Applied Mathematics and Computation 218 (2011) 10–21
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Two-dimensional quaternion wavelet transform Mawardi Bahri a, Ryuichi Ashino b,⇑, Rémi Vaillancourt c a b c
Department of Mathematics, Hasanuddin University, KM 10 Tamalanrea, Makassar, Indonesia Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan Department of Mathematics and Statistics, University of Ottawa, 585 Kind Edward Ave., Ottawa ON, Canada KIN 6N5
a r t i c l e
i n f o
Keywords: Quaternion Fourier transform Admissible quaternion wavelets
a b s t r a c t In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The quaternion Fourier transform (QFT), which is a nontrivial generalization of the real and complex Fourier transform (FT) using quaternion algebra has been of interest to researchers for some years (see e.g. [1,2,7,16]). It was found that many FT properties still hold but others have to be modified. Based on the (right-sided) QFT, one can extend the classical wavelet transform (WT) to quaternion algebra while enjoying the same properties as in the classical case. He [21] and Zhao and Peng [23] constructed the continuous quaternion wavelet transform of quaternion-valued functions. They also demonstrated a number of properties of these extended wavelets using the classical Fourier transform (FT). In [6], using the (two-sided) QFT Traversoni proposed a discrete quaternion wavelet transform which was applied by Bayro-Corrochano [12] and Zhou et al. [13]. Recently, in [18,19], we introduced an extension of the WT to Clifford algebra by means of the kernel of the Clifford Fourier transform [8]. The purpose of this paper is to construct the 2-D continuous quaternion wavelet transform (CQWT) based on quaternion algebra. We emphasize that our approach is significantly different from previous work in the definition of the exponential kernel. Our construction uses the kernel of the (right-sided) QFT which in general does not commute with quaternions. The previous papers considered the kernel of the FT which commutes with the quaternions so that the properties of the extension of the WT to quaternion algebra is a quite similar to the classical wavelets. In the present paper we use the (right-sided) QFT to investigate some important properties of the CQWT. Special attention is devoted to inner product, norm relation, and inversion formula. We show that these fundamental properties can be established whenever the admissible quaternion wavelets satisfy a particular admissibility condition. Using the properties of the CQWT and the uncertainty principle for the (right-sided) QFT [16] we establish an uncertainty principle for the CQWT. 2. Basics For convenience of further discussions, we briefly review some basic ideas on quaternions, the (right-sided) QFT and the similitude group SIM(2). The quaternion algebra over R, denoted by H, is an associative non-commutative four-dimensional algebra, ⇑ Corresponding author. E-mail addresses:
[email protected] (M. Bahri),
[email protected] (R. Ashino),
[email protected] (R. Vaillancourt). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.030
M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
H ¼ fq ¼ q0 þ iq1 þ jq2 þ kq3 jq0 ; q1 ; q2 ; q3 2 Rg;
11
ð1Þ
which obey Hamilton’s multiplication rules
ij ¼ ji ¼ k;
jk ¼ kj ¼ i;
ki ¼ ik ¼ j;
2
2
2
i ¼ j ¼ k ¼ ijk ¼ 1:
ð2Þ
The quaternion conjugate of a quaternion q is given by
¼ q0 iq1 jq2 kq3 ; q
q0 ; q1 ; q2 ; q3 2 R
ð3Þ
and it is an anti-involution, i.e.
: q qp ¼ p
ð4Þ
From (3) we obtain the norm of q 2 H defined as
jqj ¼
pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ q20 þ q21 þ q22 þ q23 : qq
ð5Þ
It is not difficult to see that
8p; q 2 H:
jqpj ¼ jqjjpj;
ð6Þ 2
It is convenient to introduce the inner product of two quaternion functions, f ; g : R ! H, as follows:
ðf ; gÞL2 ðR2 ;HÞ ¼
Z
2
2
f ðxÞgðxÞd x:
ð7Þ
R
In particular, if f = g, then we obtain the associated norm 1=2
kf kL2 ðR2 ;HÞ ¼ ðf ; f ÞL2 ðR2 ;HÞ ¼
Z R2
1=2 2 jf ðxÞj2 d x :
ð8Þ
As a consequence of the inner product (7) we obtain the quaternion Cauchy–Schwarz inequality
Z
R
Z 2 f gd x 6 2
1=2 Z 2 jf j2 d x
R2
1=2 2 jgj2 d x ;
R2
8f ; g 2 L2 ðR2 ; HÞ:
ð9Þ
Based on quaternions we can define the (right-sided) QFT. Definition 1. The QFT of f 2 L1 ðR2 ; HÞ is the function F q ff g : R2 ! H given by
F q ff gðxÞ ¼ ^f ðxÞ ¼
Z
2
R2
f ðxÞeix1 x1 ejx2 x2 d x;
ð10Þ
where x = x1e1 + x2e2, x = x1e1 + x2e2, and the quaternion exponential product eix1 x1 ejx2 x2 is called the quaternion Fourier kernel. Theorem 1. Suppose that f 2 L2 ðR2 ; HÞ and F q ff g 2 L1 ðR2 ; HÞ. Then the QFT of f is an invertible transform and its inverse is given by
F 1 q ½F q ff gðxÞ ¼ f ðxÞ ¼
1 ð2pÞ2
Z
2
R2
F q ff gðxÞejx2 x2 eix1 x1 d x:
ð11Þ
As in the classical case, we obtain Plancherel’s formula, specific to the (right-sided) QFT [2,7,16],
ðf ; gÞL2 ðR2 ;HÞ ¼
1 ð2pÞ2
ðF q ff g; F q fggÞL2 ðR2 ;HÞ :
ð12Þ
In particular, if f = g we get Parseval’s formula,
kf k2L2 ðR2 ;HÞ ¼
1 ð2pÞ2
kF q ff gk2L2 ðR2 ;HÞ :
ð13Þ
Table 1 presents some useful properties of the (right-sided) QFT. Detailed information about the QFT and its properties can be found in [1,7,16]. Following Antoine et al. [3,4], we consider the similitude group SIM(2) denoted by G on R2 associated with wavelets as follows.
G ¼ Rþ SOð2Þ R2 ¼ fða; r h ; bÞja 2 Rþ ; rh 2 SOð2Þ; b 2 R2 g; 2
where SO(2) is the special orthogonal group of R . The multiplication defined by (14) follows the group law
ð14Þ
12
M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21 Table 1 Properties of the (right-sided) QFT of f ; g 2 L2 ðR2 ; HÞ, where a; b 2 H; a 2 R n f0gare constants, x0 ¼ x0 e1 þ y0 e2 2 R2 and n 2 N. Property
Quaternion function
Quaternionic Fourier transform
Left linearity Scaling
af(x) + bg(x)
aF q ff gðxÞ þ bF q fggðxÞ
f(ax)
1 jaj2
F q ff gðxa Þ ix1 x0
Shift
f(x x0)
F q ffe
Rotation
ðxÞÞ f ðr 1 h n n @ f ðxÞi @x1 n @ f ðxÞ @x1 n @ f ðxÞ @x2
F q ff gðr 1 h ðxÞÞ
Formula Plancherel’s formula
ðf1 ; f2 ÞL2 ðR2 ;HÞ ¼
Parseval’s formula
kf kL2 ðR2 ;HÞ ¼
1 ðF q ff1 g; F q ff2 gÞL2 ðR2 ;HÞ ð2pÞ2 1 2p kF q ff gkL2 ðR2 ;HÞ
Part. deriv.
0
gðxÞejx2 y0
xn1 F q ff gðxÞ; f 2 L2 ðR2 ; HÞ ðix1 Þn F q ff gðxÞ f ¼ f0 þ if1 F q ff gðxÞðjx2 Þn ; f 2 L2 ðR2 ; HÞ
0
fa; b; r h gfa0 ; b ; rh0 g ¼ faa0 ; b þ arh b ; r hþh0 g:
ð15Þ
2
The rotation rh 2 SO(2) acts on x 2 R as usual,
rh ðxÞ ¼ ðx1 cos h x2 sin h; x1 sin h þ x2 cos hÞ;
0 6 h < 2p:
ð16Þ
The left Haar measure on G is given by 2
dkða; h; bÞ ¼ dlða; hÞd b; where dlða; hÞ ¼ dadh and dh is the Haar measure on SO(2). For the sake of simplicity, we write dl = dl(a, h) and dk = dk(a, h, b). a3 3. Construction of 2-D quaternion wavelets Based on quaternions and the (right-sided) QFT, one can extend the real (or complex) wavelet transform to a quaternion wavelet transform. This section constructs the 2-D CQWT from a group theoretical point of view. We shall characterize the admissibility condition in terms of the (right-sided) QFT and define the CQWT in terms of an admissible quaternion wavelet. 3.1. Admissible quaternion wavelet Definition 2 (Admissible quaternion wavelet). Let AQW denote the class of admissible quaternion wavelets w 2 L2 ðR2 ; HÞ which satisfy the following admissibility condition, i.e.
Z
Z Rþ
SOð2Þ
^ h ðxÞÞj2 jwðar
dadh a
ð17Þ
is a real positive constant independent of x satisfying jxj = 1. Denote by Cw, the real positive constant. Remark 1. If w 2 AWQ, then (17) is a real positive constant independent of x for x 2 R2 . Let us show this fact. Denote x = jxjx0, where jx0j = 1. Since rh is linear and da/a is the Haar measure of the multiplicative group Rþ ,
Z
Z Rþ
SOð2Þ
^ h ðxÞÞj2 jwðar
dadh ¼ a ¼
Z SOð2Þ
Z
SOð2Þ
Z
^ h ðjxjx0 ÞÞj2 jwðar
Rþ
Z
^ h ðx0 ÞÞj2 jwðar
Rþ
dadh ¼ a
Z SOð2Þ
Z Rþ
^ xjr h ðx0 ÞÞj2 jwðaj
dadh a
dadh : a
^ is continuous at x = 0, and Remark 2. Assume that w 2 L2 ðR2 ; HÞ is radially symmetric, that is, rotation invariant, jwj ^ jwð0Þj ¼ 0. Then w 2 AWQ. Let us show this fact. Denote e = (1, 0). For x satisfyingjxj = 1, there exists g 2 [0, 2p) such that ^ is also radially symmetric. Then, we have x = rg(e). Since w is radially symmetric, w
^ h ðxÞÞ ¼ wðar ^ hþg ðeÞÞ ¼ wðaeÞ; ^ wðar which implies
Z SOð2Þ
Z Rþ
^ h ðxÞÞj2 jwðar
dadh ¼ a
Z SOð2Þ
dh
Z Rþ
2 ^ jwðaeÞj
da : a
^ The condition jwð0Þj ¼ 0 ensures the integrability of the right-hand side of (18).
ð18Þ
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M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
Notice that according to (5) Cw is an invertible real constant. Using (1) we may decompose w 2 AQW into the following form
wðxÞ ¼ w0 ðxÞ þ iw1 ðxÞ þ jw2 ðxÞ þ kw3 ðxÞ;
ð19Þ
2
where ws 2 L ðR2 ; RÞ for s = 0, 1, 2, 3. Using (10) and the linearity of the (right-sided) QFT we may write (19) in the quaternionic frequency domain in the form
F q fwgðxÞ ¼
Z
2
R2
ðw0 ðxÞ þ iw1 ðxÞ þ jw2 ðxÞ þ kw3 ðxÞÞeix1 x1 ejx2 x2 d x
¼ F q fw0 gðxÞ þ iF q fw1 gðxÞ þ jF q fw2 gðxÞ þ kF q fw3 gðxÞ; 2
ð20Þ
2
where we assume that F q fws g 2 L ðR ; RÞ, for s = 0, 1, 2, 3. Like for classical wavelets [15,20], the zeroth moment of w 2 AQW vanishes,
Z
Z
2
wðxÞd x ¼
R2
2
R2
ðw0 ðxÞ þ iw1 ðxÞ þ jw2 ðxÞ þ kw3 ðxÞÞd x ¼ 0:
ð21Þ
It means that the integral of every component ws of the quaternion mother wavelet is zero
Z
R2
2
ws d x ¼ 0;
s ¼ 0; 1; 2; 3:
ð22Þ
Definition 3. For w 2 L2 ðR2 ; HÞ; a 2 Rþ ; b 2 R2 , and rh 2 SO(2), we define the unitary linear operator
U a;h;b : L2 ðR2 ; HÞ ! L2 ðG; HÞ; as
ðU a;h;b ðwÞÞ ¼ wa;h;b ðxÞ ¼
1 xb : w r h a a
ð23Þ
The family of wavelets wa,h,b are called daughter quaternion wavelets where a is a dilation parameter, b a translation vector parameter, and h an SO(2) rotation parameter. By straightforward calculations we obtain the following lemma. Lemma 1. Let w be an admissible quaternion function. Daughter quaternion wavelets (23) can be written in terms of the (rightsided) QFT as
n o n o c ðar h ðxÞÞ þ i w c ðar h ðxÞÞ ejx2 b2 þ aeix1 b1 j w c ðar h ðxÞÞ þ k w c ðarh ðxÞÞ ejx2 b2 : F q fwa;h;b gðxÞ ¼ aeix1 b1 w 0 1 2 3
ð24Þ
Proof. Definition 1 gives
F q fwa;h;b gðxÞ ¼
1 xb 2 eix1 x1 ejx2 x2 d x: w r h a a
Z R2
Performing the change of variables
F q fwa;h;b gðxÞ ¼
Z
xb a
¼ y into the above expression, we immediately obtain
1 2 wðrh yÞeix1 ðb1 þay1 Þ ejx2 ðb2 þay2 Þ a2 d y ¼ a a
R2
ð25Þ Z
2
R2
wðr h ðyÞÞeix1 b1 eiax1 y1 ejax2 y2 d yejx2 b2 :
Observe, first, that w = (w0 + iw1) + (jw2 + kw3) and use the fact that jw2eix1 above identity leads to
F q fwa;h;b gðxÞ ¼ a ¼a ¼a
Z
b 1
ð26Þ
= w2eix1b1j and kw3 eix1b1 = w3eix1b1k. The 2
ZR
2
R2
Z
R2
fw0 ðrh ðyÞÞ þ iw1 ðrh ðyÞÞ þ jw2 ðr h ðyÞÞ þ kw3 ðrh ðyÞÞg eix1 b1 eiax1 y1 ejax2 y2 d yejx2 b2 ix b 2 e 1 1 fw0 ðr h ðyÞÞ þ iw1 ðr h ðyÞÞg þ eix1 b1 fjw2 ðr h ðyÞÞ þ kw3 ðr h ðyÞÞg eiax1 y1 ejax2 y2 d yejx2 b2 n o 2 eix1 b1 fw0 ðr h ðyÞÞ þ iw1 ðr h ðyÞÞgeiax1 y1 ejax2 y2 d yejx2 b2
Z
n o 2 eix1 b1 fjw2 ðr h ðyÞÞ þ kw3 ðr h ðyÞÞgeiax1 y1 ejax2 y2 d yejx2 b2 R2 Z Z 2 2 ix1 b1 w0 ðr h ðyÞÞeiax1 y1 ejax2 y2 d yejx2 b2 þ aeix1 b1 iw1 ðr h ðyÞÞeiax1 y1 ejax2 y2 d yejx2 b2 ¼ ae R2 R2 Z Z 2 2 þ aeix1 b1 jw2 ðr h ðyÞÞeiax1 y1 ejax2 y2 d yejx2 b2 þ aeix1 b1 kw3 ðr h ðyÞÞgeiax1 y1 ejax2 y2 d yejx2 b2 þa
R2
ix1 b1
¼ ae
c ðarh ðxÞÞejx2 b2 þ aeix1 b1 w c ðar h ðxÞÞejx2 b2 ; w 0l 1l
R2
ð27Þ
14
M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
where we write
c ðar h ðxÞÞ ¼ w c ðar h ðxÞÞ þ i w c ðarh ðxÞÞ; w 0l 0 1 c ðar h ðxÞÞ ¼ j w c ðarh ðxÞÞ þ k w c ðar h ðxÞÞ: w 1l 2 3
ð28Þ
This proves the lemma. h Remark 3. Notice that if we assume that iw = wi, i.e.
w ¼ w0 þ iw1 ;
w0 ; w1 2 R:
ð29Þ
Then Lemma 1 takes the following form
b h ðxÞÞejx2 b2 : F q fwa;h;b gðxÞ ¼ aeix1 b1 wðar
ð30Þ
3.2. 2-D continuous quaternion wavelet transform (CQWT) Definition 4 (CQWT). The CQWT of a quaternion-valued function f 2 L2 ðR2 ; HÞ with respect to w 2 AQW in 2 dimensions is defined by
T w : L2 ðR2 ; HÞ ! L2 ðR2 ; HÞ f # T w f ða; h; bÞ ¼ ðf ; wa;h;b ÞL2 ðR2 ;HÞ ¼
1 xb 2 d x: f ðxÞ w r h a a R2
Z
ð31Þ
It must be remarked that the order of the terms in (31) is fixed because of the non-commutativity of the product of quaternions. Changing the order yields another quaternion valued function which differs by the signs of the terms. Eq. (31) clearly shows that the CQWT can be regarded as the inner product of a quaternion-valued signal f with daughter quaternion wavelets. Lemma 2. Suppose that w 2 AQW. If w 2 L2 ðR2 ; HÞ, then the CQWT (31) has a quaternion Fourier representation of the form
T w f ða; h; bÞ ¼
Z
a 2
ð2pÞ
R2
h i ^f ðxÞejb2 x2 w c ðar h ðxÞÞeib1 x1 þ w c ðarh ðxÞÞeib1 x1 d2 x; 0l 1l
ð32Þ
c ðar h ðxÞÞ and w c ðar h ðxÞÞ are defined in (28). where w 0l 1l Proof. We have ð12Þ
T w f ða; h; bÞ ¼ ðf ; wa;h;b ÞL2 ðR2 ;HÞ ¼ Z
ð^f ; wd a;h;b ÞL2 ðR2 ;HÞ
1 ^f ðxÞ wd ðxÞd2 x ð23Þ ¼ a;h;b ð2pÞ R2 ð2pÞ2 Z h i c ðar h ðxÞÞejx2 b2 þ eix1 b1 w c ðarh ðxÞÞejx2 b2 d2 x a^f ðxÞ eix1 b1 w 0l 1l R2 Z h i 1 ^f ðxÞ eix1 b1 w c ðarh ðxÞÞejx2 b2 d2 x þ 1 ¼ a 0l ð2pÞ2 R2 ð2pÞ2 Z Z h i 1 c ðar h ðxÞÞejx2 b2 d2 x ð4Þ c ðar h ðxÞÞeib1 x1 d2 x þ 1 a^f ðxÞ eix1 b1 w ¼ a^f ðxÞejb2 x2 w 1l 0l ð2pÞ2 R2 ð2pÞ2 R2 Z c ðar h ðxÞÞeib1 x1 d2 x: a^f ðxÞejb2 x2 w 1l ¼
1
1 ð2pÞ2
2
ð33Þ
R2
This proves (32). h Lemma 3. Let w 2 L2 ðR2 ; HÞ be a quaternion valued wavelet. If F q fwg ¼ F q fw0 g þ kF q fw3 g, then Eq. (32) can be expressed as
c ðarh ðÞÞ ðbÞ: c ðar h ðÞÞ ðbÞ þ F 1 a^f ðÞk w T w f ða; h; bÞ ¼ F 1 abf ðÞ w 0 3 q q
ð34Þ
Proof. For F q fwg ¼ F q fw0 g þ kF q fw3 g we have
c ðarh ðxÞÞejx2 b2 þ aeix1 b1 k w c ðar h ðxÞÞejx2 b2 : F q fwa;h;b gðxÞ ¼ aeix1 b1 w 0 3
ð35Þ
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M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
In view of (35), Eq. (32) takes the following form
T w f ða; h; bÞ ¼ ¼
Z
a ð2pÞ2 a ð2pÞ
R2
Z
2
R2
^f ðxÞejb2 x2 w c ðar h ðxÞÞeib1 x1 d2 x þ 0
^f ðxÞ w c ðar h ðxÞÞejb2 x2 eib1 x1 d2 x þ 0
Z
a ð2pÞ2 a ð2pÞ
2
R2
Z
R2
^f ðxÞejb2 x2 k w c ðarh ðxÞÞeib1 x1 d2 x 3
^f ðxÞk w c ðar h ðxÞÞejb2 x2 eib1 x1 d2 x ; 3
ð36Þ
where the second equality we have used the fact that
c ðar h ðxÞÞejb2 x2 ¼ ejb2 x2 k w c ðarh ðxÞÞ: kw 3 3
ð37Þ
Next, applying the inverse of the (right-sided) QFT (11) yields
c ðarh ðÞÞ ðbÞ þ F 1 abf ðÞk w c ðarh ðÞÞ ðbÞ: T w f ða; h; bÞ ¼ F 1 abf ðÞ w 0 3 q q
ð38Þ
Remark 4. It is easy to see that for F q fwg 2 R Eq. (35) reduces to
^ h ðxÞÞejx2 b2 ; F q fwa;h;b gðxÞ ¼ aeix1 b1 wðar
ð39Þ
and for F q fwg ¼ kF q fw3 g Eq. (35) takes the form
b h ðxÞÞejx2 b2 : F q fwa;h;b gðxÞ ¼ aeix1 b1 wðar
ð40Þ
The following proposition is a particular case of the lemma proved above. Proposition 1. Let w 2 L2 ðR2 ; HÞ be a quaternion valued wavelet. (i) If F q fwg 2 R, then Eq. (32) has the form
T w f ða; h; bÞ ¼
a
Z
ð2pÞ2
R2
^f ðxÞwðar ^ h ðxÞÞejb2 x2 eib1 x1 d2 x:
ð41Þ
Or, equivalently,
^ h ðxÞÞ: F q ðT w f ða; h; :ÞÞðxÞ ¼ a^f ðxÞwðar
ð42Þ
(ii) If F q fwg ¼ kF q fw3 g, then we may rewrite Eq. (32) in the form
T w f ða; h; bÞ ¼
a
Z
ð2pÞ2
R2
^f ðxÞ wðar b h ðxÞÞejb2 x2 eib1 x1 d2 x:
ð43Þ
Or, equivalently,
b h ðÞÞ ðbÞ: T w f ða; h; bÞ ¼ F 1 a^f ðÞ wðar q
ð44Þ
3.3. Examples of 2-D quaternion wavelets As examples of AQW we first take the difference of Gaussian (DOG) wavelet which the mother wavelet w obtained by subtracting a wide Gaussian from a narrow Gaussian. Example 1. Consider the two-dimensional DOG wavelets or difference-of-Gaussian wavelets (see [3]):
wðxÞ ¼
1
c2
2
2
2
2
2
eðx1 þx2 Þ=2c eðx1 þx2 Þ=2 ;
0 < c < 1:
ð45Þ
The DOG wavelet for c = 7/25 is illustrated in Fig. 1. Notice that F q fwg 2 R. When h = 0, the representation (39) implies
^ 0 ðxÞÞejx2 b2 ¼ aeix1 b1 2peðacÞ2 ðx21 þx22 Þ=2 2pea2 ðx21 þx22 Þ=2 ejx2 b2 F q fwa;0;b gðxÞ ¼ aeix1 b1 wðar 2 2 2 2 2 2 ¼ aeix1 b1 ejx2 b2 2peðacÞ ðx1 þx2 Þ=2 2pea ðx1 þx2 Þ=2 ;
ð46Þ
where we used the fact that the (right-sided) QFT of the Gaussian function is another Gaussian function (see [16]). The quaternion Fourier transform F q fwa;h;b g of the DOG wavelet is illustrated in Fig. 2 for c = 1/2, h = 0, b1 = b2 = 1 and a = 1.
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M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
12 10 8 6 4 2 0 0.8
0.4
x2
0 -0.4
0.4
0
-0.4
-0.8 -0.8
0.8
x1
Fig. 1. The DOG wavelet w for c = 7/25.
Now, we take
( f ðxÞ ¼
eðx1 þx2 Þ ; if x1 > 0 and x2 > 0; 0;
ð47Þ
otherwise:
It is known (see [17]) that the (right-sided) QFT of f is given by
F q ff gðxÞ ¼
1 ix1 jx2 kx1 x2 ð2pÞ2 ð1 þ x21 þ x22 þ x21 x22 Þ
ð48Þ
:
The CQWT with respect to the DOG wavelets (45) are obtained as follows:
T w f ða; h; bÞ ¼ F q ðT w f ða; h; ÞÞðxÞ ¼ aF q ff gðxÞF q fwa;h;b gðxÞ ¼
a2 ð1 ix1 jx2 kx1 x2 Þð2peðacÞ 2
2
ðx21 þx22 Þ=2 jx2 b2
ð2pÞ ð1 þ x þ x þ x x
2 1
2 2
2 1
a2 ð1 ix1 jx2 kx1 x2 Þð2pea 2
e
2 ðx2 þx2 Þ=2 1 2
ð2pÞ ð1 þ x þ x þ x x 2 1
2 2
2 1
Þ
2 2Þ
2 2Þ
eix1 b1 Þ
:
ð49Þ
Example 2. The two-dimensional quaternionic Hermite wavelets (compare to [9,10]) are defined by 2
2
2
2
wl ðxÞ ¼ eðx1 þx2 Þ=2 Hl ðxÞ ¼ ð1Þn @ l ðeðx1 þx2 Þ=2 Þ;
l ¼ 1; 2;
ð50Þ
where the two-dimensional quaternionic Hermite polynomials Hn and Dirac operators @ are given by, respectively, 2
2
2
2
Hl ðxÞ ¼ ð1Þn eðx1 þx2 Þ=2 @ l eðx1 þx2 Þ=2
and @ ¼ i
@ @ þj : @x1 @x2
ð51Þ
It is easy to see that Eq. (50) are alternatively real or quaternion-valued. In the following we show that in terms of the QFT them are real-valued. Notice that for l = 1 we have
Z
@ ðx2 þx2 Þ=2 @ ðx2 þx2 Þ=2 ix1 x1 jx2 x2 2 ði e 1 2 þj e 1 2 Þe e d x @x1 @x2 Z Z @ ðx2 þx2 Þ=2 ix1 x1 jx2 x2 2 @ ðx2 þx2 Þ=2 ix1 x1 jx2 x2 2 e e i e 1 2 e d x j e 1 2 e d x ¼ @x1 @x2 R2 R2 Z Z Z Z 2 2 2 2 2 2 ¼ i x1 ex1 =2 eix1 x1 dx1 ex2 =2 ejx2 x2 dx2 þ j x2 ex1 =2 eix1 x1 dx1 ex2 =2 ejx2 x2 dx2 R R R R 2 2
F q fw1 gðxÞ ¼
R2
¼ 2pðx1 þ x2 Þe
x
x
1þ 2 2 2
2
ð52Þ
: 2
@ @ For l = 2 we first observe that @ ¼ @ @ ¼ @x 2 þ @x2 . Using the properties of the (right-sided) QFT in Table 1 we have 2
1
2
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M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
1
3
0.5
2
0
1
-0.5 0
-1 -1.5
-1
-2
-2
-2.5 4
-3 4 2
2 0
ω2
-2 -2
-4 -4
ω1
0
0
4
2
ω2
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3 4
-3 4 2
-2 -4
-4
-2
-4
-4
-2
ω1
0
2
4
0
2
4
2 0
ω2
0 -2 -4
-2
-4
ω1
0
4
2
ω2
-2
ω1
Fig. 2. The quaternion Fourier transform F q fwa;h;b g of the DOG wavelet: the real part and imaginary part i (top row), j, and k (bottom row) of (46), for the scale parameter values c = 1/2, h = 0, b1 = b2 = 1 and a = 1.
F q fw2 gðxÞ ¼ ¼
Z R2
Z R2
! @ 2 ðx2 þx2 Þ=2 @ 2 ðx2 þx2 Þ=2 ix1 x1 jx2 x2 2 e e 1 2 þ 2e 1 2 e d x @x21 @x2 Z @ 2 ðx2 þx2 Þ=2 ix1 x1 jx2 x2 2 @ 2 ðx2 þx2 Þ=2 ix1 x1 jx2 x2 2 1 2 e e e d x þ e 1 2 Þe e d x 2 @x21 R2 @x2 2 2 2 2 2 2
¼ 2pðix1 Þ2 e
x
x
1þ 2 2 2
þ 2pðjx2 Þ2 e
x
x
1þ 2 2 2
¼ 2pðx21 þ x22 Þe
x
x
1þ 2 2 2
:
ð53Þ
3.4. Basic properties Some basic properties of the CQWT are summarized in the following proposition. The properties correspond to classical wavelet transform properties. Their proofs are verified by straightforward calculations and can be found in [15,18,20]. Proposition 2. Suppose that w, / 2 AQW. If w = w0 + iw1 + jw2 + kw3 and / = /0 + i/1 + j/2 + k/3 and if f, gare two quaternion functions belonging to L2 ðR2 ; HÞ, then for every ða; bÞ 2 Rþ R2 we have the following properties. (i) (Left linearity)1
½T w ðaf þ bgÞða; h; bÞ ¼ aT w f ða; h; bÞ þ bT w gða; h; bÞ; where a and b are quaternion constants in H. 1
Restricting the constants to a; b 2 R we get right linearity of the CQWT.
18
M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
(ii) (Translation covariance)
½T w f ð x0 Þða; h; bÞ ¼ T w f ða; h; b x0 Þ for any constant x0 2 R2 . (iii) (Dilation covariance)
½T w f ðcÞða; h; bÞ ¼
1 T w f ðac; h; bcÞ; c
where c is a real positive constant. (iv) (Rotation covariance)
½T w f ðrh0 Þða; h; bÞ ¼ T w f ða; h0 ; rh0 bÞ with rh0 ¼ r h0 rh . (v) (Parity)
½T Pw Pf ða; h; bÞ ¼ T w f ða; h; bÞ; where P is the parity operator defined by Pf(x) = f(x). (vi) (Antilinearity)
þ T / f ða; h; bÞb; ½T awþb/ f ða; h; bÞ ¼ T w f ða; h; bÞa for any quaternion constants a, b in H. (vii) If we introduce the translation operator M x0 wðxÞ ¼ wðx x0 Þ, then
½T Mx0 w f ða; h; bÞ ¼ T w f ða; h; b þ x0 aÞ: (viii) Consider the dilation operator Dc wðxÞ ¼ c12 w xc , c > 0. Then we have
½T Dc w f ða; h; bÞ ¼
1 T w f ðac; h; bÞ: c
3.5. Reproducing formula In this section we show that the quaternion function f can be recovered from its CQWT whenever the quaternion wavelets satisfy the following admissibility condition. Theorem 2 (Inner product relation). Suppose that w = w0 + iw1 + jw2 + kw3 2 AQW be a quaternion admissible wavelet which defines the CQWT (31). If F q fwg 2 L2 ðR2 ; RÞ satisfies the admissibility condition defined by (17). Then for every f ; g 2 L2 ðR2 ; HÞ \ L1 ðR2 ; HÞ we have
Z
Z
Z Rþ
SOð2Þ
dadh 2 T w f ða; h; bÞT w gða; h; bÞd b ¼ C w ðf ; gÞL2 ðR2 ;HÞ : a R2
ð54Þ
Proof. Applying Placherel’s formula for the (right-sided) QFT (12) to the b-integral into the left side of (54) yields (compare €chenig [5]) to Gro
Z
SOð2Þ
Z
Z
Rþ
R2
2 T w f ða; h; bÞT w gða; h; bÞd b dl ¼
Z
1
Z
Z
2 F q ðT w f ða; h; :ÞÞðxÞF q ðT w gða; h; ÞÞðxÞd x dl
ð2pÞ2 SOð2Þ Rþ R2 Z Z Z 1 ð41Þ ^ h ðxÞÞwðar ^ h ðxÞÞg^ðxÞd2 x dl ¼ a2^f ðxÞwðar 2 ð2pÞ SOð2Þ Rþ R2 ! Z Z Z 1 ^f ðxÞ ^ h ðxÞÞj2 dadh g^ðxÞd2 x ¼ j wðar a ð2pÞ2 R2 SOð2Þ Rþ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ Cw
ð2pÞ
C w is a real constant
Z
1 2
^f ðxÞg^ðxÞd x ð12Þ ¼ Cw 2
R2
Z
2
R2
f ðxÞgðxÞd x ¼ C w ðf ; gÞL2 ðR2 ;HÞ : ð55Þ
In the third equality we applied Fubini’s theorem to reverse the integration order. h In particular, if f = g in (54) we have
kT w f k2L2 ðG;HÞ ¼ C w kf k2L2 ðR2 ;HÞ : This shows that, except for the factor Cw, the CQWT is an isometry from L2 ðR2 ; HÞ to L2 ðG; HÞ.
ð56Þ
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M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
Theorem 3 (Inversion formula). Under the assumptions of Theorem 2, any quaternion function f 2 L2 ðR2 ; HÞ can be decomposed as
f ðxÞ ¼
1 Cw
Z
T w f ða; b; hÞwa;h;b dk;
ð57Þ
G
where the integral converges in the weak sense. Proof. An application of Theorem 2 gives for every g 2 L2 ðR2 ; HÞ
Z
C w ðf ; gÞL2 ðR2 ;HÞ ¼
Z
SOð2Þ
Z Z
Rþ
Z R2
Z 2 T w f ða; h; bÞT w gða; h; bÞd b dl ¼ T w f ða; h; bÞT w gða; h; bÞdk 2
T w f ða; h; bÞwa;h;b ðxÞgðxÞd xdk ¼ Z ¼ T w f ða; h; bÞwa;h;b dk; g : ¼
Z
R2
G
Z R2
G 2
T w f ða; h; bÞwa;h;b ðxÞgðxÞdkd x
G
ð58Þ
L2 ðR2 ;HÞ
G
Because the inner product identity holds for every g 2 L2 ðR2 ; HÞ we conclude that
C w f ðxÞ ¼
Z
T w f ða; b; hÞwa;b;h ðxÞdk;
ð59Þ
G
which completes the proof. h Theorem 4 (Reproducing kernel). Suppose that w 2 AQW. If 0
0 K w ða; h; b; a0 ; h0 ; b Þ ¼ C 1 w ðwa;h;b ; wa0 ;h0 ;b ÞL2 ðR2 ;HÞ ;
ð60Þ
then Kw(a, h, b; a0 , h0 , b0 ) is a reproducing kernel in L2 ðG; dkÞ, i.e., 0
T w f ða0 ; h0 ; b Þ ¼
Z
0
T w f ða; h; bÞK w ða; h; b; a0 ; h0 ; b Þdk:
ð61Þ
G
Proof. By inserting (57) into the definition of the CQWT (31) we have 0
T w f ða0 ; h0 ; b Þ ¼ ¼
Z
2
R2
Z
G
f ðxÞwa0 ;h0 ;b0 ðxÞd x ¼
Z T w f ða; h; bÞ C 1 w
R2
Z R2
Z 2 C 1 T f ða; h; bÞw ðxÞdk wa0 ;h0 ;b0 ðxÞd x h w a ;b w G
Z 2 0 wa;h;b ðxÞwa0 ;h0 ;b0 ðxÞd x dk ¼ T w f ða; b; hÞK w ða; h; b; a0 ; h0 ; b Þdk:
ð62Þ
G
The proof is complete. h
4. Uncertainty principle for the CQWT The classical uncertainty principle of harmonic analysis states that a non-trivial function and its FT cannot both be simultaneously sharply localized [22]. 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). That is, increasing the knowledge of the position decreases the knowledge of the velocity or momentum of an electron. This section extends the uncertainty principle which is valid for the QFT to the setting of the CQWT. Let us now formulate an uncertainty principle for the CQWT. This principle describes how the CQWT relates to the (rightsided) QFT of a quaternion function. Theorem 5. Let w 2 L2 ðR2 ; HÞ be an admissible quaternion wavelet that satisfies the admissibility condition (17). If w = w0 + iw1 + jw2 + kw3and assume that F q fwg 2 R, then for every f 2 L2 ðR2 ; HÞ we have the inequality (no summation over k)
1 kbk T w f ða; h; bÞkL2 ðG;HÞ kxk ^f kL2 ðR2 ;HÞ P 2
qffiffiffiffiffiffi C w kf k2L2 ðR2 ;HÞ ;
k ¼ 1; 2:
ð63Þ
In order to prove this theorem, we need to introduce the following lemma. Lemma 4.
Z SOð2Þ
Z Rþ
kxk F fT w f ða; h; Þgk2L2 ðR2 ;HÞ dl ¼ C w kxk ^f k2L2 ðR2 ;HÞ ;
k ¼ 1; 2:
ð64Þ
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M. Bahri et al. / Applied Mathematics and Computation 218 (2011) 10–21
Proof. We observe that
Z
Z Rþ
SOð2Þ
kxk F fT w f ða; h; Þgk2L2 ðR2 ;HÞ dl ¼
Z
Z R2
Z
Z
SOð2Þ
Rþ
ð41Þ
xk F fT w f gF fT w f gxk dld2 x ¼
Z R2
Z SOð2Þ
^ h ðxÞÞwðar ^ h ðxÞÞbf ðxÞxk dadh d2 x ð17Þ a xk ^f ðxÞwðar ¼ a3 Rþ Z ^ h ðxÞÞj2 dadh bf ðxÞd2 x x2k ^f ðxÞjwðar þ a R 2 ^ ¼ C w kxk f kL2 ðR2 ;HÞ : 2
Z R2
Z SOð2Þ
ð65Þ
We begin with the proof of Theorem 5. Proof. Using the uncertainty principle for the (right-sided) QFT (see [16] for more details), we get
h i1=2 h i1=2 1 kbk T w f ða; h; Þk2L2 ðR2 ;HÞ kxk F fT w f ða; h; Þgk2L2 ðR2 ;HÞ P kT w f ða; h; Þk2L2 ðR2 ;HÞ : 2
ð66Þ
Now integrating both sides of (66) with respect to the Haar measure dl, we obtain
Z
h
Z Rþ
SOð2Þ
kbk T w f ða; h; Þk2L2 ðR2 ;HÞ
i1=2 h
kxk F fT w f ða; h; Þgk2L2 ðR2 ;HÞ
Z Z i1=2 1 dl P kT w f ða; h; Þk2L2 ðR2 ;HÞ dl: 2 SOð2Þ Rþ
ð67Þ
By applying the quaternion Cauchy–Schwartz inequality (9) on the left-hand side of (67), we see that
Z
Z SOð2Þ
P
1 2
Z
Rþ
kbk T w f ða; h; Þk2L2 ðR2 ;HÞ ; d
!1=2 Z
l
Z
SOð2Þ
Rþ
SOð2Þ
Z
!1=2 kx
Rþ
2 k ; F fT w f ða; h; ÞgkL2 ðR2 ;HÞ d
l
kT w f ða; h; Þk2L2 ðR2 ;HÞ dl:
ð68Þ
Then, inserting (64) into the second term of (68), we easily obtain
Z
Z SOð2Þ
Rþ
!1=2 kbk T w f ða; h; Þk2L2 ðR2 ;HÞ d
l
Z Z 1=2 1 C w kxk^f k2L2 ðR2 ;HÞ P kT w f ða; h; Þk2L2 ðR2 ;HÞ dl: 2 SOð2Þ Rþ
ð69Þ
We recognize that the first and third terms of (69) are L2 ðG; HÞ-norms. This implies that
kbk T w f ða; h; bÞkL2 ðG;HÞ
qffiffiffiffiffiffi 1 C w kxk ^f kL2 ðR2 ;HÞ P kT w f k2L2 ðG;HÞ : 2
ð70Þ
Substituting (56) into the right-hand side of (70) and simplifying it we finally get
1 kbk T w f ða; h; bÞkL2 ðG;HÞ kxk ^f kL2 ðR2 ;HÞ P 2
qffiffiffiffiffiffi C w kf k2L2 ðR2 ;HÞ ;
ð71Þ
which concludes the proof of Theorem 5. h References [1] T.A Ell, Quaternionic-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: Proceedings of 32nd IEEE Conference on Decision and Control, pp. 148–1841, San Antonio, TX, 1993. [2] S.C. Pei, J.J. Ding, J.H. Chang, Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT, IEEE Trans. Signal Process 49 (11) (2001) 2783–2797. [3] J.P. Antoine, R. Murenzi, Two-dimensional directional wavelet and the scale-angle representation, Signal Process. 52 (3) (1996) 259–281. [4] J.P. Antoine, P. Vandergheynst, Two-dimensional directional wavelet in imaging processing, Int. J. Imag. Syst. Technol. 7 (3) (1996) 152–165. [5] K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser, Boston, 2001. [6] L. Traversoni, Imaging analysis using quaternion wavelet, in geometric algebra with applications, in: E.B. Corrochano, G. Sobczyk (Eds.), Science and Engineering, Birkhäuser, Boston, 2001. [7] E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr. 17 (3) (2007) 497–517. [8] E. Hitzer, B. Mawardi, Clifford Fourier transform on multivector fields and uncertainty principle for dimensions n = 2 (mod 4) and n = 3 (mod 4), Adv. Appl. Clifford Algebr. 18 (3–4) (2008) 715–736. [9] F. Brackx, R. Delange, F. Sommen, Clifford–Hermite wavelets in Euclidean space, J. Fourier Anal. Appl. 8 (3) (2000) 299–310. [10] F. Brackx, F. Sommen, Benchmarking of three-dimensional Clifford wavelet functions, Complex Variables: Theory and Applications 47 (7) (2002) 577– 588. [12] E. Bayro-Corrochano, The theory and use of the quaternion wavelet transform, J. Math. Imag. Vision 24 (1) (2006) 19–35. [13] J. Zhou, Y. Xu, X. Yang, Quaternion wavelet phase based stereo matching for uncalibrated images, Pattern Recogn. Lett. 28 (12) (2007) 1509–1522. [15] S. Mallat, A Wavelet Tour of Signal Processing, second ed., Academic Press, San Diego, CA, 1999. [16] B. Mawardi, E. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl. 56 (9) (2008) 2411–2417. [17] B. Mawardi, E. Hitzer, R. Ashino, R. Vaillancourt, Windowed Fourier transform of two-dimensional quaternionic signals, App. Math. Comput. 216 (8) (2010) 2366–2379.
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[18] B. Mawardi, S. Adji, J. Zhao, Clifford algebra-valued wavelet transform on multivector fields, Adv. Appl. Clifford Algebr. 21 (1) (2011) 13–30. [19] B. Mawardi, E. Hitzer, Clifford algebra Cl3,0-valued wavelet transformation, Clifford wavelet uncertainty inequality and Clifford Gabor wavelets, Int. J. Wavelets Multiresolut. Inf. Process. 5 (6) (2007) 997–1019. [20] L. Debnath, Wavelet Transforms and Their Applications, Birkhäuser, Boston, 2002. [21] J.X. He, Continuous wavelet transform on the space L2 ðR; H; dxÞ, Appl. Math. Lett. 17 (1) (2001) 111–121. [22] H. Weyl, The Theory of Groups and Quantum Mechanics, second ed., Dover, New York, 1950. [23] J. Zhao, L. Peng, Quaternion-valued admissible wavelets associated with the 2-dimensional Euclidean group with dilations, J. Nat. Geom. 20 (1) (2001) 21–32.
