Denoising of an Image Using Discrete Stationary Wavelet Transform and Various Thresholding Techniques ()
1. Introduction
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) [1]. Wavelet transform, due to its excellent localization property, has rapidly become an indispensable signal and image processing tool for a variety of applications, including compression and de-noising. Wavelet denoising attempts to remove the noise present in the signal while preserving the signal characteristics, regardless of its frequency content.
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. Figure 1 shows the block diagram of denoising using Wavelet transformation and thresholding techniques.
Denoising Procedure:
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.
2. Theoretical Aspects of Image Denoising Techniques
2.1. Discrete Wavelet Transform (DWT) [6-8]
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
Figure 1. Block diagram of denoising using wavelet transformation and thresholding techniques.
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
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 Figure 2. The resolution of the signal is changed by filtering operations, and the scale is changed by down sampling operations. Down sampling a signal corresponds to reducing the sampling rate, which is equivalent to removing some of the samples of the signal.
Where, cAx is the approximation coefficients at decomposition level x, cDx is the detail coefficients at decomposition level x. S is the original signal. From Figure 2, you can see the original signal is broken down into different levels of decomposition. In the above case, it is a 3-level decomposition. Every time the newly scaled wavelet is applied to the signal, the information captured by the coefficients remains stored at that level. Thus the remaining information contains the higher frequencies of the signal, if the scaling factor decreases.
2.2. Stationary Wavelet Transform
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 Figure 3.
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.
Figure 4 shows the decomposition of Discrete and Stationary wavelet transform. The Discrete Wavelet Transform (DWT) [11,12] is the simplest way to implement MRA. It necessitates a decimation by a factor 2N, where N stands for the level of decomposition, of the transformed signal at each stage of the decomposition. As a result, DWT is not translation invariant which leads to block artifacts and aliasing during the fusion process between the wavelet coefficients. For this reason, we use the Stationary Wavelet Transform (SWT) (Holschneider, 1988). For the SWT scheme the output signals at each stage are redundant because there is no signal downsampling; insertion of zeros between taps of the filters are used instead of decimation. Figure 5 shows the decomposition of an image using SWT at level 1.