2ω22)dx1dx2.
These equations clearly show how the QLCTs separate real signals into four quaternion components, that is, the even-even, odd-even, even-odds and odd-odd components of f.Let us give an example to illustrate expression (45).Example 13.
Consider the quaternionic distribution signal, that is, the QLCT kernel of (45)
(51)f(x1,x2)=KA2-j(x2,u0)KA1-i(x1,v0).
It is easy to see that the QLCT off is a Dirac quaternionic function; that is,
(52)ℒri,j(f)(u1,u2)=(2π)2δ(u1,u2-t_),t_=t1e1+t2e2.


## 4.2. Properties

This subsection describes important properties of the QLCTs that will be used to establish the uncertainty principles for the QLCTs.We now establish a relation between the right-sided LCTs and the right-sided QLCTs of 2D quaternion-valued signals.Lemma 14.
LetAi=[aibicidi]∈ℝ2×2 be a matrix parameter such that det(Ai)=1,  bi≠0 for i=1,2. For f∈L1(ℝ2;ℍ), one has
(53)ℒri,j(f)(u1,u2)Lrj(Lri‍(f))(u1,u2).Proof.
By using the defnition of right-sided QLCTs (45),
(54)ℒri,j(f)(u1,u2)=∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2=∫ℝLri‍(f)(u1,x2)KA2j(x2,u2)dx2=Lrj(Lri‍(f))(u1,u2).We then establish the Plancherel theorems, specific to the right-sided QLCTs.Theorem 15 (the Plancherel theorems of QLCTs).
Fori=1,2, let fi∈𝕊; the inner product (8) of two quaternionic module functions and their QLCTs is related by
(55)〈f1,f2〉L2(ℝ2;ℍ)=〈ℒri,j(f1),ℒri,j(f2)〉L2(ℝ2;ℍ).
In particular, with f1=f2=f, we get the Parseval identity; that is,
(56)∥f∥L2(ℝ2;ℍ)2=∥ℒri,j(f)∥L2(ℝ2;ℍ)2.Proof.
By the inner product (8) and definition of right-sided QLCTs (45), a straightforward computation and Fubini’s theorem show that
(57)〈ℒri,j(f1),ℒri,j(f2)〉=∫ℝ2‍(∫ℝ2‍f1(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2)×(∫ℝ2‍f2(y1,y2)KA1i(y1,u1)KA2j(y2,u2)dy1dy2-)du1du2=∫ℝ6‍f1(x1,x2)KA1i(x1,u1)2×(KA2j(x2,u2)KA2j(y2,u2)-)×f2(y1,y2)KA1i(y1,u1)-dx1dx2dy1dy2du1du2=∫ℝ6‍f1(x1,x2)KA1i(x1,u1)2×(12πbej(a2/2b2)(x22-y22)e-j(1/b2)u2(x2-y2))2×f2(y1,y2)KA1i(y1,u1)-du1dx1dx2dy1dy2du2=∫ℝ5‍f1(x1,x2)KA1i(x1,u1)2×(ej(a2/2b2)(x22-y22)δ(y2-x2))2×f2(y1,y2)KA1i(y1,u1)-dx1dx2dy1dy2du2=〈Lri(f1),Lri(f2)〉=〈f1,f2〉,
where we have used the Plancherel theorem of right-sided LCTs (24) and formula (14).Remark 16.
Note that the Plancherel theorem is not valid for the two-sided or left-sided QLCT of quaternionic signals. For this reason, we choose to apply the right-sided QLCT of 2D quaternionic signals in the present paper.Theorem15 shows that the total signal energy computed in the spatial domain equals the total signal energy in the quaternionic domain. The Parseval theorem allows the energy of a quaternion-valued signal to be considered on either the spatial domain or the quaternionic domain and the change of domains for convenience of computation.To proceed with, we prove the following derivative properties.Lemma 17.
Fori=1,2, let Ai=[aibicidi]∈ℝ2×2 be a matrix parameter,bi≠0, and aidi-bici=1.  If  f∈𝕊, then
(58)∫ℝ2‍ui2|ℒri,j(f)(u1,u2)|2du1du2=bi2∫ℝ2‍|∂∂xif(x1,x2)|2dx1dx2.Proof.
Fori=1, using (14), (15), and Fubini’s theorem, we have
(59)∫ℝ2‍u12|ℒri,j(f)(u1,u2)|2du1du2=∫ℝ2‍u12(∫ℝ2‍f(s1,s2)KA1i(s1,u1)KA2j(s2,u2)ds1ds2)×(∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2-)du1du2=∫ℝ6‍u12f(s1,s2)KA1i(s1,u1)(KA2j(s2,u2)KA2j(x2,u2)-)2×KA1i(x1,u1)-f(x1,x2)-du2ds1ds2dx1dx2du1=∫ℝ5‍u12f(s1,s2)KA1i(s1,u1)(ej(a2/2b2)(s22-x22)δ(x2-s2))×KA1i(x1,u1)-f(x1,x2)-ds1ds2dx1dx2du1=∫ℝ4‍f(s1,s2)(u12KA1i(s1,u1)KA1i(x1,u1)-)2×f(x1,x2)-du1ds1dx1dx2=-b12∫ℝ3‍f(s1,x2)222×(ei(a1/2b1)(s12-x12)∂2∂x12δ(x1-s1))f(x1,x2)-ds1dx1dx2=-b12∫ℝ2‍f(x1,x2)∂2∂x12f(x1,x2)-dx1dx2=b12∫ℝ2‍|∂∂x1f(x1,x2)|2dx1dx2.
To prove the case i=2, we argue in the same spirit as in the proof of the case i=1. Applying (14), (15), and Fubini’s theorem, we have
(60)∫ℝ2‍u22|ℒri,j(f)(u1,u2)|2du1du2=∫ℝ2‍u22(∫ℝ2‍f(s1,s2)KA1i(s1,u1)KA2j(s2,u2)ds1ds2)×(∫ℝ2‍f(x1,x2)KA1i(x1,u1)KA2j(x2,u2)dx1dx2-)du1du2=∫ℝ6‍f(s1,s2)KA1i(s1,u1)(u22KA2j(s2,u2)KA2j(x2,u2)-)2×f(x1,x2)KA1i(x1,u1)-du2ds1ds2dx1dx2du1=-b22∫ℝ5‍f(s1,s2)KA1i(s1,u1)×(ej(a2/2b2)(s22-x22)∂2∂x22δ(x2-s2))×f(x1,x2)KA1i(x1,u1)-ds1ds2dx1dx2du1=-b22∫ℝ5‍f(s1,x2)KA1i(s1,u1)×(∂2∂x22f(x1,x2))KA1i(x1,u1)-ds1dx1dx2du1=-b22∫ℝ4‍f(s1,x2)(KA1i(s1,u1)KA1i(x1,u1)-)×∂2∂x22f(x1,x2)-du1ds1dx1dx2=-b22∫ℝ3‍f(s1,x2)(ei(a1/2b1)(s12-x12)δ(x1-s1))×∂2∂x22f(x1,x2)-ds1dx1dx2=-b22∫ℝ2‍f(x1,x2)∂2∂x22f(x1,x2)-dx1dx2=b22∫ℝ2‍|∂∂x2f(x1,x2)|2dx1dx2.Some properties of the QLCT are summarized in Table1. Let f1 and f2∈𝕊, the constants α and β∈ℝ, Ai=[aibicidi]∈ℝ2×2,  bi≠0, and aidi-bici=1.Table 1
Properties of the QLCT.
Property
Function
QLCT
Real linearity
α
f
1
(
x
1
,
x
2
)
+
β
f
2
(
x
1
,
x
2
)
α
ℒ
r
i
,
j
(
f
)
(
u
1
,
u
2
)
+
β
ℒ
r
i
,
j
(
f
2
)
(
u
1
,
u
2
)
Formula
Plancherel
〈
f
1
,
f
2
〉
L
2
(
ℝ
2
;
ℍ
)
=
〈
ℒ
r
i
,
j
(
f
1
)
,
ℒ
r
i
,
j
(
f
2
)
〉
L
2
(
ℝ
2
;
ℍ
)
Parseval
∥
f
∥
L
2
(
ℝ
2
;
ℍ
)
=
∥
ℒ
r
i
,
j
(
f
)
∥
L
2
(
ℝ
2
;
ℍ
)
Derivatives
∫
ℝ
2
u
i
2
|
ℒ
r
i
,
j
(
f
)
(
u
1
,
u
2
)
|
2
d
u
1
d
u
2
=
b
i
2
∫
ℝ
2
|
∂
∂
x
i
f
(
x
1
,
x
2
)
|
2
d
x
1
d
x
2


## 5. Uncertainty Principles for QLCTs

In signal processing much effort has been placed in the study of the classical Heisenberg uncertainty principle during the last years. Shinde and Gadre [9] established an uncertainty principle for fractional Fourier transforms that provides a lower bound on the uncertainty product of real signal representations in both time and frequency domains. Korn [50] proposed Heisenberg-type uncertainty principles for Cohen transforms which describe lower limits for the time frequency concentration. In the meantime, Hitzer et al. [51–54] investigated a directional uncertainty principle for the Clifford-Fourier transform, which describes how the variances (in arbitrary but fixed directions) of a multivector-valued function and its Clifford-Fourier transform are related. On our knowledge, a systematic work on the investigation of uncertainty relations using the QLCT of a multivector-valued function has not been carried out.In the following we explicitly prove and generalize the classical uncertainty principle to quaternionic module functions using the QLCTs. We also give an explicit proof for the Gaussian quaternionic functions (the Gabor filters) to be indeed the only functions that minimize the uncertainty. We further emphasize that our generalization is nontrivial because the multiplication of quaternions and the quaternionic linear canonical kernel are both noncommutative. For this purpose we introduce the following definition.Definition 18.
Fork=1,2, let f,xkf∈L2(ℝ2;ℍ) and ℒri,j(f), ukℒri,j(f)∈L2(ℝ2;ℍ). Then the effective spatial width or spatial uncertainty Δxk of f is evaluated by
(61)Δxk:=Vark(f),
where Vark(f) is the variance of the energy distribution of f along the xk-axis defined by
(62)Vark(f)≔∥xkf∥L2(ℝ2;ℍ)2∥f∥L2(ℝ2;ℍ)2=∫ℝ2‍xk2|f(x1,x2)|2dx1dx2∫ℝ2‍|f(x1,x2)|2dx1dx2.
Similarly, in the quaternionic domain we define the effective spectral width as
(63)Δuk:=Vark(ℒri,j(f)),
where Vark(ℒri,j(f)) is the variance of the frequency spectrum of f along the uk frequency axis given by
(64)Vark(ℒri,j(f))=∥ukℒri,j(f)∥L2(ℝ2;ℍ)2∥ℒri,j(f)∥L2(ℝ2;ℍ)2=∫ℝ2‍uk2|ℒri,j(f)(u1,u2)|2du1du2∫ℝ2‍|ℒri,j(f)(u1,u2)|2du1du2.Example 19.
Let us consider a 2D Gaussian quaternionic function (Figures1, 2, 3, and 4) of the form
(65)f(x1,x2)=Ce-(α1x12+α2x22),
where C=Ci0+iCi1+jCi2+kCi3∈ℍ, for i=1,2, are quaternionic constants and α1,α2∈ℝ are positive real constants.
Then the QLCT off is given by
(66)ℒi,j(f)(u1,u2)=C(∫ℝ‍KA1i(x1,u1)e-α1x12dx1)×(∫ℝ‍e-α2x22KA2j(x2,u2)dx2)=C2b1π2α1b1-a1i×e(((4α1b1-2a1i)d1+2)/(2b1(4α1b1-2a1i)))u12×e(((4α2b2-2a2j)d2+2)/(2b2(4α2b2-2a2j)))u22×2b2π2α2b2-a2j.
This shows that the QLCT of the Gaussian quaternionic function is another Gaussian quaternionic function.
Figures1 and 2 visualize the quaternionic Gaussian function, for α1=α2=3, and α1=3 and α2=1 in the spatial domain. Figures 3 and 4 visualize the quaternionic Gaussian function, for α1=1 and α2=3, and α1=α2=1/2 in the spatial domain.Figure 1Figure 2Figure 3Figure 4Now let us begin the proofs of two uncertainty relations.Theorem 20.
Fork=1,2, let f∈𝕊. Then the following uncertainty relations are fulfilled:
(67)Δx1Δu1≥b12,Δx2Δu2≥b22.
The combination of the two spatial uncertainty principles above leads to the uncertainty principle for the 2D quaternionic signal f(x1,x2) of the form
(68)Δx1Δx2Δu1Δu2≥b1b24.
Equality holds in (68) if and only if f is a 2D Gaussian function; that is. 							(69)f(x1,x2)=βe-(C1x12+C2x22)/2,
where C1, C2 are positive real constants and β=∥f∥L2(ℝ2;ℍ)(C1C2/π2)1/4.Proof.
Applying (58) in Lemma 17 and using the Schwarz inequality (10), we have
(70)(∫ℝ2‍xk2|f(x1,x2)|2dx1dx2)×(∫ℝ2‍uk2|ℒri,j(f)(u1,u2)|2du1du2)=(∫ℝ2‍xk2|f(x1,x2)|2dx1dx2)×(bk2∫ℝ2‍|∂∂xkf(x1,x2)|2dx1dx2)≥bk2|∫ℝ2‍xkf(x1,x2)-∂∂xkf(x1,x2)dx1dx2|2.
Using the exponential form of a 2D quaternionic signal (6), let
(71)f(x1,x2)=f0(x1,x2)+f_(x1,x2)=|f(x1,x2)|ee_α,
where e_=f_(x1,x2)/|f_(x1,x2)| and α=arctan(|f_(x1,x2)|/f0(x1,x2)); then
(72)xkf(x1,x2)-∂∂xkf(x1,x2)=xk|f(x1,x2)|e-e_α∂∂xk(|f(x1,x2)|ee_α)=xk|f(x1,x2)|e-e_α×[(∂∂xk|f(x1,x2)|)ee_α+|f(x1,x2)|(∂∂xkee_α)]=xk|f(x1,x2)|(∂∂xk|f(x1,x2)|)+xk|f(x1,x2)|2(∂∂xk(e_α))=12∂∂xk(xk|f(x1,x2)|2)-12|f(x1,x2)|2+xk|f(x1,x2)|2(∂∂xk(e_α)).
Therefore,
(73)bk2|∫ℝ2‍xkf(x1,x2)-∂∂xkf(x1,x2)dx1dx2|2=bk2|∫ℝ2‍12∂∂xk(xk|f(x1,x2)|2)-12|f(x1,x2)|22+xk|f(x1,x2)|2(∂∂xk(e_α))dx1dx2|2.
The first term is a perfect differential and integrates to zero. The second term gives minus one half of the energy ∥f∥L2(ℝ2;ℍ)2.
Hence(74)(∫ℝ2‍xk2|f(x1,x2)|2dx1dx2)×(∫ℝ2‍uk2|ℒri,j(f)(u1,u2)|2du1du2)≥bk2|-12∥f∥L2(ℝ2;ℍ)2|2=bk24∥f∥L2(ℝ2;ℍ)4.
By definitions of Δxk, Δuk, and Parseval theorem (56), we have
(75)(ΔxkΔuk)2=((∫ℝ2‍xk2|f(x1,x2)|2dx1dx2)2×(∫ℝ2‍uk2|ℒri,j(f)(u1,u2)|2du1du2))×((∫ℝ2‍|f(x1,x2)|2dx1dx2)2222×(∫ℝ2‍|ℒri,j(f)(u1,u2)|2du1du2))-1=((∫ℝ2‍xk2|f(x1,x2)|2dx1dx2)2×(∫ℝ2‍uk2|ℒri,j(f)(u1,u2)|2du1du2))×(∥f∥L2(ℝ2;ℍ)4)-1≥bk24
and therefore we have the uncertainty principle as given by (67) and (68).
We finally show that the equality in (67) and (68) is satisfied if and only if f is a Gaussian quaternionic function.
Since the minimum value for the uncertainty product isbk/2, we can ask what signals have that minimum value. The Schwarz inequality (10) becomes an equality when the two functions are proportional to each other. Hence, we take g=-Cf, where C is a quaternionic constant and the -1 has been inserted for convenience. We therefore have
(76)∂∂xkf(x1,x2)=-Ckxkf(x1,x2).
This is a necessary condition for the uncertainty product to be the minimum. But it is not sufficient since we must also have the term(77)∫ℝ2‍xk|f(x1,x2)|2(∂∂xk(e_α))dx1dx2=0,
because by (73), we see that is the only way we can actually get the value of bk/2.
SinceCk is arbitrary we can write it in terms of its scalar and nonscalar parts, Ck:=Sc(Ck)+NSc(Ck). The solution of (76) is hence
(78)f(x1,x2)=βe-(C1x12+C2x22)/2f(x1,x2)=βe-(Sc(C1)x12+Sc(C2)x22)/2f(x1,x2)×e-(NSc(C1)x12+NSc(C2)x22)/2,
for some constant β. Since e_α=-(NSc(C1)x12+NSc(C2)x22)/2, it follows that
(79)∂∂xk(e_α)=-NSc(Ck)xk.
We have
(80)∫ℝ2‍xk|f(x1,x2)|2(∂∂xk(e_α))dx1dx2=-NSc(Ck)β2×∫ℝ2‍xk2e-(Sc(C1)x12+Sc(C2)x22)dx1dx2.
The only way this can be zero is if NSc(Ck)=0 and hence Ck must be a real number. We then have
(81)f(x1,x2)=βe-(C1x12+C2x22)/2,
where C1, C2 are positive real constants since f∈𝕊 and we have included the appropriate normalization β=∥f∥L2(ℝ2;ℍ)(C1C2/π2)1/4.Since the 2D Gaussian functionf(x1,x2) of (81) achieves the minimum width-bandwidth product, it is theoretically a very good prototype waveform. One can therefore construct a basic waveform using spatially or frequency-scaled versions of f(x1,x2) to provide multiscale spectral resolution. Such a wavelet basis construction derived from a Gaussian quaternionic function prototype waveform has been realized, for example, in the quaternionic wavelet transforms in [55]. The optimal space-frequency localization is also another reason why 2D Clifford-Gabor bandpass filters were suggested in [56].


## 6. Conclusion

In this paper we developed the definition of QLCT. The various properties of QLCT such as partial derivative, the Plancherel, and Parseval theorems are discussed. Using the well-known properties of the classical LCT, we established an uncertainty principle for the QLCT. This uncertainty principle states that the product of the variances of quaternion-valued signals in the spatial and frequency domains has a lower bound. It is shown that only a 2D Gaussian signal minimizes the uncertainty. With the help of this principle, we hope to contribute to the theory and applications of signal processing through this investigation and to develop further general numerical methods for differential equations. The results in this paper are new in the literature. Further investigations on this topic are now under investigation and will be reported in a forthcoming paper.



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