**9. Wavelet-based applications**

Recently, The wavelet transform is being increasingly used, not only in the field of image and signal processing applications but also in many other different areas, ranging from mathematics, physics, astronomy to statistics and economics. In image processing based applications, image compression, image denoising and image watermarking are at the cutting edge, and as such, a brief description of these wavelet-based applications is given in the following subsections (Strang & Nguyen, 1996; Burrus et al., 1998; Stromme, 1999; Ebrahimi et

al, 2002; Nibouche et al., 2000, 2001a, 2001b, 2001c, 2001d, 2002, 2003; Smith, 2003; Do & Vetterli, 2003, 2005; Hankerson et al., 2005; Nai-Xiang Yap-Peng, 2005; Xiong & Ramchandran, 2005; Chappelier & Guillemot, 2006; Cunha et al., 2006; Nai-Xiang et al., 2006; Raviraj & Sanavullah, 2007; Hernandez-Guzmane et al., 2008; Firoiu et al., 2009; Mallat, 2009; Brislawn, 2010; Oppenheim & Schafer, 2010; Ruikar & Doye, 2010 & Chen & Qian, 2011).

The Wavelet Transform for Image Processing Applications 415

baneful effects when applied indiscriminately to an image. In fact, if it is not the whole

A solution to overcome this problem has been introduced by Denoho and Johnstone (Donoho & Johnstone, 1994). Instead of exploiting either linear or non-linear filtering, their technique consists of using the DWT followed by a thresholding operation. This method exploits the energy compaction ability of the wavelet transform to separate the image from the added noise. The role of the threshold is to eliminate the noise present in the image. Finally, the enhanced "denoised" image is recovered by applying the inverse DWT. This method is also known as the wavelet shrinkage denoising, and is classified as a nonlinear processing technique due to the thresholding operation involved in the process as illustrated

> *DWT IDWT Nonlinear Operator*

Another method, which achieves better performances when compared to the previous one, consists of using an undecimated version of the DWT (Donoho & Johnstone, 1995) This choice is motivated by the fact that originally, the DWT is not a shift-invariant transform, and as such, visual artifacts can be spanned by the transform. This like-noise is more apparent around discontinuities in the image. However, in this particular case the inverse transform is not unique. As a solution, it is appropriate to take the average of the possible

Image watermarking emerged in the mid 90s as a discipline, among the wide range of multidisciplinary field of data hiding, as a methodology of protecting digital images from any piracy act. It consists of embedding a watermark (a trace) within a digital image before using or publishing it. The efficiency of a watermarking method lies generally in its ability

Watermarking techniques can be classified into two categories; spatial domain methods and transform-based methods. The wavelet-based watermarking technique falls into the latter. In (Kundur & Dimitrios, 1997, 1998 & Hernandez-Guzman et al., 2008) both the original image and the watermark are first transformed to the wavelet domain, then the resulting image pyramids are fused according to certain rules, which take into account the characteristics of the Human Visual System (HVS). The wavelet in this case facilitates a simultaneous spatial localisation and frequency spread of the watermark within the source image. It has been shown that the method is robust under compression, additive noise and

To the best of our knowledge, there is no general baseline framework for a wavelet-based watermarking system. However, in most cases, the multiresolution feature of the transform is exploited to achieve robust image watermarking implementations (Kundur & Dimitrios, 1997, 1998; Tsekeridou & Pitas, 2000; Wu et al., 2000 & Hernandez-Guzman et al., 2008).

reconstruction. The computational complexity of this approach is O(nlog(n)).

to fulfil three requirements: robustness, security and invisibility.

*De-noised Image*

image that is blurred, some of its important features (e.g. edges) are.

in Figure 22.

*Noisy Image*

**9.3 Image watermarking** 

Fig. 22. Wavelet-based denoising system

filtering (Kundur & Dimitrios, 1997, 1998)

## **9.1 Image compression**

Even though the wavelet transforms have been widely used in image coding since the late 80s, they only gained their notoriety in the field by the adoption of the first wavelet-based compression standard scheme, known as the FBI fingerprint compression standard Bradley, et al., 1993). Recently, what did Sweldens state in (Sweldens, 1996) as a need of standardising a wavelet-based compression scheme under the header "problems not sufficiently explored with wavelets", has seen the day, by the adoption of the JPEG2000 new compression standard (Ebrahimi et al., 2002). The block diagram of the JPEG2000 standard does not really differ from the JPEG standard one. The discrete wavelet transform, which replaces the DCT, is applied first to the source image. The transformed coefficients are then quantised. Finally, the output coefficients from the quantiser are encoded (using either Huffman coding or arithmetic coding techniques) to generate the compressed image (Smith, 2003; Do & Vetterli, 2005; Hankerson et al., 2005; Xiong & Ramchandran, 2005; Chappelier & Guillemot, 2006; Nai-Xiang et al., 2006; Raviraj & Sanavullah, 2007; Mallat, 2009; Oppenheim & Schafer, 2010). To recover the original image the inverse process is applied. Figure 21 shows the basic JPEG2000 Encoding Scheme (Ebrahimi et al., 2002).

Fig. 21. Wavelet-based encoding scheme
