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Image denoising has remained a fundamental problem in the field of image processing. With Wavelet transforms, various algorithms for denoising in wavelet domain were introduced. Wavelets gave a superior performance in image denoising due to its properties such as multi-resolution. The problem of estimating an image that is corrupted by Additive White Gaussian Noise has been of interest for practical and theoretical reasons. Non-linear methods especially those based on wavelets have become popular due to its advantages over linear methods. Here I applied non-linear thresholding techniques in wavelet domain such as hard and soft thresholding, wavelet shrinkages such as Visu-shrink (non-adaptive) and SURE, Bayes and Normal Shrink (adaptive), using Discrete Stationary Wavelet Transform (DSWT) for different wavelets, at different levels, to denoise an image and determine the best one out of them. Performance of denoising algorithm is measured using quantitative performance measures such as Signal-to-Noise Ratio (SNR) and Mean Square Error (MSE) for various thresholding techniques.

In many applications, image denoising is used to produce good estimates of the original image from noisy observations. The restored image should contain less noise than the observations while still keeping sharp transitions (i.e. edges) [

Wavelet thresholding [2-5] (first proposed by Donoho) is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. In our project, the wavelet thresholding techniques are applied to an image. It removes noise by killing coefficients that are insignificant relative to some threshold, and turns out to be simple and effective, depends heavily on the choice of a thresholding parameter and the choice of this threshold determines, to a great extent the efficacy of denoising.

The procedure to denoise an image is given as follows:

De-noised image = W^{−1} [T{W (Original Image + Noise)}]

Step 1: Apply forward wavelet transform to a noisy image to get decomposed image.

Step 2: Apply non-linear thresholding to decomposed image to remove noise.

Step 3: Apply inverse wavelet transform to thresholded image to get a denoised image in spatial domain.

The DWT of an image x is calculated by passing it through a series of filters. First the samples are passed through a low pass filter with impulse response g resulting in a convolution of the two:

The image is also decomposed simultaneously using a high-pass filter h. The outputs give the detail coefficients (from the high-pass filter) and approximation coefficients (from the low-pass filter). It is important that the two

filters are related to each other and they are known as a quadrature mirror filter. However, since half the frequencies of the signal have now been removed, half the samples can be discarded according to Nyquist’s rule. The filter outputs are then down sampled by 2: [9,10]

This decomposition has halved the time resolution, since only half of each filter output characterizes the signal. However, each output has half the frequency band of the input, so the frequency resolution has been doubled. This is in keeping with the Heisenberg uncertainty principle.

With the down sampling operator ↓ the above summation can be written more concisely.

The Discrete Wavelet Transform provides sufficient information both for analysis and reconstruction of the original signal, with a reduction in the computation time.

Sub-band coding is a method for calculating the Discrete Wavelet Transform. The whole sub-band process consists of a filter bank, and filters of different cut-off frequencies are used to analyze the signal at different scales.

The procedure starts by passing the signal through a half band high-pass filter and a half band low-pass filter. A half band low-pass filter eliminates exactly half the frequencies from the low end of the frequency scale. For example, if a signal has a maximum of 1000 Hz component, then half band low-pass filter removes all the frequencies above 500 Hz. The filtered signal is then down sampled, meaning some sample of the signal is removed. Then the resultant signal from the down sampled half band low-pass filter is then processed in the same way again. This process will produce sets of wavelet transform coefficients that can be used to reconstruct the signal. An example of this process is illustrated in

Where, cA_{x} is the approximation coefficients at decomposition level x, cD_{x} is the detail coefficients at decomposition level x. S is the original signal. From

The Stationary wavelet transform (SWT) is similar to the DWT except the signal is never sub-sampled and instead the filters are up sampled at each level of decomposition.

Each level’s filters are up-sampled versions of the previous as shown in

The SWT is an inherent redundant scheme, as each set of coefficients contains the same number of samples as the input. So for a decomposition of N levels, there is a redundancy of 2N.