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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1
Fractional Krawtchouk Transform With an Application to Image Watermarking
1
2
Xilin Liu, Guoniu Han, Jiasong Wu, Zhuhong Shao, Gouenou Coatrieux, and Huazhong Shu
3
Abstract—This paper proposes a novel fractional transform,
4
denoted as the fractional Krawtchouk transform (FrKT), a gen-
5
eralization of the Krawtchouk transform. The derivation of the
6
FrKT uses the eigenvalue decomposition method. We determine the
7
eigenvalues and the corresponding multiplicity of the Krawtchouk
8
transform matrix. Moreover, the orthonormal eigenvectors of the
9
transform matrix are derived. For validation purpose only and as
10
a first illustration of the interest of FrKT, a watermarking exam-
11
ple was chosen. Experimental results show that better watermark
12
robustness and imperceptibility are achieved by adjusting the frac-
13
tional orders in the FrKT.
14
Index Terms—Krawtchouk transform, fractional Krawtchouk
15
transform, weighted Krawtchouk polynomials, watermarking.
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I. INTRODUCTION 17
T
HE continuous fractional Fourier transform (FrFT), a gen-18
eralization of the Fourier transform, depends on an ad-
19
ditional parameter and can be interpreted as a rotation in the
20
time-frequency plane. The FrFT has been investigated and ap-
21
plied in quantum mechanics [1], [2] and signal processing fields
22
[3]–[5]. Such applications require the derivation of the discrete
23
fractional Fourier transform (DfrFT). Pei et al. [6], [7] proposed
24
a definition of the DfrFT based on the eigenvalue decompo-
25
sition of the transform matrix. Their method provides similar
26
transform as that of the continuous case. Moreover, the proposed
27
Manuscript received July 19, 2016; revised November 20, 2016 and December 26, 2016; accepted January 4, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Sergio Lima Netto.
This work was supported in part by the National Natural Science Foundation of China under Grants 61201344, 61271312, 61401085, 81101104, and 61073138, in part by the Ministry of Education of China under Grants 20110092110023 and 20120092120036, the Project-sponsored by SRF for ROCS, SEM, and in part by the Natural Science Foundation of Jiangsu Province under Grants BK 2012329, BK2012743, DZXX-031, and BY2014127-11, in part by the ‘333’ project under Grant BRA2015288, in part by the Qing Lan Project and Young Core Personal Project of Beijing Outstanding Talent Training Project 2016000020124G088.
X. Liu, J. Wu, and H. Shu are with the Laboratory of Image Science and Technology, School of Computer Science and Engineering, Southeast Univer- sity, Nanjing 210018, China, and also with the Centre de Recherche en Informa- tion Biom´edicale sino-franc¸ais, Nanjing 210096, China (e-mail: xilinliu168@
163.com; jswu@seu.edu.cn; shu.list@seu.edu.cn).
G. Han is with the Institute de Recherche en Math´ematiques et Applica- tion, Universit´e de Strasbourg et CNRS, Strasbourg 67084, France (e-mail:
guoniu.han@unistra.fr).
Z. Shao is with the College of Information Engineering, Capital Normal University, Beijing 100048, China (e-mail: shaozh2015@163.com).
G. Coatrieux is with the Latim, Inserm U 1101, Institut Mines- Telecom, Telecom Bretagne, Brest 29238, France (e-mail: gouenou.coatrieux@
Q1
telecom-bretagne.eu).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2017.2652383
DfrFT has important unitary and rotation properties. To achieve 28
better DfrFT results, the eigenvectors of the Fourier transform 29
matrix should better approximate the Hermite-Gaussian func- 30
tions which are the eigenfunctions of the FrFT [8]. Some other 31
definitions of DfrFT can be found in [9]–[11]. The DfrFT has 32
been applied in optical image encryption [12]. An overview 33
of the DfrFT in signal processing can be found in [13]. Some 34
other forms of fractional transforms have been developed, such 35
as the discrete fractional Hadamard transform, discrete frac- 36 tional Hilbert transform, discrete fractional cosine and sine 37
transforms [14]–[19]. 38
Yap et al. [20] introduced the Krawtchouk transform (also 39
known as Krawtchouk moments), another orthogonal trans- 40
form using the weighted Krawtchouk polynomials. Since the 41
weighted Krawtchouk polynomials are discrete, there is no nu- 42
merical approximation in deriving the transform coefficients. 43
By adjusting the parameters in the weighted two dimensional 44
(2-D) Krawtchouk polynomials, local image features can be 45
located and described. The Krawtchouk transform has been 46
successfully applied in image reconstruction and image wa- 47
termarking [21]–[23]. Yap et al. [21] used the Krawtchouk 48
transform for image reconstruction and it was shown that the 49
Krawtchouk transform outperforms other moments in terms of 50
reconstruction error. Venkataramana et al. [22] designed a wa- 51
termarking algorithm which is robust to geometric attack. Pa- 52
pakostas et al. In [23], a transform domain watermarking al- 53
gorithm based on Krawtchouk transform was proposed which 54
embeds the watermark in local regions of the image. Atakishiyev 55
et al. [24] derived a new transform known as fractional Fourier- 56
Krawtchouk transform, where the kernel function is the product 57
of an exponential function with parameterαand the Krawtchouk 58
function. 59
Our goal in this paper is to derive the fractional Krawtchouk 60
transform (FrKT) by using the eigenvalue decomposition 61
method of the transform matrix. Noticing that the eigenval- 62
ues of the transform matrix are 1 and –1, we determine the 63
multiplicity of each eigenvalue. Furthermore, the orthonormal 64
eigenvectors corresponding to the eigenvalues are derived us- 65 ing spectral decomposition and SVD. Finally, we provide the 66
definition of FrKT. The FrKT, which is a generalization of the 67
Krawtchouk transform, has two additional parameters, known as 68
the fractional orders. As an illustration of the FrKT applications, 69
a watermarking scheme for copyright protection is investigated. 70
Our intent is to assess the interest of FrKT and the benefits 71
that can be expected through a proper choice of the fractional 72
orders. 73
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The rest of the paper is organized as follows. In Section II,
74
some preliminaries about Krawtchouk transform are provided.
75
The detailed procedure of deriving the FrKT and some prop-
76
erties of FrKT are elaborated in Section III. The image water-
77
marking scheme is reported in Section IV. Experiments to test
78
its robustness are carried out in Section V. Finally, Section VI
79
concludes the paper.
80
II. PRELIMINARIES 81
The 1-D Krawtchouk transform in terms of weighted
82
Krawtchouk polynomial is defined as [20]:
83
Qn =
N−1 x= 0
Kn(x;p, N−1)f(x), n = 0, 1, . . . , N− 1 (1) where f(x) is an 1-D signal of length N, and Kn(x;p,
84
N−1) is the n-th order weighted Krawtchouk polynomial,
85
defined as
86
Kn(x;p, N−1) =κn(x;p, N−1)
w(x;p, N−1) ρ(n;p, N−1) (2) with
87
w(x;p, N−1) =
N−1 x
px(1−p)N−1−x, (3) ρ(n;p, N−1) =
p−1 p
n n!
(−N+ 1)n. (4) Andκn(x;p, N −1)is the classical Krawtchouk polynomial
88
κn(x;p, N−1) =2F1
−n,−x;−N+ 1;1 p
, p∈(0, 1) (5) where2F1is the hypergeometric function, defined as
89
2F1(a, b;c;z) = ∞ k= 0
(a)k(b)k (c)k
zk
k! (6)
and(a)k is the Pochhammer symbol given by
90
(a)k =a(a+ 1). . .(a+k−1) = Γ (a+k)
Γ (a) . (7) The weighted Krawtchouk polynomialKn(x;p, N−1)sat-
91
isfies the following orthogonal property
92
N−1 x= 0
Kn(x;p, N−1)Km(x;p, N −1) =δn m. (8) This leads to the following inverse transform
93
f(x) =
N−1 n= 0
QnKn(x;p, N−1). (9)
For anN×N imageg(x, y), the forward and inverse two 94
dimensional transformations are respectively given by 95
Qn m =
N−1 x= 0
N−1 y= 0
Kn(x;p, N−1)Km(y;q, N−1)g(x, y) (10) g(x, y) =
N−1
n=0 N−1
m=0
Qn mKn(x;p, N−1)Km(y;q, N−1). (11)
III. CONSTRUCTION OF THEFRACTIONALKRAWTCHOUK 96
TRANSFORM 97
In this section, we propose a novel discrete fractional 98
transform known as fractional Krawtchouk transform (FrKT). 99
Similarly to the development of DfrFT, the eigenvalue decom- 100
position of Krawtchouk transform matrix will be used to define 101
the discrete FrKT. The eigenvalues and eigenvectors of FrKT 102
are derived and some properties of FrKT are also investigated. 103
A. Eigenvalues and Eigenvectors of the Krawtchouk 104
Transform Matrix 105
The 1-D Krawtchouk transform for signalf(x)of lengthN 106
defined in (1) can be written in the following matrix form 107
Q=Kf (12)
where the transform matrixKis defined by 108
Kn ,x =Kn(x;p, N−1), 0≤n, x≤N−1. (13) The Krawtchouk transform matrix K has the following 109
properties: 110
Property 1: Kis symmetric. 111
This is straightforward from the definition of Krawtchouk 112
polynomial (2) and (13). 113
Property 2: K is orthogonal, that is KTK=KKT =I, 114
withIthe identity matrix [20]. 115
Property 3: The eigenvalues ofKare 1 and –1. 116
Proof: Letλbe an eigenvalue of K andu∈RN×1 be its 117
corresponding eigenvector, that is 118
Ku=λu. (14) Then, with properties 1 and 2, we have 119
u=KKu=λKu=λ2u. (15)
Thus 120
λ2−1
u= 0 (16) and the two eigenvalues ofKareλ0 = 1,λ1 =−1. 121
With these properties ofK, we have the following theorem 122
to determine the multiplicities of its eigenvalues. 123
Theorem 1: For the Krawtchouk transform matrixKof size 124
N×N, if N is even, then the multiplicities of eigenvalues 125
λ0 =1 andλ1 =−1 are equal. IfN is odd, the multiplicity of 126
eigenvalueλ0=1 is equal to one more than that of eigenvalue 127
λ1 =−1. 128
Proof: See Appendix. 129
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LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 3
Theorem 1 specifies the eigenvalue multiplicity ofK. Next,
130
we construct a set of orthonormal eigenvectors of matrix K
131
corresponding to the two eigenvalues.
132
From properties 1 and 2, we know thatKis a symmetric and
133
orthogonal real matrix. According to the spectral theorem [25],
134
we know thatKhas the following spectral decomposition
135
K=λ0P0+λ1P1 (17) wherePi, i = 0,1, is the orthogonal projection matrix on the
136
ith eigenspace ofK, andλiis the ith eigenvalue ofK. Then, the
137
expression of projection matrices can be derived as follows.
138
Since for any integerm, there is Kmx = λmx, hence the
139
matrixKm has the same eigenvectors and consequently pro-
140
jection matrices as matrixK. Note thatK0 = I, therefore, we
141
have
142
Km =λm0 P0+λm1 P1, m= 0,1,2, ... (18) To obtain the two matricesP0 andP1, we write the above
143
equation form = 0, 1 in matrix form as
144
A P0
P1 = I
K (19)
with
145
A=
I I
λ0I λ1I . (20) Notice that
146
AAT =
2I (λ0+λ1)I (λ0+λ1)I (λ20+λ21)I = 2
I 0
0 I . (21) From (21), we deduce that the inversion ofAis
147
A−1= 0.5AT. (22) MultiplyingA−1 in (22) on both left sides of (19), we have
148
P0
P1 = 0.5AT I
K = 0.5
I λ0I I λ1I
I
K . (23) Thus
149
P0 = 0.5 (I+K) (24) P1 = 0.5 (I−K). (25) From (24) and (25), we can get some properties of P0
150
andP1.
151
Property 4: PTi =Pi, i=0, 1.
152
Property 5: [26]:P20 =P0,P21 =P1.
153
Property 6: P0 andP1 are orthogonal, that isP0P1 =0,
154
where0denotes the zero matrix.
155
Lemma 1: Both matrices P0 and P1 have eigenvalues
156
0 and 1. Moreover, the multiplicity of eigenvalue 1 forP0 is
157
equal to the multiplicity of eigenvalue 1 ofK; the multiplicity
158
of eigenvalue 1 forP1is equal to the multiplicity of eigenvalue
159
–1 ofK.
160
Proof: The proof of the first part of this lemma can be found
161
in [27]. Let γ, ηandλbe respectively the eigenvalues of the
162
matricesP0,P1 andK, using (24) and (25), we have 163
|γI−P0| =|γI−0.5(K+I)|
=|(γ−0.5)I−0.5K|
= 0.5N |(2γ−1)I−K|
= 0. (26)
Similarly, we can obtain 164
|ηI−P1|= 0.5N |(2η−1)I+K|= 0 (27)
and 165
|λI−K|= 0. (28)
From (26)–(28), we have 166
2γ−1 =λ (29)
−(2η−1) =λ. (30) Hence, ifλ=1, there isγ=1, η=0, and ifλ=−1, then 167
γ=0, η=1. 168
The proof of Lemma 1 has now been completed. 169 Lemma 2: The eigenvectors corresponding to nonzero 170
eigenvalues of P0 are orthogonal to those corresponding to 171
nonzero eigenvalues ofP1. 172
Proof: Let the size of P0 be N×N, and uiandvj be 173
eigenvectors corresponding to nonzero eigenvalues ofP0 and 174
P1, respectively (from Lemma 1 and Theorem 1, we know 175
that if N is even, i.e., N = 2m, then i, j=1, 2, ..., m, and 176
if N is odd, i.e.,N = 2m+1 then i=1, 2, ..., m+1, j= 177
1, 2, ..., m), then we have 178
P0ui=ui (31)
and 179
P1vj = vj. (32)
With (31) and (32), we can write 180
(ui)Tvj = (P0ui)T (P1vj) = (ui)TPT0P1vj. (33) SincePT0P1 =0from property 6, it follows from (33) that 181
(ui)Tvj =0. (34) The proof of Lemma 2 is thus obtained. 182
Lemma 3: The eigenvectors corresponding to nonzero 183
eigenvalues of P0 andP1 are the eigenvectors of K, corre- 184
sponding to eigenvaluesλ0 =1,λ1 =−1 ofK, respectively. 185
Proof: Letui andvj be the eigenvectors corresponding to 186
nonzero eigenvalues ofP0andP1, respectively. Then, we have 187
Kui = (λ0P0+λ1P1)ui =λ0P0ui+λ1P1ui
=λ0P0ui+λ1P1P0ui=λ0P0ui+λ1PT1P0ui
=λ0P0ui=λ0ui. (35) Kvj =λ0P0vj+λ1P1vj =λ0P0P1vj +λ1P1vj
=λ1vj. (36)
The proof of Lemma 3 has now been completed. 188
IEEE Proof
We are now ready to derive a set of orthonormal eigenvec-
189
tors ofKby using the approach reported in [8]. The detailed
190
procedure is described as follows.
191
By performing the SVD decomposition of P0 andP1, we
192
have
193
P0=U0S0V0T (37) P1=U1S1V1T. (38) Since the singular values ofP0andP1are square root of non-
194
negative eigenvalues ofP0PT0 andP1PT1,respectively, using
195
properties 1 and 2, and Lemma 1, we have
196
P0 =PT0P0=
U0S0VT0T
U0S0VT0
=V0S0UT0U0S0V0T =V0S20VT0 =V0S0VT0. (39) Similarly, for P1, we have
197
P1=V1S1VT1. (40) It can be observed from (39) and (40) that
198
P0V0=V0S0, P1V1=V1S1. (41) The above equation shows that V0 andV1 are a set of or-
199
thonormal eigenvectors ofP0andP1, respectively.
200
According to Lemma 1 and Theorem 1, ifN is even (N =
201
2m), the multiplicity of the eigenvalue 1 forP0andP1is both
202
m. IfN is odd (N = 2m+1), the multiplicity of eigenvalue
203
1 forP0 ism+1, and the multiplicity of eigenvalue 1 forP1
204
ism. With this property and Lemma 3, we can takeui andvj
205
be the ith and jth column ofV0,V1respectively. Then, we can
206
claim that if N is even, i.e., N = 2m, a set of orthonormal
207
eigenvectorsVofKcan be obtained by
208
V= [u1,u2,...,um,v1,v2,...,vm], (42) and if N is odd, i.e., N=2m+1, a set of orthonormal eigen-
209
vectorsVofKcan be written as
210
V= [u1,u2,...,um,um+ 1,v1,v2,...,vm]. (43) B. The Construction of One Dimensional Fractional
211
Krawtchouk Transform
212
From the previous subsection, a set of orthonormal eigen-
213
vectors ofK can be constructed. Then, we can rearrange the
214
columns ofVto match the eigenvectors to the eigenvalues of
215
Ksuch that
216
K=VDVT (44) whereDis a diagonal matrix with diagonal entries the eigen-
217
values ofK, andVis a set of orthonormal eigenvectors,
218
V=
[u1,v1,u2,v2,...,um,vm],ifN is even
[u1,v1,u2,v2,...,um,vm,um+ 1],ifN is odd (45)
The eigenvalues inDare arranged in the following form if 219
the size ofKis even: 220
D=
⎡
⎢⎢
⎢⎢
⎢⎢
⎣ 1
−1 1
. ..
−1
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
(46)
and if the size of K is odd as: 221
D=
⎡
⎢⎢
⎢⎢
⎢⎢
⎣ 1
−1 1
. ..
1
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
. (47)
Since the diagonal elements of Dcan be written as e−j k π 222
withk = 0, 1, . . . , N−1, as the generalization of DfrFT [6], 223
[7], we take the fractional order as the power of eigenvalues 224
inD. Finally, the FrKT transform matrixKa of sizeN with 225
order a corresponding to an angle α where α = πa can be 226
defined as 227
Ka=VDaVT =
N−1 k= 0
e−j k αvkvTk (48) wherevk(k=0, 1, ..., N−1)is the kth column ofV, andDa 228
is defined as 229
Da=
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣ e−j0α
e−j α
e−j2α . ..
e−j(N−1)α
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
. (49)
Thus, the 1D forward FrKT of signalf(x)of lengthN with 230
order a can be expressed by 231
Qa =Kaf. (50) The corresponding inverse FrKT can be written as 232
f =K−aQa. (51)
C. Generalization of the Fractional Krawtchouk Transform to 233
2-D 234
The 1-D FrKT presented in the previous section can be easily 235
generalized to 2-D situations. The definition of 2-D FrKT with 236
fractional order (a, b) corresponding to angle (α, β) where 237
α = πa, β = πbof an imageg(x, y)can be achieved by firstly 238
performing the FrKT on each column of the image, and then on 239
each row of the transformation. It can be expressed as 240
Qa,b =KagKb. (52) Notice that the two matricesKaandKb generated from the 241
Krawtchouk matrix defined in (13) may have different param- 242
eters for the weighted Krawtchouk polynomials. In our paper, 243
IEEE Proof
LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 5
we define the weighted Krawtchouk polynomial parameters as
244
pforKaand asqforKb withp, q∈(0, 1).
245
The corresponding inverse FrKT can be generated as
246
g=K−aQa,bK−b. (53) From (53), it can be observed that there are two more extra
247
parameters(a, b) in the transformation when compared with
248
the traditional 2-D Krawtchouk transform. They come in addi-
249
tion to the two parameters(p, q)in the weighted Krawtchouk
250
polynomials.
251
D. Property of Fractional Krawtchouk Transform Matrix
252
For the discrete fractional Fourier transform, the basic re-
253
quirements of the transform matrix to define the DfrFT are:
254
(1) unitary, (2) index additive, and (3) approximating the con-
255
tinuous Fourier transform [8], [9]. In this section, we will inves-
256
tigate some properties of FrKT transform matrixKadefined in
257
(48). As to the properties of DfrFT transform matrix, we will
258
present some properties ofKa, such as unitary, index additivity,
259
and reduction to the Krawtchouk transform when the order is
260
equal to 1.
261
It is obvious from (49) that ifa = 0,Dareduces to the iden-
262
tity matrix and the FrKT to the identity operator. Moreover,
263
ifa = 1,Da=Dand then the FrKT is reduced to the tradi-
264
tional Krawtchouk transform. We focus on the unitary and index
265
additivity properties of FrKT transform matrix in the following.
266
Property 7: Additivity
267
KaKb =Ka+b. (54) Proof: From (48) we have
268
KaKb =VDaVT
VDbVT
=VDaDbVT. (55) Since it follows from (49) that
269
DaDb =Da+b. (56) Therefore, by substituting (56) into (55), we have
270
KaKb =VDa+bVT =Ka+b. (57) The additivity property is thus shown.
271
Property 8: Unitarity
272
K−a= (Ka)−1. (58) The proof of property 8 can be achieved by makingb = −a
273
in (54) and noticing the fact thatK0 = I.
274
IV. APPLICATION TOIMAGEWATERMARKING 275
The DfrFT has been applied in many fields such as image
276
fusion [28], image copy-move forgery detection [29] and image
277
encryption [30]. The proposed FrKT can also be applied to these
278
fields. Due to the limited paper length, our objective is only to
279
provide a watermarking illustration. A brief introduction is pro-
280
vided and we refer the interested readers to recent reviews [31],
281
[32]. Schematically, digital image watermarking is a technique
282
mainly devoted to the protection of intellectual property rights
283
by embedding a digital watermark (or a message) into the im-
284
ages. The watermark is robustly and imperceptibly embedded
285
into the host image so as to allow for example verifying the 286 origin and the destination of the data [33]. Image watermarking 287
techniques can be classified into two categories: the spatial do- 288
main methods and the transform domain methods [34]. Because 289
space-based approaches are relatively weak in case of image at- 290
tacks (i.e. image filtering, compression, etc.), transform domain 291
methods have been more extensively investigated. They include 292
for instance the discrete Fourier transform (DFT) [35], discrete 293
cosine transform (DCT) [36], [37], discrete wavelet transform 294
[38], [39], fractional Fourier transform [40], quaternion Fourier 295
transform [41], SVD transform [42], [43], and moment trans- 296
form [44]. 297
In [45], the Krawtchouk transform coefficients have been 298
used as host coefficients to embed the watermark owing to their 299
robust behavior. As a generalization of the Krawtchouk trans- 300 form, FrKT has two additional fractional orders. By adjusting 301
the fractional orders in the transform, different transform domain 302
coefficients can be obtained. So we use the FrKT coefficients to 303
embed the watermark and the fractional orders can then serve 304
as extra secret keys to enhance the security of the watermark- 305
ing scheme (refer to [46] for more details). In the following, 306
the block based watermarking approach described in [45] has 307
been applied in the FrKT domain. It has the following features 308
[47]: (i) partitioning the host image into small blocks fulfills 309
the un-detectability and imperceptibility requirements; (ii) the 310
ability to handle each block separately allows using multiple 311
secrete keys for secret block selection, improving consequently 312
the watermarking security; (iii) the watermark capacity (i.e. the 313
size of the embedded message expressed in bits of message per 314
image pixel) will vary from one block to another according to 315 their properties while establishing a compromise between the 316
watermark robustness and imperceptibility. The watermark em- 317
bedding and extraction procedures are shown in Figs. 1 and 2, 318
respectively. 319
A. Watermark Embedding Procedure 320
Let us consider an original grayscale image g of N×N 321
pixels, and anl×lwatermarkW(see Fig. 3). It is embedded 322
according to the following steps: 323
Step 1: To enhance the security of the scheme and eliminate 324
the pixel correlation in spatial domain, the watermark is scram- 325
bled fromWintoW1at first with the Arnold transform, which 326
is defined as follows [48]: 327
x∗ y∗
=
c d e f
x y
+ s
t mod (l) (59) where (x, y),(x∗, y∗)are the coordinates of the original and 328
scrambled watermark pixels, respectively. The scrambling pa- 329
rametersc, d, eandf are such that 330
c d e f
= 1. (60) We choosec = 1, d = 1, e = 1, f = 2, s = 0 andt = 331
0 in the following. 332
Step 2: Divide the original image g into 8×8 non- 333
overlapping blocks. Thus, there will beN/8 × N/8blocks 334
IEEE Proof
Fig. 1. The diagram for watermark embedding.
Fig. 2. The diagram for watermark extraction.
Fig. 3. Some original images - (a) to (j) - and watermarks - (k) and (l) - from our test database.
for the original image, wherexdenotes the lower integer part
335
ofx. To ensure the security of the watermark, we secretly select
336
l2(l≤N/8) blocks. The secret keyKEY1corresponds to the
337
position of the selected blocks. It will be necessary to know it
338
to extract the watermark.
339
Step 3: Perform FrKT on each block. The transform matrix
340
of one block is denoted by C. Note that the two parameters
341
p, q, and the fractional orders a and b are extra key values that
342
have also to be known at the extraction stage. These values are
343
denoted asKEY2.This contributes to reinforce the security of
344
the watermarking scheme. For each block, the element (k1, k2)
345
in the real partC0ofCis used to embed one bit of the watermark
346
using the Dither modulation method [49], [50]:
347
|C0(k1, k2)|
=
⎧⎪
⎨
⎪⎩
2Δ×round
|C0(k1,k2)| 2Δ
+Δ2, if W1(i, j) = 1 2Δ×round
|C0(k1,k2)|
2Δ
−Δ2, if W1(i, j) = 0 (61)
where Δ is the quantization step controlling the embedding
348
strength of the watermark bit, | · | is the absolute operator,
349
round(·) denotes the rounding operation to the nearest inte-
350
ger, W1(i, j)is the scrambled watermark bit at the position
351
(i, j)andC0 is the modified block.
352
Step 4: The inverse FrKT is applied on each modified block 353
C0to obtain the watermarked image. 354
B. Watermark Extraction Procedure 355
To extract the watermark from a received imageg∗, one has 356
to know the secret keys(KEY1,KEY2)and then to apply the 357 watermark extraction procedure as follows: 358
Step 1: Divide the test imageg∗into 8×8 non-overlapping 359
blocks, and compute the FrKT coefficients of each block. 360
Step 2: For one block FrKT coefficients C∗, the element 361
(k1, k2)in the real partC∗0 is used to extract one bit of water- 362
mark at position(i, j)using the minimum distance decoder in 363
the following way: 364
W∗1(i, j) = argσ∈{0,1}min(|C0(k1, k2)|σ− |C∗0(k1, k2)|) (62) whereW∗1is the extracted scrambled watermark, and 365
|C0(k1, k2)|σ=
⎧⎪
⎨
⎪⎩
2Δ×round
|C∗0(k1,k2)|
2Δ
+Δ2, if σ= 1 2Δ×round
|C∗0(k1,k2)| 2Δ
−Δ2 ,if σ= 0 (63) Step 3: Perform the inverse Arnold transform onW∗1to obtain 366
the extracted watermark informationW∗. 367
IEEE Proof
LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 7
Finally, the extracted watermarkW∗can be used to identify
368
the ownership and copyright.
369
V. EXPERIMENTALRESULTS 370
Experiments have been carried out to assess the validity of
371
the watermarking scheme using FrKT for image copyright pro-
372
tection. To conduct these experiments, we consider 96 gray
373
images of size 512×512 from the image database of the Com-
374
puter Vision Group, University of Granada [51]. It can be seen
375
from the embedding procedure that the maximum watermark
376
capacity, which can be embedded into the host image with size
377
N×N, is N2. We used two binary images of size 64×64
378
from the MPEG-7 database [52]: “Deer” and “Cup” as water-
379
marks in the experiments. That is, 4096 bits were embedded
380
into each host image. Some of the test images and watermarks
381
are shown in Fig. 3. One bit of watermark is embedded into
382
each block of the original image. The first row and first col-
383
umn position in the transformed block is selected to embed
384
the watermark due to the fact the low order Krawtchouk trans-
385
form coefficients have been shown more robust to attacks [45].
386
This FrKT based watermarking scheme is compared with the
387
Krawtchouk transform approach reported in [45], as well as
388
the watermarking using DCT [37], DWT [38], LWT [39], SVD
389
[42], and Tchebichef moment (TM) [44], the discrete fractional
390
cosine transform (DfrCT), the discrete fractional sine transform
391
(DfrST), and DfrFT. In the following experiments, the parame-
392
ters in the Krawtchouk polynomial arep = q = 0.5. The wa-
393
termark imperceptibility is evaluated quantitatively through the
394
Peak Signal-to-Noise Ratio (PSNR) defined as [53]
395
PSNR= 10log102552
MSE (64)
where MSE is the mean square error between the original image
396
g(x, y)and the watermarked imagegw(x, y), given by
397
MSE=
N−1 x= 0
N−1 y= 0
(g(x, y)−gw(x, y))2. (65) The performance in terms of watermark robustness is mea-
398
sured through the bit error rate (BER) expressed by [43]
399
BER= l
i= 1
l
j= 1|W∗(i, j)−W(i, j)|
l×l (66)
where W∗ is the extracted watermark and W is the original
400
binary watermark of sizel×l.
401
In a first experiment, the watermark imperceptibility was
402
tested and we determined the proper quantization step for wa-
403
termark embedding (see Dither modulation in Section IV-A).
404
Using the 96 original images and the 2 watermarks, 192 host
405
image and watermark pairs are generated. For each pair, the
406
watermark is embedded with the fractional ordera = b = 0.4
407
and the quantization step is increasing from 1 to 45 with an in-
408
crement equal to 1. Then, the PSNR of the watermarked image
409
is calculated. Fig. 4 shows the average PSNR value of the water-
410
marked images of the 192 pairs for these different quantization
411
steps. We also made a comparison with the Krawtchouk trans-
412
form based watermarking scheme. It can be seen from Fig. 4 that
413
Fig. 4. Average PSNR value obtained over our image test database depending on the watermark embedding strength (i.e. the quantization step of the dither modulation).
the PSNR value of the watermarked image decreases with the 414
increase of the quantization step. Moreover, the PSNR value of 415
the watermarked image is higher when using the FrKT transform 416
instead of the Krawtchouk transform for the same quantization 417
step. Generally, a larger quantization step is required for better 418
robustness, meanwhile the quality of watermarked image de- 419
creases. To get watermark imperceptibility, the PSNR value is 420
expected to be higher than 40 dB. Therefore, the quantization 421
step of the Krawtchouk transform based scheme could be se- 422
lected to 25, leading to an average PSNR value of 40.72 dB. 423
For the FrKT, with quantization steps 25 and 40 respectively, 424
the PSNR averages of the watermarked images are equal to 425
46.43 dB and 42.25 dB. 426
A second experiment was conducted to evaluate the robust- 427 ness of these two approaches with fractional ordera =b = 0.4, 428
and the quantization step defined in the previous experiment us- 429
ing all original images and the two watermarks shown in Fig. 3. 430
The watermark attacks included the most common signal pro- 431
cessing and geometric attacks (see Table I). Each watermarked 432
image used in the previous experiment was distorted consid- 433
ering various attacks. Table II shows some examples of the 434
extracted watermarks and their corresponding BER values. The 435
change of BER for the geometric attack is due to the inter- 436
polation error and the truncation error that occur when cor- 437
recting the geometric attack (i.e. inversely transforming the 438
geometric transformation to neutralize the attacks [41]). The 439
mean BER values of the proposed FrKT and Krawthouk trans- 440
form based approaches are displayed in Fig. 5. The comparison 441
of the FrKT based watermarking method with DCT, DWT, LWT, 442 SVD, TM, DfrST, DfrCT, and DfrFT based algorithms is shown 443
in Fig. 6. To make a fair comparison, the quantization steps in 444
these methods were adjusted such that the PSNR of the water- 445
marked image is about 40 dB. To consider the situation when 446
the fractional orders are different, we also make a comparison 447
with the proposed watermarking scheme with fractional orders 448
(a, b) = (0.4,0.5)in Fig. 5. It can be observed from Figs. 5 449
and 6 that: (1) The BER values increase with the enhance- 450
ment of the attack power of the filter, noise, JPEG compres- 451
sion, and sharpening attacks. (2) By increasing the quantization 452
strengthΔfrom 25 to 40, the proposed schemes are much more 453
IEEE Proof
TABLE I
IMAGEATTACKS AND THEIRPARAMETERIZATION
TABLE II
SOMEEXAMPLES OFEXTRACTEDWATERMARKS AND THEIRBER VALUESUNDERDIFFERENTATTACKS
robust to different attacks. (3) The FrKT based schemes with
454
(a, b) = (0.4,0.4) and(a, b) = (0.4,0.5)have lower BER
455
values for most attacks than the Krawtchouk one, that is, the
456
watermarking scheme using FrKT provides a better robustness
457
by a sound selection of the fractional orders. Some better choice
458
of these parameters will be presented in the last experiment of
459
this section. (4) The proposed scheme can achieve better wa-
460
termark robustness in most attack situations than the classical
461
algorithms, such as the DCT, DWT, LWT, SVD, DfrST, DfrCT,
462
and DfrFT based algorithms. All the methods are not robust to
463
the shifting attack because they used a block based principle,
464
and a change of the block position in the extraction procedure
465
leads to the failure of the watermark extraction. Moreover, the 466
proposed approach and most of the compared methods are not 467
robust to the histogram equalization attacks because they mod- 468
ify the pixel values in each block. In contrast, the methods of 469
[37] and [39] are robust to the histogram equalization. We think 470
this is due to their embedding strategies. The difference between 471
two coefficients of adjacent blocks is used to embed the water- 472
mark bit in [37] and the watermark is embedded into the edge 473
part of the image in [39]. 474
The third conducted experiment shows that the watermark 475
security is enhanced by the fractional orders in the FrKT. The 476
objective here is to prevent the attackers from generating the 477
IEEE Proof
LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 9
Fig. 5. Mean BER values of the FrKT and Krawtchouk transform based watermarking schemes under different attacks.
counterfeit key to extract the watermark. The Deer watermark
478
was thus embedded into the original images of Fig. 3(a) with
479
fractional ordersa = b = 0.4. In the extraction procedure, we
480
separately extract the watermark without attacks with fractional
481
ordersa = bincreasing from 0.2 to 2.0 by 0.2. The extracted
482
watermark BER values with various fractional orders are shown
483
in Fig. 7. As it can be seen, the BER of the extracted watermark
484
with wrong fractional orders is about 0.5. This indicates that the
485
watermark information is not properly extracted or equivalently
486
that, without the good parameter values, it is not possible to
487
access the embedded watermark. The watermark examples dis-
488
played in Fig. 7 confirm that any watermark information cannot
489
be recovered when using the wrong fractional orders.
490
In the last experiment, we analyze the influence of fractional
491
orders variation whena = bin the embedding and extraction
492
stage changes from 0.1 to 1 on the watermark imperceptibility 493
and robustness. The Deer watermark is separately embedded 494
into the 96 images using a quantization step equal to 25 and 495
40 respectively. Then, the PSNR of each watermarked image 496
is computed. Fig. 8 shows the average PSNR of the 96 images 497
while varying the fractional orders a=b from 0.1 to 1 by 0.1. 498
It can be seen that the smallest PSNR value corresponds to 499
the fractional ordersa = b = 1 when the FrKT reduces to the 500
Krawtchouk transform. Besides, the highest PSNR values (about 501
46 dB, much higher than the required 40 dB mentioned above 502
for imperceptibility) are obtained when a = b = 0.4 or 0.6. 503
To show the influence of the fractional orders on the robustness, 504
and to guide the choice of the fractional orders, the watermarked 505
images have been submitted to different attacks to which our 506
proposed method can better resist, such as filter, noise, JPEG, 507
IEEE Proof
Fig. 6. Mean BER values comparison of the proposed FrKT based watermarking method and some available watermarking algorithms.
Fig. 7. Extracted watermark and corresponding BER value with fractional orders different from the embedding stage in the extraction procedure (with a = b = 0.4 in the embedding procedure).
rotation, scaling, and sharpening shown in the second ex-
508
periment. Fig. 9 depicts the BER variations of the extracted
509
watermarks when using different fractional orders. One can
510
see for instance that FrKT achieves either better robustness
511
Fig. 8. Average PSNR values of watermarked image with Deer watermark under various fractional orders in FrKT (with quantization stepΔ = 25,40).
(filtering, noise and JPEG compression attacks) than the spe- 512
cial case Krawtchouk transform (a = b = 1) or similar perfor- 513
mance (scaling and rotation attacks). However, the BER values 514
corresponding to the fractional ordersa = b = 0.1 and 0.7 are 515
relatively high. This is because the absolute value of the modi- 516
fied FrKT coefficients under these orders are very small and can 517
IEEE Proof
LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 11
Fig. 9. Average BER of extracted Deer watermark with various fractional orders used in FrKT of the embedding and extraction process under different attacks.
(a) Median filter 5×5, (b) Average filter 5×5, (c) Salt & Pepper noise with density 0.02, (d) Gaussian noise with variance 0.02, (e) JPEG compression with quality factor 30, (f) Rotation with angle 25o, (g) Scaling with factor 0.9, (h) Gaussian blur with standard derivation 1, (i) Sharpening with radius 2.
be easily changed by an attack. In fact, most of the FrKT coeffi-
518
cients are in the interval(−Δ,Δ), which leads to the modified
519
value−Δ/2 if the watermark bit is 0 whileΔ/2 if the watermark
520
bit is 1. Subsequently, the watermark bit cannot be accurately
521
extracted from these modified coefficients in the watermark ex-
522
traction procedure. Moreover, it can be seen from Fig. 9 that the
523
BER of the extracted watermark from the watermarking using
524
quantization step 40 always achieves better results than that us-
525
ing quantization step 25. Notice that the BER for quantization
526
step 40 fora = b = 0.1 and 0.7 is relatively higher than for
527
quantization step 25. This is due to the fact that the watermark
528
bit 0 cannot be accurately extracted even if no attack is per-
529
formed on the watermarked image because of the small modified
530
coefficients. It can be observed from Fig. 9 that a better water-
531
mark robustness can be achieved by an appropriate choice of
532
the fractional orders, such asa = b = 0.3, 0.4,0.6,0.8,0.9.
533
However, we have pointed out in Fig. 5 that a better performance
534
can also be achieved ifa = b, such asa = 0.4, b = 0.5. Nev-
535
ertheless, up to now, it is not easy to give a standard method for
536
choosing the fractional orders. We plan to study the optimization
537
of fractional orders selection by means of adaptive watermark-
538
ing [43], [54]. Such technique can be further combined with the
539
approach reported in [55] on Human Visual System (HVS) in
540
order to adaptively determine the quantization step and to im- 541
prove the performance of the proposed watermarking scheme. 542
Beyond that, feature point detection can be applied to design a 543
local image watermarking system capable to face the watermark 544
shifting attack situation [56]. 545
VI. CONCLUSION 546
This paper makes the following main contributions: firstly, it 547
determined the eigenvalues and the corresponding multiplicity 548
of each eigenvalue of the FrKT transform matrix. Secondly, it 549
presented a method for deriving a set of orthonormal eigen- 550 vectors corresponding to each eigenvalue of the Krawtchouk 551
transform matrix. Lastly, the definition of FrKT from the eigen- 552
value decomposition of the transform matrix was given and 553
some important properties of FrKT were demonstrated, such as 554
the unitary, the index addition, and the approximation of the 555
Krawtchouk matrix with particular fractional orders. 556
For a first assessment of this theoretical study, we used a 557
watermarking application and we compared its performance 558
with the classical Krawtchouk transform and other transforms. 559
It has been shown that more watermark imperceptibility and 560
robustness under most attacks for the same capacity can be 561
