# A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

Authors: Mawardi Bahri; Ryuichi Ashino
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.


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## Body

## 1. Introduction

As it is known, the classical wavelet transform (WT) is a very useful mathematical tool. It has been discussed extensively in the literature and has been proven to be powerful and useful in the communication theory, quantum mechanics, and many other fields [1–4]. Of great interest is the study of the quaternion wavelet transform, which can be considered as a generalization of the WT using quaternion algebra. Some research papers on the continuous and discrete quaternion wavelet transforms have been published. In [5–7], the authors constructed the continuous quaternion wavelet transform (CQWT) using the quaternionic affine group and similitude group, respectively. Several fundamental properties of this extended wavelet transform, which correspond to classical continuous wavelet transform properties, were also investigated. Further, in regard to a numerical concept of the quaternion wavelet transforms, Bayro-Corrochano [8] developed the discrete quaternion wavelet transform and applied it for optical flow estimation. In [9, 10], the authors studied the discrete reduced biquaternion wavelet transform and applied it to multiscale texture classification. Another approach of the CQWT based on a natural convolution of quaternion-valued functions was recently proposed by Akila and Roopkumar [11, 12]. The essential part in the study of the quaternion wavelet transform, as usual, is to establish its Heisenberg type uncertainty principle. It plays an important role in quaternionic signal processing. Based on the Heisenberg type uncertainty principle for the quaternion Fourier transform (QFT) [13, 14], the authors in [7] proposed a component-wise uncertainty principle associated with the CQWT.Motivated by the authors in [15–17], in the present paper, we propose the directional uncertainty principle related to the CQWT and then apply this uncertainty to obtain a variation on the Heisenberg type uncertainty principle and the logarithmic uncertainty principle in the context of the CQWT. The uncertainty principle describes the relation between the QFT of a quaternion function and its CQWT. These principles are more general forms of Heisenberg’s uncertainty principle related to the CQWT [7]. To achieve the results, our first step is to derive the directional uncertainty principle for the QFT using the component-wise QFT uncertainty principle. Due to this principle, we can easily derive directional QFT uncertainty principle using a representation of polar coordinate from the ones proposed in [18]. We then study an important theorem which describes interactions between the CQWT and QFT in frequency domain. Applying the cyclic multiplication of quaternion, we obtain some useful properties of the CQWT. Based on the relationship between the extended Heisenberg uncertainty principle and properties related to the CQWT, we finally establish the logarithmic uncertainty principles associated with the CQWT.


## 2. Preliminaries

The concept of the quaternion algebra [19, 20] was introduced by Sir Hamilton in 1842 and is denoted by H in his honor. It is an extension of the complex numbers to a four-dimensional (4-D) algebra. Every element of H is a linear combination of a real scalar and three imaginary units i,  j, and k with real coefficients,(1)H=q=q0+iqi+jqj+kqk∣q0,qi,qj,qk∈R,which obey Hamilton’s multiplication rules(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 called thescalar (orreal) part of q denoted by Sc⁡(q) and iq1+jq2+kq3 is called thevector (orpure) part of q. The vector part of q is conventionally denoted by q or Vec⁡(q)=iq1+jq2+kq3.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.The conjugate q¯ of the quaternion q is the quaternion given by(5)q¯=q0-iq1-jq2-kq3,q0,q1,q2,q3∈R.It is an anti-involution; that is, (6)qp¯=p¯q¯.From (5), we obtain the norm or modulus of q∈H defined as(7)q=qq¯=q02+q12+q22+q32.It is not difficult to see that(8)Sc⁡q≤q,q≤q.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.Now we notice that(10)Sc⁡pq¯=12pq¯+qp¯=q0p0+q·p.These will lead to the cyclic multiplication; that is,(11)Sc⁡pqr=Sc⁡rqp=Sc⁡qpr,∀p,q,r∈H.Any quaternionq may be split up into(12)q=q++q-,q±=12q±iqj.The above gives(13)q±=q0±q3+iq1∓q21±k2=1±k2q0±q3+jq2∓q1.This leads to the following modulus identity:(14)q2=q-2+q+2.It is convenient to introduce an inner product for two quaternion functionsf,g:R2→H as follows:(15)f,g=∫R2fxgx¯dx,where the overline indicates the quaternion conjugation of the function. In particular, for f=g, we obtain the Lp(R2;H)-norm(16)fp=∫R2fxpdx1/p,1≤p≤2,where(17)f=f02x+f12x+f22x+f32x.As a consequence of the inner product (15), we obtain thequaternion Cauchy-Schwarz inequality(18)∫R2f¯gdx2≤∫R2f2dx∫R2g2dx,∀f,g∈L2R2;H.Definition 1.
A coupleα=(α1,α2) of nonnegative integers is called a multi-index. One denotes(19)α=α1+α2,α!=α1!α2!,
and, forx∈R2,(20)xα=x1α1x2α2.
Derivatives are conveniently expressed by multi-indices:(21)∂α=∂α∂x1α1∂x2α2.
Next, we obtain the Schwartz space as (compared to [21])(22)SR2;H=f∈C∞R2,H:supx∈R2⁡1+xk∂αfx<∞,where C∞(R2,H) is the set of smooth functions from R2 to H. Elements in the dual space S′(R2;H) of S(R2;H) are called tempered distribution.


## 3. Quaternion Fourier Transform and Its Heisenberg Uncertainty Principle

### 3.1. QFT and Properties

In the following, we introduce the (right-sided) QFT and some of its fundamental properties such as Riemann-Lebesgue lemma and continuity.Definition 2 (right-sided QFT).
The (right-sided) quaternion Fourier transform (QFT) off∈L1(R2;H) is the transform FQ{f}: R2→H given by(23)FQfω=f^ω=∫R2fxe-iω1x1e-jω2x2dx,dx=dx1dx2,where x=x1e1+x2e2,ω=ω1e1+ω2e2, and the quaternion exponential product e-iω1x1e-jω2x2 is the quaternion Fourier kernel.Theorem 3 (inverse QFT).
Suppose thatf∈L1(R2;H) and FQ{f}∈L1(R2;H). Then, the QFT of f is an invertible transform and its inverse is given by(24)FQ-1FQfx=fx=12π2∫R2FQfωejω2x2eiω1x1dω,where the quaternion exponential product ejω2x2eiω1x1 is called the inverse (right-sided) quaternion Fourier kernel.SinceS(R2;H)⊂L1(R2;H), the definition of QFT (23) may be extended to the Schwartz space. It is important to note that FQ{f} is not necessary in L1(R2;H) even if f is in L1(R2;H), so in general FQ{f} might not be well defined. However, the QFT of a Schwartz quaternion function is also in the Schwartz space.Applying (23), we have(25)FQfω=∫R2f0x+if1x+jf2x+kf3xe-iω1x1e-jω2x2dx=FQf0ω+iFQf1ω+jFQf2ω+kFQf3ω.Now, we define a module of FQ{f}(ω) as(26)FQfωQ=FQf0ω2+FQf1ω2+FQf2ω2+FQf3ω21/p.Furthermore, we obtain the Lp(R2;H)-norm(27)FQfQ,p=∫R2FQfωQpdω1/p.Remark 4.
It is worth noting here that ifFQ{fi}, i=0,2,3, is real-valued, (26) can be written in the form(28)FQfQ,p=FQfωp,where(29)FQfωQ=FQf0ω2+FQf1ω2+FQf2ω2+FQf3ω21/p.Some important properties of the QFT are stated in the following lemmas.Lemma 5 (QFT Plancherel).
Iff∈L1(R2;H)∩L2(R2;H), then(30)12π2∫R2FQfω2dω=∫R2fx2dx.Moreover,(31)12π2∫R2FQfωQ2dω=∫R2fx2dx.Proof.
We prove expression (31) of Lemma 5. Using (26), we immediately get (32)12π2∫R2FQfωQ2dω=12π2∫R2FQf0ω2+FQf1ω2+FQf2ω2+FQf3ω2dω=12π2∫R2FQf0ω2dω+∫R2FQf1ω2dω+∫R2FQf2ω2dω+∫R2FQf3ω2dω.Applying (30) into the right-hand side of the above identity gives (33)12π2∫R2FQfωQ2dω=∫R2f0x2dx+∫R2f1x2dx+∫R2f2x2dx+∫R2f3x2dx.Since fi(x), i=0,1,2,3, is real-valued, the above equation can be written in the form (34)12π2∫R2FQfωQ2dω=∫R2f02x+f12x+f22x+f32xdx,which completes the proof of the theorem.Remark 6.
Equation (30) shows that the QFT is a bounded linear operator on L1(R2;H)∩L2(R2;H). Hence, using standard density arguments, one may extend the QFT in a unique way to the whole of L2(R2;H).Lemma 7 (see [14]).
Iff∈L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then one has for every n∈N(35)FQ∂n∂x1nfω=ω1nFQfiω,(36)FQ∂n∂x2nfω=FQfωjω2n.By Riesz’s interpolation theorem, we get that the Hausdorff-Young inequality (see [15])(37)FQfQ,p′≤fpholds for 1≤p≤2 with 1/p+1/p′=1. Using inversion formula of the QFT, (37) can be rewritten in the form(38)fp′≤FQfQ,p.The following theorem is an extension of the Riemann-Lebesgue lemma in the QFT domain.Theorem 8 (Riemann-Lebesgue lemma of QFT).
For a function inf∈L1(R2;H), one has that(39)limω1→∞⁡FQfω=0,limω2→∞⁡FQfω=0.Proof.
Notice first that(40)e-iω1x1=-e-iω1x1+π/ω1,e-jω2x2=-e-jω2x2+π/ω2.Applying (40) gives (41)FQfω=∫R2fxe-iω1x1e-jω2x2dx=-∫R2fxe-iω1x1+π/ω1e-jω2x2dx.Representing FQ{f}=(1/2)FQ{f}+FQ{f} and changing variable x1+π/ω1=t1 in the above identity, we immediately obtain(42)FQfω=∫R2ft1-πω1,t2e-iω1t1e-jω2t2dt=12∫R2ft1,t2e-iω1t1e-jω2t2dt-∫R2ft1-πω1,t2e-iω1t1e-jω2t2dt=12∫R2ft1,t2-ft1-πω1,t2e-iω1t1e-jω2t2dt.This means that (43)limω1→∞⁡FQfω≤12limω1→∞∫R2ft1,t2-ft1-πω1,t2dt=0.Analogously, it can be shown that (44)limω2→∞FQfω=0.The proof is complete.Theorem 9 (continuity).
Iff∈L1(R2;H), then the quaternion Fourier transform FQ{f}(ω) is continuous on R2. Moreover,(45)fCR2;H=maxx∈R2fx≤f1,where C(R2;H) is the space of continuous quaternion functions from R2 to H.Proof.
From the QFT definition (23), we readily see that(46)FQfω+h-FQfω=∫R2fxe-iω1+h1x1e-jω2+h2x2dx-∫R2fxe-iω1x1e-jω2x2dx=∫R2fxe-iω1x1e-ih1x1e-jω2x2e-jh2x2dx-∫R2fxe-iω1x1e-jω2x2dx=∫R2fxe-iω1x1e-ih1x1e-jω2x2e-jh2x2-e-iω1x1e-jω2x2dx≤∫R2fxe-iω1x1e-ih1x1e-jh2x2e-jω2x2-e-iω1x1e-jω2x2dx=∫R2fxe-iω1x1e-ih1x1e-jh2x2-1e-jω2x2dx=∫R2fxe-ih1x1e-jh2x2-1dx.Using the triangle inequality for quaternions, we easily get (47)e-ih1x1e-jh2x2-1≤e-ih1x1e-jh2x2+1=2.This means that we have(48)FQfω+h-FQfω≤2∫R2fxdx.The quaternion function f(x) is integrable and the Lebesgue dominated convergence theorem with (46) then gives(49)limh→0⁡FQfω+h-FQfω=0.This proves that FQ{f}(ω) is continuous on R2. Again, since (48) is independent of ω,FQ{f}(ω) is, in fact, uniformly continuous on R2.


### 3.2. Uncertainty Principle for QFT

In what follows, we investigate the uncertainty principles associated with the QFT. These results will be needed in the next section.Theorem 10 (QFT component-wise uncertainty principle [14]).
Iff∈L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then(50)∫R2xk2fx2dx∫R2ωk2FQfω2dω≥2π24∫R2fx2dx2,k=1,2.Remark 11.
An alternative form of Theorem10 is(51)∫R2xk2fx2dx∫R2ωl2FQfω2dω≥2π24∫R2fx2dx2.Notice that for1≤p≤2 we can replace the L2 norms to Lp norms on the left-hand side of (50) and obtain the following theorem.Theorem 12.
Under the assumptions of Theorem10, one has(52)∫R2x1pfxpdx1/p∫R2ω1pFQifωQpdω1/p≥12∫R2fx2dx,∫R2x2pfxpdx1/p∫R2ω2pFQfωQpdω1/p≥12∫R2fx2dx.Proof.
It is not difficult to check that(53)∫R2fx2dx=-2Sc⁡∫Rxkfx∂∂xkfx¯dx.Using Holder’s inequality, we further get (54)∫R2fx2dx≤2∫Rxkfx∂∂xkfx¯dx≤2∫Rxkfxpdx1/p∫R∂∂xkfxp′dx1/p′=2xkfp∂∂xkfp′.By application of the Hausdorff-Young inequality (38) and then integration by parts, we obtain (55)∂∂xkfp′≤FQ∂∂xkQ,p=∫R2FQ∂∂xkf0p+FQ∂∂xkf1p+FQ∂∂xkf2pFQ∂∂xkf3pdω1/p.We set k=1. Using (35) gives (56)FQ∂∂x1Q,p=∫R2ω1FQif0p+ω1FQif1p+ω1FQif2p+ω1FQif3pdω1/p=∫R2ω1pFQif0p+ω1pFQif1p+ω1pFQif2p+ω1pFQif3pdω1/p=∫R2ω1pFQifQpdω1/p.For k=2, we can take similar steps as above using (36) and get(57)FQ∂∂x2Q,p=∫R2ω2pFQfQpdω1/p.This concludes the proof of the theorem.A generalized version of Theorem10 is directional uncertainty principle for the QFT given by the following.Theorem 13 (QFT directional uncertainty principle).
Iff∈L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then(58)∫R2x2fx2dx∫R2ω2FQfω2dω≥2π2∫R2fx2dx2.Proof.
We have(59)∫R2x2fx2dx∫R2ω2FQfω2dω=∫R2x12+x22fx2dx∫R2ω12+ω22FQfω2dω=∫R2x12fx2dx∫R2ω12FQfω2dω+∫R2x12fx2dx∫R2ω22FQfω2dω+∫R2x22fx2dx∫R2ω12FQfω2dω+∫R2x22fx2dx∫R2ω22FQfω2dω.Using (51) gives(60)∫R2x2fx2dx∫R2ω2FQfω2dω≥2π24∫R2fx2dx2+2π24∫R2fx2dx2+2π24∫R2fx2dx2+2π24∫R2fx2dx2=2π2∫R2fx2dx2.The result follows.Remark 14.
A different proof of Theorem13 using the logarithmic uncertainty principle for the QFT can be found in [15].Using the polar coordinate form of quaternion functionf, Yang et al. [18] obtained an alternative form of the directional uncertainty principle for the QFT as follows.Theorem 15.
Iff∈L1(R2;H)∩L2R2;H and (∂/xk)f exists and is also in L2(R2;H), then(61)∫R2xk2fx2dx∫R2ωk2FQfω2dω≥2π214+COVxkωk2,∫R2fx2dx=1,where COVxkωk is the absolute covariance.Applying (51), we can easily generalize the uncertainty principle (61) to the directional QFT uncertainty principle; that is,(62)∫R2x2fx2dx∫R2ω2FQfω2dω≥2π2414+COVxkωk2≥2π21+COVxkωk2.It is obvious that the result is the same as Theorem 4.3 in [18].


## 3.1. QFT and Properties

In the following, we introduce the (right-sided) QFT and some of its fundamental properties such as Riemann-Lebesgue lemma and continuity.Definition 2 (right-sided QFT).
The (right-sided) quaternion Fourier transform (QFT) off∈L1(R2;H) is the transform FQ{f}: R2→H given by(23)FQfω=f^ω=∫R2fxe-iω1x1e-jω2x2dx,dx=dx1dx2,where x=x1e1+x2e2,ω=ω1e1+ω2e2, and the quaternion exponential product e-iω1x1e-jω2x2 is the quaternion Fourier kernel.Theorem 3 (inverse QFT).
Suppose thatf∈L1(R2;H) and FQ{f}∈L1(R2;H). Then, the QFT of f is an invertible transform and its inverse is given by(24)FQ-1FQfx=fx=12π2∫R2FQfωejω2x2eiω1x1dω,where the quaternion exponential product ejω2x2eiω1x1 is called the inverse (right-sided) quaternion Fourier kernel.SinceS(R2;H)⊂L1(R2;H), the definition of QFT (23) may be extended to the Schwartz space. It is important to note that FQ{f} is not necessary in L1(R2;H) even if f is in L1(R2;H), so in general FQ{f} might not be well defined. However, the QFT of a Schwartz quaternion function is also in the Schwartz space.Applying (23), we have(25)FQfω=∫R2f0x+if1x+jf2x+kf3xe-iω1x1e-jω2x2dx=FQf0ω+iFQf1ω+jFQf2ω+kFQf3ω.Now, we define a module of FQ{f}(ω) as(26)FQfωQ=FQf0ω2+FQf1ω2+FQf2ω2+FQf3ω21/p.Furthermore, we obtain the Lp(R2;H)-norm(27)FQfQ,p=∫R2FQfωQpdω1/p.Remark 4.
It is worth noting here that ifFQ{fi}, i=0,2,3, is real-valued, (26) can be written in the form(28)FQfQ,p=FQfωp,where(29)FQfωQ=FQf0ω2+FQf1ω2+FQf2ω2+FQf3ω21/p.Some important properties of the QFT are stated in the following lemmas.Lemma 5 (QFT Plancherel).
Iff∈L1(R2;H)∩L2(R2;H), then(30)12π2∫R2FQfω2dω=∫R2fx2dx.Moreover,(31)12π2∫R2FQfωQ2dω=∫R2fx2dx.Proof.
We prove expression (31) of Lemma 5. Using (26), we immediately get (32)12π2∫R2FQfωQ2dω=12π2∫R2FQf0ω2+FQf1ω2+FQf2ω2+FQf3ω2dω=12π2∫R2FQf0ω2dω+∫R2FQf1ω2dω+∫R2FQf2ω2dω+∫R2FQf3ω2dω.Applying (30) into the right-hand side of the above identity gives (33)12π2∫R2FQfωQ2dω=∫R2f0x2dx+∫R2f1x2dx+∫R2f2x2dx+∫R2f3x2dx.Since fi(x), i=0,1,2,3, is real-valued, the above equation can be written in the form (34)12π2∫R2FQfωQ2dω=∫R2f02x+f12x+f22x+f32xdx,which completes the proof of the theorem.Remark 6.
Equation (30) shows that the QFT is a bounded linear operator on L1(R2;H)∩L2(R2;H). Hence, using standard density arguments, one may extend the QFT in a unique way to the whole of L2(R2;H).Lemma 7 (see [14]).
Iff∈L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then one has for every n∈N(35)FQ∂n∂x1nfω=ω1nFQfiω,(36)FQ∂n∂x2nfω=FQfωjω2n.By Riesz’s interpolation theorem, we get that the Hausdorff-Young inequality (see [15])(37)FQfQ,p′≤fpholds for 1≤p≤2 with 1/p+1/p′=1. Using inversion formula of the QFT, (37) can be rewritten in the form(38)fp′≤FQfQ,p.The following theorem is an extension of the Riemann-Lebesgue lemma in the QFT domain.Theorem 8 (Riemann-Lebesgue lemma of QFT).
For a function inf∈L1(R2;H), one has that(39)limω1→∞⁡FQfω=0,limω2→∞⁡FQfω=0.Proof.
Notice first that(40)e-iω1x1=-e-iω1x1+π/ω1,e-jω2x2=-e-jω2x2+π/ω2.Applying (40) gives (41)FQfω=∫R2fxe-iω1x1e-jω2x2dx=-∫R2fxe-iω1x1+π/ω1e-jω2x2dx.Representing FQ{f}=(1/2)FQ{f}+FQ{f} and changing variable x1+π/ω1=t1 in the above identity, we immediately obtain(42)FQfω=∫R2ft1-πω1,t2e-iω1t1e-jω2t2dt=12∫R2ft1,t2e-iω1t1e-jω2t2dt-∫R2ft1-πω1,t2e-iω1t1e-jω2t2dt=12∫R2ft1,t2-ft1-πω1,t2e-iω1t1e-jω2t2dt.This means that (43)limω1→∞⁡FQfω≤12limω1→∞∫R2ft1,t2-ft1-πω1,t2dt=0.Analogously, it can be shown that (44)limω2→∞FQfω=0.The proof is complete.Theorem 9 (continuity).
Iff∈L1(R2;H), then the quaternion Fourier transform FQ{f}(ω) is continuous on R2. Moreover,(45)fCR2;H=maxx∈R2fx≤f1,where C(R2;H) is the space of continuous quaternion functions from R2 to H.Proof.
From the QFT definition (23), we readily see that(46)FQfω+h-FQfω=∫R2fxe-iω1+h1x1e-jω2+h2x2dx-∫R2fxe-iω1x1e-jω2x2dx=∫R2fxe-iω1x1e-ih1x1e-jω2x2e-jh2x2dx-∫R2fxe-iω1x1e-jω2x2dx=∫R2fxe-iω1x1e-ih1x1e-jω2x2e-jh2x2-e-iω1x1e-jω2x2dx≤∫R2fxe-iω1x1e-ih1x1e-jh2x2e-jω2x2-e-iω1x1e-jω2x2dx=∫R2fxe-iω1x1e-ih1x1e-jh2x2-1e-jω2x2dx=∫R2fxe-ih1x1e-jh2x2-1dx.Using the triangle inequality for quaternions, we easily get (47)e-ih1x1e-jh2x2-1≤e-ih1x1e-jh2x2+1=2.This means that we have(48)FQfω+h-FQfω≤2∫R2fxdx.The quaternion function f(x) is integrable and the Lebesgue dominated convergence theorem with (46) then gives(49)limh→0⁡FQfω+h-FQfω=0.This proves that FQ{f}(ω) is continuous on R2. Again, since (48) is independent of ω,FQ{f}(ω) is, in fact, uniformly continuous on R2.


## 3.2. Uncertainty Principle for QFT

In what follows, we investigate the uncertainty principles associated with the QFT. These results will be needed in the next section.Theorem 10 (QFT component-wise uncertainty principle [14]).
Iff∈L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then(50)∫R2xk2fx2dx∫R2ωk2FQfω2dω≥2π24∫R2fx2dx2,k=1,2.Remark 11.
An alternative form of Theorem10 is(51)∫R2xk2fx2dx∫R2ωl2FQfω2dω≥2π24∫R2fx2dx2.Notice that for1≤p≤2 we can replace the L2 norms to Lp norms on the left-hand side of (50) and obtain the following theorem.Theorem 12.
Under the assumptions of Theorem10, one has(52)∫R2x1pfxpdx1/p∫R2ω1pFQifωQpdω1/p≥12∫R2fx2dx,∫R2x2pfxpdx1/p∫R2ω2pFQfωQpdω1/p≥12∫R2fx2dx.Proof.
It is not difficult to check that(53)∫R2fx2dx=-2Sc⁡∫Rxkfx∂∂xkfx¯dx.Using Holder’s inequality, we further get (54)∫R2fx2dx≤2∫Rxkfx∂∂xkfx¯dx≤2∫Rxkfxpdx1/p∫R∂∂xkfxp′dx1/p′=2xkfp∂∂xkfp′.By application of the Hausdorff-Young inequality (38) and then integration by parts, we obtain (55)∂∂xkfp′≤FQ∂∂xkQ,p=∫R2FQ∂∂xkf0p+FQ∂∂xkf1p+FQ∂∂xkf2pFQ∂∂xkf3pdω1/p.We set k=1. Using (35) gives (56)FQ∂∂x1Q,p=∫R2ω1FQif0p+ω1FQif1p+ω1FQif2p+ω1FQif3pdω1/p=∫R2ω1pFQif0p+ω1pFQif1p+ω1pFQif2p+ω1pFQif3pdω1/p=∫R2ω1pFQifQpdω1/p.For k=2, we can take similar steps as above using (36) and get(57)FQ∂∂x2Q,p=∫R2ω2pFQfQpdω1/p.This concludes the proof of the theorem.A generalized version of Theorem10 is directional uncertainty principle for the QFT given by the following.Theorem 13 (QFT directional uncertainty principle).
Iff∈L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then(58)∫R2x2fx2dx∫R2ω2FQfω2dω≥2π2∫R2fx2dx2.Proof.
We have(59)∫R2x2fx2dx∫R2ω2FQfω2dω=∫R2x12+x22fx2dx∫R2ω12+ω22FQfω2dω=∫R2x12fx2dx∫R2ω12FQfω2dω+∫R2x12fx2dx∫R2ω22FQfω2dω+∫R2x22fx2dx∫R2ω12FQfω2dω+∫R2x22fx2dx∫R2ω22FQfω2dω.Using (51) gives(60)∫R2x2fx2dx∫R2ω2FQfω2dω≥2π24∫R2fx2dx2+2π24∫R2fx2dx2+2π24∫R2fx2dx2+2π24∫R2fx2dx2=2π2∫R2fx2dx2.The result follows.Remark 14.
A different proof of Theorem13 using the logarithmic uncertainty principle for the QFT can be found in [15].Using the polar coordinate form of quaternion functionf, Yang et al. [18] obtained an alternative form of the directional uncertainty principle for the QFT as follows.Theorem 15.
Iff∈L1(R2;H)∩L2R2;H and (∂/xk)f exists and is also in L2(R2;H), then(61)∫R2xk2fx2dx∫R2ωk2FQfω2dω≥2π214+COVxkωk2,∫R2fx2dx=1,where COVxkωk is the absolute covariance.Applying (51), we can easily generalize the uncertainty principle (61) to the directional QFT uncertainty principle; that is,(62)∫R2x2fx2dx∫R2ω2FQfω2dω≥2π2414+COVxkωk2≥2π21+COVxkωk2.It is obvious that the result is the same as Theorem 4.3 in [18].


## 4. Continuous Quaternion Wavelet Transform

This section briefly introduces the continuous quaternion wavelet transform (CQWT). We shall derive two theorems of the CQWT which will be used in proving the main theorem. A more complete and detailed discussion of the properties of the CQWT can be found in [5, 7–9].Definition 16.
A quaternion-valued function is admissible if and only if it satisfies the following admissibility condition:(63)Cψ=∫R+FQψaω2daa<∞.Here, Cψ is a real positive constant independent of ω satisfying |ω|=1.Letψ∈L1(R2;H) be a quaternion mother wavelet. We consider the family of the wavelets ψa,b defined by(64)ψa,bx=Tbψax=1aψx-ba,a∈R+,where Tbf(x)=f(x-b) and ψa(x)=(1/a)ψ(x/a). Here, a is a dilation parameter and b is a translation vector parameter.The relationship between (64) and its QFT is given in the following lemma.Lemma 17.
Letψ be an admissible quaternion function. The family of the wavelets (64) can be written in terms of the QFT as(65)FQψa,bω=ae-iω1b1FQψ0aω+iFQψ1aωe-jω2b2+aeiω1b1jFQψ2aω+kFQψ3aωe-jω2b2.If we assume thatFQ{ψi}(ω), i=0,1,2,3, is real-valued (in the next section, we will always assume that the QFT of quaternion mother wavelet, i.e., FQ{ψ}(ω)=ψ^(ω), is real-valued), (65) can be rewritten in the form(66)FQψa,bω=aψ0^aω+iψ1^aω+jψ2^aω+kψ3^aωe-iω1b1e-jω2b2=25aFQψaωe-iω1b1e-jω2b2.Definition 18 (CQWT).
The CQWT of a quaternion functionf∈L2(R2;H) with respect to the quaternion mother wavelet ψ is defined by(67)Tψfa,b=∫R2fx1aψx-ba¯dx.As an easy consequence of the above definition, we further obtain the following useful theorem.Theorem 19.
Letψ∈L2(R2;H) be a quaternion admissible wavelet; then, CQWT (67) can be expressed as(68)Tψfa,b=a2π2∫R2f^ωejb2ω2eib1ω1ψ^aω¯dω.We need the following two important results, which will be useful in proving the logarithmic uncertainty principle for the CQWT.Theorem 20.
Letψ∈L2(R2;H) be a quaternion admissible wavelet; then, CQWT (67) has a quaternion Fourier representation of the form(69)FQTψfa,bω=af^ωψ^aω¯.Proof.
From the definition of QFT (23), it follows that (70)FQTψfa,bω′=a2π2∫R2∫R2f^ωejb2ω2eib1ω1ψ^aω¯e-ib1ω1′e-jb2ω2′dωdb.Using the assumption that the QFT of quaternion mother wavelet is real-valued and then applying Fubini’s theorem, we obtain(71)FQTψfa,bω′=a2π2∫R2∫R2f^ωψ^aω¯ejb2ω2eib1ω1e-ib1ω1′e-jb2ω2′dωdb=a2π2∫R2f^ωψ^aω¯∫R2ejb2ω2eib1ω1-ω1′e-jb2ω2′dbdω=a∫R2f^ωψ^aω¯δω-ω′dω=af^ω′ψ^aω′¯,∀ω′∈R2,where δ(ω-ω′)=δ(ω1-ω1′)δ(ω2-ω2′). The proof is complete.Theorem 21.
Letψ∈L2(R2;H) be a quaternion admissible wavelet which satisfies the admissibility condition defined by (63). Then, for every f∈L2(R2;H), one has(72)∫R+∫R2Tψfa,b2dbdaa3=Cψ∫R2fx2dx.Proof.
Applying Plancherel’s theorem for QFT (30) to the b-integral into the left-hand side of (72) yields (73)∫R+∫R2Tψfa,b2dbdaa3=12π2∫R+∫R2FQTψfa,bω2dωdaa3=6912π2∫R+∫R2af^ωψ^aω¯2dωdaa3.Taking into consideration Fubini’s theorem about the inversion of order of integration and applying Plancherel’s theorem (30), we get(74)∫R+∫R2Tψfa,b2dbdaa3=12π2∫R+∫R2f^ω2ψ^aω¯2dωdaa=12π2∫R+f^ω2dω∫R2ψ^aω¯2daa=Cψ2π2∫R2f^ω2dω=Cψ∫R2fx2dx,which gives the desired result.


## 5. Logarithmic Uncertainty Principle for CQWT

The simplest formulation of the uncertainty principle in harmonic analysis is Heisenberg-Weyl inequality, which gives us the information that a nontrivial function and its Fourier transform cannot both be simultaneously sharply localized [1, 22]. In this section, we first derive a variation on uncertainty principle associated with the CQWT. From this, we establish the logarithmic uncertainty principle which is valid for the QFT [14] to the setting of the CQWT.Due to the uncertainty principle for QFT (58), we have the logarithmic uncertainty principle for the QFT [15] as follows.Theorem 22 (QFT logarithmic uncertainty principle).
Forf∈S(R2;H),(75)∫R2ln⁡xfx2dx+∫R2ln⁡ωFQfω2dω≥Γ′tΓt-ln⁡π∫R2fx2dx,where Γ′(t)=(d/dt)[Γ(t)] and Γ(t) is Gamma function. Here, S(R2;H) denotes the Schwartz class on quaternion function.Applying Plancherel’s theorem for QFT (30) to the right-hand side of (75), we easily obtain(76)∫R2ln⁡xfx2dx+∫R2ln⁡ωFQfω2dω≥12π2Γ′tΓt-ln⁡π∫R2FQfω2dω.It is proved that, for everyf,ψ∈S(R2;H), the Heisenberg type uncertainty principle for the CQWT is given [7]:(77)∫R+∫R2bk2Tψfa,b2dbdaa31/2∫R2ωk2f^ω2dω1/2≥Cψ2∫R2fx2dx.A generalization of the above uncertainty principle is given in the following theorem.Theorem 23.
Letψ,f∈S(R2;H) be a quaternion admissible wavelet. Let Tψf(a,b) be the CQWT of f. If 1≤p≤2, then(78)∫R+∫R2bkpTψfa,bpdbdaa31/p∫R2ωkpf^ωpdω1/p≥a6-3p/2pCψ2∫R2fx2dx,k=1,2.For the proof of Theorem23, we use the following lemma.Lemma 24.
Suppose thatf,ψ∈S(R2;H). Then,(79)∫R+∫R2ωkpFQTψfa,bpdωdaa3≤∫R2ωkpf^ωpa3p-6/2Cψp/2dω.Proof.
A straightforward computation yields(80)∫R+∫R2ωkpFQTψfa,bpdωdaa3=69∫R2∫R+ωkpaf^ωψ^aω¯pdaa3dω=∫R2∫R+ωkpf^ωpψ^aω¯pdaa3-pdω≤∫R2ωkpf^ωp∫R+ψ^aω¯2daa6/p-2p/2dω=∫R2ωkpf^ωpa3p-6/p∫R+ψ^aω¯2daap/2dω=∫R2ωkpf^ωpa3p-6/2Cψp/2dω.The proof is complete.Proof.
Using the uncertainty principle in Theorem12, we immediately obtain(81)∫R2bkpTψfa,bpdb1/p∫R2ωkFQTψfa,bpdω1/p≥12∫R2Tψfa,b2db.Now, integrating both sides of (81) with respect to the Haar measure da/a3, we obtain(82)∫R+∫R2bkpTψfa,bpdbdaa31/p∫R+∫R2ωkpFQTψfa,bpdωdaa31/p≥12∫R+∫R2Tψfa,b2dbdaa3.Then, inserting Lemma 24 into the second term of (82), we easily obtain(83)∫R+∫R2bkpTψfa,bpdbdaa31/pa3p-6/2pCψ∫R2ωkpf^ωpdω1/p≥12∫R+∫R2Tψfa,b2dbdaa3.Substituting (72) into the right-hand side of (83) and simplifying it, we finally get(84)∫R+∫R2bkpTψfa,bpdbdaa31/p∫R2ωkpf^ωpdω1/p≥a6-3p/2pCψ2∫R2fx2dx,which concludes the proof of Theorem 23.Let us derive a logarithmic uncertainty principle associated with the continuous quaternion wavelet transform (CQWT).Theorem 25 (CQWT logarithmic uncertainty principle).
Letψ∈S(R2;H) be a quaternion admissible wavelet. Let Tψf(a,b) be the CQWT of f∈S(R2;H). Then, the following inequality is satisfied:(85)Cψ∫R2ln⁡ωf^ω2dω+∫R+∫R2ln⁡bTψfa,b2dbdaa3≥Cψ2Γ′tΓt-ln⁡π∫R2fx2dx.Next, we need the following lemma to assist the proof of the above theorem.Lemma 26.
Under the same conditions as in Theorem25, one has(86)∫R+∫R2ln⁡ωFQTψfa,b2dωdaa3=Cψ∫R2ln⁡ωf^ω2dω.Proof.
A simple calculation reveals(87)∫R+∫R2ln⁡ωFQTψfa,b2dωdaa3=69∫R2∫R+ln⁡ωaf^ωψ^aω¯2daa3dω=∫R2∫R+a2ln⁡ωf^ωψ^aωψ^aω¯f^ω¯daa3dω=∫R2∫R+ln⁡ωf^ω2ψ^aω2daadω=63Cψ∫R2ln⁡ωf^ω2dω.The proof is complete.Proof.
It is known that(88)∫R2ln⁡xfx2dx+∫R2ln⁡ωFQfω2dω≥Γ′tΓt-ln⁡π∫R2fx2dx.Notice that f,ψ∈S(R2;H). This implies that Tψf(a,b)∈S(R2;H). Therefore, we may replace f by Tψf(a,b) on both sides of (88) and get (89)∫R2ln⁡ωFQTψfa,b2dω+∫R2ln⁡bTψfa,b2db≥Γ′tΓt-ln⁡π∫R2Tψfa,b2db.Integrating both sides of this equation with respect to da/a3 yields(90)∫R+∫R2ln⁡ωFQTψfa,b2dωdaa3+∫R+∫R2ln⁡bTψfa,b2dbdaa3≥Γ′tΓt-ln⁡π∫R+∫R2Tψfa,b2dbdaa3.Inserting Lemma 26 into the first term on the left-hand side of (90), we obtain(91)Cψ∫R2ln⁡ωf^ω2dω+∫R+∫R2ln⁡bTψfa,b2dbdaa3≥Γ′tΓt-ln⁡π∫R+∫R2Tψfa,b2dbdaa3.Finally, substituting (72) into the right-hand side of (91), we have (92)Cψ∫R2ln⁡ωf^ω2dω+∫R+∫R2ln⁡bTψfa,b2dbdaa3≥Cψ2Γ′tΓt-ln⁡π∫R2fx2dx,which was to be proved.Remark 27.
It is worth nothing that, following the steps of the proof of the above theorem, we can also obtain Theorem25 using (76).


## 6. Conclusion

Based on the logarithmic uncertainty principle in the quaternion Fourier domain, we have established a logarithmic uncertainty principle related to the CQWT. It is a more general form of component-wise uncertainty principle associated with the CQWT, which describes the relationship between the QFT of a quaternion function and its CQWT. We also presented a variation on uncertainty principle related to the QFT and then found the variation on uncertainty principle related to the CQWT.



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