ABSTRACT

Image watermarking provides a means to enforce ownership of image data by altering a raw or compressed image secretly and imperceptibly. It consists of embedding inside an image information that should be undeletable by an opponent, easily and securely detectable by the owner of the contents, perceptually and statistically invisible and resistant against any type of usual processing like compression, filtering or noising. In this paper, an image-watermarking scheme based on S-transform is proposed. The S-transform (ST) is an extension of the ideas of the continuous wavelet transform (CWT), and is based on a moving and scalable localizing Gaussian window. This transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates. This transform is advantageous in that the inverse frequency dependence of the localizing Gaussian window is an improvement over the fixed width window used in short time Fourier transform (STFT). The phase of the ST also provides useful and supplementary information about spectra, which is not available from locally, referenced phase information in CWT. In this work, the focus is on embedding watermarking bits in the phase of the time-frequency representation of the host image.

KEY WORDS

Image Watermarking, Data Hiding, Steganography, Phase Watermarking, S-Transform

1. INTRODUCTION

Steganography refers to the science of invisible communication. Unlike cryptography, where the goal is to secure communications from an eavesdropper, steganographic techniques strive to hide the very presence of the message itself from an observer. The general idea of hiding some information in digital content has a widerclass of applications that go beyond steganography [1] as in Figure1. The techniquesinvolved in such applications are collectively referred to as information hiding. A special case of information hiding is digital watermarking. Digital watermarking is the process of embedding information into digital multimedia content such that the information (the watermark) can later be extracted or detected for a variety of purposes including copy prevention and control. The increasing importance of digital media brings also new challenges, as it is now straightforward to duplicate and manipulate multimedia content. So there is a strong need for security services in order to keep the distribution of digital media work more profitable for the document owner and reliable for the customer. Watermarking technology plays an important role in securing the business as it allows placing an imperceptible mark in the multimedia data to identify the legitimate owner, track authorized owners via fingerprinting or detect malicious tampering of the document.

Figure 1: Different applications beyond steganography

Previous research indicates that significant portions of the host image, for example the low frequency components, have to be modified in order to embed the information in a reliable and robust way. This led to the development of watermarking schemes embedding in the frequency domain. Robust watermarking in spatial domain can also be achieved at the cost of explicitly modeling the local image characteristics. Many image transforms have been considered; most prominent among them is the discrete cosine transform (DCT). Hence, there are a large number of watermarking algorithms that use either a block-based or global DCT.

Image watermarking imperceptibly embeds data into a host image. The original host image is modified using the signature data to create the watermarked image. The watermarked image is then distributed and circulated from legitimate to illegitimate customers. Thereby, it is subjected to various kinds of image distortion. The extraction process may or may not, depending on the nature of the application, require knowledge of the original host image to estimate the hidden signature from the distorted image that is received. The watermark is recovered from the host image. It is desired that the difference between the extracted and the original signature is as low as possible.

The signature data or the message can be some binary data, a small image or logo or a seed value to a pseudo random number generator to produce a sequence of numbers with a certain distribution. In the extraction stage, the secret key used to embed the watermark as well as the original image might be available. These systems are non-blind, non-oblivious or private watermarking systems. The other extreme is where neither the private key nor the original image is available during the extraction process. These systems are public key watermarking systems. Asymmetric watermarking schemes use different keys for embedding and detecting the watermark. Blind or oblivious watermarking schemes allow extracting the signature data without reference to the original, unwatermarked image. Semi-blind or semi-oblivious watermarking schemes rely on some data or features derived from original host image.

In the decision stage, the watermarking system analyses the extracted data. In the simplest case, the result is just a yes/no decision indicating if the copyright holder’s mark has been found in the received image data. More complex systems return the embedded logo image or the textual copyright information that was placed into the host image.

Data embedding in spatial domain is simple and computationally efficient. But these approaches failed to achieve good robustness and sacrifice strength under compression attacks. All robust watermarking algorithms operate in the transform domain that offers access to the frequency components of the host image. The host image is therefore first transformed to a domain that facilitates data embedding. Some commonly used frequency domain representations are DFT, DCT, DWT, CWT, fractal transform, Fourier Mellin transform, Fresnel transform etc [2]. The subset of the transform coefficients is modified with the prepared signature data. By choosing a suitable frequency transform domain and selecting only certain coefficients typically in low to mid frequency range, a lot of HVS modeling can be done implicitly. Finally, the inverse transformation is applied on the modified transform domain coefficients to produce the watermarked image.

Transform domain watermarking algorithms possess a number of desirable properties. Since the watermark embedded in the transform domain is irregularly distributed over the area of local support after the inverse transformation, these methods make it more difficult for an attacker to read or modify the mark. Furthermore, the frequency representation of images allows selecting only certain bands of the host signal for watermarking.

2. S-TRANSFORM

The time-frequency analyses are broadly used in image processing, medical imaging, acoustics, etc. A powerful method is the S-transform by Stockwell [3]. It is extension of the STFT, which uses frequency dependent scaling windows in analogy to WT. This permits a frequency dependent resolution with narrower windows at higher frequencies and wider windows atlower frequencies. Stockwell use Gaussian windows, but other window functions can also beemployed. Thetimefrequency representation obtained with the S-transform is unique and invertible. When averaged over time, the S-transform becomes the Fourier transform of the original time series.

An image is a function of the independent space variables x and y. The global Fourier spectrum of the image is a complex function of the frequency variables (wave numbers) k_x and k_y. The global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of the image, leading to the concept of a local spectrum at every point of the image. The 2D S-transform is a method of computation of the local spectrum at every point on the image. In addition to the variables x and y, the 2D S-transform retains the variables k_x and k_y, being a complex function of four variables. Images with strictly periodic patterns are best analysed with a global Fourier spectrum and the 2D S-transform is more useful in spectral characterization of aperiodic or random patterns.

In a 1D S-transform, a Gaussian window dilates and translates along the time axis and maps the 1D time series into a complex function of both time and frequency. There is complete and lossless invertibility between the time t to time-frequency (t, f) to frequency f and back to time domains. Thus an event in a time series can be examined not only in time domain, but also in time-frequency and frequency domains.

In wavelet transform, the high frequency signals are located in the pixel domain, while low frequency signals are located in frequency domain. The spatial resolution increases with frequency while the frequency resolution is inversely proportional to frequency. Frequency resolution is independent of frequency in the DCT domain. The S transform is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. This technique is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates. The Gaussian window is chosen because it is the most compact in time and frequency.

The S transform of a function is defined as a CWT with a specific mother wavelet multiplied by a phase factor where the mother wavelet is defined as. The dilation factor is the inverse of the frequency. This wavelet does not satisfy the condition of zero mean for an admissible wavelet, so it is not strictly a CWT. The S transform is

.

This is a representation of the local spectrum and so, a simple operation of averaging the local spectra over time gives the Fourier spectrum i.e., and is exactly recoverable from by . This relationship is clearly distinct from the concepts of CWT. ST is in fact a phase correction of the definition of the WT. It eliminates the concept of wavelet analysis by separating the mother wavelet into two parts, the slowly varying envelope i.e. the Gaussian function that localizes in time, and the oscillatory exponential kernel, which selects the frequency being localized.

The linear property of S transform ensures that for the case of additive noise, it gives. This is advantageous over the bilinear class of time-frequency representations where the presence of cross terms makes it difficult to reliable estimate the signal. The S transform can be written as operations on the Fourier spectrum of as

.

The discrete analog is used to compute the discrete S transform by taking advantage of the efficiency of the FFT and the convolution theorem. By not translating the co-sinusoid basis functions, the S transform localizes the real and imaginary components of the spectrum independently, localizing the phase as well as the amplitude spectrum.

The 2D S-transform is defined as

The average of over the variables x, y gives the 2D Fourier spectrum. The Fourier Transform is limited in that, although it can determine the spectral components of a time series, there is no information on when these components exist in the time series. The time-local information is obscured in the phase information of the Fourier transform. The STFT is the result of repeatedly multiplying the time series with shifted short time windows and performing a discrete Fourier transform. This windowing is a constant-width windowing. The choice of the window is completely arbitrary. The time and frequency resolution is the same for all spectral components. If the window function is chosen to be a Gaussian function, the STFT is called Gabor transform. The parameter is related to the width of the Gaussian function. In S-transform, is allowed to vary and is made proportional to the period of the analyzing sinusoid, where c is proportionality constant. The S-transform performs multiresolution analysis on the signal, because the width of its window varies. The width decreases as frequency increases. This gives high time resolution at high frequencies. The frequency resolution is highest at low frequencies.

The effectiveness of S-transform in providing a time-frequency representation is demonstrated by the following example. Consider a chirp signal, which is a time series that contains an oscillatory component whose phase is a parabolic function of time such as . Figures 2(a) and 2(b) show two time series, one with an increasing frequency as a function of time and the second with a decreasing frequency. When Fourier analysis is performed, the amplitude spectra are identical for both time series, indicating that there is no information about when the different frequencies existed in the time series. The time local information is hidden in the value of the phases of the spectrum as indicated in Figures 3(a) and 3(b). A contour plot of amplitude of the STFT of the time series with a Gaussian window is shown in Figure 4(a) and the ST is shown in Figure 4(b).

The comparison is performed for another 128-point time series consisting of discrete spectral components and a high frequency burst. The time series is depicted in Figure 5(a). The Fourier spectrum is shown in Figure 5(b) and the amplitude of the STFT in shown in Figure 5(c). In FT, the time of occurrence of the frequencies is not tracked and in STFT the constant resolution is apparent from the fact that the high frequency component is poorly resolved as the constant width window is averaging over a long period of time. Figure 5(d) shows the result of performing the Wigner Distribution. The cross terms in the Wigner distribution heavily contaminate the signal, and the high frequency burst is not detected. The CWT also localizes the power spectrum as a function of time. It does not retain the absolutely referenced phase information that the ST contains and therefore is not directly invertible to the FT spectrum. The ST improves on the STFT in that it has better resolution in phase space i.e. a more narrow time window for higher frequencies, giving a fundamentally more sound time frequency representation. Figure 5(e) shows the amplitude of the ST and it is clear that it accurately represents all three components. The low frequency component has very good frequency resolution, while the high frequency transient signal has very good time resolution, and is reliably detected.

3. WATERMARKING BASED ON S-TRANSFORM

In additive watermarking algorithms, the signature data is a sequence of random numbers of length that is embedded in a suitably selected subset of the host signal data coefficients, . The commonly used embedding formula is where is the weighting factor and is the resulting modified host data coefficient carrying the watermark information. Inverse embedding function can be easily derived to compute from given the original host possibly altered, image that might contain the watermark. The extracted watermark sequence is compared to the originally embedded watermark using the normalized correlation of the sequences as a similarity measure. The similarityvaries in the interval [-1, 1]; a value well above 0 and close to 1 indicates the extracted sequence matching the embedded sequence. A detection threshold can be established to make the detection decision, . The weighting factor does not necessarily have to be constant over the entire watermark sequence, but can be chosen adaptively to capture local properties of the host signal. This allows to have more energy in the watermark signal and thus have a more robust watermark. One example is where takes into account the local properties.

The ST coefficients are generally complex valued and, this leads to a magnitude and phase representation for the image. Experimentally analyzing the effect of phase on the intelligibility of the image, it is conclusive that the phase is more important than the magnitude of the DFT values. As the relative importance of the phase components of the DFT is already known, it is proposed to embed the watermark information in the phase of the ST coefficients.

A watermark that is embedded in the phase would be quite robust to tampering. The core information contained in watermarks is almost always encoded with a high degree of redundancy. So, phase distortions deliberately introduced by an enemy to impede transmission of the watermark would have to be noticeably large in order to be successful. This would cause unacceptable damage to the quality of the image. Also, phase modulation possesses superior noise immunity compared to amplitude modulation [5]. Phase based watermarking is relatively robust to changes in image contrast as contrast change affects only the magnitude of the frequency domain representation, not the phase.

In the proposed algorithm, the watermark is embedded in the maximum ST coefficient of a particular voice. A voice is a 2D function of the spatial variables x and y and fixed parameters and . The watermark is added additively to that phase so that it can be recovered easily in the extraction phase. The watermark data is a

sequence of Gaussian distributed random numbers of length with zero mean and unit variance that is embedded in a suitably selected phase of the subset of the host signal data coefficients,. The embedding is based on the additive algorithm discussed above. The length is chosen to be the size of the image and the weighting factor is chosen depending on the strength of the watermark that is to be embedded. As increases, the strength of the watermark increases and the image quality after watermarking, measured in terms of the PSNR decreases. After watermark embedding, the inverse ST is performed and the resultant image is the watermarked image, which can be circulated. Watermark can be detected and recovered by simply comparing the marked image with an unmarked original. So, this watermarking algorithm requires the original image for extracting the watermark information.

4. SIMULATION RESULTS

Standard images like Cameraman, Baboon, Barbara, Boats, Peppers and Lena were watermarked using the ST. Figure 6(a) shows the original gray scale image of 32 x 32 pixels. The watermarked image is shown in Figure 6(b). Only the ST phase is used to embed the watermark information. A total of 32 x 32 = 1024 bits is embedded. Despite the presence of the watermark the watermarked image does not contain any visible artifacts. The maximum absolute error is 45. Figure 6(c) shows the absolute difference between the original image and the marked image scaled by a factor of 64. It is interesting to note that the difference image contains smoothed contours near the edges of the image as the inverse ST is calculated by averaging different voices i.e. averaging 2D spatial domain functions and then taking the inverse DFT. The results for other images are tabulated in terms of the PSNR after watermark embedding in Table 1 for =0.9. Lena image gave the highest PSNR as it is an image with high frequencies. So, not much watermark information was embed in it. Barbara is a low frequency image and provides the lowest PSNR. Cameraman and Boats images are mid frequency images while Baboon and Peppers are slightly high frequency images.

Figure 6: (a) Original image (b) Watermarked image

(c) Absolute error image

Image (64 x 64)	PSNR (dB)
Lena	18.8
Cameraman	15.5
Barbara	14.5
Baboon	18.2
Boats	15.6
Peppers	17.5

Table 1: Image quality for different images

5. CONCLUSION

In this paper, a new watermarking algorithm is proposed based on the S-transform. In direct contrast to many other techniques, the method presented here places the watermark on the phase of the most significant i.e. maximum magnitude coefficient of an image. This can be defended by the fact that any operation that is intentionally performed to damage the watermark in some way also will unavoidably damage the image. This idea is consistent with the use of phase in the time-frequency domain as a method of conveying information, as phase information is more important to the viewer than the magnitude information.

The S-transform is considered for watermarking, as it is a time-frequency representation in which a mapping from 2D image space, to 4D space-frequency to 2D Fourier domains are performed. Features or patterns barely discernible in one domain may be prominent in any of the other domains. S-transform permits incremental benefit in using spectral localization. The only disadvantage is the computational complexity. The ST of a 2D image becomes a complex function of four variables. An N x N image occupies N² memory locations. The Fourier spectrum requires 2 x N² locations and the ST requires 2 x N⁴ points. Symmetry properties of ST and FT permit some economies. Hence, computer resources and visualization set the limits on possible applications of the 2D ST. The solution is to examine the ST piecewise by removing one degree of freedom. So, only 1D example plots are provided for the ST.

Future work will concentrate on integrating aspects of the HVS into watermark embedding. In addition, a detailed study of the effects of image distortion on a watermark will be performed. This will improve watermark detection. Finally, new techniques will be thought of to make it possible to detect a watermark without requiring the original unmarked image.

REFERENCES

1. Mehdi Kharrazi, Husrev T. Sencar & Nasir Memon, Image Steganography: Concepts and Practice (WSPC/Lecture Notes Series, 2004).

2. Peter Meerwald, Digital Image Watermarking in the Wavelet Transform Domain (Master’s Thesis, University of Salzburg, 2001).

3. R. G. Stockwell, L. Mansinha & R. P. Lowe, Localization of the Complex Spectrum: The S Transform, IEEE Transactions on Signal Processing, 44(4), 1996, 998 – 1001.

4. Robert G. Stockwell, S-Transform Analysis of Gravity Wave Activity from a Small Scale Network of Airglow Imagers (Ph.D Thesis, University of Western Ontario, 1999).

5. J. J. K. O Ruanaidh, W. J. Dowling & F. M. Boland, Phase Watermarking of Digital Images, Proc. IEEE International Conference on Image Processing, ICIP ’96, Vol. 3, 239 – 242.

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