Geometrically invariant and high capacity image watermarking scheme using accurate radial transform

Optics & Laser Technology 54 (2013) 176–184

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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec

Geometrically invariant and high capacity image watermarking scheme using accurate radial transform Chandan Singh 1, Sukhjeet K. Ranade n Department of Computer Science, Punjabi University, Patiala 147002, India

art ic l e i nf o

a b s t r a c t

Article history: Received 17 December 2012 Received in revised form 7 May 2013 Accepted 17 May 2013 Available online 21 June 2013

Angular radial transform (ART) is a region based descriptor and possesses many attractive features such as rotation invariance, low computational complexity and resilience to noise which make them more suitable for invariant image watermarking than that of many transform domain based image watermarking techniques. In this paper, we introduce ART for fast and geometrically invariant image watermarking scheme with high embedding capacity. We also develop an accurate and fast framework for the computation of ART coefficients based on Gaussian quadrature numerical integration, 8-way symmetry/anti-symmetry properties and recursive relations for the calculation of sinusoidal kernel functions. ART coefficients so computed are then used for embedding the binary watermark using dither modulation. Experimental studies reveal that the proposed watermarking scheme not only provides better robustness against geometric transformations and other signal processing distortions, but also has superior advantages over the existing ones in terms of embedding capacity, speed and visual imperceptibility. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Angular radial transform Embedding capacity Invariant watermarking

1. Introduction Digital watermarking refers to a class of methods used for a variety of forensic applications such as copyright protection, content authentication, and integrity verification. The basic aim of digital watermarking is to encode new information or watermark, into the host data in a way that if the host data is copied, the watermark is carried as well. The host data here refers to audio, video, or images, the latter being more popular as most of the image watermarking techniques can be easily adapted to accommodate audio and video implementations [1]. An efficient image watermarking scheme must satisfy some basic requirements including visual imperceptibility, robustness, embedding capacity, uniqueness, and minimum computational load for embedding or detecting the watermark. Among these, the requirement of ensuring watermark robustness is considered the most challenging one as various geometric and signal processing transformations destroy the synchronization of watermark resulting in serious problems especially when the original image is not available at the time of extraction. In order to solve this synchronization problem various watermarking schemes prefer to embed the watermark

n Corresponding author. Tel.: +91 175 3046312, mobile: +91 9876662685; fax: +91 175 3046313. E-mail addresses: [email protected] (C. Singh), [email protected] (S.K. Ranade). 1 Tel.: +91 175 3046302, mobile: +91 9872043209; fax: +91 175 3046313.

0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.05.016

signal in the transform domain. Some of the popularly used transform domains for robust watermarking are discrete cosine transform (DCT) [1,2], Fourier transform [3], discrete wavelet domain (DWT) [4], bandelet transform [5], and more recently random fraction Fourier transform (RFrFT) [6]. Embedding capacity refers to the amount of information that can be embedded into the host signal without degrading the quality of watermarked images. Few researchers work toward increasing the capacity of watermarking system as it adversely affects the watermark robustness and visual imperceptibility. High capacity systems, however, are desirable for a variety of applications such as secure media distribution [2], thumbnail embedding for authentication, auxiliary data embedding [7], and medical image watermarking [8]. Another challenge that is still open and needs to be addressed in digital watermarking is the speed of embedding and detecting or extracting watermark. Since most of the digital transmission takes place via Internet, designing a fast watermarking system that can work in limited computing environment and perform in near real-time environment can be of immense value for a variety of applications. One such application can be an on-line certification system that validates authenticated images watermarked from the unauthenticated ones at the router or server itself to prevent the circulation of unauthenticated data and nip the evil of digital content piracy. Moment invariants are widely proposed for geometrically robust image watermarking due to special properties of their kernel functions in multi-distortion of an image such as

C. Singh, S.K. Ranade / Optics & Laser Technology 54 (2013) 176–184

translation, scaling, rotation, and intensity variations. Alghoniemy and Tewfik [9] are the pioneers to incorporate geometric moment invariants for robust image watermarking. They embed a noninformative watermark robust against translation, scaling and filtering. However, the instability due to nonlinear nature of the invariants and difficulty in embedding multi-bit watermark restrict its usability for informative watermarking. Since then, numerous moment invariant based watermarking schemes have been designed using different moment families such as Zernike and pseudo-Zernike moments (ZMs/PZMs) [10–15], Fourier–Mellin moments [16], complex moments [17], wavelet moments [18]. Among these, orthogonal rotation invariant moments (ORIMs) like ZMs/PZMs are considered superior to other moment invariants because of their better reconstruction capabilities and excellent robustness against affine transformations and other signal processing distortions. Notwithstanding their wide usefulness in digital watermarking, these moments are often faced with the problem of inaccuracy, high computation complexity and numerical instability at the high order of moments. A survey of various moment based watermarking techniques can be found in [19]. In this paper, we propose the use of recently introduced angular radial transform (ART) for the fast and high capacity robust image watermarking. ART is a rotation-invariant transform adopted in MPEG-7 as a region-based shape descriptors [20]. Similar to ORIMs, the transform offers magnitude invariance to rotation and excellent resilience to noise. Two of the major advantages of the ART over other moment invariants are its low computation complexity and better numerical stability. Thus, high order transform coefficients can be obtained accurately making them more suitable for invariant watermarking. ART has also been found useful for applications like intelligent video security system [21], shape retrieval [22] and logo recognition systems [23]. Although ART is computationally more efficient than moments, its derivation still requires the computation of sinusoidal terms present in its kernel functions. This makes the computation of ART time consuming and unsuitable for real-time applications that need to be performed in the limited computing environment. Further, the traditional computation method based on zeroth order approximation produces geometric error and numerical integration error in the computation of ART. These errors adversely affect the magnitude invariance properties of the transform and make some of the low order coefficients unsuitable for invariant watermarking while the high order ART coefficients become numerically unstable. In order to alleviate these problems, a novel approach for the accurate computation of ART based on Gaussian quadrature integration technique has been proposed. Among the various numerical integration techniques available, we prefer Gaussian quadrature because it provides more accurate solution for the same number of sampling points and has been successfully applied to improve the accuracy of ZMs, orthogonal Fourier–Mellin moments (OFMMs), and radial harmonic Fourier moments (RHFMs) [24–26]. Inspired by its success, we extend the method for improving the accuracy of ART. We also derive fast algorithms for the computation of ART based on recursive relations for sinusoidal terms involved in the computation of radial and angular kernel functions and their 8-way symmetry/anti-symmetry properties. The binary watermark is embedded by quantizing the magnitudes of accurate ART coefficients. The extraction process is capable of retrieving the watermark sequence directly from the magnitudes of watermarked signal using minimum distance decoder. Simulation results show that the proposed scheme provides better robustness against various geometric and signal processing attacks, higher embedding capacity, and is more suitable for real-time applications compared to the existing ZMs/ PZMs based methods.

177

Rest of the paper is organized as follows. Section 2 presents the mathematical description of ZMs, PZMs and ART. The existing computational framework and proposed methods for improving the accuracy and speed of ART computations are given in Section 3. A brief overview of the watermarking scheme is given in Section 4. Detailed experimental results and discussions are presented in Section 5, followed by concluding remarks in Section 6.

2. Mathematical description of rotational invariant moments 2.1. Zernike moments (ZMs) ZMs belong to a class of ORIMs defined on the basis of Zernike polynomials. The ZMs, Zpq, of order p≥0 and repetition jqj≥0 for a continuous image function, f(r,θ), in the polar domain are defined over a unit disk as follows: Z 2π Z 1 pþ1 Z pq ¼ f ðr; θÞ V npq ðr; θÞ r dr dθ ð1Þ π 0 0 where V npq ðr; θÞ is the complex conjugate of the complex Zernike polynomial V pq ðr; θÞ given by jqθ V pq ðr; θÞ ¼ RZM pq ðrÞe

The Zernike radial polynomial, ðp−jqjÞ 2

ð2Þ RZM pq ðrÞ,

is defined as

 ð−1Þs ðp−sÞ!ðrÞp−2s   ; p−qj ¼ even ðp−jqjÞ s ¼ 0 s! ! 2 −s !

RZM pq ðrÞ ∑



ð3Þ

ðpþjqjÞ −s 2

The Zernike polynomials, V npq ðr; θÞ, given in Eq. (2) are orthogonal over the unit disk as follows: Z 2π Z 1 π δpp ; δqp V npq ðr; θÞ V p′q′ ðr; θÞr dr dθ ¼ ð4Þ pþ1 0 0 where δ is known as the Kronecker delta defined as  1 for i ¼ j δij ¼ 0 otherwise

ð5Þ

For a given maximum order pmax the total number of moments is given by (1+pmax)(2+pmax)/2. 2.2. Pseudo-Zernike moments (PZMs) Another class of ORIMs based on Zernike polynomials are PZMs. These moments differ from ZMs only in terms of their real-valued radial polynomial function which is defined as follows: RPZM pq ðrÞ ¼

ðp−jqjÞ

∑

s¼0

ð−1Þs ð2p þ 1−sÞ!ðrÞp−s ; p≥0 s!ðp þ jqj þ 1−sÞ!ðp−jqj−sÞ!

and

  q≤p

ð6Þ

For given maximum order pmax the total number of moments is given by (1+pmax)2. 2.3. Angular radial transform (ART) ART is a non-orthogonal rotation invariant transform. The ART coefficient, Apq , of order p≥0 and repetition jqj≥0 for a continuous image function, f ðr; θÞ, in the polar domain are defined over a unit disk as follows: Z 2π Z 1 1 Apq ¼ f ðr; θÞ V npq ðr; θÞ r dr dθ ð7Þ 2π 0 0 where V npq ðr; θÞ is the complex conjugate of ART kernel function, V pq ðr; θÞ and jqθ V pq ðr; θÞ ¼ RART p ðrÞe

ð8Þ

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C. Singh, S.K. Ranade / Optics & Laser Technology 54 (2013) 176–184

The radial kernel function of ART RART p ðrÞ is a harmonic function in r defined as ( 1 p¼0 RART ðrÞ ¼ ð9Þ p 2 cos ðπprÞ; otherwise For given maximum order, pmax , and repetition, qmax , the total number of ART coefficients is given by ð1 þ pmax Þð1 þ 2qmax Þ.

form Apq ¼

1 N−1 N−1 ∑ ∑ f ði; kÞ∬x2 þy2 ≤1 V npq ðx; yÞ dx dy i k 2π i ¼ 0 k ¼ 0

It is difficult to derive an analytical solution to the double integration involved in the right hand side of Eq. (14). Normally, the double integration is approximated by the zeroth order approximation (ZOA) method, i.e., ∬x2 þy2 ≤1 V npq ðx; yÞ dx dy≅ V npq ðxi ; yk ÞΔxi Δyk i

3. Computational framework for ART

ð14Þ

ð15Þ

k

Therefore, Eq. (14) is approximated by ART is a rotation invariant transform defined in the continuous space over the unit circular domain. However, the digital image function, f , of dimension N  N is discrete and which is defined in a rectangular domain with f ði; kÞ representing the intensity value of a pixel at location ði; kÞ for i; k ¼ 0; 1; 2; …; N−1. Thus, we map the digital image into a continuous unit disk inscribed in the square ½−1; 1  ½−1; 1 using the following normalization transformation: xi ¼

2i þ 1−N 2k þ 1−N ; yk ¼ ; N N

i; k ¼ 0; 1; :::; N−1

ð10Þ

The above transformation maps the center of the image to the center of the unit disk and each pixel ði; kÞ has the center ðxi ; yk Þ     covering a square domain dik ¼ ai ; aiþ1  bk ; bkþ1 where ai ¼

2i−N 2k−N ; bk ¼ ; N N

i; k ¼ 0; 1; :::; N

ð11Þ

The domain dik can also be represented with respect to the     Δy Δy Δx center of the pixel ði; kÞ, i.e., dik ¼ xi − Δx 2 ; xi þ 2  yk − 2 ; yk þ 2 where 2 Δx ¼ Δy ¼ N

ð12Þ

The continuous unit disk D ¼ ff ðr; θÞ; 0≤r≤1; 0≤θ≤2πg is now approximated by D0 ¼ ff ði; kÞ; x2i þ y2k ≤1g and only those pixels which satisfy the condition x2i þ y2k ≤1 contribute to the computation of ART as shown in Fig. 1, in which (a) represents an 8  8 image grid and (b) depicts the mapping of image pixels whose centers lie inside the unit disk. Consequently, Eq. (7) for the computation of ART in the Cartesian domain can now be rewritten as 1 ∬ 2 2 f ðx; yÞ V npq ðx; yÞ dx dy ð13Þ 2π xi þyk ≤1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y2 and θ ¼ tan −1 ðy=xÞ, θ∈½0; 2π. Assuming that the image function f ði; kÞ is a constant over a pixel grid dik , the discrete implementation of Eq. (12) assumes the Apq ¼

Apq ¼

1 N−1 N−1 ∑ ∑ f ði; kÞ V npq ðxi ; yk Þ Δxi Δyk 2π i ¼ 0 k ¼ 0

ð16Þ

x2i þy2k ≤1

Since Δxi ¼ Δyk ¼ 2=N, which is obtained from Eq. (12) , the above equation simplifies to Apq ¼

2 N−1 N−1 ∑ ∑ f ði; kÞ V npq ðxi ; yk Þ πN2 i ¼ 0 k ¼ 0

ð17Þ

x2i þy2k ≤1

The ZOA method produces geometric error and numerical integration error in the computation of ART. Geometric error arises because some of the pixels which intersect the circle but whose centers fall outside the disk are left out in the process of moment calculation. The numerical integration error in the computation of ART occurs due to the approximation of double integral in Eq. (15) with summation. The numerical integration error is more prominent at high values of p and q. 3.1. Proposed framework for accurate computation of ART The ZOA method makes the computation of some ORIMs and transforms with repetition q ¼ 4m; m∈Ζ less accurate as compared to other moments and transforms thus affecting their invariance property. This was first observed by Xin et al. [12] for ZMs/PZMs and later by Li et al. [27] for polar harmonic transforms (PHTs). Therefore, their watermarking techniques discard all the moments or transforms with q ¼ 4m; m∈Ζ. This reduces the number of moments or transforms coefficients by approximately 25% thus affecting the embedding capacity. Since the computational framework of ART is similar to ZMs/PZMs and PHTs, we expect similar problems when zeroth order approximation is used for image watermarking. Therefore in this section, we propose a framework for accurate computation of ART. The method is similar to the method given by Singh and Upneja [24] for ZMs computation. Its brief description with reference to the accurate computation of ART is described here.

Fig. 1. (a) An 8  8 image grid and (b) inscribed circle approximated by square grids.

C. Singh, S.K. Ranade / Optics & Laser Technology 54 (2013) 176–184

The image function, f ðx; yÞ, is a discrete function f ði; kÞ and is assumed to be a constant for a pixel grid dik . The kernel function, V npq ðx; yÞ, changes rapidly with respect to an increase in p along radial direction because it has p zeroes given by r ¼ 1=2p; p 4 0. The high data embedding capacity requires a large number of transform coefficients, therefore, high order transforms are required. The zeroth order approximation of the double integration makes the high order transform coefficients erroneous and numerically instable because the variations in the kernel function at high order of transform cannot be represented adequately by the zeroth order approximation of the kernel function. Therefore, its accurate computation is essential. For this purpose we resort to Gaussian quadrature numerical integration which is one of the best numerical integration methods [28]. Using the method, double integration in Eq. (14) is approximated by an n  n-point Gaussian numerical integration formula as follows [28]: ∬x2 þy2 ≤1 V npq ðx; yÞ dx dy≅ i

k

1

n−1 n−1

∑ ∑ wl wm V npq ðxli ; ymk Þ; N2 l ¼ 0 m ¼ 0

x2li þ y2mk ≤1

ð18Þ

where xh ¼

t l þ 2i þ 1−N t m þ 2k þ 1−N ; ymk ¼ N N

ð19Þ

and wl and wm are the given weights computed according to procedure given in [28] and t l ; t m ∈½−1; þ 1 determine the location of the sampling points in the rectangular grid xi −ðΔx=2Þ;  xi þ ðΔx=2Þ  yk −ðΔy=2Þ; yk þ ðΔy=2Þ . The parameter n is the order of the numerical integration such that the zeroth order approximation of the integration can be obtained for n ¼ 1 for which t 1 ¼ 0:0 and w1 ¼ 2:0. The pre-computed values of weights and location of sampling points up to locations n≤10 are presented in Table 1 for a quick reference. Thus, Eq. (14) is approximated to provide more accurate ART using Eq. (18) as follows: Apq ¼

1

N−1 N−1

n−1 n−1

n

∑ ∑ f ði; kÞ ∑ ∑ wl wm V pq ðxli ; ymk Þ 2πN 2 i ¼ 0 k ¼ 0 l¼0m¼0

x2li

þ

Clearly, the inequality (18) allows all those grids to participate in the computations of ART that has at least one sampling point inside the unit disk. The contribution of a sampling point is proportional to its weight and thus the pixel grids lying on circular boundary are approximated in a better way as compared to the existing ZOA method. The numerical integration method reduces geometric error as well as numerical integration error. The geometric error is reduced because the boundary of the unit disk is

Table 1 Weights and locations of sampling points for n  n-point Gaussian integration. n

wi

ti

n

wi

ti

1 2 3

2.0 1.0 0.5555555556 0.8888888889 0.3478548451 0.6521451549 0.2369268851 0.4786286705 0.5688888889 0.1713244924 0.3607615730 0.4679139346 0.1294849662 0.2797053915 0.3818300505 0.4179591837

0.0 7 0.5773502692 7 0.7745966692 0.0 7 0.8611363116 7 0.3399810436 7 0.9061798459 7 0.5384693101 0.0 7 0.9324695142 7 0.6612093865 7 0.2386191861 7 0.9491079123 7 0.7415311856 7 0.4058451514 0.0

8

0.1012285363 0.2223810345 0.3137066459 0.3626837834 0.0812743883 0.1806481607 0.2606106964 0.3123470770 0.3302393550 0.0666713443 0.1494513492 0.2190863625 0.2692667193 0.2955242247

7 0.9602898665 7 0.7966664774 7 0.5255324099 7 0.1834346425 7 0.9681602395 7 0.8360311073 7 0.6133714327 7 0.3242534234 0.0 7 0.9739065285 7 0.8650633667 7 0.6794095683 7 0.4333953941 7 0.1488743390

4 5

6

7

9

10

approximated by more number of sampling points providing a better approximation of the circle boundary. The numerical integration error is reduced because the kernel function V npq ðx; yÞ is computed at n2 sampling points thus providing much better approximation to the true integration than the ZOA. The rapid changes in the kernel function due to high order transforms are properly accounted by the n  n-point Gaussian integration. The improvement in the accuracy of ART can be demonstrated through a constant image function. Theoretically, for a constant image function f ðr; θÞ ¼ K, its ART coefficients must evaluate to 8 K=2 > > < 2K ½ð−1Þp −1 Apq ¼ ðπpÞ2 > > : 0

if p ¼ q ¼ 0 if p≠0 and q ¼ 0

ð21Þ

otherwise

In order to demonstrate the accuracy of the proposed computational framework over ZOA, we compute ART coefficients for a constant image function f ðr; θÞ ¼ K ¼ 1:0. A square grid of 128  128 is used. The ART coefficients upto moment order and repetition pmax ¼ qmax ¼ 10 are computed for 3  3-, 5  5-, and 7  7-point Gaussian numerical integration. The theoretical values of ART with repetition q ¼ 4m; m ¼ 0; 1; 2, obtained with ZOA method and different orders of integration are presented in Table 2. The other values of ART for which q≠4m; m ¼ 0; 1; 2, are computed accurately which are zero for both the ZOA method and for all orders of integration. Table 2 Magnitudes of ART coefficients up to order p≤10 and repetition q¼ 4m, m¼ 0,1,2 for a constant gray scale image (K¼ 1.0). Order and Exact value repetition (theoretical) p

ZOA method

q

Proposed method with different orders of integration 33

55

77

0

0 4 8

0.500000 0.000000 0.000000

0.500934 0.000119 0.000100

0.500063 0.500043 0.500000 0.000078 0.000018 0.000000 0.000050 0.000023 0.000000

1

0 4 8

0.405312 0.000000 0.000000

0.407153 0.000258 0.000199

0.405411 0.405372 0.405312 0.000134 0.000034 0.000000 0.000010 0.000037 0.000000

2

0 4 8

0.000000 0.000000 0.000000

0.001868 0.000237 0.000202

0.000127 0.000087 0.000000 0.000156 0.000036 0.000000 0.000099 0.000046 0.000000

3

0 4 8

0.045059 0.000000 0.000000

0.046900 0.000258 0.000196

0.045158 0.045118 0.045059 0.000134 0.000034 0.000000 0.000010 0.000037 0.000000

4

0 4 8

0.000027 0.000000 0.000000

0.001869 0.000236 0.000208

0.000127 0.000087 0.000027 0.000156 0.000036 0.000000 0.000099 0.000046 0.000000

5

0 4 8

0.016239 0.000000 0.000000

0.018080 0.000257 0.000189

0.016338 0.016298 0.016239 0.000134 0.000034 0.000000 0.000010 0.000037 0.000000

6

0 4 8

0.000027 0.000000 0.000000

0.001870 0.000234 0.000217

0.000127 0.000087 0.000027 0.000156 0.000036 0.000000 0.000099 0.000046 0.000000

7

0 4 8

0.008298 0.000000 0.000000

0.010141 0.000255 0.000180

0.008398 0.008358 0.008298 0.000134 0.000034 0.000000 0.000010 0.000037 0.000000

8

0 4 8

0.000027 0.000000 0.000000

0.001871 0.000231 0.000231

0.000127 0.000087 0.000027 0.000156 0.000036 0.000000 0.000099 0.000046 0.000000

9

0 4 8

0.005031 0.000000 0.000000

0.006875 0.000253 0.000167

0.005130 0.005090 0.005031 0.000134 0.000034 0.000000 0.000010 0.000037 0.000000

10

0 4 8

0.000027 0.000000 0.000000

0.001872 0.000227 0.000247

0.000127 0.000087 0.000027 0.000156 0.000036 0.000000 0.000098 0.000046 0.000000

ð20Þ

y2mk ≤1

179

180

C. Singh, S.K. Ranade / Optics & Laser Technology 54 (2013) 176–184

These values of ART are the same as obtained analytically and hence they are not shown in the table. As evident from the table, the accuracy in the computation of ART improves significantly with an increase in the order of integration. We also observe that all moments are computed accurately up to six decimal positions with order of integration 7  7 making them more useful for watermark embedding. 3.2. Fast computation of accurate ART The improvement in the accuracy of ART is achieved at the cost of n  n number of extra computations of the kernel function. This makes the calculation of ART slower by Oðn2 Þ. A number of researchers have contributed toward the development of approaches for fast computation of rotation invariant moments and transforms. These approaches are mainly based on 8-way symmetry/anti-symmetry of the radial and the angular functions, and the recurrence relations for the sinusoidal functions. Details of these methods can be found in [29–32]. In this section we present a brief overview of these techniques for the fast computation of ART. 3.2.1. 8-way symmetry/anti-symmetry properties Recently, Singh and Walia [29] proposed the use of 8-way symmetry/anti-symmetry property for fast computation of ZMs kernel functions which can be extended to ART. Accordingly, if ðxi ; yk Þ is the coordinate of an image pixel at location ði; kÞ in the first octant, then the values of the radial kernel functions Rp ðr ik Þ is the same for pixels at qall eight ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi locations ð 7 xi ; 7 yk Þ and ð 7 yk ; 7 xi Þ, where r ik ¼ x2i þ y2k . Further, the sinusoidal terms cos ðqθik Þ and sin ðqθik Þ, where θik ¼ tan −1 ðyk =xi Þ, involved in angular kernel function are also computed only in the first octant and values at seven other locations, nπ=2 7θik n ¼ 1; 2; 3; 4(2π þ θik is excluded) can be obtained by using periodicity of sinusoidal functions. The use of 8-way symmetry/anti symmetry property for the computation of kernel function reduces the number of computations from n2 N2 locations to n2 NðN þ 2Þ=8. 3.2.2. Recursive relations for sinusoidal kernel functions For radial kernel function, cos ð0Þ ¼ 1:0 for p ¼ 0, and forp ¼ 1, cos ðπr ik Þis computed directly while for higher order (p 4 1) the proposed recursive equation is cos ðπðp þ 1Þr ik Þ ¼ cos ðπpr ik ÞC− cos ðπðp−1Þr ik Þ

ð22Þ

where p ¼ 1; …; pmax −1 and C ¼ 2 cos ðπr ik Þ. Similarly, for angular kernel function, cos ð0Þ ¼ 1:0, sin ð0Þ ¼ 0:0 for q ¼ 0, and for q ¼ 1 we can initialize cos ðθik Þ ¼ xi =r ik and sin ðθik Þ ¼ yk =r ik . For q 4 1, these functions can be computed using following equations: cos ððq þ 1Þθik Þ ¼ cos ðqθik ÞC′− cos ððq−1Þθik Þ

ð23Þ

sin ððq þ 1Þθik Þ ¼ sin ðqθik ÞC′− sin ððq−1Þθik Þ

ð24Þ

where q ¼ 1; …; qmax −1 and C′ ¼ 2 cos ðθik Þ. The above recursive relations improve the speed of computation substantially with the additional memory requirement of only pmax þ 2qmax þ 3 words used for saving the values of sine and cosine functions.



sequence B ¼ bi ; i ¼ 1; 2; …; L is a randomly generated sequence and consists of L number of information bits such that bi ∈f0; 1g. Each bit bi ∈B is embedded into the host image by modulating some selected ART coefficient through quantization. The important steps for watermark embedding and extraction are as follows: (i) Selection of ART coefficients: as demonstrated in Section 3.1 all moments are computed accurately with the order of integration 7  7 making all of them useful for information encoding. However, taking into account only transforms having

unique magnitude we define S ¼ Apq ; 0≤p≤pmax ; 0≤q≤qmax as set of all ART coefficients, Apq , that can   be used for watermarking.   The cardinality of the set S, S, determines the maximum number of bits that can be embedded. The coefficient A00 is ignored for embedding as it is half of the average intensity.   Depending on the length of watermark bit sequence,L ð≤jSÞ, number of ART coefficients are selected to form the feature vector A ¼ Ap1 ;q1 ; Ap2 ;q2 ; …; ApL ;qL . (ii) Quantization of ART coefficients: each bit bi from B is embedded into the corresponding ART coefficient, Api ;qi of A to obtain the     o n       by modified feature vector A~ ¼ A~ p1 ;q1 ; A~ p2 ;q2 ; …; A~ pL ;qL    ~  using dither modulation where A pi ;qi  is the quantized version     of Api ;qi  computed as follows:  2 3    A −di ðbi Þ p ;q i i ~  4 5Δ þ di ðbi Þ; i ¼ 1; 2; …; L ð25Þ A pi ;qi  ¼ Δ where ½: denotes the rounding operation, Δ is the step size of quantization and di ð:Þ is the dither value for the ith quantizer with the relation di ð1Þ ¼ di ð0Þ þ Δ=2. The elements of dither vector ðd1 ð0Þ; d2 ð0Þ; …; dL ð0ÞÞ are generated pseudo randomly   and uniformly distributed over 0; Δ=2 (iii) Formation of watermarked image: the spatial watermark signal, wðxi ; yk Þ at pixel ði; kÞ can be derived using following equation: L   wðxi ; yk Þ ¼ ∑ εpi ;qi V pi ;qi ðxi ; yk Þ þ εpi ;−qi V pi ;−qi ðxi ; yk Þ ; i¼1

i; k ¼ 0; 1; 2; …; N−1

ð26Þ

where             εpi ; qi  ¼ A~ pi ; qi −Api ; qi ; i ¼ 1; 2; …; L

ð27Þ

The final watermarked image f wat ðx; yÞ is formed as f wat ðxi ; yk Þ ¼ f ðxi ; yk Þ þ wðxi ; yk Þ;

i; k ¼ 0; 1; 2; …; N−1

ð28Þ

(iv) Watermark extraction: in order to decode the embedded sequence b^ ¼ ðb^ 1; b^ 2; …; b^ L Þ, the feature vector A′ ¼ A′p1 ;q1 ; A′p2 ;q2 ; …; A′pL ;ql g is recomputed from the received image f ′wat ðx; yÞ which may be a distorted version of f wat ðx; yÞ. The magnitude of each A′pi ;qi is re-quantized with both dither     functions di ð0Þ and di ð1Þ using Eq. (25) to compute A′pi ;qi  and   0   A′pi ;qi  , respectively. A minimum distance decoder is used to 1 decode the bit embedded in M′pi ;qi as follows: b^ i ¼ argminðjA′pi ;qi jj −jA′pi ;qi jÞ2 ;

i ¼ 1; 2; …; L

ð29Þ

j∈f0;1g

4. Proposed watermarking scheme Among various moment-based watermarking schemes, we observe that the method given by Xin et al. [12] using ZMs/PZMs is one of the best methods. Therefore, we adopt the method given in [12] to design ART-based watermarking scheme. Our watermark bit

5. Experimental results and discussions This section presents the experimental results obtained after evaluating the proposed watermarking scheme in terms of visual

C. Singh, S.K. Ranade / Optics & Laser Technology 54 (2013) 176–184

181

Fig. 2. Twelve standard images used in experiments.

Fig. 3. Watermarked images f wat ðx; yÞ for the sample image Lena of size 256  256 pixel with Δ¼ 1.0, L ¼ 128.

imperceptibility, watermark robustness, embedding capacity, and time complexity. All experiments are performed on twelve standard gray scale images of size 256  256 pixels shown in Fig. 2. These images are normally used in image processing applications and given in [25]. 5.1. Visual imperceptibility One of the basic requirements while developing a watermarking scheme is to maintain good quality of the output image f wat ðx; yÞ which can be measured quantitatively using peak signal-to-noise ratio (PSNR) as follows: 0

1

B B PSNRðf ; f wat Þ ¼ 10log10 B @

C 255  N C C N N  2 A ∑ ∑ f wat ði; kÞ−f ði; kÞ 2

i¼1k¼1

2

ð30Þ

Fig. 3 shows the watermarked images along with their PSNR values after embedding a pseudo randomly chosen 128-bit watermark sequence into sample Lena image of size 256  256 pixels using four watermarking approaches with quantization step Δ ¼ 1:0. Next, we observe the average PSNR of the 12 standard images of size 256  256 pixels watermarked using ART and accurate ART as a function of quantization step Δ and compare them with the existing methods using ZMs and PZMs [12]. The ideal value of PSNR found in the literature for invisible watermarking is 44 db. As evident from Fig. 4, the quality of watermarked images obtained with proposed ART based watermarking scheme is better than existing ZMs based watermarking scheme and comparable with PZMs based watermarking scheme while the accuracy in the computation of ART further enhances the quality of the watermarked images. It can also be observed that higher values of PSNR are achieved for lower values of quantization step, however, larger values of Δ are recommended for better robustness. Keeping in view the trade-off between robustness and visual

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Fig. 5. Embedding capacity with different moments and transforms. Fig. 4. Visual imperceptibility of various watermarking schemes as a function of quantization step (Δ).

imperceptibility, we choose quantization step Δ ¼ 1:0 for further experiments. 5.2. Embedding capacity The embedding capacity of a moment or transform based watermarking scheme depends on two major factors. First, the number of magnitude independent coefficients at the given order of moment or transform determines maximum data size that can be embedded. Second, the maximum order up to which moment or transform coefficients can be computed accurately without exhibiting numerical instability. We first compute the number of moments or transform coefficients available for embedding using ZMs, PZMs, ART and accurate ART up to given order pmax according to following conditions:         ZMs: q p−q ¼ even; q≠4m  ≤p;    PZMs: q≤p; q≠4m ART:0≤q≤qmax ; q≠4m Accurate ART: 0≤q≤qmax In the above methods, the coefficient corresponding to ðp; qÞ ¼ ð0; 0Þ is avoided because it represents the average intensity. The algebraic equations for computing the embedding capacity for ZMs/PZMs in terms of maximum order, pmax , are given in [12] and those for ART and accurate ART these values can be derived directly as follows: 8 3q ðp þ1Þ max max if qmax ¼ 4m > 4 > > > > ð3q   < max þ1Þðpmax þ1Þ if qmax ¼ 4m þ 1 4   ð31Þ ¼ ð3qmax þ2Þðpmax þ1Þ S > ART if qmax ¼ 4m þ 2 > 4 > > > : ð3qmax þ3Þðpmax þ1Þ if q ¼ 4m þ 3 4

    S

accurate ART

Fig. 6. Embedding capacity vs. BER for different moments and transform with pmax ≤50.

Fig. 7. Watermark robustness against rotation.

max

¼ ðpmax þ 1Þðqmax þ 1Þ−1

ð32Þ

Fig. 5 shows the total embedding data size as a function of order pmax ¼ qmax ≤50 for the transform and moments. We observe that accurate ART provides the highest data embedding sizes followed by ART computed using ZOA method for all orders of moments and transforms. ZMs/PZMs provide lower data embeddings sizes because their repetition, q, is restricted by the condi   tion q≤p which limits the size of host media and embedding sizes, whereas there is no such restriction for the transform. Further, ZMs provide the least   data embedding sizes due to the   additional constraint of p−q ¼ even. Second factor affecting the embedding capacity is the accuracy and numerical stability in the computation of transform

Fig. 8. Watermark robustness against scaling.

C. Singh, S.K. Ranade / Optics & Laser Technology 54 (2013) 176–184

coefficients or moments. The fact that numerical instability affects the embedding capacity of a watermarking system can be observed through bit error rate (BER), which is defined as the ratio of number of bits extracted inaccurately to the total number of bits embedded. The value of BER lies between 0 and 1 with BER≥0.25 indicating failure to extract the watermark. We embed randomly generated watermarks of varying lengths L which is equal to the total number of moments or transforms computed up to order pmax ≤50 for various watermarking schemes and plot average BER of 12 images as a function L as shown in Fig. 6. It is observed that while ZMs, PZMs, and ART with ZOA method provide acceptable values of BER (≤0.25) for L ¼385, 216, and 494, respectively, the accurate ART provides very low values of BER even for L ¼2600. ZMs/PZMs moments have inherent numerical instability problems due to the presence of factorial terms in the radial part of their kernel functions. Thus, there exist a certain value of pmax ( ¼44 for ZMs and 22 for PZMs) beyond which they cannot be computed accurately even with double precision [12]. This means the maximum number of bits embedded using Xin et al. method [12] is 385 for ZMs and 216 for PZMs. ART, on the other hand, are free from factorial terms and, therefore, numerically more stable. We observe no numerical instability in the computation of ART up to pmax ¼ 25 when computed using ZOA method and up to pmax ¼ 52 using Gaussian integration method with order of integration n ¼ 7. The improvement in the numerical stability due to numerical integration is attributed to the fact that the number of sampling points is increased from one sample to 49 samples. The high numbers of samples not only improve the accuracy in the transform computation, but also they increase the numerical stability. Similar phenomenon has been observed in the accurate computation of ZMs, OFMMs and RHFMs [24–26]. It is worth mentioning here that an average BER≈0:5 is obtained using ZOA method as statistically about 50% of the bits extracted are turned out to be correct due to random behavior. Therefore, the proposed framework results in large data embedding capacity for image watermarking.

becomes high and the scale invariance property of ART is better preserved for high resolution. Next, we analyze the performance of proposed watermarking scheme against various image processing distortions such as compression, median filtering, additive noise, cropping and shearing transformation. JPEG compression is a popular image compression technique which is commonly used for exchange of images and video over Internet. We observe the watermark robustness of various watermarking schemes against JPEG compression. Graphs in Fig. 9 show the average BER obtained after extracting the watermark from watermarked images attacked with 10 different levels of JPEG compression with quality factors ranging from 10 to 100. As evident from the figure, the performance of the ZOA ART based watermarking scheme is comparable with existing ZMs/PZMs based watermarking schemes while the accurate ART based scheme provides the least values of average BER and ensures maximum robustness. The average BER values obtained after applying other attacks are listed in Table 3. As seen from the table, both the proposed ART based and existing ZMs/PZMs based watermarking schemes are resilient to common signal processing distortions with similar performances while the accuracy in the computation of ART significantly enhances the watermark robustness.

Fig. 9. Watermark robustness to JPEG compression.

5.3. Watermark robustness The robustness of a watermarking scheme is measured quantitatively using BER. We analyze the performance of proposed scheme against various attacks and compare it with the Xin et al. method [12] using ZMs/PZMs. First, we analyze the robustness of proposed watermarking scheme against two common geometric transformations namely, rotation and scaling. For rotation, twelve standard watermarked images are rotated with 9 different angles of rotation from 01 to 451 at an interval of 51 after embedding pseudo random watermark signal of length 128bits. Because of the 8-way symmetry/anti-symmetry properties, rotations beyond 451 will produce identical trends. As shown in Fig. 7, the proposed watermarking scheme provides much lower values of average BER under rotation attack compared to ZMs/ PZMs based watermarking schemes. Thus, the proposed scheme extracts the watermark more accurately and is more robust irrespective of the angle of rotation. For analyzing scaling attack, the watermarked images of size 256  256 pixels are resized with eight scaling factors ranging from 0.25 to 2.0 and average BER curves are plotted as shown in Fig. 8. As seen from the figure, the proposed watermarking scheme is completely robust for all scaling factors ≥0.50 whereas ZMs/ PZMs based schemes can resist only small scaling factors. At low scale factors, the BER values are much more than their values at higher scales. In fact, at scale factor more than one, the BER values are almost zero. This trend is due to the fact that the ART values are computed more accurately when the image resolution

183

Table 3 Average BER for various watermarking schemes under various attacks. Attack

Average BER ZMs

Median filter Window size 3  3 Window size 5  5

PZMs

ART (ZOA)

Accurate ART

0.0198 0.0253 0.0199 0.0215 0.0384 0.0253

0.0156 0.0223

0.0374 0.0453 0.0412 0.1253 0.2435 0.2236 0.1632 0.2017 0.2328

0.0353 0.1223 0.1823

Shearing x: 0.00, y: 0.05 x: 0.05, y: 0.00 x: 0.05, y: 0.05

0.0077 0.0187 0.0153 0.0876 0.0328 0.0845 0.0945 0.0978 0.0978

0.0045 0.0654 0.0764

Cropping 5 Rows and 5 columns 10 Rows and 10 columns

0.0483 0.1823 0.1043 0.0525 0.2374 0.1832

0.0876 0.1491

Noise Gaussian (μ ¼ 0, s2 ¼ 0:2) Speckle (μ ¼ 0, s2 ¼ 0:1) Salt and pepper (μ ¼ 0, s2 ¼ 0:1)

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References

Fig. 10. Comparative analysis of the time taken for the computation of moments and transform for a gray scale image of size 64  64 pixels.

5.4. Time complexity Time complexity is another major issue related to the performance of a high capacity watermarking system. Considering the fact that the computation of transform coefficients or moments is the key operation in watermark embedding and extraction, we investigate the time taken for computing all ART coefficients and ZM/PZMs up to maximum order pmax ¼ qmax ¼ 30 for a sample image of Lena of size 64  64 pixels. It may be noted here that the time taken does not depend on the contents of gray scale images, therefore, only one image is considered for time analysis. We compute ART with traditional framework using ZOA method, proposed fast and accurate method for ART, and ZMs/PZMs using one of the fastest methods available (modified Kintner's fast method [29]). All methods are implemented in VC++ 6.0 under Windows 7 environment on Intel 2.50 GHz processor with 1 GB RAM. As seen from Fig. 10, the computation of ART takes much lesser time than ZMs/PZMs. ZMs/ PZMs are more computation intensive as their radial kernel functions are polynomials in degree p while those of ART consist of lowcomplexity sinusoidal functions. Further speed enhancement is achieved by applying fast methods for the computation of accurate ART. These methods are based on 8-way symmetry/anti-symmetry properties of the kernel functions and recursive relations for sinusoidal functions. The accurate and fast computational framework makes ART more suitable for applications that need to perform in real-time environment. 6. Conclusions In this paper, we introduce accurate ART for the fast and high capacity robust image watermarking system with low computation time. In contrast to conventionally used ZMs/PZMs, the transform is computationally more efficient and exhibits better numerical stability. We also develop a computational framework to enhance the accuracy and time complexity of ART computation. Experimental studies establish that the proposed watermarking scheme using accurate ART not only improves watermark robustness, embedding capacity, and visual imperceptibility of the watermarked image, but also has speed advantage compared to existing moment based watermarking schemes. Thus, the proposed method is more suitable for watermarking applications that need to perform with specified time-constraint or with large data sizes. Acknowledgments We are thankful to anonymous reviewers for their constructive and useful comments to raise the standard of the paper.

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