## Wavelet transform based watermark for digital images

Optics Express, Vol. 3, Issue 12, pp. 497-511 (1998)

http://dx.doi.org/10.1364/OE.3.000497

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### Abstract

In this paper, we introduce a new multiresolution watermarking method for digital images. The method is based on the discrete wavelet transform (DWT). Pseudo-random codes are added to the large coefficients at the high and middle frequency bands of the DWT of an image. It is shown that this method is more robust to proposed methods to some common image distortions, such as the wavelet transform based image compression, image rescaling/stretching and image halftoning. Moreover, the method is hierarchical.

© Optical Society of America

## 1. Introduction

*advantages*to the approach in the wavelet transform domain. The first advantage is that the watermarking method has multiresolution characteristics and is hierarchical. In the case when the received image is not distorted significantly, the cross correlations with the whole size of the image may not be necessary, and therefore much of the computational load can be saved. The second advantage lies in the following argument. It is usually true that the human eyes are not sensitive to the small changes in edges and textures of an image but are very sensitive to the small changes in the smooth parts of an image. With the DWT, the edges and textures are usually well confined to the high frequency subands, such as HH, LH, HL etc. Large coefficients in these bands usually indicate edges in an image. Therefore, adding watermarks to these large coefficients is difficult for the human eyes to perceive. The third advantage is that this approach matches the emerging image/video compression standards. Our numerical results show that the watermarking method we propose is very robust to wavelet transform based image compressions, such as the embedded zero-tree wavelet (EZW) image compression scheme, and as well as to other common image distortions, such as additive noise, rescaling/stretching, and halftoning. The intuitive reason for the advantage of the DWT approach over the DCT approach in rescaling is as follows. The DCT coefficients for the rescaled image are shifted in two directions from the ones for the original image, which degrades the correlation detection for the watermark. Since the DWT are localized not only in the time but also in the frequency domain [9–15], the degradation for the correlation detection in the DWT domain is not as serious as the one in the DCT domain.

*τ*= 0 offset) of the watermark and the difference in the DCT domain of the watermarked image and the original image. Even though both the difference and the watermark are normalized, the inner product may be small if the difference significantly differs from the watermark although there may be a watermark in the image. In this case, it may fail to detect the watermark. In this paper, we propose to take the correlation at all offsets

*τ*of the watermark and the difference in the DWT domain the watermarked image and the original image in different resolutions. The advantage of this new approach is that, although the peak correlation value may not be large, it is much larger than all other correlation values at other offsets if there is a watermark in the image. This ensures the detection of the watermark even though there is a significant distortion in the watermarked image. The correlation detection method in this paper is a relative measure rather than an absolute measure as in [2].

## 2. Discrete Wavelet Transform (DWT): A Brief Review

*x*[

*n*] can be decomposed recursively as

*j*=

*J*+1,

*J*,…,

*J*

_{0}where

*c*

_{J+1,k}=

*x*[

*k*],

*k*∊

**Z**,

*J*+1 is the high resolution level index, and

*J*

_{0}is the low resolution level index. The coefficients

*c*

_{j0,k},

*d*

_{J0,k},

*d*

_{J0+1,k},…,

*d*

_{j,k}are called the DWT of signal

*x*[

*n*], where

*c*

_{J0,k}is the lowest resolution part of

*x*[

*n*] and

*d*

_{j,k}are the details of

*x*[

*n*] at various bands of frequencies. Furthermore, the signal

*x*[

*n*] can be reconstructed from its DWT coefficients recursively

*x*[

*n*]. To ensure the above IDWT and DWT relationship, the following orthogonality condition on the filters

*H*(

*ω*) and

*G*(

*ω*) is needed:

*H*(

*ω*) and

*G*(

*ω*) is given by

*x*[

*n*] can be also described in the form of two channel tree-structured filterbanks as shown in Fig. 1. The DWT and IDWT for two dimensional images

*x*[

*m*,

*n*] can be similarly defined by implementing the one dimensional DWT and IDWT for each dimension

*m*and

*n*separately: DWT

_{n}[DWT

_{m}[

*x*[

*m*,

*n*]]], which is shown in Fig. 2. An image can be decomposed into a pyramid structure, shown in Fig. 3, with various band information: such as low-low frequency band, low-high frequency band, high-high frequency band etc. An example of such decomposition with two levels is shown in Fig. 4, where the edges appear in all bands except in the lowest frequency band, i.e., the corner part at the left and top.

## 3. Watermarking in the DWT Domain

*y*[

*m*,

*n*] denote the DWT coefficients, which are not located at the lowest frequency band, of an image

*x*[

*n*,

*m*]. We add a Gaussian noise sequence

*N*[

*m*,

*n*] with mean 0 and variance 1 to

*y*[

*m*,

*n*]:

*α*is a parameter to control the level of the watermark, the square indicates the amplification of the large DWT coeffcients. We do not change the DWT coefficients at the lowest resolution. Then, we take the two dimensional IDWT of the modified DWT coefficients

*y*͂ and the unchanged DWT coefficients at the lowest resolution. Let

*x*͂[

*m*,

*n*] denote the IDWT coefficients. For the resultant image to have the same dynamic range as the original image, it is modified as

*x*͂[

*m*,

*n*] be the same dynamic range as the original image

*x*[

*m*,

*n*]. The resultant image

*x*͂[

*m*,

*n*] is the watermarked image of

*x*ˆ [

*m*,

*n*]. The encoding part is illustrated in Fig. 5(a).

*LL*

_{1}) band, low-high (

*LH*

_{1}) band, high-low (

*HL*

_{1}) band, and high-high (

*HH*

_{1}) band, respectively. We then compare the signature added in the

*HH*

_{1}band and the difference of the DWT coefficients in

*HH*

_{1}bands of the received and the original images by calculating their cross correlations. If there is a peak in the cross correlations, the signature is called detected. Otherwise, compare the signature added in the

*HH*

_{1}and

*LH*

_{1}bands with the difference of the DWT coefficients in the

*HH*

_{1}and

*LH*

_{1}bands, respectively. If there is a peak, the signature is detected. Otherwise, we consider the signature added in the

*HL*

_{1},

*LH*

_{1}, and

*HH*

_{1}bands. If there is still no peak in the cross correlations, we continue to decompose the original and the received signals in the

*LL*

_{1}band into four additional subbands

*LL*

_{2},

*LH*

_{2},

*HL*

_{2}and

*HH*

_{2}and so on until a peak appears in the cross correlations. Otherwise, the signature can not be detected. The decoding method is illustrated in Fig. 5(b).

## 4. Numerical Examples

*the total energies of the watermark values in these two approaches are the same*. It should be noted that we have also implemented the DCT watermarking method when the pseudorandom sequence is added to the DCT values at the same positions as the ones in the above DWT approach, i.e., the middle frequencies. We found that the performance is not as good as the one by adding watermarks in all the frequencies in the DCT domain.

*HH*

_{1}band with the DWT approach, where the cross correlations are shown in Fig. 9(a) and a peak can be clearly seen. When the variance of the additive noise is large, such as the one shown in Fig. 8(a), the

*HH*

_{1}band information is not good enough with the DWT approach, where the cross correlations are shown in Fig. 9(b) and no clear peak can be seen. However, the signature can be detected by using the information in the

*HH*

_{1}and

*LH*

_{1}bands with the DWT approach, where the cross correlations are shown in Fig. 9(d) and a peak can be clearly seen. For the second noisy image, we have also implemented the DCT approach. In this case, the signature with the DCT approach can not be detected, where the correlations are shown in Fig. 9(c) and no clear peak can be seen. Similar results hold for the “car” image and the correlations are shown in Fig. 10.

*rescaling*, an image, x, is reduced to 3/4 of the original size. The method of the rescaling is from the MATLAB function called “imresize.” as imresize(x, 1-1/4, ‘method’) where ‘method’ indicates one of the methods in the interpolations between pixels: piecewise constant, bilinear spline, and cubic spline. With the received smaller size image, for the watermark detection we extend it to the normal size, i.e., 512 × 512, by using the same Matlab function “imresize” as imresize(y, 1+1/3, ‘method’), where ‘method’ is also one of the above interpolation methods. In this experiment, we implemented two different interpolation methods in imresize in the rescaling distortion: the piecewise constant method and the cubic spline method. In the detection, we alway use the cubic spline as imresize(y, 1+1/3, ‘bicubic’). Similar results also hold for other combinations of these interpolation methods. Fig. 11 illustrate the detection results for the “peppers” image: Fig. 11(a),(c) show the cross correlations with the DWT approach while Fig. 11(b),(d) show the cross correlations with the DCT approach. In Fig. 11(a), (b), the rescaling method is imresize(x,1-1/4,‘nearest’), i.e., the piecewise constant interpolation is used. In Fig. 11(c),(d), the rescaling method is imresize(x,1-1/4,‘bicubic’), i.e., the cubic spline interpolation is used. One can see the better performance of the DWT approach over the DCT approach. Similar results hold for the “car” image and are shown in Fig. 12.

*bpp*. With these two compressed images, the correlations are shown in Fig. 17 (a) and (b), where a peak in the middle can be clearly seen in Fig. 17(a) with the DWT approach, but no clear peaks can be seen in Fig. 17(b) with the DCT approach. This is not very surprising because the compression scheme is not suitable for the DCT approach. It should be noticed that the wavelet filters in the EZW compression are the commonly used Daubechies “9/7” biorthogonal wavelet filters while the wavelet filters in the watermarking are the simpliest Haar wavelet filters mentioned in Section 2.

*x*[

*m*,

*n*] be an image with 8 bit levels. To halftone it, we do the nonuniform thresholding through the Bayer‘s dither matrix

*T*[17]:

*x*[

*m*,

*n*]. If

*x*[

*m** 4 +

*j*,

*n** 4 +

*k*] ≥

*T*

_{j,k}, then it is quantized to 1, and otherwise it is quantized to 0. Both DWT and DCT watermarking methods are tested. Surprisingly, we found that the watermarking method based on DWT we proposed in this paper is more robust than the method based on the DCT in [2–3]. The correlations are shown in Fig. 18(a) and (b), where (a) corresponds to the DWT approach while (b) corresponds to the DCT approach. One can clearly see a peak in the middle in Fig. 18(a) while no any clear peak in the middle can be seen in Fig. 18(b). In this experiment, the watermark was added to the middle frequencies in the DCT approach and no inverse halftoning was used.

## 5. Conclusion

## 6. Acknowledgements

## References

1. | R. G. van Schyndel, A. Z. Tirkel, and C. F. Osborne, “A digital watermark,” Proc. ICIP’94 , |

2. | I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon, “Secure spread spectrum watermarking for images, audio and video,” Proc. ICIP’96 , |

3. | J. Zhao and E. Koch, “Embedding robust labels into images for copyright protection,” Proceedings of the International Congress on Intellectual Property Rights for Specialized Information, Knowledge and New Technologies, Vienna, Austria, August 21-25, 242–251 (1995). |

4. | R. B. Wolfgang and E. J. Delp, “A watermark for digital images,” Proc. ICIP‘96 , |

5. | I. Pitas, “A method for signature casting on digital images,” Proc. ICIP’96 , |

6. | N. Nikolaidis and I. Pitas, “Copyright protection of images using robust digital signatures,” Proceedings of ICASSP’96, Atlanta, Georgia, May, 2168–2171 (1996). |

7. | M. D. Swanson, B. Zhu, and A. H. Tewfik, “Transparent robust image watermarking,” Proc. ICIP’96 , |

8. | M. Schneider and S.-F. Chang, “A robust content based digital signature for image authentication,” Proc. ICIP’96 , |

9. | S. Mallat, “Multiresolution approximations and wavelet orthonormal bases of |

10. | I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. on Pure and Appl. Math. , |

11. | O. Rioul and M. Vetterli, “Wavelets and signal processing,” IEEE Signal Processing Magazine , 14–38, (1991). [CrossRef] |

12. | I. Daubechies, Ten Lectures on Wavelets, (SIAM, Philadelphia, 1992). |

13. | P. P. Vaidyanathan, Multirate Systems and Filter Banks, (Prentice Hall, Englewood Cliffs, NJ, 1993). |

14. | M. Vetterli and J. Kovačević, Wavelets and Subband Coding, (Prentice Hall, Englewood Cliffs, NJ, 1995). |

15. | G. Strang and T. Q. Nguyen, Wavelets and Filter Banks, (Wellesley-Cambridge Press, Cambridge, 1996). |

16. | J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. on Signal Processing , |

17. | R. Ulichney, Digital Halftoning, (MIT Press, Massachusetts, 1987). |

18. | S. Craver, N. Memon, B-L Yeo, and M. M. Yeung, “Resolving rightful ownerships with invisible watermarking techniques: limitations, attacks, and implications,” IBM Research Report (RC 20755), March 1997. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(110.2960) Imaging systems : Image analysis

**ToC Category:**

Focus Issue: Digital watermarking

**History**

Original Manuscript: October 28, 1998

Published: December 7, 1998

**Citation**

Xiang Gen Xia, Charles Boncelet, and Gonzalo Arce, "Wavelet transform based watermark for digital images," Opt. Express **3**, 497-511 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-12-497

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### References

- R. G. van Schyndel, A. Z. Tirkel, and C. F. Osborne, "A digital watermark," Proc. ICIP'94, 2, 86-90 (1994).
- I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon, "Secure spread spectrum watermarking for images, audio and video," Proc. ICIP'96, 3, 243-246 (1996).
- J. Zhao and E. Koch, "Embedding robust labels into images for copyright protection," Proceedings of the International Congress on Intellectual Property Rights for Specialized Information, Knowledge and New Technologies, Vienna, Austria, August 21-25, 242-251 (1995).
- R. B. Wolfgang and E. J. Delp, "A watermark for digital images," Proc. ICIP'96, 3, 219-222 (1996).
- I. Pitas, "A method for signature casting on digital images," Proc. ICIP'96, 3, 215-218 (1996).
- N. Nikolaidis and I. Pitas, "Copyright protection of images using robust digital signatures," Proceedings of ICASSP'96, Atlanta, Georgia, May, 2168-2171 (1996).
- M. D. Swanson, B. Zhu, and A. H. Tewfik, "Transparent robust image watermarking," Proc. ICIP'96, 3, 211-214 (1996).
- M. Schneider and S.-F. Chang, "A robust content based digital signature for image authentication," Proc. ICIP'96, 3, 227-230 (1996).
- S. Mallat, "Multiresolution approximations and wavelet orthonormal bases of L 2 (R)," Trans. Amer. Math. Soc., 315, 69-87 (1989).
- I. Daubechies, "Orthonormal bases of compactly supported wavelets," Comm. on Pure and Appl. Math., 41, 909-996 (1988). [CrossRef]
- O. Rioul and M. Vetterli, "Wavelets and signal processing," IEEE Signal Processing Magazine, 14-38, (1991). [CrossRef]
- I. Daubechies, Ten Lectures on Wavelets, (SIAM, Philadelphia, 1992).
- P. P. Vaidyanathan, Multirate Systems and Filter Banks, (Prentice Hall, Englewood Cliffs, NJ, 1993).
- M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, (Prentice Hall, Englewood Cliffs, NJ, 1995).
- G. Strang and T. Q. Nguyen, Wavelets and Filter Banks, (Wellesley-Cambridge Press, Cambridge, 1996).
- J. Shapiro, "Embedded image coding using zerotrees of wavelet coefficients," IEEE Trans. on Signal Processing, 41, 3445-3462 (1993). [CrossRef]
- R. Ulichney, Digital Halftoning, (MIT Press, Massachusetts, 1987).
- S. Craver, N. Memon, B-L Yeo, and M. M. Yeung, "Resolving rightful ownerships with invisible watermarking techniques: limitations, attacks, and implications," IBM Research Report (RC 20755), March 1997.

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