**9.2 Image denoising**

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Image manipulation, includes a wide range of operations like digitising, copying, transmitting, displaying … etc. Unfortunately, such manipulations generally degrade the image quality by spanning many types of noise. Hence, to recover the original structure of the image, the undesired added noise needs to be localised and then removed. In image processing, noise removal is achieved through the usage of filtering-based denoising techniques (Nai-Xiang & Yap-Peng, 2005; Chappelier & Guillemot, 2006; Firoiu et al., 2009; Mallat, 2009; Nafornita et al., 2009; Ruikar & Doye, 2010; Oppenheim & Schafer, 2010 & Chen & Qian, 2011). Traditionally, image denoising or image enhancement is performed using either linear filtering or non-linear filtering. Linear filtering is achieved either by using spatial techniques, as low pass filtering, or frequency techniques, as the Fast Fourier Transform (FFT). On the other hand, statistical and morphological filters are typical examples of non-linear filtering. However, the filtering techniques lead in some cases to

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

**DWT Quantisation Encoding**

**Storage Device**

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**IDWT De-quantisation Decoding**

Image manipulation, includes a wide range of operations like digitising, copying, transmitting, displaying … etc. Unfortunately, such manipulations generally degrade the image quality by spanning many types of noise. Hence, to recover the original structure of the image, the undesired added noise needs to be localised and then removed. In image processing, noise removal is achieved through the usage of filtering-based denoising techniques (Nai-Xiang & Yap-Peng, 2005; Chappelier & Guillemot, 2006; Firoiu et al., 2009; Mallat, 2009; Nafornita et al., 2009; Ruikar & Doye, 2010; Oppenheim & Schafer, 2010 & Chen & Qian, 2011). Traditionally, image denoising or image enhancement is performed using either linear filtering or non-linear filtering. Linear filtering is achieved either by using spatial techniques, as low pass filtering, or frequency techniques, as the Fast Fourier Transform (FFT). On the other hand, statistical and morphological filters are typical examples of non-linear filtering. However, the filtering techniques lead in some cases to

shows the basic JPEG2000 Encoding Scheme (Ebrahimi et al., 2002).

**9.1 Image compression** 

**Source Image**

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**Recovered Image**

**9.2 Image denoising** 

Fig. 21. Wavelet-based encoding scheme

baneful effects when applied indiscriminately to an image. In fact, if it is not the whole image that is blurred, some of its important features (e.g. edges) are.

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 in Figure 22.

Fig. 22. Wavelet-based denoising system

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