**7. Conclusion and further development**

Recently, there has been considerable interest in using the wavelet transform as a powerful tool for recovering SAR images from noisy data. The main reason for the choice of multiscale bases of decompositions is that the statistics of many natural signals, when decomposed in such bases are significantly simplified. When multiplicative contamination is concerned, multiscale methods involve a preprocessing step consisting of a logarithmic transform to separate the noise from the original image. However, thresholding methods have two main drawbacks: i) the choice of the threshold, arguably the most important design parameter, is made in an *ad hoc* manner; and ii) the specific distributions of the signal and noise may not be well matched at different scales. To address these disadvantages, Bayesian theory can be introduced , which outperform classical linear processors and simple thresholding estimators in removing noise from visual images.

Denoising should not be confused with smoothing. Smoothing removes high frequencies and retains low frequencies whereas denoising attempts to remove whatever noise is present and retain whatever signal is present regardless of the spectral content of the noisy signal.Wavelet shrinkage denoising is denoising by shrinking (i.e., nonlinear soft thresholding) coefficients in the wavelet transform domain. It consists of three steps: 1) a linear forward wavelet transform, 2) a nonlinear shrinkage denoising, and 3) a linear inverse wavelet transform. Because of the nonlinear shrinking of coefficients in the transform domain, this procedure is distinct from those denoising methods that are entirely linear. Moreover, it is considered as a nonparametric method. Thus, it is distinct from parametric

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methods, including both linear and nonlinear regression, in which parameters must be estimated for a particular model that must be assumed a priori. If the common sense approach to practical problem solving is adopted, then the practitioner should exploit any and all *a priori* information available for his particular problem, and use an appropriate denoising procedure as determined by the most relevant outcome measure. Determining the most appropriate procedure necessarily involves experiments to compare the performance of a wavelet shrinkage denoising method with any other methods under consideration. In addition, issues of computational complexity must be considered. Complexity of algorithms may be measured according to CPU computing time and flops, or the number and kind of algorithm steps and their impact on firmware or hardware requirements. Here a new statistical representation for the wavelet decomposition coefficients of SAR images is introduced and it is found that shrinkage is to be more effective than traditional methods both in terms of speckle reduction and signal detail preservation. The SAR images evaluated all are coded in eight-bit. The motivation is that as wavelet transform is good at energy compaction, the small coefficients are more likely due to noise and large coefficient due to important signal features . The proposed technique is based upon the analysis of wavelet transform which uses a soft thresholding method for thresholding the small coefficients without affecting the significant features of the image. In the chapter, image denoising is studied using various wavelets for different images at various levels of decomposition and comparison are done between the families and wavelet shrinkage techniques. It is unlikely that one particular wavelet shrinkage denoising procedure will be suitable, no less optimal, for all practical problems. However, it is likely that there will be many practical problems, for which after appropriate experimentation, wavelet-based denoising with either hard or soft thresholding proves to be the most effective procedure. Estimation of the power spectrum by wavelet-based denoising of the log-periodogram may prove to be one such important application with great promise for further development.

#### **8. References**


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**1. Introduction**

digital signal.

With the widespread use of Internet and digital multimedia technologies, the interest in copyright protection of digital content has been rapidly increased. Digital watermarking has emerged as a possible solution for intellectual property rights protection. Watermarking has also proven to be a promising tool in many applications such as broadcast monitoring, fingerprinting, authentication and device control. In digital watermarking, additional information, called the *watermark*, is *imperceptibly* embedded into the original digital content. Different applications pose different requirements on watermarking. For example, fragile watermarking is required in content authentication applications, while in applications such as copyright control the watermark should be robust to attacks1. In each application, the watermarking method makes a trade-off between the perceptual invisibility, robustness, security, data capacity and availability of side information. For instance, to increase the robustness of a watermark, the watermark strength needs to be increased, which in turn may make the watermark more visible. The invisibility requirement of watermarking limits the maximum amount of watermark bits (watermarking capacity) that can be embedded into a

**Image Watermarking in** 

*The University of British Columbia, Vancouver,* 

*Canada* 

**20**

**Higher-Order Gradient Domain** 

Ehsan N. Arya, Z. Jane Wang and Rabab K.••Ward

In the last two decades, a lot of work has been done in the field of image watermarking. The reader may refer to (Cox, 2008) for a survey of watermarking methods. Watermarking approaches can generally be classified into two categories (Wu & Liu, 2003): *spread spectrum* (SS) based watermarking (Cox et al., 1997; Podilchuk & Zeng, 1998) and *quantization* based watermarking (Chen & Wornell, 2001; Kundur & Hatzinakos, 2001; Moulin & Koetter, 2005).

In general, any watermarking system that spreads the host signal over a wide frequency band can be called *spread spectrum* watermarking (Barni, 2003). In most SS type methods, a pseudo-random noise-like watermark is added (or multiplied) to the host feature sequence (Cox et al., 1997). While SS watermarking methods are robust to many types of attacks, they suffer from the host interference problem (Cox et al., 1999). This is because the host signal itself acts as a source of interference when extracting the watermark, and this may

<sup>1</sup> The attacks are defined as the processes that may impair the detection of the watermark.

Below, these two approaches are discussed with some detail.

**1.1 Spread spectrum watermarking**

reduce the detector's performance.
