Fractional Krawtchouk Transform with an Application to Image Watermarking 3

IEEE TRANSACTIONS ON SIGNAL PROCESSING 1

1 Fractional Krawtchouk Transform With

2 an Application to Image Watermarking

3 Xilin Liu, Guoniu Han, Jiasong Wu, Zhuhong Shao, Gouenou Coatrieux, and Huazhong Shu

4 Abstract—This paper proposes a novel fractional transform, DfrFT has important unitary and rotation properties. To achieve 28 5 denoted as the fractional Krawtchouk transform (FrKT), a gen- better DfrFT results, the eigenvectors of the Fourier transform 29 6 eralization of the Krawtchouk transform. The derivation of the matrix should better approximate the Hermite-Gaussian func- 30 7 FrKT uses the eigenvalue decomposition method. We determine the 8 eigenvalues and the corresponding multiplicity of the Krawtchouk tions which are the eigenfunctions of the FrFT [8]. Some other 31 9 transform matrix. Moreover, the orthonormal eigenvectors of the definitions of DfrFT can be found in [9]Ð[11]. The DfrFT has 32 10 transform matrix are derived. For validation purpose only and as been applied in optical image encryption [12]. An overview 33 11 a first illustration of the interest of FrKT, a watermarking exam- of the DfrFT in signal processing can be found in [13]. Some 34 12 ple was chosen. Experimental results show that better watermark other forms of fractional transforms have been developed, such 35 13 robustness and imperceptibility are achieved by adjusting the frac- 14 tional orders in the FrKT. as the discrete fractional Hadamard transform, discrete frac- 36 tional Hilbert transform, discrete fractional cosine and sine 37 15 Index Terms—Krawtchouk transform, fractional Krawtchouk transforms [14]Ð[19]. 38 16 transform, weighted Krawtchouk polynomials, watermarking. Yap et al. [20] introduced the Krawtchouk transform (also 39 known as Krawtchouk moments), another orthogonal trans- 40 17 I. INTRODUCTION form using the weighted Krawtchouk polynomials. Since the 41 weighted Krawtchouk polynomials are discrete, there is no nu- 42 18 HE continuous fractional Fourier transform (FrFT), a gen- merical approximation in deriving the transform coefficients. 43 19 eralization of the Fourier transform, depends on an ad- T By adjusting the parameters in the weighted two dimensional 44 20 ditional parameter and can be interpreted as a rotation in the (2-D) Krawtchouk polynomials, local image features can be 45 21 time-frequency plane. The FrFT has been investigated and ap- located and described. The Krawtchouk transform has been 46 22 plied in quantum mechanics [1], [2] and signal processing fields successfully applied in image reconstruction and image wa- 47 23 [3]Ð[5]. Such applications require the derivation of the discrete termarking [21]Ð[23]. Yap et al. [21] used the Krawtchouk 48 24 fractional Fourier transform (DfrFT). Pei et al. [6], [7] proposed transform for image reconstruction and it was shown that the 49 25 a definition of the DfrFT based on the eigenvalue decompo- Krawtchouk transform outperforms other moments in terms of 50 26 sition of the transform matrix. Their method provides similar reconstruction error. Venkataramana et al. [22] designed a wa- 51 27 transform as that of the continuous case. Moreover, the proposed termarking algorithm which is robust to geometric attack. Pa- 52 pakostas et al. In [23], a transform domain watermarking al- 53 Manuscript received July 19, 2016; revised November 20, 2016 and December 26, 2016; accepted January 4, 2017. The associate editor coordinating the review gorithm based on Krawtchouk transform was proposed which 54 of this manuscript and approving it for publication was Dr. Sergio Lima Netto. embeds the watermark in local regions of the image. Atakishiyev 55 This work was supported in part by the National Natural Science Foundation of et al. [24] derived a new transform known as fractional Fourier- 56 China under Grants 61201344, 61271312, 61401085, 81101104, and 61073138, in part by the Ministry of Education of China under Grants 20110092110023 and Krawtchouk transform, where the kernel function is the product 57 20120092120036, the Project-sponsored by SRF for ROCS, SEM, and in part by of an exponential function with parameter α and the Krawtchouk 58 the Natural Science Foundation of Jiangsu Province under Grants BK 2012329, function. 59 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 Our goal in this paper is to derive the fractional Krawtchouk 60 Project of Beijing Outstanding Talent Training Project 2016000020124G088. transform (FrKT) by using the eigenvalue decomposition 61 X. Liu, J. Wu, and H. Shu are with the Laboratory of Image Science and method of the transform matrix. Noticing that the eigenval- 62 Technology, School of Computer Science and Engineering, Southeast Univer- sity, Nanjing 210018, China, and also with the Centre de Recherche en Informa- ues of the transform matrix are 1 and Ð1, we determine the 63 tion Biomedicale« sino-français, Nanjing 210096, China (e-mail: xilinliu168@ multiplicity of each eigenvalue. Furthermore, the orthonormal 64 163.com; [email protected]; [email protected]). eigenvectors corresponding to the eigenvalues are derived us- 65 G. Han is with the Institute de Recherche en Mathematiques« et Applica- tion, Universite« de Strasbourg et CNRS, Strasbourg 67084, France (e-mail: ing spectral decomposition and SVD. Finally, we provide the 66 [email protected]). definition of FrKT. The FrKT, which is a generalization of the 67 Z. Shao is with the College of InformationIEEE Engineering, Capital Normal Krawtchouk Proof transform, has two additional parameters, known as 68 University, Beijing 100048, China (e-mail: [email protected]). the fractional orders. As an illustration of the FrKT applications, 69 Q1 G. Coatrieux is with the Latim, Inserm U 1101, Institut Mines- Telecom, Telecom Bretagne, Brest 29238, France (e-mail: gouenou.coatrieux@ a watermarking scheme for copyright protection is investigated. 70 telecom-bretagne.eu). Our intent is to assess the interest of FrKT and the benefits 71 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. that can be expected through a proper choice of the fractional 72 Digital Object Identifier 10.1109/TSP.2017.2652383 orders. 73

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74 The rest of the paper is organized as follows. In Section II, For an N × N image g(x, y), the forward and inverse two 94 75 some preliminaries about Krawtchouk transform are provided. dimensional transformations are respectively given by 95 76 The detailed procedure of deriving the FrKT and some prop- N−1 N−1 77 erties of FrKT are elaborated in Section III. The image water- Qnm = Kn (x; p, N − 1) Km (y; q,N − 1) g (x, y) 78 marking scheme is reported in Section IV. Experiments to test x=0 y=0 79 its robustness are carried out in Section V. Finally, Section VI (10) 80 concludes the paper. N−1 N−1 g(x, y)= QnmKn (x; p, N −1)Km (y; q,N−1). (11) 81 II. PRELIMINARIES n=0 m=0 82 The 1-D Krawtchouk transform in terms of weighted III. CONSTRUCTION OF THE FRACTIONAL KRAWTCHOUK 96 83 Krawtchouk polynomial is defined as [20]: TRANSFORM 97 N −1 In this section, we propose a novel discrete fractional 98 − − Qn = Kn (x; p, N 1) f (x),n =0, 1, ...,N 1 transform known as fractional Krawtchouk transform (FrKT). 99 x=0 Similarly to the development of DfrFT, the eigenvalue decom- 100 (1) position of Krawtchouk transform matrix will be used to define 101 84 where f(x) is an 1-D signal of length N, and Kn (x; p, the discrete FrKT. The eigenvalues and eigenvectors of FrKT 102 85 N − 1) is the n-th order weighted Krawtchouk polynomial, are derived and some properties of FrKT are also investigated. 103 86 defined as A. Eigenvalues and Eigenvectors of the Krawtchouk 104 w (x; p, N − 1) K (x; p, N − 1) = κ (x; p, N − 1) (2) Transform Matrix 105 n n ρ (n; p, N − 1) The 1-D Krawtchouk transform for signal f(x) of length N 106 87 with defined in (1) can be written in the following matrix form 107 Q = Kf (12) N − 1 − − w (x; p, N − 1) = px (1 − p)N 1 x , (3) x where the transform matrix K is defined by 108 p − 1 n n! − ≤ ≤ − − Kn,x = Kn (x; p, N 1) , 0 n, x N 1. (13) ρ (n; p, N 1) = − . (4) p ( N +1)n The Krawtchouk transform matrix K has the following 109 properties: 110 88 And κn (x; p, N − 1) is the classical Krawtchouk polynomial Property 1: K is symmetric. 111 1 This is straightforward from the definition of Krawtchouk 112 κ (x; p, N − 1) = F −n, −x; −N +1; ,p∈ (0, 1) n 2 1 p polynomial (2) and (13). 113 T T (5) Property 2: K is orthogonal, that is K K = KK = I, 114 115 89 where 2 F1 is the hypergeometric function, defined as with I the identity matrix [20]. Property 3: The eigenvalues of K are 1 and –1. 116 ∞ N ×1 (a) (b) zk Proof: Let λ be an eigenvalue of K and u ∈ R be its 117 F (a, b; c; z)= k k (6) 2 1 (c) k! corresponding eigenvector, that is 118 k=0 k Ku = λu. (14) 90 and (a)k is the Pochhammer symbol given by Then, with properties 1 and 2, we have 119 Γ(a + k) λ λ2 (a) = a (a +1)...(a + k − 1) = . (7) u = KKu = Ku = u. (15) k Γ(a) Thus 120 − 91 The weighted Krawtchouk polynomial Kn (x; p, N 1) sat- λ2 − 1 u =0 (16) 92 isfies the following orthogonal property λ λ − IEEE andProof the two eigenvalues of K are 0 =1, 1 = 1. 121 − N1 With these properties of K, we have the following theorem 122 − − Kn (x; p, N 1) Km (x; p, N 1) = δnm. (8) to determine the multiplicities of its eigenvalues. 123 x=0 Theorem 1: For the Krawtchouk transform matrix K of size 124 N × N,ifN is even, then the multiplicities of eigenvalues 125 93 This leads to the following inverse transform λ0 = 1 and λ1 = −1 are equal. If N is odd, the multiplicity of 126 λ N−1 eigenvalue 0 = 1 is equal to one more than that of eigenvalue 127 λ = −1. 128 f(x)= Qn Kn (x; p, N − 1). (9) 1 n=0 Proof: See Appendix. 129 LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 3

130 Theorem 1 speciﬁes the eigenvalue multiplicity of K.Next, matrices P0 , P1 and K, using (24) and (25), we have 163 131 we construct a set of orthonormal eigenvectors of matrix K |γI − P0 | = |γI − 0.5(K + I)| 132 corresponding to the two eigenvalues. 133 From properties 1 and 2, we know that K is a symmetric and = |(γ − 0.5) I − 0.5K| 134 orthogonal real matrix. According to the spectral theorem [25], =0.5N |(2γ − 1) I − K| 135 we know that K has the following spectral decomposition =0. (26) K = λ0 P0 + λ1 P1 (17) Similarly, we can obtain 164 136 where P i ,i= 0, 1, is the orthogonal projection matrix on the | − | N | − | 137 ith eigenspace of K, and λi is the ith eigenvalue of K. Then, the ηI P1 =0.5 (2η 1) I + K =0 (27) 138 expression of projection matrices can be derived as follows. m m and 165 139 Since for any integer m, there is K x = λ x, hence the m 140 matrix K has the same eigenvectors and consequently pro- |λI − K| =0. (28) 0 141 jection matrices as matrix K. Note that K = I, therefore, we From (26)Ð(28), we have 166 142 have 2γ − 1=λ (29) m λm λm K = 0 P0 + 1 P1 ,m=0, 1, 2, ... (18) − (2η − 1) = λ. (30) 143 To obtain the two matrices P0 and P1 , we write the above λ λ − 144 equation for m = 0, 1 in matrix form as Hence, if = 1, there is γ = 1,η = 0, and if = 1, then 167 γ = 0,η = 1. 168 P0 I The proof of Lemma 1 has now been completed. 169 A = (19) P1 K Lemma 2: The eigenvectors corresponding to nonzero 170 eigenvalues of P0 are orthogonal to those corresponding to 171 145 with nonzero eigenvalues of P1 . 172 II Proof: Let the size of P0 be N × N, and ui and vj be 173 A = . (20) P 174 λ0 I λ1 I eigenvectors corresponding to nonzero eigenvalues of 0 and P1 , respectively (from Lemma 1 and Theorem 1, we know 175 146 Notice that that if N is even, i.e., N = 2m, then i, j = 1, 2, ..., m, and 176 2I (λ + λ )I I0 if N is odd, i.e., N = 2m + 1 then i = 1, 2, ..., m + 1,j = 177 AAT = 0 1 =2 . (21) λ λ λ2 λ2 1, 2, ..., m), then we have 178 ( 0 + 1 )I ( 0 + 1 )I 0I P0 ui = ui (31) 147 From (21), we deduce that the inversion of A is and 179 A−1 =0.5AT . (22) P1 vj =vj . (32) −1 148 Multiplying A in (22) on both left sides of (19), we have With (31) and (32), we can write 180 P I I λ I I 0 T 0 T T T T =0.5A =0.5 . (23) (ui) vj =(P0 ui ) (P1 vj )=(ui ) P0 P1 vj . (33) P1 K I λ1 I K T Since P0 P1 = 0 from property 6, it follows from (33) that 181 149 Thus T (ui ) vj = 0. (34) P0 =0.5(I + K) (24) The proof of Lemma 2 is thus obtained. 182 P1 =0.5(I − K) . (25) Lemma 3: The eigenvectors corresponding to nonzero 183 184 150 From (24) and (25), we can get some properties of P0 eigenvalues of P0 and P1 are the eigenvectors of K, corre- λ λ − 185 151 and P1 . sponding to eigenvalues 0 = 1, 1 = 1ofK, respectively. T Proof: Let ui and vj be the eigenvectors corresponding to 186 152 Property 4: Pi = Pi ,i= 0, 1. [ ] 2 2 nonzero eigenvalues of P and P , respectively. Then, we have 187 153 Property 5: 26 : P0 = P0 , P1 = P1 . 0 1 154 Property 6: P0 and P1 are orthogonal,IEEE that is P0 P1 = 0, KuProof=(λ P + λ P ) u = λ P u + λ P u i 0 0 1 1 i 0 0 i 1 1 i 155 where 0 denotes the zero matrix. λ λ λ λ T 156 Lemma 1: Both matrices P0 and P1 have eigenvalues = 0 P0 ui + 1 P1 P0 ui = 0 P0 ui + 1 P1 P0 ui 157 0 and 1. Moreover, the multiplicity of eigenvalue 1 for P0 is = λ0 P0 ui = λ0 ui . (35) 158 equal to the multiplicity of eigenvalue 1 of K; the multiplicity 159 of eigenvalue 1 for P1 is equal to the multiplicity of eigenvalue Kvj = λ0 P0 vj + λ1 P1 vj = λ0 P0 P1 vj + λ1 P1 vj 160 –1ofK. = λ1 vj . (36) 161 Proof: The proof of the ﬁrst part of this lemma can be found 162 in [27]. Let γ, η and λ be respectively the eigenvalues of the The proof of Lemma 3 has now been completed. 188 4 IEEE TRANSACTIONS ON SIGNAL PROCESSING

189 We are now ready to derive a set of orthonormal eigenvec- The eigenvalues in D are arranged in the following form if 219 190 tors of K by using the approach reported in [8]. The detailed the size of K is even: 220 ⎡ ⎤ 191 procedure is described as follows. 1 192 By performing the SVD decomposition of P and P ,we ⎢ ⎥ 0 1 ⎢ −1 ⎥ 193 have ⎢ ⎥ ⎢ 1 ⎥ D = ⎢ ⎥ (46) T ⎢ . ⎥ P0 = U0 S0 V0 (37) ⎣ .. ⎦ T − P1 = U1 S1 V1 . (38) 1 and if the size of K is odd as: 221 194 Since the singular values of P0 and P1 are square root of non- ⎡ ⎤ T T 195 negative eigenvalues of P P and P P , respectively, using 1 0 0 1 1 ⎢ ⎥ 196 properties 1 and 2, and Lemma 1, we have ⎢ −1 ⎥ ⎢ ⎥ ⎢ 1 ⎥ T T T T D = ⎢ ⎥ . (47) P0 = P0 P0 = U0 S0 V0 U0 S0 V0 ⎢ . ⎥ ⎣ .. ⎦ T T 2 T T = V0 S0 U0 U0 S0 V0 = V0 S0 V0 = V0 S0 V0 . (39) 1

−jkπ 197 Similarly, for P1 ,wehave Since the diagonal elements of D can be written as e 222 with k = 0, 1, ..., N − 1, as the generalization of DfrFT [6], 223 T P1 = V1 S1 V1 . (40) [7], we take the fractional order as the power of eigenvalues 224 a in D. Finally, the FrKT transform matrix K of size N with 225 198 It can be observed from (39) and (40) that order a corresponding to an angle α where α = πa can be 226 deﬁned as 227 P0 V0 = V0 S0 , P1 V1 = V1 S1 . (41) N−1 a a T −jkα T K = VD V = e vk v (48) 199 The above equation shows that V0 and V1 are a set of or- k k=0 200 thonormal eigenvectors of P0 and P1 , respectively. a 201 According to Lemma 1 and Theorem 1, if N is even (N = where vk (k = 0, 1, ..., N − 1) is the kth column of V, and D 228 202 2m), the multiplicity of the eigenvalue 1 for P and P is both is deﬁned as 229 0 1 ⎡ ⎤ 203 m.IfN is odd (N = 2m + 1), the multiplicity of eigenvalue e−j0α 204 1forP is m + 1, and the multiplicity of eigenvalue 1 for P ⎢ ⎥ 0 1 ⎢ e−jα ⎥ 205 is m. With this property and Lemma 3, we can take ui and vj ⎢ ⎥ ⎢ −j2α ⎥ 206 be the ith and jth column of V , V respectively. Then, we can a ⎢ e ⎥ 0 1 D = ⎢ ⎥ . (49) 207 claim that if N is even, i.e., N = 2m, a set of orthonormal ⎢ . ⎥ ⎣ .. ⎦ 208 eigenvectors V of K can be obtained by e−j(N −1)α V =[u1 ,u2 ,...,um ,v1 ,v2 ,...,vm ] , (42) Thus, the 1D forward FrKT of signal f(x) of length N with 230 order a can be expressed by 231 209 and if N is odd, i.e., N = 2m + 1, a set of orthonormal eigen- 210 vectors V of K can be written as Qa = Ka f. (50)

V =[u1 ,u2 ,...,um ,um +1, v1 ,v2 ,...,vm ] . (43) The corresponding inverse FrKT can be written as 232 f = K−a Qa . (51) 211 B. The Construction of One Dimensional Fractional 212 Krawtchouk Transform C. Generalization of the Fractional Krawtchouk Transform to 233 2-D 234 213 From the previous subsection, a set of orthonormal eigen- 214 vectors of K can be constructed. Then, we can rearrange the The 1-D FrKT presented in the previous section can be easily 235 215 columns of V to match the eigenvectors to the eigenvalues of generalized to 2-D situations. The definition of 2-D FrKT with 236 216 K such that IEEE fractionalProof order (a, b) corresponding to angle (α, β) where 237 α = πa, β = πbof an image g(x, y) can be achieved by firstly 238 T K = VDV (44) performing the FrKT on each column of the image, and then on 239 each row of the transformation. It can be expressed as 240 217 where D is a diagonal matrix with diagonal entries the eigen- a,b a b 218 values of K, and V is a set of orthonormal eigenvectors, Q = K gK . (52) a b Notice that the two matrices K and K generated from the 241 [u1 ,v1 ,u2 ,v2 ,...,um ,vm ] , if N is even V= (45) Krawtchouk matrix defined in (13) may have different param- 242 [u1 ,v1 ,u2 ,v2 ,...,um ,vm ,um +1] , if N is odd eters for the weighted Krawtchouk polynomials. In our paper, 243 LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 5

244 we define the weighted Krawtchouk polynomial parameters as into the host image so as to allow for example verifying the 286 a b 245 p for K and as q for K with p, q ∈ (0, 1). origin and the destination of the data [33]. Image watermarking 287 246 The corresponding inverse FrKT can be generated as techniques can be classified into two categories: the spatial do- 288 − − main methods and the transform domain methods [34]. Because 289 g = K a Qa,bK b . (53) space-based approaches are relatively weak in case of image at- 290 247 From (53), it can be observed that there are two more extra tacks (i.e. image filtering, compression, etc.), transform domain 291 248 parameters (a, b) in the transformation when compared with methods have been more extensively investigated. They include 292 249 the traditional 2-D Krawtchouk transform. They come in addi- for instance the discrete Fourier transform (DFT) [35], discrete 293 250 tion to the two parameters (p, q) in the weighted Krawtchouk cosine transform (DCT) [36], [37], discrete wavelet transform 294 251 polynomials. [38], [39], fractional Fourier transform [40], quaternion Fourier 295 transform [41], SVD transform [42], [43], and moment trans- 296 252 D. Property of Fractional Krawtchouk Transform Matrix form [44]. 297 In [45], the Krawtchouk transform coefficients have been 298 253 For the discrete fractional Fourier transform, the basic re- used as host coefficients to embed the watermark owing to their 299 254 quirements of the transform matrix to define the DfrFT are: robust behavior. As a generalization of the Krawtchouk trans- 300 255 (1) unitary, (2) index additive, and (3) approximating the con- form, FrKT has two additional fractional orders. By adjusting 301 256 tinuous Fourier transform [8], [9]. In this section, we will inves- a the fractional orders in the transform, different transform domain 302 257 tigate some properties of FrKT transform matrix K defined in coefficients can be obtained. So we use the FrKT coefficients to 303 258 (48). As to the properties of DfrFT transform matrix, we will a embed the watermark and the fractional orders can then serve 304 259 present some properties of K , such as unitary, index additivity, as extra secret keys to enhance the security of the watermark- 305 260 and reduction to the Krawtchouk transform when the order is ing scheme (refer to [46] for more details). In the following, 306 261 equal to 1. a the block based watermarking approach described in [45] has 307 262 It is obvious from (49) that if a = 0, D reduces to the iden- been applied in the FrKT domain. It has the following features 308 263 tity matrix and the FrKT to the identity operator. Moreover, a [47]: (i) partitioning the host image into small blocks fulfills 309 264 if a = 1, D = D and then the FrKT is reduced to the tradi- the un-detectability and imperceptibility requirements; (ii) the 310 265 tional Krawtchouk transform. We focus on the unitary and index ability to handle each block separately allows using multiple 311 266 additivity properties of FrKT transform matrix in the following. secrete keys for secret block selection, improving consequently 312 267 Property 7: Additivity the watermarking security; (iii) the watermark capacity (i.e. the 313 a b a+b K K = K . (54) size of the embedded message expressed in bits of message per 314 image pixel) will vary from one block to another according to 315 268 Proof: From (48) we have their properties while establishing a compromise between the 316 a b a T b T a b T K K = VD V VD V = VD D V . (55) watermark robustness and imperceptibility. The watermark em- 317 bedding and extraction procedures are shown in Figs. 1 and 2, 318 269 Since it follows from (49) that respectively. 319 Da Db = Da+b . (56) A. Watermark Embedding Procedure 320 270 Therefore, by substituting (56) into (55), we have Let us consider an original grayscale image g of N × N 321 Ka Kb = VDa+b VT = Ka+b . (57) pixels, and an l × l watermark W (see Fig. 3). It is embedded 322 271 The additivity property is thus shown. according to the following steps: 323 272 Property 8: Unitarity Step 1: To enhance the security of the scheme and eliminate 324 − the pixel correlation in spatial domain, the watermark is scram- 325 K−a =(Ka ) 1 . (58) bled from W into W1 at first with the Arnold transform, which 326 273 The proof of property 8 can be achieved by making b = −a is defined as follows [48]: 327 0 274 in (54) and noticing the fact that K = I. x∗ cd x s = + mod (l) (59) y∗ ef y t 275 IV. APPLICATION TO IMAGE WATERMARKING ∗ ∗ 276 The DfrFT has been applied in many fields such as image where (x, y), (x ,y ) are the coordinates of the original and 328 scrambled watermark pixels, respectively. The scrambling pa- 329 277 fusion [28], image copy-move forgeryIEEE detection [29] and image Proof 278 encryption [30]. The proposed FrKT can also be applied to these rameters c, d, e and f are such that 330 279 fields. Due to the limited paper length, our objective is only to cd 280 provide a watermarking illustration. A brief introduction is pro- =1. (60) ef 281 vided and we refer the interested readers to recent reviews [31], 282 [32]. Schematically, digital image watermarking is a technique We choose c = 1,d= 1,e= 1,f = 2,s= 0 and t = 331 283 mainly devoted to the protection of intellectual property rights 0 in the following. 332 284 by embedding a digital watermark (or a message) into the im- Step 2: Divide the original image g into 8 × 8 non- 333 285 ages. The watermark is robustly and imperceptibly embedded overlapping blocks. Thus, there will be N/8×N/8 blocks 334 6 IEEE TRANSACTIONS ON SIGNAL PROCESSING

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.

335 for the original image, where x denotes the lower integer part Step 4: The inverse FrKT is applied on each modified block 353 336 of x. To ensure the security of the watermark, we secretly select C 0 to obtain the watermarked image. 354 2 337 l (l ≤ N/8) blocks. The secret key KEY1 corresponds to the 338 position of the selected blocks. It will be necessary to know it B. Watermark Extraction Procedure 355 339 to extract the watermark. ∗ 340 Step 3: Perform FrKT on each block. The transform matrix To extract the watermark from a received image g , one has 356 341 of one block is denoted by C. Note that the two parameters to know the secret keys (KEY1 , KEY2 ) and then to apply the 357 342 p, q, and the fractional orders a and b are extra key values that watermark extraction procedure as follows: 358 ∗ 343 have also to be known at the extraction stage. These values are Step 1: Divide the test image g into 8 × 8 non-overlapping 359 344 denoted as KEY2 . This contributes to reinforce the security of blocks, and compute the FrKT coefficients of each block. 360 ∗ 345 the watermarking scheme. For each block, the element (k1 ,k2 ) Step 2: For one block FrKT coefficients C , the element 361 ∗ 346 in the real part C0 of C is used to embed one bit of the watermark (k1 ,k2 ) in the real part C0 is used to extract one bit of water- 362 347 using the Dither modulation method [49], [50]: mark at position (i, j) using the minimum distance decoder in 363 the following way: 364 |C (k ,k )| 0 1 2 ∗ | | −| ∗ | ⎧ W1 (i, j)=argσ∈{0,1} min( C 0 (k1 ,k2 ) σ C0 (k1 ,k2 ) ) ⎪ |C 0 (k1 ,k2 )| Δ ⎨2Δ × round + , if W1 (i, j)=1 (62) 2Δ IEEE2 Proof∗ = (61) where W1 is the extracted scrambled watermark, and 365 ⎪ | | ⎩ × C 0 (k1 ,k2 ) − Δ 2Δ round 2Δ 2 , if W1 (i, j)=0 ⎧ |C ∗ (k ,k )| ⎨⎪ × 0 1 2 Δ 2Δ round 2Δ + 2 , if σ =1 |C (k ,k )| = 348 where Δ is the quantization step controlling the embedding 0 1 2 σ ∗ ⎪ |C (k ,k )| ⎩ × 0 1 2 − Δ 349 strength of the watermark bit, |·| is the absolute operator, 2Δ round 2Δ 2 , if σ =0 350 round(·) denotes the rounding operation to the nearest inte- (63) ∗ 351 ger, W1 (i, j) is the scrambled watermark bit at the position Step 3: Perform the inverse Arnold transform on W1 to obtain 366 ∗ 352 (i, j) and C 0 is the modified block. the extracted watermark information W . 367 LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 7

∗ 368 Finally, the extracted watermark W can be used to identify 369 the ownership and copyright.

370 V. E XPERIMENTAL RESULTS

371 Experiments have been carried out to assess the validity of 372 the watermarking scheme using FrKT for image copyright pro- 373 tection. To conduct these experiments, we consider 96 gray 374 images of size 512 × 512 from the image database of the Com- 375 puter Vision Group, University of Granada [51]. It can be seen 376 from the embedding procedure that the maximum watermark 377 capacity, which can be embedded into the host image with size 2 378 N × N,isN . We used two binary images of size 64 × 64 379 from the MPEG-7 database [52]: “Deer” and “Cup” as water- 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 380 marks in the experiments. That is, 4096 bits were embedded modulation). 381 into each host image. Some of the test images and watermarks 382 are shown in Fig. 3. One bit of watermark is embedded into 383 each block of the original image. The first row and first col- the PSNR value of the watermarked image decreases with the 414 384 umn position in the transformed block is selected to embed increase of the quantization step. Moreover, the PSNR value of 415 385 the watermark due to the fact the low order Krawtchouk trans- the watermarked image is higher when using the FrKT transform 416 386 form coefficients have been shown more robust to attacks [45]. instead of the Krawtchouk transform for the same quantization 417 387 This FrKT based watermarking scheme is compared with the step. Generally, a larger quantization step is required for better 418 388 Krawtchouk transform approach reported in [45], as well as robustness, meanwhile the quality of watermarked image de- 419 389 the watermarking using DCT [37], DWT [38], LWT [39], SVD creases. To get watermark imperceptibility, the PSNR value is 420 390 [42], and Tchebichef moment (TM) [44], the discrete fractional expected to be higher than 40 dB. Therefore, the quantization 421 391 cosine transform (DfrCT), the discrete fractional sine transform step of the Krawtchouk transform based scheme could be se- 422 392 (DfrST), and DfrFT. In the following experiments, the parame- lected to 25, leading to an average PSNR value of 40.72 dB. 423 393 ters in the Krawtchouk polynomial are p = q = 0.5. The wa- For the FrKT, with quantization steps 25 and 40 respectively, 424 394 termark imperceptibility is evaluated quantitatively through the the PSNR averages of the watermarked images are equal to 425 395 Peak Signal-to-Noise Ratio (PSNR) defined as [53] 46.43 dB and 42.25 dB. 426 2 255 A second experiment was conducted to evaluate the robust- 427 PSNR = 10log10 (64) MSE ness of these two approaches with fractional order a=b = 0.4, 428 396 where MSE is the mean square error between the original image and the quantization step defined in the previous experiment us- 429 ing all original images and the two watermarks shown in Fig. 3. 430 397 g(x, y) and the watermarked image gw (x, y), given by The watermark attacks included the most common signal pro- 431 N−1 N−1 2 cessing and geometric attacks (see Table I). Each watermarked 432 MSE = (g(x, y) − gw (x, y)) . (65) image used in the previous experiment was distorted consid- 433 x=0 y=0 ering various attacks. Table II shows some examples of the 434 398 The performance in terms of watermark robustness is mea- extracted watermarks and their corresponding BER values. The 435 399 sured through the bit error rate (BER) expressed by [43] change of BER for the geometric attack is due to the inter- 436 polation error and the truncation error that occur when cor- 437 l l |W ∗(i, j) − W (i, j)| i=1 j=1 recting the geometric attack (i.e. inversely transforming the 438 BER = × (66) l l geometric transformation to neutralize the attacks [41]). The 439 ∗ 400 where W is the extracted watermark and W is the original mean BER values of the proposed FrKT and Krawthouk trans- 440 401 binary watermark of size l × l. form based approaches are displayed in Fig. 5. The comparison 441 402 In a first experiment, the watermark imperceptibility was of the FrKT based watermarking method with DCT, DWT, LWT, 442 403 tested and we determined the proper quantization step for wa- SVD, TM, DfrST, DfrCT, and DfrFT based algorithms is shown 443 404 termark embedding (see Dither modulation in Section IV-A). in Fig. 6. To make a fair comparison, the quantization steps in 444 405 Using the 96 original images andIEEE the 2 watermarks, 192 host these Proof methods were adjusted such that the PSNR of the water- 445 406 image and watermark pairs are generated. For each pair, the marked image is about 40 dB. To consider the situation when 446 407 watermark is embedded with the fractional order a = b = 0.4 the fractional orders are different, we also make a comparison 447 408 and the quantization step is increasing from 1 to 45 with an in- with the proposed watermarking scheme with fractional orders 448 409 crement equal to 1. Then, the PSNR of the watermarked image (a, b)=(0.4, 0.5) in Fig. 5. It can be observed from Figs. 5 449 410 is calculated. Fig. 4 shows the average PSNR value of the water- and 6 that: (1) The BER values increase with the enhance- 450 411 marked images of the 192 pairs for these different quantization ment of the attack power of the filter, noise, JPEG compres- 451 412 steps. We also made a comparison with the Krawtchouk trans- sion, and sharpening attacks. (2) By increasing the quantization 452 413 form based watermarking scheme. It can be seen from Fig. 4 that strength Δ from 25 to 40, the proposed schemes are much more 453 8 IEEE TRANSACTIONS ON SIGNAL PROCESSING

TABLE I IMAGE ATTACKS AND THEIR PARAMETERIZATION

TABLE II SOME EXAMPLES OF EXTRACTED WATERMARKS AND THEIR BER VALUES UNDER DIFFERENT ATTACKS

454 robust to different attacks. (3) The FrKT based schemes with leads to the failure of the watermark extraction. Moreover, the 466 455 (a, b)=(0.4, 0.4) and (a, b)=(0.4, 0.5) have lower BER proposed approach and most of the compared methods are not 467 456 values for most attacks than the Krawtchouk one, that is, the robust to the histogram equalization attacks because they mod- 468 457 watermarking scheme using FrKT provides a better robustness ify the pixel values in each block. In contrast, the methods of 469 458 by a sound selection of the fractionalIEEE orders. Some better choice [37]Proof and [39] are robust to the histogram equalization. We think 470 459 of these parameters will be presented in the last experiment of this is due to their embedding strategies. The difference between 471 460 this section. (4) The proposed scheme can achieve better wa- two coefﬁcients of adjacent blocks is used to embed the water- 472 461 termark robustness in most attack situations than the classical mark bit in [37] and the watermark is embedded into the edge 473 462 algorithms, such as the DCT, DWT, LWT, SVD, DfrST, DfrCT, part of the image in [39]. 474 463 and DfrFT based algorithms. All the methods are not robust to The third conducted experiment shows that the watermark 475 464 the shifting attack because they used a block based principle, security is enhanced by the fractional orders in the FrKT. The 476 465 and a change of the block position in the extraction procedure objective here is to prevent the attackers from generating the 477 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.

478 counterfeit key to extract the watermark. The Deer watermark stage changes from 0.1 to 1 on the watermark imperceptibility 493 479 was thus embedded into the original images of Fig. 3(a) with and robustness. The Deer watermark is separately embedded 494 480 fractional orders a = b = 0.4. In the extraction procedure, we into the 96 images using a quantization step equal to 25 and 495 481 separately extract the watermark without attacks with fractional 40 respectively. Then, the PSNR of each watermarked image 496 482 orders a = b increasing from 0.2 to 2.0 by 0.2. The extracted is computed. Fig. 8 shows the average PSNR of the 96 images 497 483 watermark BER values with various fractional orders are shown while varying the fractional orders a = b from 0.1 to 1 by 0.1. 498 484 in Fig. 7. As it can be seen, the BERIEEE of the extracted watermark It canProof be seen that the smallest PSNR value corresponds to 499 485 with wrong fractional orders is about 0.5. This indicates that the the fractional orders a = b = 1 when the FrKT reduces to the 500 486 watermark information is not properly extracted or equivalently Krawtchouk transform. Besides, the highest PSNR values (about 501 487 that, without the good parameter values, it is not possible to 46 dB, much higher than the required 40 dB mentioned above 502 488 access the embedded watermark. The watermark examples dis- for imperceptibility) are obtained when a = b = 0.4 or 0.6. 503 489 played in Fig. 7 confirm that any watermark information cannot To show the influence of the fractional orders on the robustness, 504 490 be recovered when using the wrong fractional orders. and to guide the choice of the fractional orders, the watermarked 505 491 In the last experiment, we analyze the influence of fractional images have been submitted to different attacks to which our 506 492 orders variation when a = b in the embedding and extraction proposed method can better resist, such as filter, noise, JPEG, 507 10 IEEE TRANSACTIONS ON SIGNAL PROCESSING

Fig. 6. Mean BER values comparison of the proposed FrKT based watermarking method and some available watermarking algorithms.

Fig. 8. Average PSNR values of watermarked image with Deer watermark IEEE underProof various fractional orders in FrKT (with quantization step Δ=25, 40). 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). (filtering, noise and JPEG compression attacks) than the spe- 512 cial case Krawtchouk transform (a = b = 1) or similar perfor- 513 508 rotation, scaling, and sharpening shown in the second ex- mance (scaling and rotation attacks). However, the BER values 514 509 periment. Fig. 9 depicts the BER variations of the extracted corresponding to the fractional orders a = b = 0.1 and 0.7 are 515 510 watermarks when using different fractional orders. One can relatively high. This is because the absolute value of the modi- 516 511 see for instance that FrKT achieves either better robustness fied FrKT coefficients under these orders are very small and can 517 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.

518 be easily changed by an attack. In fact, most of the FrKT coeffi- order to adaptively determine the quantization step and to im- 541 519 cients are in the interval (−Δ, Δ), which leads to the modified prove the performance of the proposed watermarking scheme. 542 520 value −Δ/2 if the watermark bit is 0 while Δ/2 if the watermark Beyond that, feature point detection can be applied to design a 543 521 bit is 1. Subsequently, the watermark bit cannot be accurately local image watermarking system capable to face the watermark 544 522 extracted from these modified coefficients in the watermark ex- shifting attack situation [56]. 545 523 traction procedure. Moreover, it can be seen from Fig. 9 that the 524 BER of the extracted watermark from the watermarking using VI. CONCLUSION 546 525 quantization step 40 always achieves better results than that us- 526 ing quantization step 25. Notice that the BER for quantization This paper makes the following main contributions: firstly, it 547 527 step 40 for a = b = 0.1 and 0.7 is relatively higher than for determined the eigenvalues and the corresponding multiplicity 548 528 quantization step 25. This is due to the fact that the watermark of each eigenvalue of the FrKT transform matrix. Secondly, it 549 529 bit 0 cannot be accurately extracted even if no attack is per- presented a method for deriving a set of orthonormal eigen- 550 530 formed on the watermarked image because of the small modified vectors corresponding to each eigenvalue of the Krawtchouk 551 531 coefficients. It can be observed from Fig. 9 that a better water- transform matrix. Lastly, the definition of FrKT from the eigen- 552 532 mark robustness can be achievedIEEE by an appropriate choice of value Proof decomposition of the transform matrix was given and 553 533 the fractional orders, such as a = b = 0.3, 0.4, 0.6, 0.8, 0.9. some important properties of FrKT were demonstrated, such as 554 534 However, we have pointed out in Fig. 5 that a better performance the unitary, the index addition, and the approximation of the 555 535 can also be achieved if a = b, such as a = 0.4,b= 0.5. Nev- Krawtchouk matrix with particular fractional orders. 556 536 ertheless, up to now, it is not easy to give a standard method for For a first assessment of this theoretical study, we used a 557 537 choosing the fractional orders. We plan to study the optimization watermarking application and we compared its performance 558 538 of fractional orders selection by means of adaptive watermark- with the classical Krawtchouk transform and other transforms. 559 539 ing [43], [54]. Such technique can be further combined with the It has been shown that more watermark imperceptibility and 560 540 approach reported in [55] on Human Visual System (HVS) in robustness under most attacks for the same capacity can be 561 12 IEEE TRANSACTIONS ON SIGNAL PROCESSING

562 expected by adjusting the fractional orders in FrKT. The optimal It can be deduced from (A7) that 585 563 choice of different fractional orders, the embedding position and n n k n 564 the quantization step will be considered in our future work. (1 + x) ξy ξy CT = 1+ . (A8) x xk 1+y 1+y k=0 565 APPENDIX

566 To prove Theorem 1, we need two Lemmas. Before giving Substituting (A8) into (A6) yields 586 567 these two lemmas, we first define some notations and properties. n n M k n M − k ξy (1 + y) 568 Let f(t) be a formal power series, the coefficients of t in ξk = CT 1+ . k − y n 569 k n k 1+y y f(t) are denoted as [t ]f(t).Letf(t) and g(t) be two formal k=0 570 power series, then we have [57]: (A9) n On the other hand, we have 587 [tn ]f(t)g(t)= [yk ]f(y) [tn−k ]g(t) (A1) n n M − n k=0 (1 + ξ)k ∞ k n − k k=0 n k n k [t ]f (g (t)) = [y ]f (y) [t ]g(t) . (A2) − n (1 + x)n (1 + y)M n k=0 = CT (1 + ξ)k x,y xk yn−k k n n k=0 571 The coefficient of x in (1 + x) is ( k ), and the coefficient n−k M −k M −k n k 572 of y in (1 + y) is ( ). That is, ( )=[x ](1 + n n M −n n−k k (1 + x) k (1 + y) M −k − − n n k M k = CTx,y k ((1 + ξ) y) n 573 x) , and ( n−k )=[y ](1 + y) . For convenience, the x y k=0 574 binomial coefficients can also be represented as constant terms n M −n 575 CT [58]: n (1 + y) = CT ((1 + ξ) y)k n y k yn n (1 + x) k=0 = CT k (A3) k x M −n n (1 + y) − = CT (1 + (1 + ξ) y) . (A10) M − k (1 + y)M k y yn = CT (A4) n − k yn−k Lemma A1 can be concluded from (A10) and (A9). 588 576 In this context, if two variables x and y appear in the same Lemma A2: Let d be a constant and k1 + k2 = M,ifM is 589 577 expression, CTx represents the constant term with respect to x, even, set M = 2h (h = 0, 1, 2 ...), then the following for- 590 578 and CTx,y represents the constant term with respect to x and y. mula holds: 591 579 Lemma A1: Let M, n, and k be non-negative integers such ≥ ≥ M 580 that M n k, ξ is a constant independent of M, n and k, d + k d + k d + h − k1 1 2 581 then the following formula holds ( 1) = . (A11) k1 k2 d k1 =0 n n n M − k n M − n ξk = (1 + ξ)k . k n − k k n − k If M is odd, set M = 2h + 1 (h = 0, 1, 2 ...), then we 592 k=0 k=0 (A5) have: 593 582 Proof: From (A3) and (A4), we have M k1 d + k1 d + k2 n (−1) =0. (A12) k n M − k k1 k2 ξ k =0 k n − k 1 k=0 − Proof: According to the method of coefficients rules and 594 n (1 + x)n (1 + y)M k = CT ξk using the following property 595 x,y xk yn−k k=0 −r r + n − 1 n − n (1 + x)n ξy k (1 + y)M = ( 1) , (A13) = CT n n x,y xk 1+y yn k=0 IEEE weProof have 596 n M n k (1 + y) (1 + x) ξy = CT CT . (A6) d+1 y n x k 1 −(d +1) d + k y x 1+y k1 1 − k1 k=0 [t1 ] = = ( 1) . 1+t1 k1 k1 583 By using the variable elimination rule [57], if f(x) is any (A14) 584 power series and θ is independent of x, then Similarly, 597 n n d+1 f(x) k k k 1 d + k CTx θ = [x ]f(x) θ = f(θ). (A7) k2 2 xk [t2 ] = . (A15) k=0 k=0 1 − t2 k2 LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 13

598 It follows from (A14) and (A15) that From Lemma A1 we have 608 n k − − n −1 n N 1 k p − N −1 M − (1 p) k d + k d + k p k n k 1 − p (−1) 1 1 2 k=0 k1 k2 k1 =0 n k n 1 n N −1−n p − = 1 − (1 − p)N 1 . M d+1 d+1 p k n − k 1 − p k1 1 k2 1 k=0 = [t1 ] [t2 ] . (A16) 1+t1 1 − t2 (A20) k1 =0 Thus, the trace of K can be expressed as 609 599 If k1 + k2 = 2h, (A16) can be written as N−1 trace (K)= K n,n M n=0 d + k d + k − k1 1 2 ( 1) N−1 n k n k1 k2 1 n N −1−n p k1 =0 = 1 − p k n − k 1 − p 1 d+1 1 d+1 n=0 k=0 =[t2h ] − − 1+t 1 t × (1 − p)N 1 d+1 2h 1 N−1 N−1 k n =[t ] 1 n N −1−n p 1 − t2 = 1 − p k n − k 1 − p k=0 n=k d + h = . (A17) − d × (1 − p)N 1 N−1 N−1−k k 600 If k + k = 2h + 1, (A16) can be written as d=n−k 1 d + k N −1−d−k 1 2 = 1 − p k d k=0 d=0 M d + k d + k d+k − k1 1 2 ( 1) × p − N −1 k1 k2 (1 p) k1 =0 1 − p d+1 d+1 N−1 N−1−d 2h+1 1 1 d + k N −1−d−k =[t ] = (−1)k 1+t 1 − t k d d=0 k=0 d+1 2h+1 1 d =[t ] p N −1 1 − t2 × (1 − p) . (A21) 1 − p ∞ d +1+k − 1 =[t2h+1] t2k The above equation can be written as 610 k k=0 N−1 N−1−d d d + k N − 1 − d − k p (−1)k =0. (A18) k d 1 − p d=0 k=0 − 601 The proof of Lemma A2 is now completed. × (1 − p)N 1 602 With the above two lemmas, we are now ready to prove N−1 d N−1−d 603 Theorem 1. N −1 p k d + k − = (1 − p) (−1) 604 Proof of Theorem 1: Let Kn,n(n = 0, 1, ..., N 1) be the 1 − p k d=0 k=0 605 diagonal elements of Krawtchouk transform matrix K.From − − − 606 the deﬁnition of weighted Krawtchouk polynomials (2)Ð(7), we × N 1 d k 607 have d IEEE ProofN−1 d N −1−2d − p d + k n (−n) (−n) w (n; p, N − 1) = (1 − p)N 1 (−1)k K = k k · 1 − p k n,n (−(N − 1)) · pk · k! ρ (n; p, N − 1) d=0 k=0 k=0 k N − 1 − d − k n k n × 1 n N − 1 − k p . (A22) = − d p k n − k 1 − p k=0 Note that in the derivation of the last step of (A22), we 611 N −1−d−k − have used the fact that ( )=0ifN − 1 − d − kN− 1 − 2d. Letting k1 = k, and k2 = N − 1 − 613 14 IEEE TRANSACTIONS ON SIGNAL PROCESSING

614 2d − k1 in (A22), and using Lemma A2, we obtain [10] S. Pei, W. Hsue, and J. Ding, “Discrete fractional Fourier transform based 651 on new nearly tridiagonal commuting matrices,” IEEE Trans. Signal Pro- 652 − − N 1 2d cess., vol. 54, no. 10, pp. 3815Ð3828, Oct. 2006. 653 k d + k N − 1 − d − k (−1) [11] M. T. Hanna, N. P. A. Seif, and W. A. E. M. Ahmed, “Hermite- 654 k d Gaussian-like eigenvectors of the discrete Fourier transform matrix based 655 k=0 on the direct utilization of the orthogonal projection matrices on its 656 N −1−2d eigenspaces,” IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2815Ð2819, 657 d + k d + k = (−1)k1 1 2 Jul. 2006. 658 k1 k2 [12] N. Singh and A. Sinha, “Optical image encryption using fractional Fourier 659 k1 =0 transform and chaos,” Opt. Laser. Eng., vol. 46, no. 2, pp. 117Ð123, 660 ⎧ 2008. 661 ⎨ d + N −1−2d 2 , if N is odd [13] E. Sejdic, I. Djurovic, and L. Stankovic, “Fractional Fourier transform 662 = d (A23) as a signal processing tool: An overview of recent developments,” Signal 663 ⎩ Process., vol. 91, no. 6, pp. 1351Ð1369, 2011. 664 0, if N is even [14] S. C. Pei and M. H. Yeh, “Discrete fractional Hadamard transform,” in 665 Proc. IEEE Int. Symp. Circuits Syst., 1999, vol. 3, pp. 179Ð182. 666 d+ N −1−2d 615 2 − [15] S. C. Pei and M. Yeh, “Discrete fractional Hilbert transform,” IEEE Trans. 667 Noticing that ( d )=0ifd>0.5(N 1) and using 616 (A23), we have for odd value of N Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 11, pp. 1307Ð 668 1311, Nov. 2000. 669 − − N1 N1 d N −1−2d [16] S. C. Pei and M. H. Yeh, “The discrete fractional cosine and sine trans- 670 − p N 1 d + 2 forms,” IEEE Trans. Signal Process., vol. 49, no. 6, pp. 1198Ð207, 671 Kn,n = (1 − p) 1 − p d Jun. 2001. 672 n=0 d=0 [17] G. Cariolaro, T. Erseghe, and P. Kraniauskas, “The fractional discrete 673 cosine transform,” IEEE Trans. Signal Process., vol. 50, no. 4, pp. 902Ð 674 0.5(N −1) d − p 0.5(N − 1) 911, Apr. 2002. 675 = (1 − p)N 1 1 − p d [18] S. C. Pei and J. J. Ding, “Fractional cosine, sine, and Hartley transforms,” 676 d=0 IEEE Trans. Signal Process., vol. 50, no. 7, pp. 1661Ð1680, Jul. 2002. 677 [19] S. C. Pei and W. L. Hsue, “Tridiagonal commuting matrices and fraction- 678 N −1 2 alizations of DCT and DST matrices of types I, IV, V, and VIII,” IEEE 679 N −1 p = (1 − p) 1+ =1. (A24) Trans. Signal Process., vol. 56, no. 6, pp. 2357Ð2369, Jun. 2008. 680 1 − p [20] P. Yap, R. Paramesran, and S. Ong, “Image analysis by Krawtchouk mo- 681 ments,” IEEE Trans. Image Process., vol. 12, no. 11, pp. 1367Ð1377, 682 617 And if N is even Nov. 2003. 683 N−1 N−1 d [21] P. Yap, P. Raveendran, and S. Ong, “Krawtchouk moments as a new set of 684 − p K = (1 − p)N 1 0=0. (A25) discrete orthogonal moments for image reconstruction,” in Proc. Int. Joint 685 n,n 1 − p Conf. Neural Networks, 2002, vol. 1, pp. 908Ð912. 686 n=0 d=0 [22] A. Venkataramana and P. A. Raj, “Image watermarking using Krawtchouk 687 moments,” in Proc. Int. Conf. Comput. Theory Appl., 2007, pp. 676Ð680. 688 618 Because the eigenvalues of K are only equal to 1 and Ð1, from [23] G. A. Papakostas, E. D. Tsougenis, and D. E. Koulouriotis, “Near optimum 689 619 (A24) and (A25), it is straightforward to conclude that: if N = local image watermarking using Krawtchouk moments,” in Proc. IEEE 690 Int. Workshop Imag. Syst. Tech., 2010, pp. 464Ð467. 691 620 2h (h = 0, 1, 2, ...), the multiplicity of eigenvalue λ1 = 1 and λ − [24] N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk trans- 692 621 2 = 1ish, and if N = 2h + 1 (h = 0, 1, 2, ...),themul- form,” J. Opt. Soc. Amer. A, vol. 14, no. 7, pp. 1467Ð1477, 1997. 693 622 tiplicity of eigenvalue λ1 = 1ish + 1, and the multiplicity of [25] S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra. Englewood 694 Cliffs, NJ, USA: Prentice-Hall, 1979. 695 623 λ2 = −1ish. [26] T. Lawson, Linear Algebra. New York, NY, USA: Wiley, 1996, pp. 203Ð 696 624 The proof of Theorem 1 is so achieved. 206. 697 [27] G. W. Stewart, Introduction to Matrix Computations. New York,NY,USA: 698 625 REFERENCES Academic, 1973. 699 [28] L. Chang, X. Feng, X. Li, and R. Zhang, “A fusion estimation method 700 626 [1] V. Namias, “The fractional order Fourier transform and its application to based on fractional Fourier transform,” Digit. Signal Process., vol. 59, 701 627 quantum mechanics,” J. Inst. Math. Appl., vol. 25, pp. 241Ð65, 1980. pp. 66Ð75, 2016. 702 628 [2] A. C. McBride and F. H. Kerr, “On namias’s fractional Fourier transforms,” [29] Y.F. Gan, “Research on copy-move image forgery detection using features 703 629 IMA J. Appl. Math., vol. 39, no. 2, pp. 59Ð175, 1987. of discrete polar complex exponent transform,” Int. J. Bifurcat. Chaos, 704 630 [3] L. B. Almeida, “The fractional Fourier transform and time-frequency vol. 25, no. 14, pp. 1540018-1Ð1540018-15, 2015. 705 631 representations,” IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3084Ð [30] R. Tao, X. Y. Meng, and Y. Wang, “Image encryption with multiorders of 706 632 91, Nov. 1994. fractional Fourier transforms,” IEEE Trans. Inf. Forensics Security,vol.5, 707 633 [4] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier no. 4, pp. 734Ð738, Apr. 2010. 708 634 Transform: With Applications in Optics and Signal Processing. London, [31] V. S. Verma and R. K. Jha, “An overview of robust digital image water- 709 635 U.K.: Wiley, 2001. marking,” IETE Tech. Rev., vol. 32, no. 6, pp. 479Ð496, 2015. 710 636 [5] R. Tao, B. Deng, and Y. Wang, Fractional Fourier Transform and Its [32] E. D. Tsougenis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Touras- 711 637 Applications. Beijing, China: Tisinghua University Press, 2009. sis, “Performance evaluation of moment-based watermarking methods: A 712 638 [6] S. C. Pei and M. H. Yeh, “Improved discrete fractional Fourier transform,” review,” J. Syst. Softw., vol. 85, no. 8, pp. 1864Ð1884, 2012. 713 639 Opt. Lett., vol. 22, no. 14, pp. 1047Ð1049, 1997. [33] F. Hartung and M. Kutter, “Multimedia watermarking techniques,” Proc. 714 640 [7] S. Pei, M. Yeh, and C. Tseng, “DiscreteIEEE fractional Fourier transform based ProofIEEE, vol. 87, no. 7, pp. 1079Ð1107, Jul. 1999. 715 641 on orthogonal projections,” IEEE Trans. Signal Process., vol. 47, no. 5, [34] Q. Cheng and T. S. Huang, “An additive approach to transform-domain 716 642 pp. 1335Ð1348, May 1999. information hiding and optimum detection structure,” IEEE Trans. Multi- 717 643 [8] M. T. Hanna, N. P. A. Seif, and W. A. E. M. Ahmed, “Hermite-Gaussian- media, vol. 3, no. 3, pp. 273Ð284, Sep. 2001. 718 644 like eigenvectors of the discrete Fourier transform matrix based on the [35] A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform-based 719 645 singular-value decomposition of its orthogonal projection matrices,” IEEE watermarking method with an optimal implementation radius,” J. Electron. 720 646 Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 11, pp. 2245Ð2254, Imag., vol. 20, no. 3, pp. 033008-1Ð033008-8, 2011. 721 647 Nov. 2004. [36] J. R. Hernandez,« M. Amado, and F. Perez-Gonz« alez,« “DCT-domain wa- 722 648 [9] C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional termarking techniques for still images: Detector performance analysis and 723 649 Fourier transform,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1329Ð a new structure,” IEEE Trans. Image Process., vol. 9, no. 1, pp. 55Ð68, 724 650 1337, May 2000. Jan. 2000. 725 LIU et al.: FRACTIONAL KRAWTCHOUK TRANSFORM WITH AN APPLICATION TO IMAGE WATERMARKING 15

726 [37] S. A. Parah, J. A. Sheikh, N. A. Loan, and G. M. Bhat, “Robust and Guoniu Han received the B.S. degree in applied 801 727 blind watermarking technique in DCT domain using inter-block coefficient mathematics from Wuhan University, Wuhan, China, 802 728 differencing,” Digit. Signal Process., vol. 53, pp. 11Ð24, 2016. in 1987, and the Ph.D. degree in mathematics from 803 729 [38] C. L. Lei, Z. X. Zhang, Y. H. Wang, B. Ma, and D. Huang, “Dither the University of Strasbourg I, Strasbourg, France, in 804 730 modulation of significant amplitude difference for wavelet based robust 1992. Since 1993, he has been a Research Associate 805 731 watermarking,” Neurocomputing, vol. 166, pp. 404Ð415, 2015. at the French National Center for Scientific Research. 806 732 [39] V. S. Verma and R. K. Jha, “Improved watermarking technique based on 807 733 significant difference of lifting wavelet coefficients,” Signal Image Video 734 Process., vol. 9, no. 6, pp. 1443Ð1450, 2015. 735 [40] Q. Guo, Z. J. Liu, and S. T. Liu, “Image watermarking algorithm based on 736 fractional Fourier transform and random phase encoding,” Opt. Commun., 737 vol. 284, pp. 3918Ð3923, 2011. 738 [41] B. J. Chen, G. Coatrieux, G. Chen, X. M. Sun, J. L. Goatrieux, and H. Z. 739 Shu, “Full 4-D quaternion discrete Fourier transform based watermarking Jiasong Wu received the joint Ph.D. degree from 808 740 for color images,” Digit. Signal Process., vol. 28, pp. 106Ð119, 2014. the Southeast University, Nanjing, China, and the 809 741 [42] J. Guo and H. Prasetyo, “False-positive free SVD-based image water- University of Rennes 1, Rennes, France, in 2012. 810 742 marking,” J. Vis. Commun. Image R., vol. 25, no. 5, pp. 1149Ð1163, 2014. He is currently an Assistant Professor of Southeast 811 743 [43] B. Y. Lei, I. Y. Soon, and E. L. Tan, “Robust SVD-based audio watermark- University. His research interests include fast algo- 812 744 ing scheme with differential evolution optimization,” IEEE Trans. Audio, rithms of digital signal processing, compressed sens- 813 745 Speech, Language Process., vol. 21, no. 11, pp. 2368Ð2378, Nov. 2013. ing, and convolutional network. 814 746 [44] E. D. Tsougenis, G. A. Papakostas, and D. E. Koulouriotis, “Introducing 815 747 the separable moments for image watermarking in a totally moment- 748 oriented framework,” in Proc. Int. Conf. Digit. Signal Process., 2013, 749 pp. 1Ð6. 750 [45] E. D. Tsougenis, G. A. Papakostas, and D. E. Koulouriotis, “Image water- 751 marking via separable moments,” Multimedia Tools Appl., vol. 74, no. 11, 752 pp. 3985Ð4012, 2015. 753 [46] P. Bas, T. Furon, F. Cayre, G. Doerr,¬ and B. Mathon, Springer Briefs in Zhuhong Shao received the B.S. degree in biomed- 816 754 Electrical and Computer Engineering: Watermarking Security. Singapore: ical engineering from Jilin Medical University, Jilin, 817 755 Springer, 2016. China, in 2009, and the M.S. degree in electrical en- 818 756 [47] N. M. Makbol, B. E. Khoo, and T. H. Rassem, “Block-based discrete gineering from Beijing Jiaotong University, Beijing, 819 757 wavelet transform-singular value decomposition image watermarking China, in 2011 and the Ph.D. degree in computer 820 758 scheme using human visual system characteristics,” IET Image Process., science and technology from Southeast University, 821 759 vol. 10, no. 1, pp. 34Ð52, 2016. Nanjing, China, in 2015. He is currently a Lecturer in 822 760 [48] X. Wang, C. Wang, H. Yang, and P. Niu, “A robust blind color image the College of Information Engineering, Capital Nor- 823 761 watermarking in quaternion Fourier transform domain,” J. Syst. Softw., mal University, Beijing, China. His research interests 824 762 vol. 86, no. 2, pp. 255Ð277, 2013. include image analysis and pattern recognition. 825 763 [49] B. Chen and G. W. Wornell, “Quantization index modulation: a class of 826 764 provably good methods for digital watermarking and information embed- 765 ding,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1423Ð1443, May 2001. 766 [50] H. Y. Yang, X. Y. Wang, P. P. Niu, and A. L. Wang, “Robust color image 767 watermarking using geometric invariant quaternion polar harmonic trans- 768 form,” ACM Trans. Multimedia Comput. Commun. Appl., vol. 11, no. 3, 769 pp. 1Ð26, 2015. 770 [51] Test images database. [Online]. Available: http://decsai.ugr.es/cvg/ 771 dbimagenes/ Gouenou Coatrieux received the Ph.D. degree in 827 Q2 772 [52] The MPEG-7 database. [Online]. Available:http://www.dabi.temple.edu/ signal processing and telecommunication from the 828 773 ∼shape/MPEG7/dataset.html University of Rennes I, Rennes, France, in collab- 829 774 [53] L. Li, S. Li, A. Abraham, and J. Pan, “Geometrically invariant image oration with the Ecole Nationale Superieure« des 830 775 watermarking using polar harmonic transforms,” Inf. Sci., vol. 199, pp. 1Ð Tel« ecommunications,« Paris, France, in 2002. He is 831 776 19, 2012. currently a Professor in the Department of Informa- 832 777 [54] B. W. Feng, W. Lu, W. Sun, J. W. Huang, and Y. Q. Shi, “Robust im- tion and Image Processing, Institut Mines-Telecom, 833 778 age watermarking based on Tucker decomposition and adaptive-lattice Telecom Bretagne, Brest, France, and his research 834 779 quantization index modulation,” Signal Process. Image Commun., vol. 41, is conducted in the LaTIM Laboratory, INSERM 835 780 pp. 1Ð14, 2016. U1101, Brest. His primary research interests include 836 781 [55] S. P. Maity and M. K. Kundu, “Perceptually adaptive spread transform im- medical information system security, watermarking, 837 782 age watermarking scheme using Hadamard transform,” Inf. Sci., vol. 181, electronic patient records, and healthcare knowledge management. 838 783 pp. 450Ð465, 2011. 839 784 [56] X. Y. Wang, Y. N. Liu, S. Li, H. Y. Yang, P. P. Niu, and Y. Zhang, “A 785 new robust digital watermarking using local polar harmonic transform,” 786 Comput. Electr. Eng., vol. 46, pp. 403Ð418, 2015. 787 [57] D. Merlini, R. Sprugnoli, and M. C. Verri, “The method of coefficients,” 788 Amer. Math. Monthly, vol. 114, no. 1, pp. 40Ð57, 2007. 789 [58] I. M. Gessel, “The method of coefficients,” in Proc. Waterloo Workshop 790 Comput. Algebra, Waterloo, ON, Canada,IEEE 2008. Proof 791 Xilin Liu received the B.S. degree in information and Huazhong Shu received the B.S. Degree in applied 840 792 computing science and the M.S. degree in computa- mathematics from Wuhan University, Wuhan, China, 841 793 tional mathematics both from the East China Univer- in 1987, and the Ph.D. Degree in numerical analy- 842 794 sity of Technology, Nanchang, China, in 2010 and sis from the University of Rennes, Rennes, France, in 843 795 2013, respectively. He is currently working toward 1992. He is currently a Professor in the Department of 844 796 the Ph.D. degree in computer science and technology Computer Science and Engineering, Southeast Uni- 845 797 from Southeast University, Nanjing, China. His re- versity, Nanjing, China. His recent research interests 846 798 search interests include fractional order transforms, include the image analysis, pattern recognition, and 847 799 moment invariant, and digital image watermarking. fast algorithms of digital signal processing. 848 800 849 850 QUERIES

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