**2.3 Denoising**

In many signal or image processing applications, the input data is corrupted by some noise which need to be removed or at least reduced. Wavelet denoising techniques work by adjusting the wavelet coefficients of the signal in such a way that the noise is reduced while the signal is preserved (Sivakumar,2007). There are many different methods for adjusting the coefficients but the basic principle is to keep large coefficients while reducing small coefficients. This adjustment is known as thresholding the coefficients.One rationale for this approach is that often real signals can be represented by a few large wavelet coefficients, while (for standard orthogonal wavelet transforms) white noise signals are represented by white noise of the same variance in the wavelet coefficients. Therefore the reconstruction of the signal from just the large coefficients will tend to contain most of the signal energy but little of the noise energy. An alternative rationale comes from considering the signal as being piecewise stationary. For each piece the

optimum denoising method is a Wiener filter whose frequency response depends on the local power spectrum of the signal. When the signal power is high, the power is kept mostly; when the signal power is low, the signal is attenuated. The size of each wavelet coefficient can be interpreted as an estimate of the power in some time-frequency bin and set the small ones to zero in order to approximate adaptive Wiener filtering. The first wavelet transform proposed for denoising was the standard orthogonal transform. However, orthogonal wavelet transforms (DWT) produce results that substantially vary even for small translations in the input and so a second transform was proposed, the nondecimated wavelet transform (NDWT) , which produced shift invariant results by effectively averaging the results of a DWT-based method over all possible positions for the origin. Experiments on test signals show that the NDWT is superior to the DWT. The main disadvantage of the NDWT is that even an efficient implementation takes longer to compute than the DWT, by a factor of the three times the number of levels used in the decomposition.Kingsbury has proposed the use of the DT-CWT for denoising because this transform not only reduces the amount of shift-variance but also may achieve better compaction of signal energy due to its increased directionality. In other words, at a given scale an object edge in an image may produce significant energy in 1 of the 3 standard wavelet subbands, but only 1 of the 6 complex wavelet subbands.
