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Advances in Electrical and Computer Engineering
Volume 13, Number 3, 2013
Quantum Image Filtering in the
Frequency Domain
Simona CARAIMAN, Vasile I. MANTA
Department of Computer Engineering, Technical University of Iasi,
D. Mangeron 27, 700050 Iasi, Romania
sarustei@cs.tuiasi.ro
1
Abstract—In this paper we address the emerging field of
Quantum Image Processing. We investigate the use of quantum
computing systems to represent and manipulate images. In
particular, we consider the basic task of image filtering. We
prove that a quantum version for this operation can be
achieved, even though the quantum convolution of two
sequences is physically impossible. In our approach we use the
principle of the quantum oracle to implement the filter
function. We provide the quantum circuit that implements the
filtering task and present the results of several simulation
experiments on grayscale images. There are important
differences between the classical and the quantum
implementations for image filtering. We analyze these
differences and show that the major advantage of the quantum
approach lies in the exploitation of the efficient implementation
of the quantum Fourier transform.
Index Terms—quantum image processing, quantum Fourier
transform, quantum oracle, image filtering.
I. INTRODUCTION
The spectacular perspectives offered by the realization of
a quantum computer have determined an increasing interest
for research in quantum information and quantum
computation. It seems that, in certain cases, the massive
parallelism inherent in quantum computational systems can
lead to exponential speedups over the best classical
approaches [1-3]. Nevertheless, exploiting the remarkable
properties of quantum systems for developing efficient
quantum algorithms is a challenging task. This is due to the
fundamental differences between the operating modes of
quantum and classical computers.
The key role in most of the known quantum algorithms is
played by the quantum Fourier transform. It is the main
ingredient for the efficient quantum order-finding algorithm
introduced by Peter Shor [4]. Other problems such as
performing discrete logarithms and factoring, which for
large numbers are considered intractable on a classical
computer, benefit from similar speedups as they can be
reduced to order finding.
The remarkable properties of quantum systems have led
to the emergence of innovative ideas in all major fields of
computing, including graphics processing. Nevertheless,
speeding up certain signal processing tasks is a rather under
researched area. There is an important potential use of
quantum computation in this field generated by the more
efficient quantum versions of the Fourier transform, wavelet
transform [5] and of the discrete cosine transform [6-7].
1
This work was supported by the project PERFORM-ERA Postdoctoral
Performance for Integration in the European Research Area (ID-57649),
financed by the European Social Fund and the Romanian Government).
The research in quantum image processing is still
confronting with fundamental aspects such as representing
and storing an image on a quantum computer and the basic
processing operations. Representation of color information
on one qubit was proposed for the Qubit Lattice approach
[8] and was also employed in the FRQI framework [9].
Several basic processing operations were defined in the
FRQI framework: geometrical transformations [10], onequbit quantum gates applied on the color wire [11], a
similarity measure between two images based on pixel
differences [12], two strategies for quantum image
watermarking [13, 14].
Beach et al. [15] show that Grover's quantum search
algorithm [16] is applicable to image processing tasks such
as pose recognition in a model-based machine vision
system.
Other important contributions to the quantum imageprocessing field rely on the exploitation of maybe the most
valuable resource of many–qubit quantum systems,
entanglement. It was shown that it could lead to the
development of efficient methods for representing and
retrieving information about the objects in a quantum image
[17] and also to a quantum image compression scheme [18].
In this paper we focus on how to achieve a quantum
version of a rather basic classical task, that of image
filtering. A common approach is to convolve the image with
a filter function, which in the frequency domain translates
into a multiplication operation. However, there are
fundamental differences between classical and quantum
operations, the latter being necessarily invertible due to the
reversible nature of quantum computation. Therefore there
are classical processing operations that cannot be directly
applied to quantum images. Such examples are convolution
and correlation [19]. In our paper we describe a method to
achieve the filtering of a quantum image by exploiting the
quantum Fourier transform and the principle of the quantum
oracle.
Before describing the proposed quantum image-filtering
algorithm, background is given to make the paper selfcontained. We give a short introduction to the basic
concepts in quantum computing and briefly overview the
quantum version of the discrete Fourier transform. In
Section III we describe our approach for representing a
quantum image and then discuss the proposed technique for
quantum image filtering. In Section IV we provide the
results of applying various filters on quantum images by
performing simulation experiments. In Section V we
analyze the features of the proposed quantum algorithm for
image filtering with respect to its classical counterpart and
Digital Object Identifier 10.4316/AECE.2013.03013
77
1582-7445 © 2013 AECE
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Advances in Electrical and Computer Engineering
Volume 13, Number 3, 2013
to the use of other quantum image representation
approaches. In Section VI we summarize our conclusions
and discuss the prospects for further development of the
quantum image-processing field in the light of the new
contributions presented in this paper.
II. BASIC CONCEPTS IN QUANTUM COMPUTING
The quantum analogous of the classical bit is called a
qubit. The states of the qubit can be completely described by
the superposition of two orthonormal basis states, which in
Dirac notation are labelled 0 and 1 (in a Hilbert space
0 , 1,
  C 2 ). This orthonormal system
0  1 0 
with
and 1   0 1 , is called computational
T
T
basis. Any state

can be described by a linear
combination of these two states:
   0   1 with     1 .
(1)
Complex values  and  represent the probability
2
2
amplitudes of the basis states. This means that measuring
the quantum system 
yields 0 with probability 
2
and 1 with probability  .
2
The state of an n -qubit quantum computer is described
by an unit vector in Hilbert space H  C 2 :
n
 
2n 1

i 0
i
2n 1
where

i 0
(2)
i
2
i
 1 and  i
2
represents the probability of
obtaining state i when measuring the register.
A quantum register s is represented by a sequence of
qubits. If s is an n -qubit quantum register and U is an
operator in the states space H , then the operator U (s)
applied to a register is called a quantum gate. Any quantum
operator is a unitary operator and thus any computational
process can be implemented by a sequence of quantum
gates. Elementary quantum gates include single qubit gates
(Not, Hadamard, phase-shift, rotations), controlled gates
(CNOT, CPHASE) and the Toffoli gate [20]. The unitarity
of quantum operators ensures the reversibility of the
computational process. We remark that the number of
outputs of a quantum gate is equal to the number of inputs.
For example, the Not gate ( X ) and the Hadamard gate ( H )
act on a single qubit and can be described by
0 1
(3)
X 
 , X  0   1   0  1 ,
1 0


1

H 0 
0 1

1
1


1

2
H
.
(4)

,
1
2  1 1  
H 1 
0 1

2
The controlled-Not gate ( CNOT ) acts on two qubits, a
target qubit and a control qubit. The former is flipped if and
only if the latter is in state 1 :
00  00 ;
78
01  01 ;
10  11 ;
11  10 .
The action of the controlled-Not gate is x, y  x, x  y ,
where  is addition modulo two, and has the following
matrix representation:
1 0 0 0


0 1 0 0
.
(5)
CNOT  
0 0 0 1


0 0 1 0
The only irreversible operation allowed is the
measurement of quantum states, which has the effect of
collapsing the superposition into one of the computational
basis states.
A direct consequence of the principles of quantum
physics is the immense computing power of a quantum
machine compared to that of a classical one. This is due to
three remarkable quantum resources that have no classical
counterparts: quantum parallelism, quantum interference
and entanglement of quantum states.
These remarkable properties of quantum systems allowed
the formulation of optimal algorithms for two fundamental
problems: integer factorization (Shor's algorithm [4]) and
the search in an unstructured database (Grover's algorithm
[16]). Thus, two main classes of quantum algorithms have
better time complexity than their classical counterparts. The
algorithms in the first class are based on the quantum
Fourier transform and provide remarkable solutions for
solving the factorization and discrete logarithm problems,
with an exponential speedup over the best known classical
algorithms. The algorithms in the second class are based on
the mechanism of quantum amplitude amplification [21]
found in Grover's quantum search algorithm. This class of
algorithms provides a quadratic speedup with respect to the
best classical algorithms.
III. QUANTUM IMAGE FILTERING
A. Representation of Quantum Images
In order to represent the quantum image we will use a
quantum register prepared in a state that encodes both color
and position of a pixel [22]:
2n
m
1 2 1 2 1
Q  C m  P 2 n  n    ij j i .
(6)
2 i 0 j 0
Pixel positions are encoded in register P using 2n qubits,
which are enough to store an image with N  2n  2n pixels.
Register P is in the form y x where y and x encode
the row and column coordinates of a pixel respectively.
Color information for each pixel is represented using
m  log 2 L qubits encoding the L colors (gray levels)
present in the image. Coefficients  ij , with

2m 1
j 0
2
 ij  1
for all i with 0  i  22 n , are used to express the color of a
pixel with position i by means of a superposition of all
possible colors. For a given pixel i coefficients  ij take
value 1 if the color of the pixel is j , and 0 otherwise. This
is illustrated in Fig. 1 with a simple example of a 2  2
image with four colors.
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Advances in Electrical and Computer Engineering
Volume 13, Number 3, 2013
algorithm that computes the discrete Fourier transform on
2 n elements requires exponentially more operations, using
O n2 n gates. Even though the quantum Fourier transform
 
is an efficient quantum algorithm for performing a Fourier
transform of quantum amplitudes, it does not speed up any
task that requires computing the Fourier transform on
classical data. This is because the complete set of Fourier
coefficients are in fact encoded in the amplitudes of the
quantum state given by the right side of (8). As noted in
Section II, the amplitudes in a quantum computer cannot be
directly accessed by measurement. Nevertheless, a more
subtle use of the quantum Fourier transform allows for
exponential speedups to be obtained when solving several
interesting problems. These include the order-finding
problem, the factoring problem, and, in combination with
the quantum search algorithm, the problem of counting the
solutions to a search problem.
Figure 1. Example of a simple 2  2 quantum image with four possible
colors (two qubits are used to represent the color information and two
qubits encode the position of each pixel).
B. The Quantum Fourier Transform and Its Inverse
The quantum Fourier transform (QFT) on an orthonormal
basis 0 , 1 , , N  1 is a linear operator whose action on
a computational basis state is defined by
1 N 1 2 ixk N
(7)
QFTN x 
k .
e
N k 0
It can be easily verified that the transform in (7) is unitary
( QFTN x is normalized to unity and QFTN x is
orthogonal to QFTN x 
unless x  x ). Applied to a
superposition of states x with complex amplitudes  x it
produces
another
superposition
of
states
k
with
amplitudes related to  x by an appropriate discrete Fourier
transform:
 N 1
 N 1
QFTN    x x     k k ,
(8)
 x0
 k 0
where
1 N 1 2 ikx N
k 
(9)
 e x .
N x 0
From the unitarity of the quantum Fourier transform it
follows that its inverse is the Hermitian adjoint of the
quantum Fourier operator, QFT 1  QFT † . It maps the state
given by the right side of (7) to x
and its action on a
computational basis state is described by
1 N 1 2 ixk N
(10)
QFTN1 x 
k .
e
N k 0
The quantum Fourier transform and its inverse can be
efficiently implemented using one-qubit Hadamard gates
and 2-qubit controlled-phase gates. That is, a quantum
circuit that performs the n-qubit Fourier transform needs
O n 2 such quantum gates. In contrast, the best classical
 
C. Quantum Circuit for Image Filtering
In classical image processing image filtering can be
achieved by convolving the input image with a filter. This
can be achieved either in the spatial domain or in the
frequency domain but, in general, the latter approach is
computationally faster, especially as the filter size increases.
This is due to the convolution theorem, which leads to the
following steps for performing the filtering process:
1. Compute F (u, v) , the Fourier transform of the
input image f ( x, y ) ;
2. Multiply the Fourier transformed image with a
filter function H (u, v) ;
3. The filtered image is obtained by computing the
inverse Fourier transform of the result obtained in
step 2.
In order to achieve the filtering of a quantum image we
need to derive the corresponding quantum steps of the above
procedure. Nevertheless, as proved by Lomont [19], the
quantum convolution of two sequences, encoded in the
coefficients of two quantum states, cannot be achieved
directly. It also holds true for the correlation operation. This
is because, due to linearity constraints, there is no physical
process capable of realizing the multiplication step. That is,
for arbitrary quantum states  i ai i and  j b j j there is
no quantum operator to compute the state
N 1
N 1 N 1
i , j 0
k 0 j 0
P
   a j bk  j k
 ai b j ij 
where   1
 ab
i
2
j
(11)
is the normalization factor and
N  2n for some integer n  0 . In order to avoid the
consequences of this result we propose to apply the filtering
step using a quantum oracle provided as a black box. As
shown in Figure 2, at the output of the proposed quantum
filtering circuit we actually find the input image. Thus,
Lomont’s proof is not contradicted but only avoided.
Moreover,
exploiting
the
quantum
interference
phenomenon, we can use an additional qubit initially in state
0 to reinterpret the quantum image as a superposition of
two images. If for example a high pass or a low pass filter is
79
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Advances in Electrical and Computer Engineering
Volume 13, Number 3, 2013
Figure 2. The quantum circuit for image filtering in the frequency domain.
used, the output image is the sum of the image containing
the high frequencies and the image containing the
corresponding low frequencies. The additional qubit can be
used to make the distinction between the two images.
In the following we analyse this process and describe the
state of the quantum image filtering circuit at each step of
the computation as marked in Fig. 2. The input state I 0 is
represented by the input image and an additional qubit in
state 0 :
I0  Q  0 
where
Q
1
2n
2n 1 2n 1 2m 1
  
y 0 x 0 j 0
holds
the
j y x 0 ,
ij
quantum
image
(12)
using
the
representation described in Section III.A. Applying the
quantum Fourier transform on the image produces state
I1 :


I1  I m  QFT22 n Q  I 0
1
2n
2n 1 2n 1 2m 1
1
 n
2
2n 1 2n 1 2m 1


  
y 0 x 0 j 0
  
y 0 x 0 j 0
1
22 n
1
 2n
2
yxj
y
x
0
k
where I1good
(15)
is the state containing the ‘good’ frequencies
is the state containing
that the filter allows to pass, I1bad
the ‘bad’ frequencies that are suppressed by the filter and t
is the number of ‘good’ frequencies. Applying the oracle
operator U H to this superposition one can use the additional
qubit to make the distinction between the two states, where
the action of U H is
UH
kp z 
kp z  H (k , p) .
(16)
I 2  I m  U H I1

1
22 n
2n 1 2n 1 2m 1 2n 1
 
y 0 x 0 j 0 k , p 0
2 i ( yk  xp )
 yxj e
j UH  k p 0
2n

, (17)
filter H (k , p ) is
yxj

  
2n 1
yxj
j
e
2n
2 iyk
2n 1
2n
k
k 0
 
yxj
e
2
n
e
x
0
(13)
2 ixp
2n
p 0
p 0
2 iyk
2n 1 2n 1 2m 1 2n 1
y 0 x 0 j 0 k , p 0
 QFT
j QFT2n y
e
2
n
 QFT  k ik 1  i1i0
j k p 0
thus be re-written:
m
n
n
2 iyk 2 ixp
1 2 1 2 1 2 1
n
n
I 2  2 n      yxj e 2 e 2 j k p 0 
2 y  0 x  0 j  0 k , pSbad
1
 2n
2
2n 1 2n 1 2m 1
 
y  0 x  0 j  0 k , pS good
 yxj e
2 iyk
2 ixp
2n
2n
e
(19)
j k p 1
TABLE 1. THE Sgood AND Sbad FREQUENCY SETS FOR SOME
.
(14)
The next step performed by the quantum circuit is the
equivalent of the classical filtering step. Nevertheless, there
is a fundamental distinction compared to the classical
operation that allows us to avoid the result proved by
Lomont. The state of the register holding the image does not
actually change to a state representing the filtered image but
rather it undergoes an interference process with the
additional qubit initially in state 0 . This is achieved using
a quantum oracle built using the filter function H (k , p ) .
The quantum state I1
1, k , p  S good
H (k , p )  
.
(18)
0, k , p  Sbad
S good and Sbad represent the sets of coordinates for the
‘good’ and ‘bad’ frequencies classified accordingly by the
corresponding filter function. The resulting state, I 2 can
2 ixp
 QFT ik 1    QFT i0
is actually a superposition of two
states, a state representing only the filtered frequencies and a
80
22 n  t
t
I1bad  2 n I1good ,
2n
2
2
qubits unaffected. In our case z is initially 0 and the
where I and I m denote the identity operator on one and m
qubits, respectively. We also used the fact that the quantum
Fourier transform on k qubits is the k -fold tensor product
of k one-qubit quantum Fourier transforms (written
QFT  k ):
QFT2k i
I1 
U H only acts on the position qubits and leaves the color
j QFT22 n
2n 1 2n 1 2m 1
y 0 x 0 j 0
state representing the frequencies removed by the filter:
Filter
Low-pass
COMMON IDEAL FILTERS.
Frequency Sets

 S good  k , p D(k , p )  D0 

 Sbad  k , p D(k , p )  D0 

High-pass

 S good  k , p D(k , p )  D0 

 Sbad  k , p D(k , p )  D0 

Band-pass

S
 k , p DL  D(k , p )  DH 
 good
 Sbad  k , p D(k , p )  DL and D(k , p)  DL 

Band-stop

 S good  k , p D(k , p )  DL and D (k , p )  D

 Sbad  k , p DL  D(k , p )  DH 

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Advances in Electrical and Computer Engineering
Volume 13, Number 3, 2013
Various choices for the filter function H (k , p ) lead to
different values in the ‘good’ and ‘bad’ frequency sets.
Table 1 shows the different sets associated with some ideal
filters.
The last computational step in the quantum circuit in Fig.
2 represents the inverse quantum Fourier transform that
reverts from the frequency to the spatial representation of
the image. The final state of the circuit contains the
superposition of two quantum images: the image containing
the frequencies passed by the filter and the image containing
the frequencies suppressed by the filter. For example, if a
low-pass filter is used, the final image is actually a
superposition of the low-frequencies image and the highfrequencies image. Moreover, the distinction between these
two images can still be made using the additional qubit:
I3 
1  2 1 2 1 2 1
     yxj j QFT21 k QFT21 p 0 
22 n  y0 x0 j 0 k , pS
2 1 2 1 2 1

 j QFT21 k QFT21 p 1 
, (20)
      yxj
y 0 x 0 j 0 k , pS

n
m
n

n
n
bad
n
m
n
n
n
good
1  2 1 2 1 2 1
 j
      yxj
23 n  y 0 x0 j 0 k , pS
n
n

2n 1
m
e
2n 1 2n 1 2m 1
 
yxj
y 0 x 0 j 0 k , pS good
 is  yxj e
where  yxj
can interpret state
2n 1
j
e
2n
e
r
2 ikr
2n 1
2n
2  ipt
2n
t 0 
t 0
e
r
r 0
2 iyk
2n 1
2n
r 0
bad
  
2 ikr

t 1

2 ipt
2n
t 0
2 ixp
e
I3
2n
. By re-arranging the sums we
as the superposition of the two
images:
I3 
1
 3n
2
2n 1 2n 1 2m 1
1
23 n
2n 1 2n 1 2m 1
1
23 n
2n 1 2n 1 2m 1
1
23 n
2n 1 2n 1 2m 1



2 i ( kr  pt )
2n 1 2n 1
     
r  0 t  0 j  0 k , pSbad y  0 x  0
'
yxj
r  0 t  0 j  0 k , pS good y  0 x  0
  
r 0 t 0 j 0
  
r 0 t 0 j 0
2 i ( kr  pt )
2n 1 2n 1
     
j r t 0 
2n
e
'
yxj
2n
e
IV. SIMULATION OF QUANTUM IMAGE FILTERING
j r t 1
,(21)
b
rtj
j r t 0 
g
rtj
j r t 1
where the color of a pixel with position
rt
is
2m 1

j 0
b
rtj
j
in the image containing the suppressed
frequencies, Qbad , and
1
22 n
In this section we present the results of applying the
filtering operation on the gray scale images in Fig. 3. The
simulations were performed in MATLAB by representing
the quantum images in the form of an  ij -matrix according
to (6) and by applying (21) to compute the intensity values
for the pixels in the resulting quantum states that represent
the filtered images, Qbad and Qgood . Three 32 by 32
 Qbad 0  Qgood 1
1
22 n
both images in the final superposition using the quantum
storage and retrieval protocol described by VenegasAndraca and Bose [8]. The disadvantage of this protocol is
that it involves preparing several copies of the input image,
applying the same computational process on each copy and
then sampling using the measurement operator. This is in
fact a statistical procedure that serves for minimizing the
uncertainty in the retrieval process of a quantum parameter.
This uncertainty comes from the probabilistic nature of
quantum measurement.
An alternative for extracting useful information from a
quantum-transformed image is to apply further processing
steps. In fact, this is also the common procedure in classical
image processing. The filtering step is part of a preprocessing stage where the image is enhanced such that
further operations, e.g. segmentation, could be better
applied. Then, other classes of measurements could be
applied on the
processed image that could reveal its
properties without the need to actually ‘see’ the processed
image. For example, the segmentation of the high frequency
image, i.e. the image containing edge information, can
produce a quantum state representing only the pixels
belonging to the edges of an object. Variations of quantum
searching, quantum counting and/or quantum integral
estimation can then be applied to compute useful statistics
associated to the shape of the object, e.g. perimeter, area,
etc. Such information could then be determined by
measuring the final quantum state. The result of the
measurement can then be used for comparing two images in
a content-based image retrieval system. It follows that
implementing such a system quantumly can bring a
significant speedup compared to the classical variant. This is
due both to the exponential speedup of the quantum version
of the Fourier transform, but also due to the speedup brought
by the quantum search algorithm.
2m 1

j 0
g
rtj
j
in the image with
the frequencies passed through by the filter, Qgood .
In classical image processing the final stage usually
assumes extracting information from the result by looking at
the image. This, however, cannot be achieved in the
quantum domain because the resulting image(s) are stored in
quantum states. A measurement on the final state I 3 only
samples from the transformed image, revealing only the
color at a single pixel position. However, one can retrieve
pixels images were used: a synthetic image containing only
two gray levels (Fig. 3 left) and two sub-images of a
microscopy image (Fig. 3 right). For each image, the color
information for each pixel is represented using m  8 qubits
to encode 256 possible gray levels and the pixel positions
are encoded using 2n  10 qubits.
The synthetic image was filtered using a high pass filter
with D0  6.4 and the resulting high- and low-frequency
images are shown in Fig. 4, corresponding to the Qgood
and Qbad
states, respectively. Scaled versions of the 32 by
32 images are presented. As expected, the high-pass filter
emphasizes a horizontal edge in the middle of the image and
the low-pass filter has a blurring effect. The horizontal
bands in the filtered images are due to the ringing effect of
81
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Advances in Electrical and Computer Engineering
Volume 13, Number 3, 2013
the ideal filter.
Figure 3. Test images. Left – synthetic image, 32  32 pixels, with two
gray levels. Right – Microscopy image of a tissue section displaying cell
nuclei as bright round objects; A,B are 32  32 pixels sub-images used for
testing purposes.
Figure 4. Result of filtering the synthetic image in Fig. 3 with a high-pass
filter, D0  0.2  32  6.4 . Left – image containing the corresponding high
Fourier transform. For comparison and displaying purposes,
the classical representation of the quantum high- and lowfrequency images is extracted from the corresponding
quantum superpositions by multiplying the amplitudes, i.e.
the  coefficients, with the integer encoding of the state
vectors. Just like in the classical case, negative floating point
values are also obtained for the gray levels in the filtered
images. The minimum value in the image is displayed as
black, the maximum value is displayed as white and the
values in between are displayed as intermediate shades of
gray, using 256 gray levels. The gray level for pixel (3,3) in
the high-frequency image is represented as 105 and in the
low-frequency image as 246.
The results of filtering sub-images A and B of the
microscopy image are presented in Fig. 5 and Fig. 6. As
only four gray levels are present in the original sub-image
A, the quantum representation of each pixel in the filtered
images is expressed as a superposition of four basis vectors.
The gray levels and the corresponding amplitudes in the
quantum representation of pixel (3,3) of sub-image A are
detailed in Table III. The original sub-image B contains 189
different gray levels. The quantum amplitudes of the
superposition representing the intensity value of the same
pixel in sub-image B are presented in Fig. 7.
frequencies. Right – image containing the corresponding low frequencies.
TABLE II. QUANTUM REPRESENTATION OF THE PIXEL WITH COORDINATES
(3,3) IN THE SYNTHETIC IMAGE IN FIG. 3, FILTERED WITH A HIGH-PASS
FILTER WITH D0  0.2  32  6.4
input
image
high-freq.
image ( Qgood )
low-freq.
image ( Qbad )
gray level
170
-3.91
173.91
C
1 170
8
image containing the corresponding high frequencies. Right - image
containing the corresponding low frequencies.
0.0559 100
-0.0559 100
-0.0559 170
1.0559 170
In the filtered images, the intensity value of each pixel is
represented by a quantum superposition of the two basis
vectors corresponding to the gray levels in the original
image. For example, for pixel with coordinates (3,3), the
gray levels and the corresponding quantum representation in
each image – original, high- and low-frequency images – are
detailed in Table II. The resulting quantum state,
corresponding to the representation in (21), is a normalized
state:

g
rtj

b 2
rtj
1,
Figure 5. Result of filtering sub-image A of the cell image in Fig. 3 with a
high-pass filter, D0  0.2  32  6.4 . Left – original sub-image A. Center -
(22)
TABLE III. QUANTUM REPRESENTATION OF THE PIXEL WITH COORDINATES
(3,3) IN SUB-IMAGE A OF THE CELL IMAGE IN FIG. 3, FILTERED WITH A HIGHPASS FILTER WITH D0  0.2  32  6.4
input
image
high-freq.
image
low-freq.
image
gray level
0
1.3942
-1.3942
C
10
-0.0262 0
1.0262 0
-0.0008 1
0.0008 1
0.0079 51
0.0079 51
0.0191 52
0.0191 52
8
j
and is equivalent to the quantum representation of the gray
level in the original image. This is consistent with the
classical case where the summation of the corresponding
high- and low-frequency images gives the original image.
In general, the inverse Fourier transform is complex.
However, when both the input image and the filter are realvalued, the imaginary part of the inverse transform should
be zero. In practice, the imaginary components appear to be
non-zero usually due to computational approximations and
need to bee ignored. Thus, in the filtered image, the 
coefficients are taken to be the real part of the inverse
82
Figure 6. Result of filtering sub-image B of the cell image in Fig. 3 with a
high-pass filter, D0  0.2  32  6.4 . Left – original sub-image B. Center image containing the corresponding high frequencies. Right - image
containing the corresponding low frequencies.
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Advances in Electrical and Computer Engineering
Figure 7. Quantum representation of the pixel with coordinates (3,3) in subimage B of the cell image in Fig. 3 filtered with a high-pass filter with
D0  0.2  32  6.4 . The gray level of the pixel is 0.2788 and is represented
by a superposition of quantum states that correspond to all the gray levels in
the original sub-image.
V. DISCUSSION
The convolution operation in the spatial domain
corresponds to multiplication in the frequency domain.
Thus, for spatial invariant linear filters, convolution can be
implemented efficiently using the discrete Fourier
transform. Despite the obvious speedup that could be
introduced by the quantum version of the Fourier transform,
a quantum implementation for the filtering operation in the
frequency domain is not straightforward. This is due to the
fact that the quantum convolution of two sequences,
encoded in the coefficients of two quantum states, is
physically impossible. Thus, in order to develop a quantum
version of the image filtering operation we devise a
workaround to this problem. Our solution is based on the
use of a quantum oracle. An oracle is a quantum circuit that
‘recognizes’ solutions to a given problem. It is supplied as a
black box that provides a yes-no answer to a specific
question. In our case the oracle operator is used to
implement the filter function. Thus, we are able to avoid the
impossibility of the quantum convolution by working with
the image containing both ‘good’ and ‘bad’ frequencies as
classified by the filter. The trick is to reinterpret this image
as the sum of two filtered images: the image containing only
the ‘good’ frequencies and the image containing only the
‘bad’ frequencies. This way we don’t need to actually
convolve the image with the filter function but only use the
filter to distinguish between the two component images. The
output state of the quantum circuit implementing the
filtering process is a superposition of the two quantum
images.
The main advantage of our quantum implementation for
the image filtering operation is related to the efficiency of
the quantum Fourier transform. It requires exponentially less
operations than the classical fast Fourier transform.
However, this speedup is usually difficult to be exploited
because the complete set of Fourier coefficients are encoded
as amplitudes of quantum states and cannot be directly
accessed. In our method we are able to preserve the speedup
of the quantum Fourier transform as the Fourier coefficients
need not be extracted.
In classical information processing the result of image
filtering is represented by an image containing only the
frequencies passed through by the filter. The result of our
quantum image filtering method is different from the
classical counterpart under several aspects. In the classical
case, the original image cannot be reconstructed out of the
Volume 13, Number 3, 2013
filtered image. In contrast, the quantum filtering process is
reversible because only unitary operators are used to act
upon the system. Another contrast regards the retrieval of
the filtered image(s). In the classical case the process is
straightforward, while the quantum representation of the
filtered image(s) encodes this information in the amplitudes
of the quantum states. A statistical procedure would then be
necessary to extract these amplitudes by measurement. Such
a procedure involves preparing several copies of the input
image and applying the same computational process on each
copy. However, in most image processing applications,
filtering is a pre-processing step and its result is usually an
input for several other operations. Thus, the retrieval of the
filtered image is not necessary. It can be further processed
and useful information can be extracted at the end of the
quantum computational process as discussed in Section
III.C. Further processing can be applied on either of the two
images in the quantum superposition represented by the
final state of the quantum filtering circuit. This can be
achieved making use of the oracle qubit and controlled
operators. This qubit distinguishes between the two quantum
states. It is flipped if the value of the filter function
implemented by the oracle is one and remains unchanged
otherwise. Thus it can be used as a control qubit to
manipulate one image or the other.
In the quantum protocol for image filtering proposed in
this paper the color of a pixel in the quantum image is
quantized and represented with an appropriate number of
qubits, much like in the classical case ( m  log 2 L qubits are
needed to represent the L colors of a gray level image). The
same m qubits are used to store the colors of all the pixels
in the image, which is possible by efficiently exploiting the
properties of quantum superpositions. This, in fact, allows
for an overall exponentially lower memory space to be used
than in the classical case ( m  2n qubits are needed to store
a 2n  2n image with L gray levels compared to m22 n
classical bits).
Another approach for encoding the color information
would be to use a single qubit as suggested by VenegasAndraca and Bose [8] and Le et al. [9]. It relies on the
definition of a machine capable of detecting electromagnetic
waves and producing an initialized qubit that encodes the
frequency of the electromagnetic wave in the phase
parameter. Le et al. use single qubit unitary operators to
achieve basic color processing operations such as color
inversion [11]. However this representation restricts the
possible processing operations that can be defined in the
quantum domain. For example, using our representation, the
mechanism of quantum amplitude amplification can be
employed to define quantum procedures for histogram
computation and threshold-based segmentation [22]. The
one-qubit color representation is not suited for such a
purpose because the quantum states representing pixels with
a given color are not computational basis states. Even
though in our representation the necessary state space is
larger than in the representation suggested in [8], we
consider that the broader possibilities for processing
operations are much more valuable. Still, we find that the
mechanism for image filtering described in this paper is
straightforward to be employed on the one-qubit
83
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Advances in Electrical and Computer Engineering
representation. Moreover, we envision that a more advanced
quantum image processing field will exploit various image
representation models. This is much like in the classical case
where one uses different image formats and color models
depending on the purpose of the image manipulation process
(we use compressed formats for image transmission, the
RGB color model for image display, the L*a*b* color
model for color analysis, etc.).
Volume 13, Number 3, 2013
applications for quantum information processing. It would
also represent an overall justification for the immense
efforts needed for building working quantum computers.
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VI. CONCLUSIONS
In this paper we addressed the field of Quantum
Information Processing by analyzing the prospects of
applying quantum computation concepts to image
processing tasks. Specifically, we developed a quantum
version for the image filtering operation. This is an
important technique that comes up in many image
processing applications. It is usually applied in the early
stages of vision processing for image enhancement. For
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characteristics introduced in the acquisition process (e.g.
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Our quantum version for the image filtering operation
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The two components can be efficiently distinguished and
exploited using the oracle qubit.
The result obtained in this paper offers a promising
perspective on the investigation of more such applications of
quantum computing to image processing tasks. The
developments in the field of Quantum Image Processing are
still in an early stage. We believe that the contributions to
this field are not a mere digression in a hot topic as many
may think, but they could significantly impact the overall
advent of the main framework of Quantum Information
Processing. Interesting ideas could arise for the design of
new quantum algorithms and the development of novel
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