Moment-based local image watermarking via genetic optimization

Applied Mathematics and Computation 227 (2014) 222–236

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Moment-based local image watermarking via genetic optimization G.A. Papakostas a,⇑, E.D. Tsougenis b, D.E. Koulouriotis b a Human Machines Interaction (HMI) Laboratory, Department of Computer and Informatics Engineering, TEI of Eastern Macedonia and Thrace, GR-65404 Agios Loukas, Kavala, Greece b Democritus University of Thrace, Department of Production Engineering and Management, 67100 Xanthi, Greece

a r t i c l e

i n f o

Keywords: Image watermarking Locality Genetic algorithms Image moments Dither modulation

a b s t r a c t A totally optimized image watermarking methodology that manages to enhance its local behaviour by applying the Krawtchouk moments is presented through this paper. The introduced technique is making use of a simple genetic algorithm in order to optimize the set of parameters that significantly influences the locality properties alongside with the overall performance of the watermarking procedure. Herein, the watermarking process is tackled as an optimization procedure. In this context, the appropriate set of configuration parameters is being searched for ensuring high quality watermarked images and low bit error rates at the extraction stage. For this purpose, several traditional and newly introduced image quality measures are used in order to quantify the influence of the examined set of parameters, as far as the quality of the watermarked image and the accuracy of the extracted watermark are concerned. The proposed method produces watermarked images of high quality and ensures high detection rates under several non-geometric attacks, by using less prior-knowledge at the detector’s side. Extensive experiments have shown that by handling the watermarking process as an optimization problem, more robust and accurate watermarking schemes can be derived. Ó 2013 Published by Elsevier Inc.

1. Introduction The rapid growing of digital media interchange and transportation through computer networks makes the issue of the digital content’s protection a crucial subject for the copyright and the authentication of the content’s owner. This specific need motivated scientists to study and develop efficient methods able to secure digital images, videos, sounds from malicious actions which tend to distort or illegally make use of them, without taking permission from the owner authority. A popular and efficient methodology commonly used to ensure the integrity, authority and authenticity of the digital media is the well-known process of watermarking. This procedure deals with the appropriate insertion of copyright information into the media content, in order to give a copyright signature of the owner. The watermarking process has been applied successfully on several digital contents such as images [1], videos [2] and sounds [3]. As far as the watermarking of digital images is concerned, there is an open research issue that is strongly connected with the tracking of ideal positions for watermarks’ insertion. The watermark information needs to be optimal embedded, in the sense that the distortion of the resulted watermarked image should be negligible without degrading its ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (G.A. Papakostas), [email protected] (E.D. Tsougenis), [email protected] (D.E. Koulouriotis). 0096-3003/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.amc.2013.11.036

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robustness under attacking conditions. Moreover, along with the determination of the optimal image portion the overall watermarking procedure depends on a set of configuration parameters that also need to be optimized, in order to generate watermarked images of high quality and to extract high fidelity watermarks in attacking environments. The proposed methodology manages to automatically select the optimum local region of the cover image and embed the watermark information in the transform domain defined by the Krawtchouk moments of the image. The Krawtchouk moments have the inherent property of adjusting their local behaviour [4] by setting the appropriate parameters (p1, p2) of the corresponding orthogonal polynomials. The specific area determination parameters (p1, p2) in conjunction with embedding strength (D) and moments’ order (n) and repetition (m) are the configuration parameters that aimed to be optimized herein, by using a Genetic algorithm (GA) and a suitable objective function that defines the appropriateness of the problem’s solutions. Since the main requirements of a typical watermarking scheme is the high visual quality of the watermarked image and its robustness under certain attacks, a number of traditional and newly introduced image quality metrics are applied. As a matter of fact, the image quality and robustness of the watermarking process are assessed by PSNR [5], SSIM [6], Q [7], QT [8], Qk [8] indices and the well known Bit error rate (BER) respectively. These metrics are used to evaluate the usefulness of each candidate solution, through an appropriate formed objective function. As far as the locality behaviour of the Krawtchouk moments is concerned, preliminary results [5] have shown that there is a near optimum parameter set (p1, p2), which ensures high watermarking quality and robustness to certain watermarking attacks, simultaneously. Therefore, the optimum parameters (p1, p2) can be used as surrogates to the conventional ones (0.5, 0.5) which are traditionally used until now [9]. Moreover, the current work shows that the optimal selection of all free configurations parameters (embedding strength, moment order and repetition), gives more degrees of freedom in the overall watermarking process. It is worth pointing out that although the transformation of the watermarking procedure to an equivalent optimization process has already been reported in the literature [10–12], the present work constitutes the first attempt in optimizing a moment-based watermarking scheme. Moment-based watermarking methods are controlled by different configuration parameters (locality parameters, moment order and repetition) than other watermarking methods, which depend on the used moment transformation that controls the way the watermark information is inserted and therefore needs special manipulation. The paper is organized by describing the fundamentals of Krawtchouk moments in Section 2. Section 3, introduces the proposed watermarking scheme and analyses its main operational principles. An extensive experimental study regarding the performance of the introduced methodology takes place in Section 4, while the main conclusions are summarized in Section 5. 2. Orthogonal image moments Image moments have attracted the attention of the scientific community for several decades, as a powerful tool to describe the content of an image. They have been used in many research fields of the engineering life, such as pattern recognition [13], computer vision [14] and image processing [15] with considerable results. Recently, the scientists developed the orthogonal moments and moment invariants, which use as kernel functions polynomials that constitute orthogonal basis and therefore they present minimum information redundancy, meaning that different moment orders describe different image part. Some representative moment families are the Legendre [15], Zernike and Pseudo-Zernike [16], Fourier–Mellin [17], Tchebichef [18], Krawtchouk [4] moments. The main advantage of the orthogonal image moments is their ability to uniquely describe the content of an image by permitting the full reconstruction of the image they describe. The information embedment through moments’ domain in conjunction with the minimum reconstruction error of the final watermarked image, makes image moments an attractive and useful tool in watermarking applications [5,9,19–23]. Due to Krawtchouk moments’ local behaviour, which ensures the robustness in the presence of attacking conditions, the specific transformation domain is applied to this work and it will be further discussed through the following paragraph. 2.1. Krawtchouk moments Krawtchouk orthogonal moments constitute a high discriminative moment family defined in the discrete domain, introduced in image analysis by Yap et al. [4]. Krawtchouk moments are making use of the discrete Krawtchouk polynomials, having the following form,

  X N 1 ¼ K n ðx; p; NÞ ¼ 2 F 1 n; x; N; ak;n;p xk p k¼0

ð1Þ

where x, n = 0, 1, 2,. . ., N, N > 0, p e (0, 1) and 2F1 is the hypergeometric function. The computation of the Krawtchouk polynomials by using Eq. (1), presents numerical fluctuations and therefore a more stable version of them, the weighted Krawtchouk polynomials is used,

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðx; p; NÞ  K n ðx; p; NÞ ¼ K n ðx; p; NÞ qðn; p; NÞ

ð2Þ

where q(n; p, N) is the norm of the Krawtchouk polynomials,

qðn; p; NÞ ¼ ð1Þn

 n 1p n! ; p ðNÞn

n ¼ 1; . . . ; N

ð3Þ

and w (x; p, N), the weight function of the Krawtchouk moments,

wðx; p; NÞ ¼



N x

 px ð1  pÞNx

ð4Þ

In Eq. (3) the symbol ()n corresponds to the Pochhammer symbol, which for the general case is defined as ðaÞk ¼ ða þ 1Þ . . . ða þ k þ 1Þ. Based on the above definitions, the orthogonal discrete Krawtchouk image moments of (n + m)th order, of an NxM image with intensity function f(x, y) is defined as follows:

K nm ¼

N1 M1 X X

 n ðx; p ; N  1ÞK  m ðy; p ; M  1Þf ðx; yÞ K 1 2

ð5Þ

x¼0 y¼0

In practice, the computation of the weighted Krawtchouk polynomials is not performed through Eq. (2), since it is a very time consuming procedure, instead the following recursive algorithm [4] is applied.

 nþ1 ðx; p; NÞ ¼ AðNp  2np þ n  xÞK  n ðx; p; NÞ  Bnð1  pÞK  n1 ðx; p; NÞ pðN  nÞK

ð6Þ

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðN  nÞ ; A¼ ð1  pÞðn þ 1Þ

B¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ðN  nÞðN  n þ 1Þ ð1  pÞ2 ðn þ 1Þn

ð7Þ

With

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 ðx; p; NÞ ¼ wðx; p; NÞ; K qð0; p; NÞ  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðx; p; NÞ  1 ðx; p; NÞ ¼ 1  x K pN qð1; p; NÞ

ð8Þ

Krawtchouk moments are proved to be effective local descriptors, unlike the other moment families which capture the global features of the objects they describe [4,5,24]. This locality property is controlled by the appropriate adjustment of the p1, p2 parameters of Eq. (5). Recently, there is a significant increase of the scientific interest on applying image moments in watermarking applications [5,9,19–23]. This moment-based watermarking category [20] uses the image moments as containers of the embedded information and thus it is a transform domain watermarking approach. 3. Proposed methodology An optimized image watermarking methodology using Krawtchouk moments is presented forthwith. The new algorithm takes advantage of the spatial watermark insertion mechanism introduced in [5], by avoiding the reconstruction of the watermarked image by its modified Krawtchouk moments, a procedure that adds significant computational burden and undesired reconstruction errors. Additionally, the usage of the dither modulation [25] provides the opportunity of inserting the binary message into the Krawtchouk moments of the cover image and afterwards extracting it without taking into consideration the original image or its moments in contrary with other similar algorithms [5,9]. Besides the aforementioned novelties of the proposed algorithm, dealing with the watermarking processing steps, its adaptive mechanism (in selecting the most appropriate image portion for watermark embedding), in relation to some configuration parameters constitute an important contribution that enhances the method’s performance significantly. This adaptation is achieved through an optimization scheme that uses a simple Genetic algorithm (GA). The set of parameters that aimed to be optimized consists of the (p1, p2) parameters, which define the spatial location of the Krawtchouk polynomials where they are computed, the quantization step (D) of the dither modulator and the moment order (n) and repetition (m) that generate the moments components participating in watermark embedding. These parameters are provided in an optimal way by the genetic algorithm, while their appropriateness is measured according to specific fitness function incorporating different quality measures. In the following sections, a detailed description of the fundamental elements of the proposed methodology is presented.

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3.1. Genetic optimization Genetic algorithms (GAs) have played a major role in many applications of the engineering science, since they constitute a powerful tool for optimization. A simple genetic algorithm is a stochastic method that performs searching in wide search spaces, depending on some probability values that mimics the evolutionary process which characterizes the evolution of living organisms [26]. Due to the specific reasons, GA has the ability to converge to the global minimum or maximum, depending on the specific application and to skip possible local minima or maxima [27], respectively. In this paper the image watermarking procedure is defined as an optimization problem, which is described by the following formula:

min

p1 ;p2 ;Dk ;ni ;mi

ðf Þ

ð9Þ

Based on the above Eq. (9), the ultimate goal is the minimization of an objective function f which is highly dependent on the set of parameters (p1, p2, Dk, ni, mi). In the above optimization problem, p1 and p2 are the parameters of the Krawtchouk polynomials that control their locality behaviour, Dk are the quantization steps which control the embedding strength of the watermarks’ insertion and ni, mi are the orders and repetitions of the used moments. The number of the applied quantization steps (k), the orders and repetitions (i) depend on the problem definition, as explained in the experimental section. This optimization procedure is illustrated in the following Fig. 1. As it is depicted in Fig. 1, the main processing step dealing with the watermarking procedure is the fitness assignment that measures the appropriateness of a candidate solution (set of parameters). This fitness is assigned by applying specific objective function incorporating the quality of the watermarked image and the fidelity of the extracted watermark information. As a matter of fact, four different quality and correlation measures are used in all the experiments, defined according to Table 1. The image quality measures defined in Table 1 are used to evaluate the quality of the watermarked image, while BER examines the number of the missed bits at the extraction stage. In the experimental section the quality measures will be combined with BER into the fitness function in order to evaluate the suitability of the solutions provided by the genetic algorithm. It’s worth noting that while PSNR and BER have been applied frequently in watermarking, the rest of the quality measures SSIM, Q, QT and QK are firstly used in this work. In the case of SSIM and Q indices, the parameters lx, ly and rx, ry correspond to the mean and standard deviation intensity values of the compared images. 3.2. Watermark embedding The watermark embedding procedure participating to the optimization flow chart of Fig. 1, consists of the processing modules depicted in the following Fig. 2.

Fig. 1. Proposed watermarking via genetic optimization.

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Table 1 Image quality and watermark fidelity measures. Image quality measures Peak to signal noise ratio (PSNR) [5]





Lmax Lmax ; where MSE ð2lx ly þC 1 Þð2rxy þC 2 Þ ðl2x þl2y þC 1 Þðr2x þr2y þC 2 Þ

PSNR ¼ 10log10

Lmax ¼ 255

Structural similarity (SSIM) [6]

SSIMðx; yÞ ¼



C1 ¼ K 1 L K 1 ¼ 0:01 with and L ¼ 255 C2 ¼ K 2 L K 2 ¼ 0:03

Universal image quality index (Q) [7]

Q ðx; yÞ ¼ ðr2 þr2 Þ½lx 2 þy l2  x y x y P P Q T;K ðx; yÞ ¼ 19 2p¼0 2q¼0 Iðx;yÞ where Iðx;yÞ is a correlation index, relative to the used moment family pq pq

Tchebichef moments quality index (QT) [8] Krawtchouk moments quality index (QK) [8] Watermark fidelity measure Bit error rate (BER)

4rxy l

l

Number of bit errors/ total number of bits inserted

Fig. 2. Watermark embedding.

In the above Fig. 2, the original image corresponds to the cover image where the L-bit length binary message (b1, b2, . . ., bL) is inserted, by constructing the final watermarked image. All the intermediate processing stages participating in the proposed watermarking embedding phase are described in the following. Step 1: Krawtchouk moments A set of Krawtchouk moments up to an order P such that (n + m) < P and n, m – 0 is computed according to Eq. (5). The second constraint is applied in order to prohibit the distortion of the low order components, since they describe the coarse image information and a possible modification of them may lead to watermarked images of low quality. The number of computed moments in this step [9], subject to the above constraints is equal to L = (P + 1)(P + 2)/2  1. At this stage there is a need of a key set K1, which is also necessary at the detectors’ side and corresponds to the set of parameters K1:{P, p1, p2}. Step 2: Dither modulation Dither modulation constitutes a significant methodology that embeds a signal into another one, increases the embedding rate with minimum distortion of the original signal and ensures robustness under attacking conditions [25]. The dither modulation embedding scheme has been applied successfully in moment-based watermarking algorithms [21–23]. In this work the Krawtchouk moments of the original image is used as the host signal where the L-bit length binary message (b1, b2, . . ., bL) is inserted according to Eq. (10).

  ~ n m ¼ K ni mi  di ðbi Þ D þ di ðbi Þ; K i i D

i ¼ 1;

;L

ð10Þ

where [] is the rounding operator, D the quantization step (key K2) and di() the ith dither function satisfying di(1) = D/ 2 + di(0). The dither vector (d1(0), d2(0), . . ., dL(0)) is uniformly distributed in the range [0, D]. Step 3: Watermark construction ~ n m ) resulted by the application of the dither modulation according to Eq. (10) are The modified Krawtchouk moments (K i i used to construct the watermark information. However, unlike the previously presented Krawtchouk moment-based algorithms [5,9] there is not any reconstruction of the entire image, since the specific process constitutes a laborious and time consuming task due to the fact that it requires a large number of moments in order to achieve low reconstruction errors. On the contrary, the proposed method manages to isolate the watermark part of the modified moments (frequency domain) and construct a watermark image which is added to the original image (spatial domain), as depicted in the following Eqs. (11)–(13):

fw ðx; yÞ ¼

N X N N X N X X ~n m K  n ðxÞK  m ðyÞ ¼  n ðxÞK  m ðyÞ K ðK ni mi þ Dni mi ÞK i i i i i i i¼1 j¼1

i¼1 j¼1

N X N n X X  n ðxÞK  m ðyÞ þ  n ðxÞK  m ðyÞ K ni mi K Dni mi K ¼ i i i i i¼1 j¼1

i¼1

ð11Þ

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where the quantity Dni mi corresponds to the additional information inserted to the cover image through its moments, which is equal to the difference between the modified and the original Krawtchouk moments defined as:

~n m  Kn m Dni mi ¼ K i i i i

ð12Þ

By identifying that the double summation of Eq. (11) corresponds to the original image and the second part of the equation to the inserted information, Eq. (11) can be rewritten as:

fw ðx; yÞ ¼ f ðx; yÞ þ Wðx; yÞ

ð13Þ

It is worth noting that the number of the applied moments in Eq. (11) is small and equal to the binary message’s length. In fact this number is significantly smaller than that of [5,9]. Based on Eq. (13) it is evident that the watermark is embedded in spatial domain, which constitutes an advantage of the proposed algorithm since it avoids any image reconstruction that needs many image moments. 3.3. Watermark extraction The watermark extraction procedure at the detector’s side consists of almost the same processing steps with the embedding stage, as illustrated in Fig. 3. The only difference is the application of the Minimum Distance Decoder (MDD) module, which is applied in order to extract the embedded binary message from the dither modulated, modified Krawtchouk moments of the incoming (attacked) image. The operation of the MDD is defined by the following formula:

Minimum Distance Decoder

 2 ^ ¼ arg min jK 0 j  K 0 b i ni mi j ni mi j2½0;1

ð14Þ

where jK 0ni mi jj is the ith Krawtchouk moment of the attacked image which is quantized considering a bit value of j e [0, 1]. ^ is decided to be 0 or 1 regarding the distance between the corresponding quantized Krawtchouk moments Therefore a bit b i ^ value. and its original value K 0ni mi . The extracted bit is assigned depending on the j of the minimum distance b i Finally, it is worthwhile noting that the usage of the dither modulation in both watermark embedding and extraction procedures enhances the blind nature of the proposed algorithm, since there is no longer any need of the original Krawtchouk moments or the watermarked image in order to extract the hidden information. Instead, it is capable of extracting the embedded information by just applying the same quantization module in combination with the MDD. 4. Experimental study In order to investigate the performance of the proposed watermarking scheme, a set of appropriate experiments has been arranged. For the experimental purposes, specific software has been developed in C++ language, while all experiments are executed in an Intel i5 3.3 GHz PC with 8 GB RAM. For experimental purposes three well known benchmark images of 256  256 pixels size are selected and depicted in Fig. 4. Through all the experiments, a set of attacked images is generated by using the well-known Stirmark benchmark [28], in order to evaluate the robustness of the proposed algorithm in extracting the embedded information. The applied attack types are: Median filtering (2  2, 4  4, 6  6, 8  8), Gaussian noise (5%, 10%, 15%, 20%) and JPEG compression (Q = 5%, 10%, 15%, 20%, 40%, 60%, 80%). It is noted that only signal processing attacks have been selected since the geometric attacks (rotation, scaling) can be easily handled with a pre-processing step of image normalization [23]. Moreover, the selection of the appropriate image portion is independent to geometric attacks, since the interpolation process applied to these transformations, slightly affects the watermarked image. The presented settings (Table 2) are applied to the GA throughout the whole set of the experiments in order to produce the optimum results for each case. The fitness function which is used to evaluate the appropriateness of each candidate solution in each examined case takes two forms regarding to the type of the used image quality index. In this context, the fitness function (FA) in the case of the PSNR index is defined as:

Fig. 3. Watermark extraction.

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Fig. 4. Test images (a) Lena, (b) Bike and (c) Lighthouse.

Table 2 Genetic algorithm’s settings. Parameter

Value

Population size Maximum generations Crossover probability Mutation probability Selection method Crossover points

50 50 0.8 0.01 Stochastic universal approximation (SUS) 2 Points

T 1X F A ¼ SF 1  jPSNR  PSNRtarget j þ SF 2  BERj T j¼1

! ð15Þ

where T is the number of attacks encountered in the procedure, SF1, SF2 are scaling factors equal to 10 and 1 respectively, BERj is the BER of the jth attacked image and PSNRtarget is a desired PSNR value equal to 44. The incorporation of the target PSNR transforms the optimization to a constrained procedure in order to ensure a minimum of image quality that must be acquired. For the cases of the other quality images the fitness function (FB) takes the following form:

F B ¼ SF 1  j1  IQ j þ SF 2 

T 1X BERj T j¼1

!

ð16Þ

where IQ is one of the quality indices of Table 1 (except for PSNR) and the scaling factors take the same values with Eq. (15). The experimental study is divided into four parts each one defining a different watermarking optimization problem. During the first part, the selection of the best set of parameters (p1, p2) is examined when different quality indices (Table 1) are used into the fitness function regarding to the identification of the appropriate image portion for watermark embedment. Having tracked the optimum host area, the embedding strength of the watermarking process, which is actually the quantization step (D) of the dither modulation, is optimized through the second part of the experiments. The specific parameter remains the same for each quantized moment coefficient during the first two experimental parts in contrast with the third part, where the assumption that the assignment of different Ds with respect to coefficients’ magnitude size might produce better results is considered. Due to the fact that the watermarking message can be lengthy enough, the optimization task could be highly difficult, so in this study only three different Ds are applied during quantization process. For this reason the moment coefficients are sorted in descending order and they are separated linearly into three groups, each one having its quantization step D. Therefore, in this case the number of unknowns is increased to five (p1, p2, D), where D is the vector D = [D1, D2, D3]. In the fourth part, along with the five parameters of the previous experiment, the orders (n) and repetitions (m) of the participated moment coefficients are optimized. Unfortunately, since the number of orders and repetitions are equal to the message length (L) being embedded, the additional parameters are 2L in this case (p1, p2, D, n, m), with n = [n1, n2, ..., nL] and m = [m1, m2, ..., mL]. For the sake of the experimental study, the watermark message will be selected to be of 200 bits length (L = 200) for all experiments, which constitutes a typical embedding information in the area of moment-based watermarking methods. The detailed analysis as long as the experiments’ outcomes of each part are presented in the following subsections. 4.1. Case 1: unknown parameters (p1, p2) This experimental part aims to find the best set of parameters (p1, p2) which controls the image portion where the watermark information is inserted. The suitability of a set is evaluated by using the quality and fidelity indices of Table 1.

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Since only the parameters (p1, p2) are optimized in this part, the remaining configuration parameters D and P, are set to 100 and 19, respectively. The number of Krawtchouk moments computed by setting P = 19, according to step 2 of the embedding phase (Section 3.2), is equal to 209 and thus the first 200 moments are kept. Furthermore, the high value of parameter D, is justified by the high magnitudes of the Krawtchouk moments (103 order) that need a corresponding high influence in order to be affected. The watermarking detection results under various attacking conditions, along with the values of the optimized parameters separately for each applied image quality metric, are summarized in the following Tables 3–5, for the Lena, Bike and Lighthouse image respectively. By examining Tables 3–5 tables the marginal outperformance of the SSIM (0.15, 0.21, 0.12 mean BER) quality index over the rest ones can be deduced clearly. Furthermore, the Q (0.20, 0.21, 0.13 mean BER) and QK (0.16, 0.22, 0.12 mean BER) indices show similar behaviour having the next more efficient performance. Moreover, PSNR (0.21, 0.23 0.14 mean BER) and QT (0.22, 0.23 0.24 mean BER) quality measures show the worst performance among all the indices. It is worth pointing out the high performance of the QK index compared to the similar QT one. This happens due to the fact that the QK index uses the optimized set of parameters (p1, p2) derived by the optimization process and therefore the computation of its correlation index, which is strongly connected with the polynomial location, takes into account the image information located in the optimized image portion. The selected image portions by the SSIM and QK indices are similar for the case of Lena image. The p1 and p2 parameters that define the specific host areas control the horizontal and vertical direction of the watermark localization, respectively. These portions correspond to the upper, centered on axis y, image region, while in the case of the other quality indices the decided portions vary to a region sited on the lower/center side of the image. The selections of the SSIM and QK indices show remarkable robustness to the median filtering attack due to high homogeneity of the specific regions, while their behavior is similar to the other quality measures under noise and jpeg attacks. The almost identical performance of all quality indices for the Gaussian noise and jpeg attacks is justified by the fact that the former attack is applied on the entire image area following a Gaussian distribution and therefore the affection to the portion selection will be similar for all regions. The latter attack suppresses the high frequency components of the image. In cases of high compressed images (Q = 5%, 10%, 15%) these components correspond to a significant part of the initial image and therefore the jpeg compression will affect the entire image in a common way. Similar qualitative results are derived in the case of the Bike image, with the additional observation that the overall performance is lower than that of the Lena image. The Bike image is more complex compared to Lena, without any homogeneous regions and thus the performance regarding the median filtering is worst. The other attacks still affect significantly the extraction procedure with small differences among the quality indices. Finally, in the case of the Lighthouse image the detection performance is quite improved, since the genetic algorithm found more suitable image portions, almost for all quality indices (0.14, 0.12, 0.13, 0.24, 0.12 mean BERs). The optimal image portion is located at the right and centered on y axis side of the image over the wooden fence, with the characteristic of a repeated white pattern. This is the reason why the median filtering does not significantly affect the detection procedure, while the noise and jpeg compression attacks show the same behavior as for the previous two images. The above discussion can be justified optically by examining the corresponding watermarked images along with the values of each quality index, as depicted in the following Fig. 5.

Table 3 Watermarking detection results for the Lena test image. Image – Lena PSNR (44 dB)

SSIM (0.9955)

Q (0.9865)

QT (0.9614)

QK (0.9947)

Optimized Parameters

p1 = 0.7995 p2 = 0.6456

p1 = 0.1161 p2 = 0.4848

p1 = 0.5191 p2 = 0.4453

p1 = 0.5045 p2 = 0.4391

p1 = 0.1002 p2 = 0.3623

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

BER 0.27 0.29 0.36 0.47 0.16 0.26 0.27 0.41 0.41 0.20 0.05 0.01 0.00 0.00 0.00 0.21

BER 0.08 0.08 0.15 0.27 0.23 0.24 0.33 0.36 0.39 0.16 0.06 0.01 0.00 0.00 0.00 0.15

BER 0.37 0.40 0.39 0.44 0.11 0.20 0.26 0.29 0.38 0.16 0.06 0.00 0.00 0.00 0.00 0.20

BER 0.39 0.36 0.44 0.48 0.11 0.21 0.25 0.32 0.42 0.17 0.09 0.02 0.00 0.00 0.00 0.22

BER 0.06 0.11 0.18 0.27 0.20 0.30 0.34 0.38 0.36 0.19 0.04 0.00 0.00 0.00 0.00 0.16

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Table 4 Watermarking detection results for the Bike test image. Image – Bike PSNR (44 dB)

SSIM (0.9993)

Q (0.9987)

QT (0.9808)

QK (0.9986)

Optimized Parameters

p1 = 0.4280 p2 = 0.4734

p1 = 0.3470 p2 = 0.3821

p1 = 0.3443 p2 = 0.3764

p1 = 0.7473 p2 = 0.5414

p1 = 0.8153 p2 = 0.7931

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

BER 0.40 0.53 0.43 0.45 0.14 0.22 0.27 0.34 0.38 0.20 0.07 0.03 0.00 0.00 0.00 0.23

BER 0.25 0.34 0.44 0.46 0.15 0.22 0.33 0.34 0.40 0.21 0.05 0.00 0.00 0.00 0.00 0.21

BER 0.30 0.38 0.44 0.44 0.14 0.20 0.30 0.33 0.39 0.17 0.06 0.02 0.00 0.00 0.00 0.21

BER 0.40 0.46 0.43 0.47 0.12 0.20 0.31 0.29 0.42 0.21 0.06 0.03 0.00 0.00 0.00 0.23

BER 0.34 0.34 0.41 0.42 0.20 0.24 0.32 0.38 0.42 0.15 0.05 0.01 0.00 0.00 0.00 0.22

Table 5 Watermarking detection results for the Lighthouse test image. Image – Lighthouse PSNR (44 dB)

SSIM (0.9966)

Q (0.9906)

QT (0.9603)

QK (0.9985)

Optimized Parameters

p1 = 0.8763 p2 = 0.6933

p1 = 0.8239 p2 = 0.5663

p1 = 0.8359 p2 = 0.5916

p1 = 0.6278 p2 = 0.1711

p1 = 0.8370 p2 = 0.7149

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

BER 0.00 0.01 0.05 0.13 0.19 0.25 0.37 0.38 0.39 0.19 0.09 0.01 0.00 0.00 0.00 0.14

BER 0.01 0.01 0.02 0.06 0.16 0.25 0.29 0.37 0.42 0.16 0.04 0.00 0.00 0.00 0.00 0.12

BER 0.00 0.01 0.03 0.10 0.19 0.22 0.29 0.42 0.43 0.16 0.05 0.01 0.00 0.00 0.00 0.13

BER 0.38 0.44 0.51 0.49 0.16 0.22 0.30 0.40 0.43 0.14 0.05 0.01 0.00 0.00 0.00 0.24

BER 0.00 0.01 0.06 0.09 0.14 0.19 0.35 0.35 0.34 0.17 0.06 0.02 0.00 0.00 0.00 0.12

Summarizing, two are the main outcomes of this experimental case: (1) the superiority of the SSIM (QT also shows a close performance) as an image quality measure in contrast with the traditional PSNR index, which presents a weakness in measuring the images’ distortions and (2) the importance of image’s portion proper selection to the accurate extraction of the embedded information at the detector’s side. In order to better realize the value of handling the watermarking procedure as an optimization process, where the image portion for watermark insertion is optimally selected, the same experiments are conducted with the typical/non-optimal parameter set (p1, p2) = (0.5, 0.5), while the corresponding quality indices are measured. The following Fig. 6 illustrates the watermarked images and the corresponding values of the quality measures for this non-optimized case. What is deduced from Fig. 6 is that the (p1, p2) parameters do not affect significantly the PSNR index, since it takes almost the same value although the parameters are changed. Moreover, in the cases of Lena and Lighthouse images the watermark information is more visible compared to the watermarked images derived with the optimized parameter set (Fig. 5), with lower values of the quality indices. Furthermore, the visual result of the watermarked Bike image seems to be better than that of the optimized case. The multiple patterns located in the center of the image increase image’s complexity and make the watermark information highly imperceptible.

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Fig. 5. Watermarked (a) Lena, (b) Bike and (c) Lighthouse test images for optimized (p1, p2).

Except for the imperceptibility requirement that has to be maintained, the robustness of the watermarked images under attacking conditions needs also to be satisfied. For the conventional case of the non-optimized set of parameters (p1, p2) = (0.5, 0.5) the detection results are summarized in Table 6.

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Fig. 6. Watermarked Lena, Bike and Lighthouse (from left to right) test images for (p1, p2) = (0.5, 0.5).

By comparing the BERs of Table 6 with the corresponding results of Tables 3–5, it can be easily concluded that the optimization of the image portion for watermark insertion can significantly decrease the fault detection rates from 0.24 to 0.15, 0.25 to 0.21 and 0.22 to 0.12 for the case of Lena, Bike and Lighthouse benchmark images respectively. Although the quality indices highlight great differences between the optimized and non-optimized watermarked images, the optical results are inconsistent. Moreover, the detection rates of the three benchmark images are significantly improved by 37.5%, 16% and 45.5%, respectively. It is worth mentioning that only the SSIM quality index will be used to the remaining three experiments due to the fact that it presents the most stable and high performance compared to the other quality measures studied in this experimental part. 4.2. Case 2: unknown parameters (p1, p2, D) This study aims to investigate the enhancement of the watermarking performance gained by adding the quantization step (D) of the dither modulation as an additional parameter into the optimization procedure. In the moment-based watermarking methods the choosing of the D value is usually performed by trial and error, a task that is quite laborious. The selection of the appropriate D value is very crucial in satisfying the imperceptibility requirement since the specific parameter controls the amount of the embedded information and thus determines the visual quality of the resulted watermarked image.

Table 6 Watermarking detection results for all test images for (p1, p2) = (0.5, 0.5). Images

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

Lena

Bike

Lighthouse

BER 0.45 0.50 0.47 0.48 0.13 0.23 0.26 0.32 0.50 0.20 0.02 0.02 0.00 0.00 0.00 0.24

BER 0.42 0.54 0.50 0.51 0.13 0.23 0.25 0.32 0.50 0.28 0.07 0.05 0.00 0.00 0.00 0.25

BER 0.28 0.50 0.44 0.46 0.12 0.23 0.25 0.32 0.52 0.20 0.07 0.01 0.00 0.00 0.00 0.22

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The results of the conducted experiments in this case are illustrated in the following Fig. 7, while the corresponding detection rates are summarized in Table 7. By examining the watermarked images of Fig. 7 and the detection rates of Table 7, it can be deduced that by adding an extra degree of freedom to the watermarking procedure, a reduction of the detection errors can be achieved. More precisely, the BER is reduced from 0.15 to 0.11 (26.6% reduction), 0.21 to 0.07 (66.6% reduction) and 0.12 to 0.10 (16.6% reduction), for Lena, Bike and Lighthouse benchmark images, respectively. Apart from the remarkable reduction of the detection errors, the important conclusion is that the specific performance can be achieved without affecting significantly the quality of the watermarked image. More precisely, the solution produced by the genetic optimization can highly satisfy the robustness and imperceptibility requirements by pointing out the most appropriate region for watermark insertion along with an optimum embedding strength calibration. 4.3. Case 3: unknown parameters (p1, p2, Dk) In order to further improve the watermarking process, an important assumption is decided and examined in the third case herein. The assumption is that each moment coefficient should have its individual quantization step (D), since each moment has different magnitude and thus the amount of the inserted information cannot be the same. However, by assigning a separate quantization step (D) to each used moment coefficient, the number of the unknowns will be increased significantly, in fact 200 parameters have to be added for the case of the message length under testing. This increase of the total number of optimized parameters causes the growth of the search space, a circumstance that makes genetic algorithm incapable to solve this optimization problem. Due to the specific situation, it is decided to investigate the

Fig. 7. Watermarked Lena, Bike and Lighthouse (from left to right) test images for optimized (p1, p2, D).

Table 7 Watermarking detection results for all test images for optimized (p1, p2, D). Images Lena

Bike

Lighthouse

Optimized Parameters

p1 = 0.3632 p2 = 0.5006 D = 192.830

p1 = 0.1303 p2 = 0.8940 D = 465.5827

p1 = 0.4797 p2 = 0.5554 D = 196.9022

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

BER 0.12 0.16 0.21 0.31 0.24 0.17 0.14 0.09 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.11

BER 0.07 0.10 0.23 0.29 0.01 0.03 0.10 0.11 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.07

BER 0.02 0.14 0.24 0.29 0.23 0.16 0.10 0.08 0.18 0.01 0.00 0.00 0.00 0.00 0.00 0.10

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Fig. 8. Watermarked Lena, Bike and Lighthouse (from left to right) test images for optimized (p1, p2, Dk).

Table 8 Watermarking detection results for all test images for optimized (p1, p2, Dk). Images Lena

Bike

Lighthouse

Optimized Parameters

p1 = 0.3829 p2 = 0.4512 D1 = 195.0475 D2 = 185.1507 D3 = 236.0025

p1 = 0.1759 p2 = 0.8871 D1 = 479.2665 D2 = 252.4593 D3 = 346.5761

p1 = 0.5436 p2 = 0.4366 D1 = 216.0038 D2 = 281.8605 D3 = 196.7729

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

BER 0.16 0.18 0.27 0.35 0.23 0.16 0.12 0.13 0.17 0.01 0.00 0.00 0.00 0.00 0.00 0.12

BER 0.11 0.13 0.24 0.28 0.01 0.07 0.14 0.18 0.05 0.01 0.00 0.00 0.00 0.00 0.00 0.08

BER 0.01 0.08 0.18 0.25 0.25 0.18 0.15 0.10 0.11 0.01 0.00 0.00 0.00 0.00 0.00 0.09

need for different quantization step (D) per moment coefficient, by dividing the set of moments into three groups after sorting them with descend order by magnitude and assigning different D value to each group. Following this way, only three extra parameters (k = 3; D1, D2, D3) will be added to the optimization procedure, by raising the total unknowns to five. The watermarking results of this experimental case are illustrated in Fig. 8, while the corresponding detection rates are summarized in Table 8. A careful look at the above watermarked images and the corresponding detection rates of Table 8, gives the information that the usage of different quantization step (D) for each moment coefficient leads to more accurate watermarking results. As far as the robustness of the optimized watermarked images is concerned, one can conclude that the multiple Ds can improve the detection rates in the case of the Lighthouse image (from 0.10 to 0.09) while decreasing the rates of the other two images (from 0.11 to 0.12 and from 0.07 to 0.08). However, these differences to the detection performances are quite small, almost negligible. What is more remarkable in this experiment is that the decrease of the detection rates is followed by an increase to the watermarked images quality, as it can be deducted from SSIM index of Fig. 8. Based on the pre-mentioned results, the obtained solution manages to embed information into image portions where the imperceptibility requirement is highly satisfied but also maintains the robustness requirement. This first study in applying different quantization steps to the moment coefficients, shows that a more robust and accurate watermarking procedure can exist, by taking into account the significance of the moment coefficients that participate to the dither modulation process. Moreover, the applied partition of moments into three groups, seems to be incapable to improve significantly the overall performance, but still gives an indication that a more appropriate scheme for assigning quantization steps (Ds) to the moments can enhance the watermarking process.

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Fig. 9. Watermarked Lena, Bike and Lighthouse (from left to right) test images for optimized (p1, p2, Dk, ni, mi).

Table 9 Watermarking detection results for all test images for optimized (p1, p2, Dk, ni, mi). Images Lena

Bike

Lighthouse

Optimized Parameters

p1 = 0.5756 p2 = 0.4237 D1 = 135.4557 D2 = 229.2816 D3 = 193.9199 ni, mi

p1 = 0.1658 p2 = 0.8739 D1 = 233.8871 D2 = 165.5731 D3 = 203.4652 ni, mi

p1 = 0.4081 p2 = 0.1638 D1 = 186.0398 D2 = 191.9578 D3 = 271.3413 ni, mi

Attack type Median filter 2  2 Median filter 4  4 Median filter 6  6 Median filter 8  8 Gaussian noise 5% Gaussian noise 10% Gaussian noise 15% Gaussian noise 20% JPEG (Q = 5%) JPEG (Q = 10%) JPEG (Q = 15%) JPEG (Q = 20%) JPEG (Q = 40%) JPEG (Q = 60%) JPEG (Q = 80%) Mean values

BER 0.14 0.15 0.25 0.33 0.08 0.05 0.03 0.07 0.24 0.03 0.00 0.00 0.00 0.00 0.00 0.09

BER 0.14 0.21 0.34 0.37 0.11 0.15 0.21 0.22 0.14 0.02 0.00 0.00 0.00 0.00 0.00 0.12

BER 0.08 0.13 0.25 0.34 0.15 0.16 0.19 0.18 0.11 0.00 0.00 0.00 0.00 0.00 0.00 0.10

4.4. Case 4: unknown parameters (p1, p2, Dk, ni, mi) In the final experimental case, it is aimed to increase the watermarking detection rates by preserving the high quality of the watermarked images, through the addition of extra free parameters able to further optimize the overall watermarking procedure. The remaining parameters that could be optimized are the order (n) and repetition (m) of the moment coefficients which are also strongly connected to watermark information hiding. However, the addition of 2 parameters (ni, mi) for each ith moment coefficient would cause a significant increase to the problem’s unknowns that for the case of the 200 bit message length corresponds to 400 more parameters and makes the optimization problem quite difficult to solve. Based on this observation, it is decided to add only 20 parameters, 10 for the orders and 10 for the repetitions and to take all their possible combinations by forming a set of 100 (ni, mi) pairs. The remaining 10 orders from the total set of orders up to a maximum value equal to 19 in conjunction with the remaining repetitions of a same set are combined to form another set of 100 (ni, mi) pairs. As a matter of fact, the total number of unknowns are 25 after taking into account the p1, p2 and Ds parameters of the previous section. The watermarking results of this case are illustrated in Fig. 9, while the corresponding detection rates are summarized in Table 9. By comparing the watermarking results of Fig. 9 and Table 9 with the previous case, it can be deduced that the optimization of the defined watermarking procedure partially has failed. Although the detection rates of the Lena image is improved from 0.12 to 0.09, the quality of the resulted watermarked image is lower than that of Fig. 8a (SSIM value decreases from 0.9943 to 0.9906), a difference which is apparent optically. For the case of the other two benchmark images, the results are quite worst since the detection errors have been slightly increased for both images.

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The low watermarking performance of this experimental case testifies that the optimization task was not able to find a useful solution, due to the increased search space. Particularly, the number of the unknowns has increased by 400% from 5 to 25, an increase that significantly affects the ability of the genetic algorithm to locate an optimal solution to the high complex solutions space. In order to handle such a complex problem, more sophisticated genetic algorithms having adaptive crossover and mutation operators are needed. 5. Conclusion A detailed investigation of image watermarking process by handling it as an optimization procedure, under several configurations was presented in the previous sections. The proposed moment-based watermarking method is applied in an optimized framework where its performance analysis was studied in four independent cases, regarding the number and type of the free parameters being optimized, giving promising results. 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