-
On uncertainty principle for quaternionic linear
canonical transform
Kit-Ian Kou
Department of Mathematics, Faculty of Science and Technology,
University of Macau,Macau.
Jian-Yu Ou
Department of Mathematics, Faculty of Science and Technology,
University of Macau,Macau.
Joao Morais
Center for Research and Development in Mathematics and
Applications,Department of Mathematics, University of Aveiro,
Portugal.
accepted to appear in ABSTRACT AND APPLIED ANALYSIS
Abstract
In this paper, we generalize the linear canonical transform
(LCT) to quater-nion-valued signals, known as the quaternionic
linear canonical transform(QLCT). Using the properties of the LCT
we establish an uncertainty prin-ciple for the QLCT. This
uncertainty principle prescribes a lower bound onthe product of the
effective widths of quaternion-valued signals in the spatialand
frequency domains. It is shown that only a 2D Gaussian signal
minimizesthe uncertainty.
Keywords: Quaternionic analysis, quaternionic Fourier
transform,quaternionic linear canonical transform, uncertainly
principle,hypercomplex functions, quantum mechanics, Gaussian
quaternionic signal.
1. Introduction
The classical uncertainty principle of harmonic analysis states
that a non-trivial function and its Fourier transform (FT) cannot
both be sharply local-ized. The uncertainty principle plays an
important role in signal processing[17, 18, 31, 32, 52, 58, 59, 61,
72, 74, 83], and physics [3, 16, 39, 40, 41, 53,
Preprint submitted to Elsevier March 22, 2013
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54, 66, 75, 81]. In quantum mechanics an uncertainty principle
asserts thatone cannot make certain of the position and velocity of
an electron (or anyparticle) at the same time. That is, increasing
the knowledge of the positiondecreases the knowledge of the
velocity or momentum of an electron. Inquaternionic analysis some
papers combine the uncertainty relations and thequaternionic
Fourier transform (QFT) [4, 36, 60].
The QFT plays a vital role in the representation of
(hypercomplex) sig-nals. It transforms a real (or quaternionic) 2D
signal into a quaternion-valuedfrequency domain signal. The four
components of the QFT separate fourcases of symmetry into real
signals instead of only two as in the complexFT. In [67] the
authors used the QFT to proceed color image analysis. Thepaper [7]
implemented the QFT to design a color image digital watermark-ing
scheme. The authors in [6] applied the QFT to image
pre-processingand neural computing techniques for speech
recognition. Recently, certainasymptotic properties of the QFT were
analyzed and straightforward gener-alizations of classical
Bochner-Minlos theorems to the framework of quater-nionic analysis
were derived [24, 25]. In this paper, we study the
uncertaintyprinciple for the QLCT, the generalization of the QFT in
the Hamiltonianquaternionic algebra.
The classical LCT being a generalization of the FT, was first
proposedin the 1970s by Moshinsky and Collins [14, 57]. It is an
effective processingtool for chirp signal analysis, such as the
parameter estimation, samplingprogress for non-bandlimited signals
with nonlinear Fourier atoms [51] andthe LCT filtering [80, 30,
62]. The windowed LCT [46], with a local win-dow function, can
reveal the local LCT-frequency contents, and it enjoyshigh
concentrations and eliminates cross terms. The analogue of the
Poissonsummation formula, sampling formulas, series expansions,
Paley-Wiener the-orem and uncertainly relations are studied in [46,
47]. In view of numerousapplications, one is particularly
interested in higher dimensional analogues toEuclidean space. The
LCT was first extended to the Clifford analysis settingin [45]. It
was used to study the generalized prolate spheroidal wave
functionsand the connection with energy concentration problems
[45]. In the presentwork, we study the QLCT which transforms a
quaternionic 2D signal intoa quaternion-valued frequency domain
signal. Some important properties ofthe QLCT are analyzed. An
uncertainty principle for the QLCT is estab-lished. This
uncertainty principle prescribes a lower bound on the product ofthe
effective widths of quaternion-valued signals in the spatial and
frequencydomains. To the best of our knowledge, the study of a
Heisenberg type un-
2
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certainty principle for the QLCT has not been carried out yet.
The resultsin this paper are new in the literature. The main
motivation of the presentstudy is to develop further general
numerical methods for differential equa-tions and to investigate
localization theorems for summation of Fourier seriesin the
quaternionic analysis setting. Further investigations and
extensions ofthis topic will be reported in a forthcoming
paper.
The article is organized as follows. Section 2 gives a brief
introductionto some general definitions and basic properties of
quaternionic analysis.The LCT of 2D quaternionic signal is
introduced and studied in Section3. Some important properties such
as Parseval and inversion theorems areobtained. In Section 4, we
introduce and discuss the concept of QLCT,and demonstrate some
important properties that are necessary to prove theuncertainty
principle for the QLCT. The classical Heisenberg
uncertaintyprinciple is generalized for the QLCT in Section 5. This
principle prescribesa lower bound on the product of the effective
widths of quaternion-valuedsignals in the spatial and frequency
domains. Some conclusions are drawn inSection 6.
2. Preliminaries
The quaternionic algebra was invented by Hamilton in 1843 and is
de-noted by H in his honor. It is an extension of the complex
numbers to a4D algebra. Every element of H is a linear combination
of a real scalar andthree orthogonal imaginary units (denoted,
respectively, by i, j and k) withreal coefficients
H := {q = q0 + iq1 + jq2 + kq3| q0, q1, q2, q3 R},where the
elements i, j and k obey the Hamiltons multiplication rules
i2 = j2 = k2 = 1; ij = ji = k, jk = kj = i, ki = ik = j.For
every quaternionic number q = q0 + q, q = iq1 + jq2 + kq3, the
scalarand non-scalar parts of q, are defined as Sc(q) := q0 and
NSc(q) := q,respectively.
Every quaternion q = q0 +q has a quaternionic conjugate q = q0q.
Thisleads to a norm of q H defined as
|q| := qq =q20 + q
21 + q
22 + q
23.
3
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Let |q| and ( R) be polar coordinates of the point (q0, q) H
thatcorresponds to a nonzero quaternion q = q0 +q. Quaternion q can
be writtenin polar form as
q = |q|(cos + e sin ), (1)
where q0 = |q| cos , |q| = |q| sin , = arctan(|q|/q0) and e =
q/|q|. If q 0,the coordinate is undefined; and so it is always
understood that q 6= 0whenever = arg q is discussed.
The symbol e e, or exp(e), is defined by means of an infinite
series (orEulers formula) as
ee :=n=0
(e)n
n!= cos + e sin ,
where is to be measured in radians. It enables us to write the
polar form(1) in exponential form more compactly as
q = |q|e e = |q| exp(q
|q| arctan( |q|q0
)). (2)
Quaternions can be used for three- or four-entry vector
analyses. Recently,quaternions have also been used for color image
analysis. For q = q0 + iq1 +jq2 + kq3 H, we can use q1, q2 and q3
to represent, respectively, the R, G,and B values of a color image
pixel, and set q0 = 0.
For p = 1 and 2, the quaternion modules Lp(R2;H) are defined
as
Lp(R2;H) := {f |f : R2 H, fLp(R2;H) :=R2|f(x1, x2)|pdx1dx2
L2(R2;H):=
R2f(x1, x2)g(x1, x2)dx1dx2 (3)
whose associated norm is
fL2(R2;H) :=(
R2|f(x1, x2)|2dx1dx2
)1/2. (4)
4
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As a consequence of the inner product (4), we obtain the
quaternionic Cauchy-Schwarz inequality
|Sc(< f, g >L2(R2;H))| | < f, g >L2(R2;H) |
fL2(R2;H)gL2(R2;H) (5)for any f, g L2(R2;H).
In [12, 42] a Clifford-valued generalized function theory is
developed. Inthe following, we adopt the definition that T is
called a tempered distribution,if T is a continuous linear
functional from S := S(R2) to H, where S(R2) isthe Schwarz class of
rapidly decreasing functions. The set of all tempereddistributions
is denoted by S. If T S, we denote this value for a testfunction by
writing
T [] :=
RT (x1, x2)(x1, x2)dx1dx2,
using square brackets. (In literature one often sees the
notation < T, >,but we shall avoid this, since it does not
completely share the properties ofthe inner product.)
This is equivalent with the one defined in [12] using modules,
and enablesus to define Fourier transforms on tempered
distributions, by the formula
T [] = T [], for all S,which is just to perform Fourier
transform
(1, 2) =
R2(x1, x2)e
i(x11+x22)dx1dx2
on each of the components of the distribution. We will use the
followingresults:
1(1, 2) = (2pi)2(1, 2), (6)
(i||D)(1, 2) = 11 22 , (7)
where = (1, 2), || = 1 + 2, D = ( x1 )1( x2 )2 and is the
usualDirac delta function.
In the following we introduce the LCT for 2D quaternionic
signals.
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3. LCTs of 2D quaternionic signals
The LCT was first introduced in the 70s and is a four-parameter
classof linear integral transform, which includes among its many
special cases,the FT, the fractional Fourier transform (FRFT), the
Fresnel transform, theLorentz transform and scaling operations. In
a way, the LCT has moredegrees of freedom and is more flexible than
the FT and the FRFT, but withsimilar computation cost as the
conventional FT [43]. Due to the mentionedadvantages, it is natural
to generalize the classical LCT to the quaternionicalgebra.
3.1. Definition
Using the definition of the LCT [77, 80], we extend the LCT to
the 2Dquaternionic signals. Let us define the left-sided and
right-sided LCTs of 2Dquaternionic signals.
Definition 3.1. (Left-sided and right-sided LCTs)
Let Ai =
[ai bici di
] R22 be a matrix parameter such that det(Ai) = 1,
for i = 1, 2. The left-sided and right-sided LCTs of 2D
quaternionic signalsf L1(R2;H) are defined by
Lil(f)(u1, x2) :=
1i2pib1R e
i(a12b1
x21 1b1 x1u1+d12b1
u21
)f(x1, x2)dx1, b1 6= 0;
d1e
ic1d12u21f(d1u1, x2), b1 = 0,
and
Ljr(f)(x1, u2) :=
R f(x1, x2)
1j2pib2
ej(a22b2
x22 1b2 x2u2+d22b2
u22
)dx2, b2 6= 0;
f(x1, d2u2)d2e
jc2d22u22 , b2 = 0,
respectively.
Note that for bi = 0 (i = 1, 2), the LCT of a signal is
essentially a chirpmultiplication and it is of no particular
interest for our objective in this work.Hence, without loss of
generality, we set bi 6= 0 in the following sections unlessstated.
Therefore
Lil(f)(u1, x2) =
RK iA1(x1, u1)f(x1, x2)dx1, b1 6= 0 (8)
6
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and
Ljr(f)(x1, u2) =
Rf(x1, x2)K
jA2
(x2, u2)dx2, b2 6= 0, (9)
where the kernel functions
K iA1(x1, u1) :=1
i2pib1ei(a12b1
x21 1b1 x1u1+d12b1
u21
)(10)
and
KjA2(x2, u2) :=1
j2pib2ei(a22b2
x22 1b2 x2u2+d22b2
u22
), (11)
respectively.
3.2. Properties
The following proposition summarizes some important properties
of thekernel functions K iA1 (and K
jA2
) of the left-sided (and right-sided) LCTswhich will be useful
to study the properties of LCTs, such as the PlancherelTheorem.
Proposition 3.1. Let the kernel function KA be defined by (10)
or (11).Then
(i) KA(x, ) = KA(x,);(ii) KA(x,) = KA(x, );
(iii) KA(x, ) = KB(x, ), where B =
(a bc d
);
(iv)RKA1(x, )KA2(, y)d = KA1A2(x, y), where A1A2 corresponds
to
matrix product.
The proofs of properties (i) to (iii) follow from the
definitions (10) and(11). The proof of property (iv) can be found
in [62, 80].
Note that some properties of the LCT for 2D quaternionic signals
followfrom the one dimensional case [62, 77].
Proposition 3.2. Let LAi(f) (i = 1, 2) be defined by Lil in (8),
or L
jr in (9),
respectively. If f, g L1L2(R2;H), then7
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(i) Additivity
LA2(LA1)(f) = LA2A1(f), for LAi := Lil;
LA2(LA1)(f) = LA1A2(f), for LAi := Ljr.
(ii) Reversibility
LAi1(LAi)(f) = f, for LAi := Lil;
LAi1(LAi)(f) = f, for LAi := Ljr.
(iii) Plancherel Theorem (right-sided LCT) If f, g S, then
f, gL2(R2;H) = Ljr(f), Ljr(g)L2(R2;H). (12)
In particular, with f = g, we get the Parseval theorem,
i.e.,
f2L2(R2;H) = Ljr(f)2L2(R2;H).
Proof. By the Fubinis theorem, property (iv) of Proposition 3.1
establishesthe additivity property (i) of left-sided LCTs.
LA2 (LA1 (f) (u1, x2)) (u1, x2)
=
RK iA2(u1, y1)LA1 (f) (u1, x2)du1
=
R
(RK iA2(u1, y1)K
iA1
(x1, u1)du1
)f(x1, x2)dx1
= LA2A1(f)(y1, x2).
The proof of the right-sided LCT LAi := Ljr is similar.
Reversibility property (ii) is an immediate consequence of
additivity prop-erty (i) once we observe that A1 = Ai and A2 =
A
1i .
8
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To verify property (iii), applying Fubinis theorem, it suffices
to see that
Ljr(f), Ljr(g)L2(R2;H)=
R2Ljr(f)(u1, x2)L
jr(g)(u1, x2)du1dx2
=
R4f(x1, x2)K
jA1
(x1, u1)KjA1
(y1, u1) g(y1, x2)du1dx1dy1dx2
=1
2pib1
R4f(x1, x2)e
ja12b1
(x21y21)ej1b1u1(x1y1) g(y1, x2)du1dx1dy1dx2
=
R3f(x1, x2)e
ja12b1
(x21y21)(y1 x1) g(y1, x2)dx1dy1dx2= f, gL2(R2;H),
where we have used (6).
Notice that the left-sided and right-sided LCTs of quaternionic
signalsare unitary operators on L2(R2;H). In signal analysis, it is
interpreted in thesense that (right-sided) LCT of quaternionic
signal preserves the energy of asignal.
Remark 3.1. Note that Plancherel Theorem is not valid for the
two-sidednor left-sided LCT of 2D quaternionic signal. For this
reason, we study theright-sided LCT of 2D quaternionic signals in
the following.
It is worth noting that when A1 = A2 =
[0 11 0
], the left-sided and
right-sided LCTs of f reduce to the left-sided and right-sided
FTs of f . Thatis,
Lil(f)(u1, x2) =12pii
Reix1u1f(x1, x2)dx1 =
12pii
F il (f)(u1, x2)
and
Ljr(f)(x1, u2) =12pij
Rf(x1, x2)e
jx2u2dx2 =12pij
F jr(f)(x1, u2),
respectively. Here
F il (f)(u1, x2) :=
Reix1u1f(x1, x2)dx1
9
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and
F jr(f)(x1, u2) :=
Rf(x1, x2)e
jx2u2dx2 (13)
are the left-sided FT and right-sided FT of f ,
respectively.
We now formulate the linear canonical integral representation of
a 2Dquaternionic signal f .
Theorem 3.1. (Linear Canonical Inversion Theorem) Suppose that f
L1(R2;H), that f is continuous except for a finite number of finite
jumps inany finite interval, and that f(s, t) = 1
2(f(s, t+) + f(s, t)) for all t and s.
Then
f(s, t0) = lim
Lir(f)(s, )K
iA1(, t0)d (14)
for every t0 and s where f has (generalized) left and right
partial derivatives.In particular, if f is piecewise smooth (i.e.,
continuous and with a piecewisecontinuous derivative), then the
formula holds for all t0 and uniformly in s.
Proof. Put
I(s, t0;) :=
Lir(f)(s, )K
iA1(, t0)d
and rewrite this expression by inserting the definition of
Lir(f),
I(s, t0;) =
(Rf(s, t)K iA(t, )dt
)K iA1(, t0)d
=
R
f(s, t)K iA(t, )K
iA1(, t0)ddt
=
Rf(s, t)
[1
2pibei
a2b
(t2t20) ei
1b(t0t)d
]dt
=1
4piei
a2bt20
Rf(s, t)
(ei
a2bt2 sin(
b(t0 t))t0 t
)dt
=1
4piei
a2bt20
Rf(s, t0 u)
(ei
a2b
(t0u)2 sin(bu)
u
)du.
Switching the order of integration is permitted, because the
improper doubleintegral is absolutely convergent over the strip (t,
) R [, ], and in
10
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the last step we have put t0 t = u. Using the formula 0
eia2b
(t0u)2 sin(bu)
udu = 2piei
a2bt20 , for , b > 0, (15)
we can write
1
2piei
a2bt20
0
f(s, t0 u)(ei
a2b
(t0u)2 sin(bu)
u
)du f(s, t0) (16)
=1
2piei
a2bt20
0
(f(s, t0 u) f(s, t0))(ei
a2b
(t0u)2 sin(bu)
u
)du.
Now let > 0 be given. Since we have assumed that f L1(R2;H),
thereexists a number such that
1
2pi
|f(s, t0 u)| du < .
Changing the variable, we find that
eia2b
(t0u)2 sin(bu)
udu =
b
eia2b
(t0 bx )2 sinxx
dx 0, as /b. (17)
The last integral in (16) can be split into three terms:
1
2piei
a2bt20
0
f(s, t0 u) f(s, t0)u
(ei
a2b
(t0u)2 sin(
bu))du
+1
2piei
a2bt20
f(s, t0 u)(ei
a2b
(t0u)2 sin(bu)
u
)du
12piei
a2bt20f(s, t0)
(ei
a2b
(t0u)2 sin(bu)
u
)du = I1 + I2 I3.
The term I3 tends to zero as b > 0 and , because of (17). The
termI2 can be estimated:
|I2| = 12piei a2b t20
f(s, t0 u)(ei
a2b
(t0u)2 sin(bu)
u
)du
1
2pi
|f(s, t0 u)|du .
11
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In the term I1 we have the function g(s, u) = (f(s, t0 u) f(s,
t0))/u.This is continuous except for jumps in the interval R (0, ),
and it has thefinite limit g(s, 0+) =
tfL(s, t0) as u 0; this means that g is bounded
uniformly in s and thus integrable on the interval. By the
Riemann-Lebesguelemma, we conclude that I1 0 as . All this together
gives, since can be taken as small as we wish,
1
2piei
a2bt20
0
f(s, t0u)(ei
a2b
(t0u)2 sin(bu)
u
)du f(s, t0), as /b.
A parallel argument implies that the corresponding integral over
(, 0)tends to f(s, t0+) uniformly in s. Taking the mean value of
these two results,we have completed the proof of the theorem.
Remark 3.2. If Lir(f) L1(R2;H), then (14) can be written as the
absolutelyconvergent integral
f(s, t0) =
RLir(f)(s, )K
iA1(, t0)d
The following lemma gives the relationship between the
left(right)-sidedLCTs and Left(right)-sided FTs of f .
Lemma 3.1. Let Ai =
[ai bici di
] R22 be a matrix parameter such that
det(Ai) = 1, for i = 1, 2. Let f L1(R2;H), we have
Lil(f)(u1, x2) = eid12b1
u21F il
(1
i2pib1eia12b1
()2f(, x2)
)(u1b1, x2
), (18)
and
Ljr(f)(x1, u2) =
(F jr
(f(x1, ) 1
j2pib2eja22b2
()2)(
x1,u2b2
))ejd22b2
u22 . (19)
Proof. By the definition of Lil(f) in (8), a direct computation
shows that
Lil(f)(u1, x2) =
R
1i2pib1
ei(a12b1
x21 1b1 x1u1+d12b1
u21
)f(x1, x2)dx1
= eid12b1
u21
(Reix1 u1b1
(1
i2pib1eia12b1
x21f(x1, x2)
)dx1
)= e
id12b1
u21F il
(1
i2pib1eia12b1
()2f(, x2)
)(u1b1, x2
).
Similarly, by the definition of Ljr(f) in (9), we obtain
(19).
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The LCT can be further generalized into the offset linear
canonical trans-form (offset LCT) [1, 63, 80]. It has two extra
parameters which repre-sents the space and frequency offsets. The
basic theories of the LCT havebeen developed including uncertainty
principles [71, 75], convolution theorem[77, 78], Hilbert transform
[21, 83], sampling theory [51, 77], discretization[33, 43, 70] and
so on, which enrich the theoretical system of the LCT. On theother
hand, since the LCT has three free parameters, it is more flexible
andhas found many applications in radar system analysis, filter
design, phaseretrieval, pattern recognition and many other areas
[62, 77].
4. QLCTs of 2D quaternionic signals
4.1. Definition
This section leads to the quaternionic linear canonical
transforms (QLCTs).Due to the non-commutative property of
multiplication of quaternions, thereare many different types of
QLCTs: two-sided QLCTs, left-sided QLCTs andright-sided QLCTs.
Definition 4.1. (Two-sided QLCTs)
Let Ai =
[ai bici di
] R22 be a matrix parameter such that det(Ai) = 1,
bi 6= 0 for i = 1, 2. The two-sided QLCTs of signals f L1(R2;H)
are thefunctions Li,j(f) : R2 H given by
Li,j(f)(u1, u2) :=R2K iA1(x1, u1)f(x1, x2)K
jA2
(x2, u2)dx1dx2, (20)
where u1, u2 = u1e1 + u2e2, with KiA1
(x1, u1) and KjA2
(x2, u2) given by (10)and (11), respectively.
Definition 4.2. (Left-sided QLCTs)
Let Ai =
[ai bici di
] R22 be a matrix parameter such that det(Ai) = 1,
bi 6= 0 for i = 1, 2. The left-sided QLCTs of signals f L1(R2;H)
are thefunctions Li,jl (f) L(R2;H) given by
Li,jl (f)(u1, u2) :=R2K iA1(x1, u1)K
jA2
(x2, u2)f(x1, x2)dx1dx2,
where the kernels K iA1 and KjA2
are given by (10) and (11), respectively.
13
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Due to the validity of the Plancherel Theorem, we study the
right-sidedQLCTs of 2D quaternionic signals in this paper.
Definition 4.3. (Right-sided QLCTs)
Let Ai =
[ai bici di
] R22 be a matrix parameter such that det(Ai) = 1,
bi 6= 0 for i = 1, 2. The left-sided QLCTs of signals f L1(R2;H)
are thefunctions Li,jr (f) L1(R2;H) given by
Li,jr (f)(u1, u2) :=R2f(x1, x2)K
iA1
(x1, u1)KjA2
(x2, u2)dx1dx2, (21)
where K iA1 and KjA2
given by (10) and (11), respectively.
It is significant to note that when A1 = A2 =
[0 11 0
], the QLCT of f
reduces to the QFT of f . We denote it by
F i,jr (f)(u1, u2) :=R2f(x1, x2)e
ix1u1ejx2u2dx1dx2. (22)
Remark 4.1. In fact, the right-sided QLCTs defined above can be
generalizedas follows:
Le1,e2r (f)(u1, u2) =R2f(x1, x2)K
e1A1
(x1, u1)Ke2A2
(x2, u2)dx1dx2, (23)
where e1 = e1,ii + e1,jj + e1,kk and e2 = e2,ii + e2,jj + e2,kk
so that
e21,i + e21,j + e
21,k = e
22,i + e
22,j + e
22,k = 1 (i.e., e
21 = e
22 = 1)
e1,ie2,i + e1,je2,j + e1,ke2,k = 0.
Equation (21) is the special case of (23) in which e1 = i, and
e2 = j.
Remark 4.2. For bi 6= 0 (i = 1, 2) and f L1(R2;R), the
(right-sided)QLCT of a 2D signal f L1(R2;H) in (21) has the
closed-form representa-tion:
Li,jr (f)(1, 2) := 0(1, 2) + 1(1, 2) + 2(1, 2) + 3(1, 2),
14
-
where we put the integrals
0(1, 2) =
R2f(x1, x2)
1
2pi
kb1b2cos
(a12b1
x21 1
b1x11 +
d12b1
21
)cos
(a22b2
x22 1
b2x22 +
d22b2
22
)dx1dx2,
1(1, 2) =
R2f(x1, x2)
i
2pi
jb1b2sin
(a12b1
x21 1
b1x11 +
d12b1
21
)cos
(a22b2
x22 1
b2x22 +
d22b2
22
)dx1dx2,
2(1, 2) =
R2f(x1, x2)
j
2pi
ib1b2cos
(a12b1
x21 1
b1x11 +
d12b1
21
)sin
(a22b2
x22 1
b2x22 +
d22b2
22
)dx1dx2,
3(1, 2) =
R2f(x1, x2)
k
2pib1b2
sin
(a12b1
x21 1
b1x11 +
d12b1
21
)sin
(a22b2
x22 1
b2x22 +
d22b2
22
)dx1dx2.
These equations clearly show how the QLCT separate real signals
into fourquaternion components, i.e., the even-even, odd-even,
even-odd and odd-oddcomponents of f .
Let us give an example to illustrate expression (21).
Example 4.1. Consider the quaternionic distribution signal, i.e.
the QLCTkernel of (21)
f(x1, x2) = KjA2
(x2, u0)KiA1
(x1, v0). (24)
It is easy to see that the QLCT of f is a Dirac quaternionic
function, i.e.
Li,jr (f)(u1, u2) = (2pi)2(u1, u2 t), t = t1e1 + t2e2.
15
-
4.2. Properties
This subsection describes important properties of the QLCTs that
willbe used to establish the uncertainty principles for the
QLCTs.
We now establish a relation between the right-sided LCTs and the
right-sided QLCTs of 2D quaternion-valued signals.
Lemma 4.1. Let Ai =
[ai bici di
] R22 be a matrix parameter such that
det(Ai) = 1, bi 6= 0 for i = 1, 2. For f L1(R2;H), we have
Li,jr (f)(u1, u2) = Ljr(Lir(f)
)(u1, u2). (25)
Proof. By using the definition of right-sided QLCTs (21),
Li,jr (f)(u1, u2)=
R2f(x1, x2)K
iA1
(x1, u1)KjA2
(x2, u2)dx1dx2
=
RLir(f)(u1, x2)K
jA2
(x2, u2)dx2
= Ljr(Lir(f)
)(u1, u2).
We then establish the Plancherel theorems, specific to the
right-sidedQLCTs.
Theorem 4.1. (Plancherel theorems of QLCTs)For i = 1, 2, let fi
S, the inner product (3) of two quaternionic modulefunctions and
their QLCTs is related by
< f1, f2 >L2(R2;H)=< Li,jr (f1),Li,jr (f2) >L2(R2;H)
. (26)
In particular, with f1 = f2 = f , we get the Parseval identity,
i.e.
f2L2(R2;H) = Li,jr (f)2L2(R2;H). (27)
16
-
Proof. By the inner product (3) and definition of right-sided
QLCTs (21), astraightforward computation and Fubinis theorem shows
that
< Li,jr (f1),Li,jr (f2) >
=
R2
(R2f1(x1, x2)K
iA1
(x1, u1)KjA2
(x2, u2)dx1dx2
)(
R2f2(y1, y2)K iA1(y1, u1)K
jA2
(y2, u2)dy1dy2
)du1du2
=
R6f1(x1, x2)K
iA1
(x1, u1)(KjA2(x2, u2)K
jA2
(y2, u2))
f2(y1, y2)K iA1(y1, u1)dx1dx2dy1dy2du1du2
=
R6f1(x1, x2)K
iA1
(x1, u1)
(1
2pibeja22b2
(x22y22)ej1b2u2(x2y2)
)f2(y1, y2)K iA1(y1, u1)du1dx1dx2dy1dy2du2
=
R5f1(x1, x2)K
iA1
(x1, u1)(eja22b2
(x22y22)(y2 x2))
f2(y1, y2)K iA1(y1, u1)dx1dx2dy1dy2du2
= < Lir(f1), Lir(f2) >=< f1, f2 >
where we have used the Plancherel theorem of right-sided LCTs
(12) andformula (6).
Remark 4.3. Note that Plancherel Theorem is not valid for the
two-sidednor left-sided QLCT of quaternionic signals. For this
reason, we choose toapply the right-sided QLCT of 2D quaternionic
signals in the present paper.
Theorem 4.1 shows that the total signal energy computed in the
spatialdomain equals to the total signal energy in the quaternionic
domain. TheParseval theorem allows the energy of a
quaternion-valued signal to be con-sidered on either the spatial
domain or the quaternionic domain, and thechange of domains for
convenience of computation.
To proceed with, we prove the following derivative
properties.
Lemma 4.2. For i = 1, 2, let Ai =
[ai bici di
] R22 be a matrix parameter,
17
-
bi 6= 0 and aidi bici = 1. If f S, thenR2u2iLi,jr (f)(u1, u2)2
du1du2 = b2i
R2
xif(x1, x2)2 dx1dx2 (28)
Proof. For i = 1, using (6), (7) and Fubinis theorem, we
haveR2u21Li,jr (f)(u1, u2)2 du1du2
=
R2u21
(R2f(s1, s2)K
iA1
(s1, u1)KjA2
(s2, u2)ds1ds2
)(
R2f(x1, x2)K iA1(x1, u1)K
jA2
(x2, u2)dx1dx2
)du1du2
=
R6u21f(s1, s2)K
iA1
(s1, u1)(KjA2(s2, u2)K
jA2
(x2, u2))
K iA1(x1, u1) f(x1, x2)du2ds1ds2dx1dx2du1
=
R5u21f(s1, s2)K
iA1
(s1, u1)(eja22b2
(s22x22)(x2 s2))K iA1(x1, u1)
f(x1, x2)ds1ds2dx1dx2du1
=
R4f(s1, s2)
(u21K
iA1
(s1, u1)K iA1(x1, u1))f(x1, x2)du1ds1dx1dx2
= b21R3f(s1, x2)
(eia12b1
(s21x21) 2
x21(x1 s1)
)f(x1, x2)ds1dx1dx2
= b21R2f(x1, x2)
2
x21f(x1, x2)dx1dx2
= b21
R2
x1f(x1, x2)2 dx1dx2.
To prove the case i = 2, we argue in the same spirit as in the
proof of the
18
-
case i = 1. Applying (6), (7) and Fubinis theorem, we
haveR2u22Li,jr (f)(u1, u2)2 du1du2
=
R2u22
(R2f(s1, s2)K
iA1
(s1, u1)KjA2
(s2, u2)ds1ds2
)(
R2f(x1, x2)K iA1(x1, u1)K
jA2
(x2, u2)dx1dx2
)du1du2
=
R6f(s1, s2)K
iA1
(s1, u1)(u22K
jA2
(s2, u2)KjA2
(x2, u2))
f(x1, x2)K iA1(x1, u1)du2ds1ds2dx1dx2du1
= b22R5f(s1, s2)K
iA1
(s1, u1)
(eja22b2
(s22x22) 2
x22(x2 s2)
)f(x1, x2)K iA1(x1, u1)ds1ds2dx1dx2du1
= b22R5f(s1, x2)K
iA1
(s1, u1)
(2
x22f(x1, x2)
)K iA1(x1, u1)ds1dx1dx2du1
= b22R4f(s1, x2)
(K iA1(s1, u1)K
iA1
(x1, u1)) 2x22
f(x1, x2)du1ds1dx1dx2
= b22R3f(s1, x2)
(eia12b1
(s21x21)(x1 s1)) 2x22
f(x1, x2)ds1dx1dx2
= b22R2f(x1, x2)
2
x22f(x1, x2)dx1dx2
= b22
R2
x2f(x1, x2)2 dx1dx2.
Some properties of the QLCT are summarized in Table 1. Let f1
and
f2 S, the constants and R, Ai =[ai bici di
] R22, bi 6= 0, and
aidi bici = 1.
5. Uncertainty principles for QLCTs
In signal processing much effort has been placed in the study of
the classi-cal Heisenberg uncertainty principle during the last
years. Shinde et al. [72]
19
-
Table 1: Properties of the QLCT.
Property Function QLCTReal linearity f1(x1, x2) + f2(x1, x2)
Li,jr (f)(u1, u2) + Li,jr (f2)(u1, u2)FormulaPlancherel < f1, f2
>L2(R2;H)= < Li,jr (f1),Li,jr (f2) >L2(R2;H)Parseval f
L2(R2;H)= Li,jr (f) L2(R2;H)Derivatives
R2 u
2i
Li,jr (f)(u1, u2)2 du1du2 = b2i R2 xif(x1, x2)2
dx1dx2established an uncertainty principle for fractional Fourier
transforms thatprovides a lower bound on the uncertainty product of
real signal represen-tations in both time and frequency domains.
Korn [44] proposed Heisen-berg type uncertainty principles for
Cohen transforms which describe lowerlimits for the timefrequency
concentration. In the meantime, Hitzer et al.[37, 38, 55, 56]
investigated a directional uncertainty principle for the
Clifford-Fourier transform, which describes how the variances (in
arbitrary but fixeddirections) of a multivector-valued function and
its Clifford-Fourier trans-form are related. On our knowledge, a
systematic work on the investigationof uncertainty relations using
the QLCT of a multivector-valued function isnot carried out.
In the following we explicitly prove and generalize the
classical uncertaintyprinciple to quaternionic module functions
using the QLCTs. We also give anexplicit proof for Gaussian
quaternionic functions (Gabor filters) to be indeedthe only
functions that minimize the uncertainty. We further emphasize
thatour generalization is non-trivial because the multiplication of
quaternionsand the quaternionic linear canonical kernel are both
non-commutative. Forthis purpose we introduce the following
definition.
Definition 5.1. For k = 1, 2, let f, xkf L2(R2;H) and Li,jr (f),
ukLi,jr (f) L2(R2;H). Then the effective spatial width or spatial
uncertainty 4xk of fis evaluated by
4xk :=
Vark(f),
where Vark(f) is the variance of the energy distribution of f
along the xk-axis
20
-
defined by
Vark(f) :=xkf2L2(R2;H)f2L2(R2;H)
=
R2 x
2k|f(x1, x2)|2dx1dx2
R2 |f(x1, x2)|2dx1dx2. (29)
Similarly, in the quaternionic domain we define the effective
spectral widthas
4uk :=
Vark
(Li,jr (f)
),
where Vark(Li,jr (f)) is the variance of the frequency spectrum
of f along the
uk frequency axis given by
Vark(Li,jr (f)) = ukLi,jr (f)2L2(R2;H)Li,j(f)2L2(R2;H)
=
R2 u
2k|Li,jr (f)(u1, u2)|2du1du2
R2 |Li,jr (f)(u1, u2)|2du1du2. (30)
Example 5.1. Let us consider a 2D Gaussian quaternionic function
(Figs.1, 2, 3 and 4) of the form
f(x1, x2) = Ce(1x21+2x22),
where C = Ci0 + iCi1 + jCi2 + kCi3 H, for i = 1, 2, are
quaternionicconstants and 1, 2 R are positive real constants.
Then the QLCT of f is given by
Li,j(f)(u1, u2)= C
(RK iA1(x1, u1)e
1x21dx1
)(Re2x
22KjA2(x2, u2)dx2
)= C
2b1pi
21b1 a1ie(41b12a1i)d1+22b1(41b12a1i) u
21+
(42b22a2j)d2+22b2(42b22a2j) u
22
2b2pi
22b2 a2j .
This shows that the QLCT of the Gaussian quaternionic function
is anotherGaussian quaternionic function.
Figures 1 and 2 visualize the quaternionic Gaussian function for
1 =2 = 3, and 1 = 3 and 2 = 1 in the spatial domain. Figures 3 and
4visualize the quaternionic Gaussian function for 1 = 1 and 2 = 3,
and1 = 2 =
12
in the spatial domain.
21
-
2 1.5 1 0.5 0 0.5 1 1.5 2
2
1.5
1
0.5
0
0.5
1
1.5
22 1.5 1 0.5 0 0.5 1 1.5 2
2
1.5
1
0.5
0
0.5
1
1.5
2
Fig. 1 Fig. 2
2 1.5 1 0.5 0 0.5 1 1.5 2
2
1.5
1
0.5
0
0.5
1
1.5
22 1.5 1 0.5 0 0.5 1 1.5 2
2
1.5
1
0.5
0
0.5
1
1.5
2
Fig. 3 Fig. 4
Now let us begin the proofs of two uncertainty relations.
Theorem 5.1. For k = 1, 2, let f S. Then the following
uncertaintyrelations are fulfilled:
4x14u1 b12
(31)
and
4x24u2 b22. (32)
22
-
The combination of the two spatial uncertainty principles above
leads to theuncertainty principle for the 2D quaternionic signal
f(x1, x2) of the form
4x14x24u14u2 b1b24. (33)
Equality holds in (33) if and only if f is a 2D Gaussian
function, i.e.
f(x1, x2) = e(C1x21+C2x22)/2,
where C1, C2 are positive real constants and =
fL2(R2;H)(C1C2pi2
)1/4.
Proof. Applying (28) in Lemma 4.2 and using Schwarz inequality
(5), wehave (
R2x2k|f(x1, x2)|2dx1dx2
)(R2u2kLi,jr (f)(u1, u2)2 du1du2)
=
(R2x2k|f(x1, x2)|2dx1dx2
)(b2k
R2
xk f(x1, x2)2 dx1dx2
)
b2kR2xkf(x1, x2)
xkf(x1, x2)dx1dx2
2 .Using the exponential form of a 2D quaternionic signal (2),
let
f(x1, x2) = f0(x1, x2) + f(x1, x2) = |f(x1, x2)|e e,where e =
f(x1, x2)/|f(x1, x2)| and = arctan(|f(x1, x2)|/f0(x1, x2)),
then
xkf(x1, x2)
xkf(x1, x2)
= xk|f(x1, x2)|e e xk
(|f(x1, x2)|e e)
= xk|f(x1, x2)|e e[(
xk|f(x1, x2)|
)e e + |f(x1, x2)|
(
xke e)]
= xk|f(x1, x2)|(
xk|f(x1, x2)|
)+ xk|f(x1, x2)|2
(
xk(e)
)=
1
2
xk
(xk|f(x1, x2)|2
) 12|f(x1, x2)|2 + xk|f(x1, x2)|2
(
xk(e)
).
23
-
Therefore,
b2k
R2xkf(x1, x2)
xkf(x1, x2)dx1dx2
2 (34)= b2k
R2
1
2
xk
(xk|f(x1, x2)|2
) 12|f(x1, x2)|2
+xk|f(x1, x2)|2(
xk(e)
)dx1dx2
2 .The first term is a perfect differential and integrates to
zero. The secondterm gives minus one half of the energy
f2L2(R2;H).
Hence (R2x2k|f(x1, x2)|2dx1dx2
)(R2u2k|Li,jr (f)(u1, u2)|2du1du2
) b2k
12f2L2(R2;H)2 = b2k4 f4L2(R2;H).
By definitions of 4xk, 4uk and Parseval theorem (27), we
have
(4xk4uk)2 =(R2 x
2k|f(x1, x2)|2dx1dx2
) (R2 u
2k|Li,jr (f)(u1, u2)|2du1du2
)(R2 |f(x1, x2)|2dx1dx2
) (R2 |Li,jr (f)(u1, u2)|2du1du2
)=
(R2 x
2k|f(x1, x2)|2dx1dx2
) (R2 u
2k|Li,jr (f)(u1, u2)|2du1du2
)f4L2(R2;H)
b2k
4
and therefore we have the uncertainty principle as given by
(31), (32) and(33).
We finally show that the equality in (31), (32) and (33) are
satisfied ifand only if f is a Gaussian quaternionic function.
Since the minimum value for the uncertainty product is bk/2, we
can askwhat signals have that minimum value. The Schwarz inequality
(5) becomesan equality when the two functions are proportional to
each other. Hence,we take g = Cf , where C is a quaternionic
constant and the 1 has beeninserted for convenience. We therefore
have
xkf(x1, x2) = Ckxkf(x1, x2). (35)
24
-
This is a necessary condition for the uncertainty product to be
the minimum.But it is not sufficient since we must also have the
term
R2xk|f(x1, x2)|2
(
xk(e)
)dx1dx2 = 0,
because by equation (34), we see that is the only way we can
actually getthe value of bk/2.
Since Ck is arbitrary we can write it in terms of its scalar and
non-scalarparts, Ck := Sc(Ck) +NSc(Ck). The solution of equation
(35) is hence
f(x1, x2) = e(C1x21+C2x22)/2
= e(Sc(C1)x21+Sc(C2)x
22)/2e(NSc(C1)x
21+NSc(C2)x
22)/2,
for some constant . Since e = (NSc(C1)x21 + NSc(C2)x22)/2, it
followsthat
xk(e) = NSc(Ck)xk.
We have R2xk|f(x1, x2)|2
(
xk(e)
)dx1dx2
= NSc(Ck)2R2x2ke
(Sc(C1)x21+Sc(C2)x22)dx1dx2.
The only way this can be zero is if NSc(Ck) = 0 and hence Ck
must be areal number. We then have
f(x1, x2) = e(C1x21+C2x22)/2, (36)
where C1, C2 are positive real constants since f S and we have
includedthe appropriate normalization = fL2(R2;H)
(C1C2pi2
)1/4.
Since the 2D Gaussian function f(x1, x2) of (36) achieves the
minimumwidth-bandwidth product, it is theoretically a very good
prototype waveform. One can therefore construct a basic wave form
using spatially or fre-quency scaled versions of f(x1, x2) to
provide multiscale spectral resolution.Such a wavelet basis
construction derived from a Gaussian quaternionic func-tion
prototype waveform has been realized, for example, in the
quaternionicwavelet transforms in [5]. The optimal space-frequency
localization is alsoanother reason why 2D Clifford-Gabor bandpass
filters were suggested in[10].
25
-
6. Conclusion
In this paper we developed the definition of QLCT. The various
proper-ties of QLCT such as partial derivative, Plancherel and
Parseval theoremsare discussed. Using the well-known properties of
the classical LCT, we es-tablished an uncertainty principle for the
QLCT. This uncertainty principlestates that the product of the
variances of quaternion-valued signals in thespatial and frequency
domains has a lower bound. It is shown that only a 2DGaussian
signal minimizes the uncertainty. With the help of this
principle,we hope to contribute to the theory and applications of
signal processingthrough this investigation and to develop further
general numerical methodsfor differential equations. The results in
this article are new in the literature.Further investigations on
this topic are now under investigation and will bereported in a
forthcoming paper.
7. Acknowledgement
The first author acknowledges financial support from the
research grantof the University of Macau No. MYRG142(Y1-
L2)-FST11-KKIand the Sci-ence and Technology Development Fund
FDCT/094/2011A. This work wassupported by FEDER funds through
COMPETEOperational ProgrammeFactors of Competitiveness (Programa
Operacional Factores de Competi-tividade) and by Portuguese funds
through the Center for Research andDevelopment in Mathematics and
Applications (University of Aveiro) andthe Portuguese Foundation
for Science and Technology (FCTFundacaopara a Ciencia e a
Tecnologia), within project PEst-C/MAT/UI4106/2011with COMPETE
number FCOMP-01-0124-FEDER-022690. Partial supportfrom the
Foundation for Science and Technology (FCT) via the
post-doctoralgrant SFRH/BPD/66342/2009 is also acknowledged by the
third author.
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