Xiaoxiao HU, Dong CHENG, Kit Ian KOU

1The First Affiliated Hospital of Wenzhou Medical University, Wenzhou Medical University, Wenzhou 325000, China

2Research Center for Mathematics and Mathematics Education,Beijing Normal University (Zhuhai), Zhuhai 519087, China

3Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China

Abstract:The main purpose of this paper is to study different types of sampling formulas of quaternionic functions,which are bandlimited under various quaternion Fourier and linear canonical transforms. We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms. In addition, the relationships among different types of sampling formulas under various transforms are discussed. First, if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin, then the sampling formulas under various quaternion Fourier transforms are identical. If this rectangle is not symmetric about the origin, then the sampling formulas under various quaternion Fourier transforms are different from each other. Second, using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform, we derive sampling formulas under various quaternion linear canonical transforms.Third, truncation errors of these sampling formulas are estimated. Finally, some simulations are provided to show how the sampling formulas can be used in applications.

Key words: Quaternion Fourier transforms; Quaternion linear canonical transforms; Sampling theorem;Quaternion partial and total Hilbert transforms; Generalized quaternion partial and total Hilbert transforms; Truncation errors

1 Introduction

Sampling theory is one of the most important mathematical techniques used in communication engineering,information theory, signal analysis,image processing,and so on(Zayed,1993;Cheng and Kou,2019,2020). Sampling theories of multi-dimensional real signals in RNappeared in Zayed (1993). Sampling theories of high-dimensional signals in several complex variable settings(Kou and Qian,2005a)and the Clifford analysis setting (Kou and Qian, 2005b)were obtained. Recently, the quaternions, which are hyper-complex numbers, have been proved to be an effective tool in quite a few applications of multi-dimensional signal processing analysis (Jiang et al., 2016; Zou et al., 2016; Hu and Kou, 2018;Bahia and Sacchi,2020;Alon and Paran,2021),and account for correlated nature of the signal components in a natural way. Meanwhile, numerous novel tools in multi-dimensional signal and image processing have been developed using the quaternion modeling technique. The representative tools are quaternion Fourier transforms (QFTs), quaternion fractional Fourier transforms(QFrFTs),and quaternion linear canonical transforms(QLCTs)(Ell et al.,2014;Kou and Morais, 2014;Chen et al., 2015;Kou et al., 2017; Lian, 2021). Some important theorems have been studied, such as the inverse theorems associated with QFTs and QLCTs(Hu and Kou,2017)and convolution theorems associated with QFTs and QLCTs(Pei et al.,2001;Hitzer E, 2017).

The sampling expansions associated with the right-sided QFT were studied in Cheng and Kou(2018),but the main shortcoming of this approach is that only the right-sided QFT case was considered.In this study, we study not only the sampling formula associated with the right-sided QFT, but also the sampling formulas associated with the left- and two-sided QFTs. Therefore,sampling formulas associated with various QLCTs are obtained.

There are five main points in this study: (1) to study the quaternionic function sampling formulas bandlimited (BL) to a rectangle that is symmetric about the origin under various QFTs; (2) to study the quaternionic function sampling formulas BL to a rectangle that is not symmetric about the origin under various QFTs; (3) to explore not only the sampling formulas using samples of themselves, but also samples of the partial derivatives and quaternion partial and total Hilbert transforms; (4) to obtain the sampling formulas associated with various QLCTs using relationships between QFTs and QLCTs; (5) to estimate truncation errors of these sampling formulas.

2 Preliminaries of quaternion, QFTs,and QLCTs

In this section,we are devoted to the exposition of basic preliminary materials that are used extensively throughout this paper.

Let H denote the Hamiltonian skew field of quaternions,which has been proved to provide a natural framework for a unified treatment of three-and four-dimensional signals.

A quaternionic number takes the form of

whereq0-q3∈R,and i,j,k are orthogonal imaginary parts obeying the following rules: i2= j2=k2=ijk=-1. In this way,the quaternionic algebra can be regarded as a non-commutative extension of complex numbers C. Whenq0=0,qbecomes a pure quaternion. Letμ:=iμ1+jμ2+kμ3denote the unit pure quaternion such thatμ2=-1.Let Hμbe the field spanned by{1,μ}, which is the sub-field of H.That is, Hμ:={q|q=q0+μqμ,q0,qμ∈R, μ2=-1}. The quaternion conjugate of a quaternionqis defined byq*=q0-iq1-jq2-kq3,which implies the modulus ofq∈H defined asFrom Eq. (1), it follows that a quaternionic functionf: R2→H can be expressed asf(x,y)=f0(x,y)+if1(x,y)+jf2(x,y)+kf3(x,y),wherefn∈R,n=0,1,2,3.

LetLp(R2,H)(integerp ≥1)be the linear space of all quaternionic functions in R2,whose quaternion modulus isLp(R2,H) :={f|f: R2→H,‖f‖p

Based on the quaternion concept,various QFTs(Ell et al.,2014)and QLCTs(Kou et al.,2013)have been introduced. Forf∈L1(R2,H),the two-sided QFT is as follows:

The right-sided QFT is as follows:

The left-sided QFT is as follows:

QLCTs are the generalizations of QFTs. Letbe real matrix parameters with unit determinant,i.e., det(AAAi)=aidi-cibi=1,fori=1,2.

The two-sided QLCT is as follows:

The right-sided QLCT is as follows:

The left-sided QLCT is as follows:

Herein kernelsandof QLCTs are given byandNote that whenbi=0 (i=1,2), QLCT of a function is essentially a chirp multiplication and is of no particular interest to us. Hence, without loss of generality, we setbi /= 0 (i= 1,2) throughout the paper. It is significant to note that QLCT converts to its special cases when we take different matricesAAAi(i= 1,2).For example, when, QLCT is reduced to QFT, wheree-jπ/4. IfandAAA2=, QLCT becomes QFrFT multiplied by the fixed phase factors e-iα/2and e-jβ/2.QFTs are invertible, and for their inversion theorems, readers can refer to Hu and Kou(2017).

Lemma 1(Hu and Kou, 2017) Suppose that the quaternionic functionf(x,y) satisfies one of the following conditions:

(1)fand the corresponding QFT both belong toL1(R2,H).

(2)f∈L2(R2,H),if supp(Fn[f])?[σ1,σ2]×[σ3,σ4],wheren= R, L, or T, andσi(i= 1,2,3,4)are real constants,and can be chosen at will.

Then

whereD= R2for condition (1) andD= [σ1,σ2]×[σ3,σ4] for condition(2).

Lemma 2(Quaternion Cauchy-Schwarz inequality) (Kou and Qian, 2005a) Iff,g∈L2(R2,H),then H¨older’s inequality yields

Lemma 3(Parseval equality)(Hitzer EMS, 2007)Setf∈L2(R2,H). Then we have

3 Sampling theorems for bandlimited(BL)quaternionic functions

3.1 Sampling theorems for BL quaternionic functions in the QFT sense

Letfbe a quaternionic function defined on T2,where T=[0,2π]and T2is the Cartesian product of T×T. The spaceLp(T2,H) consists of all quaternionic functions such thatT2|f(x,y)|pdxdy ＜∞.For a functionf∈L2(T2,H),we can define the right-sided quaternion Fourier coefficients ascn,m=. Then the right-sided quaternion Fourier series can be written asf(x,y)～

Lemma 4(Bessel’s inequality) Letf∈L2(T2,H).Then Bessel’s inequality holds:

The proof of Lemma 4 is available in the supplementary materials.

Due to Bessel’s inequality, the right-sided quaternion Fourier serieswill converge in the sense ofL2.Indeed, it can be proved that the system{einxejmy|(n,m)∈Z2}is complete inL2(T2,H).Then with an argument similar to that in Pan(2000),we have the following lemma:

The Parseval equality holds as

Definition 1f(x,y)is said to be a BL signal(function)of[-σ1,σ1]×[-σ2,σ2] in the right-sided QFT sense, i.e.,FR[f](v,u)=0, for|v|＞σ1or|u|＞σ2.

In the following, for simplicity, we use the following abbreviated notations: (σ1,σ2):=[-σ1,σ1]×[-σ2,σ2]. To formulate our sampling formulas, we need a very important result:

Theorem 1Supposef(x,y) =f0(x,y) +if1(x,y)+jf2(x,y)+kf3(x,y),f∈L2∪L1(R2,H).

Then the following four statements are equivalent:

ProofIt is sufficient to prove the equivalence between the first and second statements becauseFR[fn](v,u) =FL[fn](v,u) =FT[fn](v,u), wherefnare real functions,n=0,1,2,3.

(2) =?(1). Asf∈L2∪L1(R2,H),thenfn∈L2∪L1(R2,R)andFR[f](v,u)=FR[f0](v,u)+iFR[f1](v,u)+jFR[f2](v,u)+kFR[f3](v,u), so the second statement can imply the first statement.

(1)=?(2). Because

it follows that

Taking the right-sided QFT on both sides, we obtain

Hence, the first statement implies the second statement.

Remark(1) We can see from Theorem 1 that iff∈L2∪L1(R2,H) is BL to (σ1,σ2) in the rightsided QFT sense, then it is also BL to (σ1,σ2) in the two- and left-sided QFT senses, and vice versa.In this case,fis said to be BL to (σ1,σ2) in the QFT sense. (2)Iff(x,y)is BL to a rectangle that is not symmetric about the origin, then the sampling formulas offunder the various types of QFTs will be different. We will show them in Theorems 3—5.

Theorem 2(Sampling theorem for QFTs) Suppose thatf∈L2(R2,H)is BL to(σ1,σ2)in the QFT sense. Thenf(x,y) can be reconstructed from its sampled values at the pointsvia the following formula:

whereand.The series converges in theL2norm, and it is absolutely and uniformly convergent on any compact subset of R2.

ProofFrom Hitzer EMS (2007), we haveFR[f](v,u)∈L2([-σ1,σ1]×[-σ2,σ2],H). Therefore,by Lemma 5,we have

where

From Lemma 1, we have

which, in view of Eq. (6),implies Eq. (5). The convergence in Eq. (7) is understood to be in the sense ofL2. However, Eq. (5) is readily seen to converge absolutely and uniformly on any compact subset of R2when we apply the quaternion Cauchy-Schwarz inequality (3)and Eqs. (4)and(5).

Iff(x,y) is BL to a rectangle that is not symmetric about the origin, then the sampling formula is as follows:

Theorem 3(Sampling theorem for two-sided QFT) Iff∈L2(R2,H) andFT[f](v,u) = 0,for|v-v0|＞σ1or|u-u0|＞σ2,we have

The series converges in theL2norm, and it is absolutely and uniformly convergent on any compact subset of R2.

ProofBecauseFT[f](v,u) = 0,for|v -v0|＞σ1or|u-u0|＞σ2,then e-iv0xf(x,y)e-ju0yis BL to (σ1,σ2).Hence from Theorem 2, by substituting e-iv0xf(x,y)e-ju0yforf(x,y)in sampling series (5),we obtain sampling formula(8).

Theorem 4(Sampling theorem for right-sided QFT) Iff∈L2(R2,H) andFR[f](v,u) = 0,for|v-v0|＞σ1or|u-u0|＞σ2,we have

The series converges in theL2norm. Moreover, it converges absolutely and uniformly on any compact subset of R2.

ProofFrom the assumption off,we have

For|u-u0|＞σ2, we have

Applyingto both sides of Eq. (11)at pointxn, we have

By substitutingf(xn,y)e-ixnv0in the above equation into Eq. (10), we obtain Eq. (9).

Theorem 5(Sampling theorem for left-sided QFT) Iff∈L2(R2,H) andFL[f](v,u) = 0,for|v-v0|＞σ1or|u-u0|＞σ2,we have

The series converges in theL2norm. Moreover, it converges absolutely and uniformly on any compact subset of R2.

ProofUsing an argument similar to the proof of Theorem 4, we can easily carry out the proof of this theorem.

Next, we will introduce the quaternion partial and total Hilbert transforms (Bulow and Sommer,2001;Kou et al.,2017)associated with QFTs,and derive the sampling formula using samples of its quaternion partial and total Hilbert transforms. In addition, the sampling formula using the samples of its partial derivatives is derived. Then, the sampling rate can be reduced by taking multiple types of samples simultaneously.

Definition 2Letf∈L2∪L1(R2,H). Then the quaternion partial Hilbert transformH1offalong thexandyaxes and the quaternion total Hilbert transformH2along thexandyaxes offare given by

Let

Using the analogous definition of the quaternion analytic signal given in Bulow and Sommer (2001)and Hahn and Snopek (2005),we have the following quaternion analytic signal associated withFR:

Lemma 6Letf∈L2(R2,R). Then we have

The quaternion analytic signal is defined by

Moreover,

Theorem 6Iff∈L2(R2,H)andFR[f](v,u)=0,for|v|＞σ1or|u|＞σ2,we have

ProofIff∈L2(R2,H), thenfk∈L2(R2,R),k=0,1,2,3. It follows that the quaternion analytic signaltakes the form of

From Lemma 6, we haveFR[fqk](v,u) = (1 +sgn(v))(1 + sgn(u))FR[fk](v,u); that is to say,fkqis BL to [0,σ1]×[0,σ2]. It follows from Theorem 4 that

Noting thatfk(x,y) is the real part offqk(x,y),with some straightforward calculations, we can derive the sampling formula offkas

Becausef=f0+if1+jf2+kf3,then sampling formula(14) holds forf.

From Theorems 1 and 6, we have the following corollary:

Corollary 1Iff∈L2(R2,H),thenf(x,y) can be reconstructed from the samples of its quaternion partial and toltal Hilbert transforms, if one of the following conditions holds:

(3)FL[f](v,u)=0,for|v|＞σ1or|u|＞σ2.

In the following, we derive the sampling formula involving the samples of the original function and its partial derivatives. Some lemmas are needed first:

Lemma 7(Hitzer EMS, 2007) Iff∈L2∩Cm+n(R2,H) andwe have

wheren,m∈Z.

Lemma 8(Marvasti,2001)

whereμ=iμ1+jμ2+kμ3is the unit pure quaternion such thatμ2=-1,for special cases ofμ=i, j, or k.

Here,∈k(v,t) =∈k(v -σ,t),forv∈(0,σ) andk= 1,2. Both functions may be expanded in theirσ-periodic boundedly converging Fourier series on(-σ,σ)/{0};that is to say,

Theorem 7Iff∈L2∩C2(R2,H) is BL to(σ1,σ2),andthen the following sampling formula holds:

ProofFrom the assumption off, we can recoverffrom its QFT domain as follows:

From Lemma 8, assume eivx=∈i1(v,x) +iv∈i2(v,x) and ejuy=∈j1(u,y)+j∈j2(u,y).After substituting these equations into Eq.(18), we obtain

Using Lemmas 7 and 8, with straightforward calculations,we have sampling formula(17).

3.2 Sampling theorems for BL quaternion functions in the QLCT sense

f∈L2(R2,H) is said to be a BL signal (function) to [-σ1,σ1]×[-σ2,σ2] (short for (σ1,σ2)) in the two-sided QLCT sense, if[f](v,u) = 0,for|v|＞σ1or|u|＞σ2.

Theorem 8Supposef∈L2(R2,H) andfor|v| ＞ σ1or|u| ＞ σ2.Assumeg= eia1x2/(2b1)f(x,y)eja2y2/(2b2). ThenFT[g](v,u)=0,for

ProofFrom the definition of the two-sided QLCT,we obtain

Theorem 9Iff∈L2(R2,H)is BL to(σ1,σ2)in the two-sided QLCT sense, then the following sampling formula forfholds:

whereandThe series converges in theL2norm. Moreover,it converges absolutely and uniformly on any compact subset of R2.

ProofTheorem 8 implies thatis BL toin the QFT sense. So, applying Theorem 2 to,we obtain the final results.

Similarly, we obtain the following sampling series offinvolving the function and its partial derivatives:

Theorem 10Iff∈L2∩C2(R2,H) is BL to(σ1,σ2)in the two-sided QLCT sense,andthen the following sampling series forfholds:

In the following, to derive the sampling series forfusing the samples of its Hilbert transform associated with QLCT, we introduce the generalized partial and total Hilbert transforms for the two-sided QLCT in Kou et al. (2017).

Definition 3Letf∈L2∪L1(R2,H). Thenfalong thexandyaxes of the generalized quaternion partial Hilbert transformsandand along thexandyaxes of the generalized quaternion total Hilbert transformare given by

Lemma 9Iff∈L2(R2,H),assumeg(x,y) =Then we have

ProofWe give only the proof of the first relationship; the other two relationships can be proved in a similar manner. By definition, we have

which completes the proof.

Theorem 11Iff∈L2(R2,H)is BL to(σ1,σ2)in the two-sided QLCT sense, then the following sampling series forfholds:

ProofFrom Theorem 8,g(x,y) =is BL toin the QFT sense. Applying sampling series (16) tog(x,y), we obtain

Substituting the relationships in Lemma 9 andin the above sampling series ofg, we obtain sampling series(19).

f∈L2(R2,H)is a BL function to(σ1,σ2)in the two-sided QLCT sense, which does not mean thatfis a BL function to(σ1,σ2)in the right-or left-sided QLCT. It is natural to ask, iffis a BL function to(σ1,σ2)in the right-sided QLCT sense,how could we reconstruct it from its sampled values at the point(n,m)∈Z2. This problem is carried out by the following theorem:

Theorem 12Iff∈L2(R2,H)is BL to(σ1,σ2)in the right-sided QLCT sense, i.e.,

then the following sampling series forfholds:

where0,i= 1,2). The series converges in theL2norm.Moreover, it converges absolutely and uniformly on any compact subset of R2.

ProofBecause

then we have

Becausef∈L2(R）2,H) andusing the relationship between kernels of QFT and QLCT, we obtain

Then, from the assumption ofis aBL function with respect toyin the QFT sense. Therefore,we can recovergv(y) from the samples at pointtmvia the following formula:

Multiplying both sides of the above equality by, we have

Taking the inverse quaternion Fourier transformon both sides at pointand using the fact that0,i=1,2),we obtain

Substituting it into Eq.(21),we obtain sampling series(20).

4 Error analysis

4.1 Truncation errors associated with QFT

Truncation error occurs naturally in applications, because only a finite number of samples are given in practice. Forf∈L2(R2,H) which is BL to(σ1,σ2) in the QFT sense, let R1N,M,R2N,M,and R3N,Mdenote the truncation errors off(x,y):

whereh1:=π/σ1andh2:=π/σ2,

Lemma 10(Splettst?sser et al., 1981)

whereσ ＞0.

Lemma 11(Jagerman,1966)

where|x|＜Nh1.

Theorem 13Forf∈L2(R2,H) which is BL to (σ1,σ2) in the QFT sense, let R1N,Mbe defined by Eq. (22),where|x|＜Nh1,|y|＜Mh2,N ≥1,M ≥1,

Then

where

The proof of Theorem 13 is available in the supplementary materials.

Further estimates of|R1N,M(x,y)|may be obtained by estimating quantitiesKN,LM, andJN,M.

An immediate consequence of R2N,M(x,y)is the following theorem which can be obtained by some arguments similar to Theorem 13:

Theorem 14Let R2N,Mbe defined by Eq.(24),|x|＜Nh1,|y|＜Mh2,N ≥1,M ≥1,

wherei= 2,3,4,5,f2=f,f3=Hx[f],f4=Hy[f],andf5=Hxy[f].Then

where

Herein,j=2,3,4,5.

Before estimating R3N,M(x,y), there is a need to point out that BL quaternionic functionf(z1,z2)has been proved to be a quaternion holomorphic function (Hu and Kou, 2018), which is holomorphic in two variables(z1,z2):

wherez1∈Hi,Hi={z1|z1=x1+ix2,x1,x2∈R},,and.Then truncation error R3N,M(x,y) is given by

HereCi, shown in Fig. 1, is a simple closed contour enclosing both the pointz1=xand the zero pointz1=nh1for all integersK1(x)-N ≤n ≤K1(x)+N.Cjshown in Fig. 1 is also a simple closed contour enclosing both the pointz2=yand the zero pointz2=mh2for all integersK2(y)-M ≤m ≤K2(y)+M.

Fig. 1 The simple closed contours Ci (a) and Cj (b)

Fig. 2 Reconstructed images for Lena, flower, and bird by Algorithm 1

Fig. 3 Reconstructed images for house, pepper, and horse by Algorithm 1

The proof of Theorem 14 is available in the supplementary materials.

Theorem 15If quaternionic functionf(x,y)is BL to (r1σ1,r2σ2) in the QFT sense, where 0≤ri ＜1 (i= 1,2) and|f(x,y)|≤C,for (x,y)∈R2andC ＞0,then an upper bound for the truncation error R3N,Mat point(x,y) is given by

Before giving the proof of Theorem 15,we need the following lemma:

Lemma 12Iff∈L2(R2,H)is BL to(σ1,σ2),then

The proof of Lemma 12 is available in the supplementary materials.

From this lemma, using the technique in Yao and Thomas(1966),we obtain Theorem 15.

4.2 Truncation errors associated with QLCT

Iff(x,y) is BL to (σ1,σ2) in the twosided QLCT sense, Lemma 8 implies that eia1x2/(2b1)f(x,y)eja2y2/(2b2)is BL toin the two-sided QFT sense. So, we have the following truncation errors in the QLCT sense from Theorems 13—15. Let,anddenote the truncation errors off(x,y) as follows (Assume:=b1π/σ1and=b2π/σ2):

Corollary 2Suppose thatf∈L2(R2,H)is BL to(σ1,σ2) in the two-sided QLCT sense. LetN,Mbe defined by Eq. (26),where|x| ＜ N,|y| ＜M,N ≥1,M ≥1,

Then

where

Corollary 3Letbe defined by Eq. (27),

whereand[f].Then we have

where

Herein,j=2,3,4,5.

Corollary 4If quaternionic functionf(x,y)is BL to (r1σ1/b1,r2σ2/b2) in the QFT sense, where 0≤ri ＜1 (i= 1,2) and|f(x,y)|≤C,for (x,y)∈R2andC ＞0, then an upper bound for the truncation error ?R3N,Mat point(x,y)is given by

5 Examples

In this section, we use mainly sampling formula (8) in Theorem 3 to produce a high-resolution image from its corresponding low-resolution version.The quality of the high-resolution image is measured by the structural similarity index measure (SSIM)and feature similarity index measure (FSIM) in Algorithm 1.

By Figs. 2 and 3, our sampling formula can recover the color image from low to high resolution.The quantitative measurements in Table 1 show the effectiveness of the proposed sampling formula.

Algorithm 1 Image reconstruction 1: Input the test color image f(t1,t2) and convert the color image into the quaternion form.2: The test image is downsampled by factor 2.3: Generate a high-resolution (HR) image from the downsampled image by Eq. (8).4: Compute the SSIM and FSIM to evaluate the quality of the generated HR image.

Table 1 SSIM and FSIM values of the reconstructed images

6 Conclusions

First, by Lemma 1, if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin in the right-sided QFTs, then it is also bandlimited to this rectangle in the left- and twosided QFTs,and vice versa. Therefore,if the quaternionic function is bandlimited to this rectangle,then the sampling formula associated with various QFTs is identical. However, if the quaternionic function is bandlimited to a rectangle that is not symmetric about the origin, then the sampling formulas associated with various QFTs are different. Second, we obtained not only the sampling formulas using the samples, but also the sampling series using samples of the partial derivatives and quaternion partial and total Hilbert transforms. Third,the sampling formulas associated with various QLCTs were obtained by the relationships of QFTs and QLCTs. Fourth, the truncation errors of those sampling formulas were derived. At last,by Algorithm 1,the sampling formula was applied to color image reconstruction.

In the future, we will apply the sampling series to color images, and multi-dimensional signals will be explored.

Contributors

Xiaoxiao HU designed the research. Xiaoxiao HU and Dong CHENG processed the data. Xiaoxiao HU drafted the paper. Kit Ian KOU helped organize the paper. Xiaoxiao HU and Dong CHENG revised and finalized the paper.

Compliance with ethics guidelines

Xiaoxiao HU, Dong CHENG, and Kit Ian KOU declare that they have no conflict of interest.

List of electronic supplementary materials

Proof S1 Proof of Lemma 4

Proof S2 Proof of Theorem 13

Proof S3 Proof of Theorem 14

Proof S4 Proof of Lemma 12