# A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

Authors: Mawardi Bahri; Resnawati; Selvy Musdalifah
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.


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## Body

## 1. Introduction

The quaternion Fourier transform (QFT) is an extension of the classical two-dimensional Fourier transform (FT) [1–4] in the framework of quaternion algebra. It plays an important role in the representation of the two-dimensional quaternion signals. A number of useful properties of the extended transform are generalizations of the corresponding properties of the FT with some modifications (see, e.g., [5–15]). The QFT has found many applications in color image processing and signal analysis; we refer the reader to [16–19] and the references mentioned therein. An extension of the QFT in the framework of the classical linear canonical transform (LCT) (see [20–22]), known as the quaternion linear canonical transform, has received much attention in recent years. Due to the several definitions of the QFT, there are basically three ways of obtaining the quaternion linear canonical transform (QLCT): the right-sided quaternion linear canonical transform, the left-sided quaternion linear canonical transform, and the two-sided quaternion linear canonical transform. The right-sided quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition, and so on. Recent works related to some important properties of the QLCT such as Parseval’s theorem, reconstruction formula, and component-wise uncertainty principles were also published [18, 23–25]. It was found that the properties of the QLCT are extensions of the corresponding version of the QFT with some modifications.On the other hand, the uncertainty principle plays one important role in signal processing. It describes a function and its FT, which cannot both be simultaneously sharply localized. One example of this is the Heisenberg uncertainty principle concerning position and momentum wave functions in quantum physics. In signal processing, an uncertainty principle states that the product of the variances of the signal in thetime andfrequency domains has a lower bound. Up till now, several attempts have been made to extend the uncertainty principles associated with the QFT and QLCT domains. The component-wise and directional uncertainty principles associated with the QFT were proposed in [11]. In [26, 27], the authors established a component-wise uncertainty principle for the QLCT and proved that the equality is achieved for optimal quaternion Gaussian function. Recently, the authors [23] proposed the logarithmic uncertainty principle associated with the QLCT which is the generalization of the logarithmic uncertainty principle for the QFT.Therefore, the main objective of the present paper is to establish full uncertainty principle for the two-sided QLCT, which is a new general form of component-wise uncertainty principle for the two-sided QLCT. This uncertainty principle is derived using the connection between the QFT and QLCT. We also obtain full uncertainty principle for the right-sided QLCT using a relation between the right-sided QLCT and two-sided QLCT. We also derive the Hausdorff-Young inequality associated with the two-sided QLCT.This paper is organized as follows. In Section2, we briefly review the basic knowledge of quaternion and its split used in the next section. In Section 3, we introduce the definition of the QFT and the QLCT. The relationship between the QFT and the QLCT is also discussed in this section. In Section 4, we derive a new version uncertainty principle associated with the QLCT, which shows that the spread of a quaternion-valued function and its QLCT are inversely proportional. Section 5 concludes this paper.


## 2. Preliminaries

In this section, let us briefly recall some basic definitions and properties of the quaternions (for more details, see [28]).


### 2.1. Quaternions

The quaternions, a generalization of complex numbers, are members of a noncommutative division algebra. The set of quaternions is denoted byH. Every element of H can be written in the following form: (1)H=q=q0+iq1+jq2+kq3;q0,q1,q2,q3∈R, with the units i, j, k, which obey the following: (2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=ijk=-1.For a quaternion q=q0+iq1+jq2+kq3∈H, q0 is simply called the scalar part of q denoted by Sc(q) and q=iq1+jq2+kq3 is called the vector part of q denoted by Vec(q).Letp,q∈H and p, q be their vector parts, respectively. Equation (2) yields the quaternionic multiplication qp as(3)qp=q0p0+q·p+q0p+p0q+q×p,where (4)q·p=-q1p1+q2p2+q3p3,q×p=iq2p3-q3p2+jq3p1-q1p3+kq1p2-q2p1. Analogously to the complex case, a quaternionic conjugation q¯ is given by(5)q¯=q0-iq1-jq2-kq3,which leads to the anti-involution; that is, (6)qp¯=p¯q¯. With the help of (5), we get the norm or modulus of q∈H as(7)q=qq¯=q02+q12+q22+q32.One can easily verify that(8)qp=qp,q+p≤q+p,∀p,q∈H.Using conjugate (5) and the modulus ofq, we can define the inverse of q∈H\{0} as (9)q-1=q¯q2, which shows that H is a normed division algebra.In quaternionic notation, we may define an inner product for quaternion-valued functionsf,g:R2→H as follows:(10)f,g=∫R2fxgx¯dx,dx=dx1dx2with symmetric real scalar part(11)f,g=f,g0=12f,g+g,f=∫R2Scfxg¯xdx.In particular, for f=g, we obtain the L2(R2;H)-norm(12)f=f,f=∫R2fx2dx1/2.A quaternion module L2(R2;H) is then defined as(13)L2R2;H=f∣f:R2⟶H,f<∞.


### 2.2. Split Quaternion and Properties

In this section, we study some of the basic formulas of split quaternion (see [9]), which will be used to prove the fundamental results in the sequel.Definition 1.
For two quaternion square rootsμ,ν such that μ2=ν2=-1, one may express a quaternion q as(14)q=q-+q+,q±=12q±μqν.For the special case ofμ=ν, any quaternion q may be split up into commuting and anticommuting parts with respect to μ; that is,(15)μq-=q-μμq+=-q+μ.It easily seems that the commuting and anticommuting parts satisfy the interesting properties:(16)μ2=μ-2+μ+2=-1μ+μ-+μ-μ+=0.We learn from the above equation that(17)q±eμθ=e∓μθq±,where(18)cos⁡θ=q0q,sin⁡θ=q12+q22+q32q. In particular, taking μ=i and ν=j, (14) becomes(19)q=q++q-,q±=12q±iqj.The above gives(20)q±=q0±q3+iq1∓q21±k2=1±k2q0±q3+jq2∓q1.This leads to the following modulus identity:(21)q2=q-2+q+2.Furthermore, one can obtain (22)Scp+q¯-=0.


## 2.1. Quaternions

The quaternions, a generalization of complex numbers, are members of a noncommutative division algebra. The set of quaternions is denoted byH. Every element of H can be written in the following form: (1)H=q=q0+iq1+jq2+kq3;q0,q1,q2,q3∈R, with the units i, j, k, which obey the following: (2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=ijk=-1.For a quaternion q=q0+iq1+jq2+kq3∈H, q0 is simply called the scalar part of q denoted by Sc(q) and q=iq1+jq2+kq3 is called the vector part of q denoted by Vec(q).Letp,q∈H and p, q be their vector parts, respectively. Equation (2) yields the quaternionic multiplication qp as(3)qp=q0p0+q·p+q0p+p0q+q×p,where (4)q·p=-q1p1+q2p2+q3p3,q×p=iq2p3-q3p2+jq3p1-q1p3+kq1p2-q2p1. Analogously to the complex case, a quaternionic conjugation q¯ is given by(5)q¯=q0-iq1-jq2-kq3,which leads to the anti-involution; that is, (6)qp¯=p¯q¯. With the help of (5), we get the norm or modulus of q∈H as(7)q=qq¯=q02+q12+q22+q32.One can easily verify that(8)qp=qp,q+p≤q+p,∀p,q∈H.Using conjugate (5) and the modulus ofq, we can define the inverse of q∈H\{0} as (9)q-1=q¯q2, which shows that H is a normed division algebra.In quaternionic notation, we may define an inner product for quaternion-valued functionsf,g:R2→H as follows:(10)f,g=∫R2fxgx¯dx,dx=dx1dx2with symmetric real scalar part(11)f,g=f,g0=12f,g+g,f=∫R2Scfxg¯xdx.In particular, for f=g, we obtain the L2(R2;H)-norm(12)f=f,f=∫R2fx2dx1/2.A quaternion module L2(R2;H) is then defined as(13)L2R2;H=f∣f:R2⟶H,f<∞.


## 2.2. Split Quaternion and Properties

In this section, we study some of the basic formulas of split quaternion (see [9]), which will be used to prove the fundamental results in the sequel.Definition 1.
For two quaternion square rootsμ,ν such that μ2=ν2=-1, one may express a quaternion q as(14)q=q-+q+,q±=12q±μqν.For the special case ofμ=ν, any quaternion q may be split up into commuting and anticommuting parts with respect to μ; that is,(15)μq-=q-μμq+=-q+μ.It easily seems that the commuting and anticommuting parts satisfy the interesting properties:(16)μ2=μ-2+μ+2=-1μ+μ-+μ-μ+=0.We learn from the above equation that(17)q±eμθ=e∓μθq±,where(18)cos⁡θ=q0q,sin⁡θ=q12+q22+q32q. In particular, taking μ=i and ν=j, (14) becomes(19)q=q++q-,q±=12q±iqj.The above gives(20)q±=q0±q3+iq1∓q21±k2=1±k2q0±q3+jq2∓q1.This leads to the following modulus identity:(21)q2=q-2+q+2.Furthermore, one can obtain (22)Scp+q¯-=0.


## 3. Relationship between Quaternion Fourier Transform (QFT) and Quaternion Linear Canonical Transform (QLCT)

In this section, we introduce the QFT and its relationship to the QLCT. We begin by introducing different types of the QFT.Definition 2 (two-sided, right-sided, and left-sided QFTs).
The two-sided, right-sided, and left-sided quaternion Fourier transforms (QFTs) off∈L1(R2;H) are given by, respectively,(23)Fqfω=∫R2e-iω1x1fxe-jω2x2dx(24)Fqrfω=∫R2fxe-iω1x1e-jω2x2dx(25)Fqlfω=∫R2e-iω1x1e-jω2x2fxdx,where x=x1e1+x2e2, ω=ω1e1+ω2e2, and the quaternion exponential product e-iω1x1e-jω2x2 is the quaternion Fourier kernel. From (24), we get the partial right-sided QFT:(26)Fqr,ifω1,x2=∫R2fxe-iω1x1dx1Fqr,jfx1,ω2=∫R2fxe-jω2x2dx2.It is not difficult to check that the relationship between the two-sided QFT and the right-sided QFT takes the form(27)Fqfω=Fqrf0+if1ω+Fqrjf2+kf3-ω1,ω2.Definition 3 (left-sided, right-sided, and two-sided QLCTs).
Suppose thatA1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) are real matrix parameters satisfying det⁡(A1)=det⁡(A2)=1. The left-sided, right-sided, and two-sided QLCTs of a quaternion signal f∈L1(R2;H) are defined by, respectively,(28)LA1,A2l,Hfω=∫R2KA1x1,ω1KA2x2,ω2fxdx(29)LA1,A2r,Hfω=∫R2fxKA1x1,ω1KA2x2,ω2dx(30)LA1,A2Hfω=∫R2KA1x1,ω1fxKA2x2,ω2dx,where the kernel functions of the QLCT above are given by(31)KA1x1,ω1=12πb1ei/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2,forb1≠0d1eic1d1/2ω12,forb1=0,KA2x2,ω2=12πb2ej/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2,forb2≠0d2ejc2d2/2ω22,forb2=0.From the definition of the QLCT, we can easily see that whenb1b2=0 and b1=b2=0, the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work, we always assume b1b2≠0. As a special case, when A1=A2=(ai,bi,ci,di)=(0,1,-1,0), for i=1,2, the QLCT definition (30) will lead to the QFT definition; that is,(32)LA1,A2Hfω=∫R2e-iπ/42πe-iω1x1fxe-jω2x2e-jπ/42πdx=e-iπ/42πFqfωe-jπ/42π.The following lemma describes the general relationship between the two-sided QFT and the two-sided QLCT of 2D quaternion-valued signals.Lemma 4.
The QLCT of a signalf with matrix parameters A1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) can be seen as the QFT of the signal f in the following form:(33)LA1,A2Hfω=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1x12fxeja2/2b2x22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22.We introduce(34)hx=eia1/2b1x12e-iπ/42πb1fxe-jπ/42πb2eja2/2b2x22.This implies that (33) can be rewritten in the form(35)Fqhω1b1,ω2b2=e-id1/2b1ω12LA1,A2Hhωe-jd2/2b2ω22.Further, we have the following lemma which describes a relation between the right-sided QLCT and the two-sided QLCT of 2D quaternion-valued signals.Lemma 5.
Forf∈L1(R2;H), one has(36)LA1,A2r,Hfω=LA1,A2Hf0+if1ω+LA1∗,A2∗Hjf2+kf3-ω1,ω2,(37)LA1,A2Hfω=LA1,A2r,Hf0+if1ω+LA1∗,A2∗r,Hjf2+kf3-ω1,ω2,where the matrix parameters A1∗=(a1,-b1,c1,d1) and A2∗=(a2,-b2,c2,d2).The following lemma allows us to represent the right-sided QLCT to the single right-sided QFT of 2D quaternion-valued signals.Lemma 6.
Forf∈L1(R2;H), one has(38)LA1,A2r,Hfω=Fqr,jFqr,igxω1b1,x2eid1/2b1ω12eja2/2b2x22-π/42πb2ω1b1,ω2b2.Proof.
Proofs of Lemmas4, 5, and 6 are straightforward and are therefore omitted for brevity.


## 4. A Version of Uncertainty Principle Associated with QLCT

The classical uncertainty principle of harmonic analysis describes that a nontrivial function and its Fourier transform cannot be sharply localized simultaneously. In quantum mechanics, the uncertainty principle asserts that one cannot at the same time be certain of the position and of the velocity of an electron (or any particle). Let us now establish a version of the uncertainty principle associated with the QLCT. However, before proceeding the statement of this main result, we need to introduce a modified uncertainty principle for the QFT as follows (see [29] for more details).Theorem 7 (the two-sided QFT uncertainty principle).
LetS(R2;H) be the quaternion Schwartz space. If the quaternion-valued function f∈S(R2;H), then the following inequality holds:(39)∫R2x12fx2x22dx∫R2ω12Fqfω2ω22dω≥2π212∫R2fx2dx-∫R2x1x2∂fx∂x2¯∂fx∂x1dx2.It is shown that the quaternionic Gabor filters minimize the above uncertainty. Analogously, we get the uncertainty principle associated with the right-sided QFT in the following lemma.Lemma 8 (the right-sided QFT uncertainty principle).
Under the hypotheses of Theorem7, one has the following inequality:(40)∫R2x12fx2x22dx∫R2ω12Fqrf0+if1ω+Fqrjf2+kf3-ω1,ω22ω22dω≥2π212∫R2fx2dx-∫R2x1x2∂fx∂x2¯∂fx∂x1dx2.Theorem7 has been recently generalized in the context of the QLCT by the authors of [30]. Our generalization is given by the following theorem.Theorem 9 (the two-sided QLCT uncertainty principle).
Under the assumptions of Theorem7, one has(41)∫R2x12fx2x22dx∫R2ω12LA1,A2Hfω2ω22dω≥b1b2212∫R2fx2dx-∫R2x1x2∂f~x∂x2¯∂f~x∂x1dx2,where f~(x) is defined by(42)f~x=eia1/2b1x12fxeja2/2b2x22.It is worth noting here that if ∫R2x1x2∂f~x/∂x2¯∂f~x/∂x1dx=0, (41) can be reduced to(43)∫R2x12fx2x22dx∫R2ω12LA1,A2Hfω2ω22dω≥b1b224∫R2fx2dx2.Proof.
By replacingf by h defined by (34) on both sides of (39), we immediately get(44)∫R2x12hx2x22dx∫R2ω12Fqhω2ω22dω≥2π212∫R2hx2dx-∫R2x1x2∂hx∂x2¯∂hx∂x1dx2.Let ω=ω/b; then, (44) becomes(45)∫R2x12hx2x22dx∫R2ω12b12Fqhωb2ω22b22dωb≥2π212∫R2hx2dx-∫R2x1x2∂hx∂x2¯∂hx∂x1dx2.In view of (34), we obtain(46)∫R2x122π2b1b22fx2x22dx∫R2ω12b12Fqhωb2ω22b22dω≥12π2b1b2212∫R2fx2dx-∫R2x1x2∂eia1/2b1x12fxeja2/2b2x22∂x2¯∂eia1/2b1x12fxeja2/2b2x22∂x1dx2.This leads to(47)∫R2x12b1b24fx2x22dx∫R2ω12Fqhωb2ω22dω≥1b1b2212∫R2fx2dx-∫R2x1x2∂eia1/2b1x12fxeja2/2b2x22∂x2¯∂eia1/2b1x12fxeja2/2b2x22∂x1dx2.Inserting (35) into (47), we easily obtain (48)∫R2x12b1b24fx2x22dx∫R2ω12e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω222ω22dω≥1b1b2212∫R2fx2dx-∫R2x1x2∂eia1/2b1x12fxeja2/2b2x22∂x2¯∂eia1/2b1x12fxeja2/2b2x22∂x1dx2. The above identity can be further simplified to(49)∫R2x12fx2x22dx∫R2ω12LA1,A2Hfω2ω22dω≥b1b2212∫R2fx2dx-∫R2x1x2∂f~x∂x2¯∂f~x∂x1dx2, where f~(x) is defined in (42). Therefore, the proof is complete.The above theorem is also valid for the right-sided QLCT as described in the following lemma.Lemma 10 (the right-sided QLCT uncertainty principle).
Under the assumptions of Theorem7, one has(50)∫R2x12fx2x22dx∫R2ω12LA1,A2r,Hf0+if1ω+LA1∗,A2∗r,Hjf2+kf3-ω1,ω22ω22dω≥b1b2212∫R2fx2dx-∫R2x1x2∂f~x∂x2¯∂f~x∂x1dx2.Proof.
The proof follows directly from (37).Now, we define a module ofFq{f}(ω) as(51)Fqfωq=Fqf0ω2+Fqf1ω2+Fqf2ω2+Fqf3ω21/2, where(52)Fqfiω=∫R2e-iω1x1fixe-jω2x2dx,i=0,1,2,3. Furthermore, we obtain the Lp(R2;H)-norm(53)Fqfq,p=∫R2Fqfωqpdx1/p.It should be noticed that ifFq{fi},i=0,2,3, is real-valued function, then (53) reduces to(54)Fqfq,p=Fqfp,where(55)Fqfp=∫R2Fqfωpdx1/p.By Riesz’s interpolation theorem, one can get the Hausdorff-Young inequality related to the QFT (see [10]):(56)Fqfq,p′≤fpholds for 1≤p≤2 with 1/p+1/p′. This gives the following important theorem.Theorem 11 (Hausdorff-Young inequality).
If1≤p≤2 and letting p′ be such that 1/p+1/p′=1, then, for all f∈Lp(R2;H), it holds that(57)LA1,A2Hfq,p′≤b1b2-1/2+1/p′2πfp,where(58)LA1,A2Hfq,p′=∫R2LA1,A2Hfωqp′dω1/p′.Proof.
From the Hausdorff-Young inequality for the QFT, we have(59)∫R2Fqfωqp′dω1/p′≤∫R2fxpdx1/p. Based on the arguments used in the proof of Theorem 9, we immediately get(60)∫R2Fqhωqp′dω1/p′≤∫R2hxpdx1/p∫R21b1b2Fqhωbqp′dω1/p′≤∫R2e-iπ/42πb1eia1/2b1x12fxe-jπ/42πb2eja2/2b2x22pdx1/p1b1b21/p′∫R2Fqhωbqp′dω1/p′≤12πb1b21/2∫R2fxpdx1/p.Hence, (61)∫R2e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22qp′dω1/p′≤b1b2-1/2+1/p′2π∫R2fxpdx1/p.Or, equivalently,(62)∫R2LA1,A2Hfωqp′dω1/p′≤b1b2-1/2+1/p′2π∫R2fxpdx1/p. Thus, the theorem is completely proved.


## 5. Conclusion

In this paper, we derived a version of the uncertainty principle for the QLCT using a relation between the QFT and the QLCT. We presented Hausdorff-Young inequality associated with the QLCT. This inequality is very useful for establishing a variation on Heisenberg’s uncertainty principle related to the QLCT.



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