Digital Image Watermarking Algorithm Based on Fast Curvelet Transform

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1 J. Software Engineering & Appications, 010, 3, doi:10.436/jsea Pubished Onine October 010 ( 939 igita Image Watermarking Agorithm Based on Fast Curveet Transform Jindong Xu, Huimin Pang, Jianping Zhao Coege of Physica Engineering, Qufu Norma University, Shandong, China. Emai: Received Juy 15 th, 010; revised August 18 th, 010; accepted August 1 st, 010. ABSTRACT A digita image ing agorithm based on fast curveet transform is proposed. Firsty, the carrier image is decomposed by fast curveet transform, and, the ing image is scrambed by Arnod transform. Secondy, the binary ing image is embedded into the medium frequency coefficients according to the human visua characteristics and curveet coefficients. Experiment resuts show that the proposed agorithm has good performance in both invisibiity and security and aso has good robustness against the noise, cropping, fitering, JPEG compression and other attacks. Keywords: igita Image Watermarking, Fast Curveet Transform, Human Visua Characters, Robustness, Invisibiity 1. Introduction In the past two decades, more and more researchers have devoted to the transform domain which shows good performance and robustness. Recenty, Candès and onoho [1,] have deveoped a new mutiscae transform which is caed the Curveet transform. It is anisotropic with strong direction, and provides optimay sparse representations of objects aong a genera curve with bounded curvature. Hence, it has been widey used in image processing fied. At the same time, many research papers proposed to embed based on the first curveet transform [3-8]. Shi et a. [3] proposed a semi-fragie ing agorithm by embedding the wartermark in the maximum modue of curveet coefficient. This agorithm keeps good toerance against JPEG compression. The method of Thai et a. [4] embedded a in curveet coefficients which are seected by a threshod. This method has good invisibiity and robustness. Thai et a. [5] embedded the in the curveet transform which contain as much edge information as possibe. It has good invisibiity but poor robustness. The impementation of the first curveet transform incudes subband decomposition, smooth partitioning, renormaize and ridgeet anaysis. So these agorithms have many probems such as impementation compexity and arge amount of redundancy. This paper proposes a digita image ing agorithm based on fast curveet transform. Firsty, we take a binary-vaued image which contains copyright information as, and adapt Arnod transform to it to improve its security. Secondy, according to the human visua characteristics, we embed the in curveet coefficients of the origina image. This agorithm is simpe and easy to impement. Experiment resuts show that our agorithm has good invisibiity and security, and is robustness to the noise, cropping, fitering, JPEG compression and other attacks.. Curveet Transform Simiar to the waveet transform and the ridgeet transform, the curveet transform theory is based on sparsity theory [8]. The idea of curveet is to cacuate the inner reationship between the signa and the curveet function to reaize the sparse representation of the signa..1. Continuous Time Curveet Transform The curveet transform can be expressed as f,, c j,, k :, j k (1) here, j = 0, 1,,, is a scae parameter; = 0, 1,,, is an orientation parameter; and k = (k 1, k ) Z is a transation parameter. The mother curveet is φ j (x), its Fourier transform is φ j (ω) = U j (ω), where U j is fre-

2 940 igita Image Watermarking Agorithm Based on Fast Curveet Transform quency window defined in the poar coordinate system such as: 3/4 j j j / U j r, W rv () W and V are radia and anguar windows respectivey and wi aways obey certain admissibiity conditions. Curveet at scae -j, orientation θ and position j, 1 j j xk R k1, k can be expressed as: j, jk,, x j R xx k (3) So, for f L (R ), curveet transform is expressed as: 1 cik,, : fˆ ˆ jk,, ( )d (4) 1 ˆ j, f U j Rθ ωexpi xk, d.. igita Curveet Transform igita curveet transform is inear and takes as input Cartesian arrays of the form f [t 1, t ], 0 t 1, t < n, which aows the output as a coection of coefficients:,, : 1, jk,. 1, c j k f t t t t 0 t1, tn In order to improve the curveet transform in the sense that they are conceptuay simper, faster and far ess redundant. Paper [] proposed the Fast iscrete Curveet Transform (FCT). There are two digita impementation of FCT. The first is based on unequay-spaced fast Fourier transform (USFFT) whie the second is based on the wrapping of speciay seected Fourier sampes. The FCT-Wrapping uses simper choice of spatia grid to transate curveets at each scae and ange. It needs ess two-dimensiona FFTs than FCT-USFFT, so it is quicky. The architecture of FCT via Wrapping is then roughy as foows: 1) Appy the FFT and obtain Fourier sampe f n1, n, -n/ n 1, n n/ (n is the size of the picture). ) For each scae/ange pair (j, ), form the product U, 1 ˆ j n, n f n1, n. 3) Wrap this product around the origin and obtain f, 1 ˆ j n, n W U j, f n1, n, where the range for n and 1 n is now 0 n1 L1,j and 0 n L, j (for in the range ( /4, /4). 4) Appy the inverse FFT to each f j,, hence coecting the discrete coefficients c j,, k. 3. Watermarking Agorithm Based on FCT 3.1. Watermark (5) In this agorithm, the is a binary-vaued image with 3 3 pixes (Figure 1). Arnod transform is used to increase the data security, and its function is defined as (6). x 1 1 x mod N (6) y k k1y Adapt n (here, n = 8) times Arnod transform to the origina W, then we obtain scrambing (Figure ). 3.. Watermark Embedding According to the Human Visua Characteristic, we choose to embed the into medium-frequency. In seected scae, the coefficients of each orientation are sorted and find criterion T i by ampitude factor λ (λ 0). Then choose the minimum T i as embedding parameter T. Finay, the is embedded to coefficients which are chosen by T. The embedding procedure (Figure 3) is described as foows. 1) Appy FCT to the origina image, get curveet coefficient C. ) Seect the curveet coefficient to embed a bit according to the foowing conditions. For 0 0 C j, k T or T C j, k3t For 1 T C j, kt (7) Then record the positions of these coefficients and the positions of 0 and 1 in the. 3) The embedded coefficients are modified by the foowing equations. Embed 0 C jk,, C jk,, mod C j, k, TT 4 (8) Embed 1 C jk,, C jk,, mod C j, k, T 3T 4 4) o inverse FCT to C, obtain the ed image Watermark Extraction The extraction procedure is composed of 4 steps and each step is described as foows. Figure 1. Watermark. Figure. Scrambed.

3 igita Image Watermarking Agorithm Based on Fast Curveet Transform 941 j 1 j j j 4 j 5 n j 3 j, k / Cj, k 3T / C j, k C j, kmodc j, k, TT / 4 0 C T T T / C, j, k T C j, k C j, k modc j, k, T 3T / 4 IFCT Figure 3. Embedding procedure. 1) Appy curveet transform to ed image, obtain curveet coefficient C. ) Locate the positions from the origina image by using embedding procedure step. Then extract the coefficients C from the ed image using those position. 3) Extract the W from C with the foowing rue 1 if mod C j, k, T T i (9) 0 if mod C j, k, T T 4) Appy T -n (T is Arnod transform period) times Arnod transform to W, obtain the binary image. 4. Experimenta Resuts In this program, the image is transformed through FCT via Wrapping. We use standard pixe image Lena (Figure 4) for evauation of our proposed method, and conduct experiments binary-vaued image ing with the noted above parameter scae = 3, λ = We have investigated the invisibiity and the robustness of our ing system, anayzed the agorithm performance by objective and subjective standards Invisibiity Tests The ed image is shown in Figure 5(a), the detected from ed image is shown in Figure 5(b). Figure 4. Origina image. (a) Figure 5. Invisibiity Tests. (a) Watermarked image [PSNR = dB]; (b) etected [NC = 1.00]. (b)

4 94 igita Image Watermarking Agorithm Based on Fast Curveet Transform Tabe 1. JPEG compression attack. Q PSNR NC etected Tabe. Gaussian ow pass fitering attack. Standard eviation (Window) 0.5(3) 1.5(3) 0.5(5) 1.5(5) 3(5) PSNR NC etected Tabe 3. Other attacks. Attacks Gaussian noise (0.001) Sat & Pepper noise (0.01) Random cropping (1/16) Random cropping (1/3) Change contrast PSNR NC etected Subjectivey, the ed image has good invisibiity. Compare Figure 5(b) with Figure, they are fuy consistent. The origina is accuratey recovered. 4.. Robustness Tests To evauate the robustness of agorithm, a the attacks are tested by the software of Stirmark [10,11]. The Peak Signa to Noise Ratio (PSNR) is empoyed to evauate the quaity of ed image after attack, and the Normaized Correation Coefficient (NC) is used to evauate the quaity of extracted for some attacks such as JPEG compression (Quaity factor Q), Gaussian ow pass fitering, adding noise, cropping and so on. The simuation resuts are shown as foowing Tabes. It can be seen from Tabe 1 and Tabe, the proposed method show very good robustness to JPEG compression and Gaussian ow pass fitering. From Tabe 3, the method aso has good robustness against noise, cropping and so on. 5. Concusions This paper proposes a method by embedding a into the origina image based on FCT. At the same time, Arnod transform is appied to improve the security of the system. The experimenta resuts show that the ed image has good invisibiity and robustness against JPEG compression and Gaussian ow pass fitering. There are many unstabe coefficients are discovered from the experiments. For instance, a sma change of the image wi arouse big changes of these coefficients. These unstabe factors can infuence the extracting of. Therefore, the proposed agorithm can not give a good performance of rotation and scaing, etc. In the future, we wi unceasing devote ourseves to the study of the robustness system against geometric attacks based on curveet transform. REFERENCES [1]. L. onoho and M. R. uncan, igita Curveet

5 igita Image Watermarking Agorithm Based on Fast Curveet Transform 943 Transform: Strategy, Impementation and Experiments, Proceeding of the SPIE on Waveet Appication VII, Orando, 000, pp [] E. J. Candes, L. emanet and. L. onoho, Fast iscrete Curveet Transforms [R], Appied and Computationa Mathematics, 005, pp [3] J. P. Shi and Z. J. Zhai, Curveet Transform for Image Authentication, Rough Set and Knowedge Technoogy, LNAI 406, 006, pp [4] T.. Hien, I. Kei, H. Harak, et a., Curveet-omain Image Watermarking Based on Edge-Embedding, Lecture Notes in Computer Science, Vo. 4693, No. 010, 007, pp [5] T.. Hien, K. Miyara, et a., Curveet Transform Based Logo Watermarking, Innovative Ago-Rithms and Techniques in Automation, Industria Eectronics and Teecommunication, 007, pp [6] H. Y. Leung, L. M. Cheng and L. L. Cheng, igita Watermarking Schemes Using Muti-Resoution Curveet and HVS Mode Lecture Notes in Computer Science, Vo. 5703, 009, pp [7] H. Y. Leung, L. M. Cheng and L. L. Cheng. A Robust Watermarking Scheme Using Seective Curveet Coefficients, Internationa Conference on Inteigient Information Hiding and Mutimedia Signa Processing, Harbin, August 008, pp [8] T.. Hien, K. Miyara, I. Kei and F. F. Ai, igita Watermarking Based on Curveet Transform, 9th Internationa Symposium on Signa Processing and Its Appications, Sharjah, Feburary 007, pp [9] J. L. Cheng and T. Shan. eveopment and Prospect of Image Mutiscae Geometric Anaysis, Acta Eectronic Sinica, Vo. 31, No. 1 A, pp [10] A. P. Fabien and Petitcoas. Watermarking Schemes Evauation, IEEE. Signa Processing, Vo. 17, No. 5, 000, pp [11] Stirmark. /stirmark/