Denoising By Wavelet Transform

DENOISING BY WAVELET TRANSFORM Junhui Qian Rice University Department of Electrical Engineering 6100 South Main St. Houston, TX 77251-1892 ABSTRACT Wavelet hard thresholding denoising and soft thresholding denoising(wavelet shrinkage denoising) provide a new way to reduce noise in signal. The author demonstrates the different merits of each method and proposes a new thresholding method, Qian thresholding, that achieves a compromise between the two, which is also demonstrated that the hard and soft thresholding methods are its extreme cases.

2. HARD THRESHOLDING AND SOFT THRESHOLDING

Let 
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 denote the forward and inverse wavelet transform operators. Let 
 ! denote the thresholding operator with threshold  . The practice of thresholding denoising consists of the following three steps:

1. INTRODUCTION Denoising, or noise reduction, is a permanent topic for engineers and applied scientists. The problem of denoising is quite varied due to variety of signals and noises. This article considers deterministic signals in zero-mean white noise, as y(n) in (1) 
 
 
 (1) where s(t) is the signal to be estimated, and n(t) is a zeromean white noise with variance  . To be exact, the problem is to estimate x(t), or denoise y(t). It is always tempting to reduce noise after some kind of signal transformation. An appropriate transform can project signal to a domain where the signal energy is concentrated in a small number of coefficients. If noise, on the other hand, is evenly distributed across this domain, this domain will be a very nice place to do denoising, for the SignalNoise Ratio (SNR) is greatly increased in some important coefficients, or, the signal is highlighted in this domain while the noise is not. In this sense, for signals composed with a number of sinusoids, it is wise to denoise in frequency domain. Similarly, for piece-wise constant signals or piecewise polynomial signals, it is advantageous to reduce noise in Wavelet transform domain, or time-scale domain, where these signals have a very sparse representation. Since a wide range of signals can be classified into piece-wise polynomial, Wavelet transform has become an essential tool for many applications, especially image processing.

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Hard thresholding and soft thresholding are only different in step 3. In the case of hard thresholding,

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In the case of soft thresholding, or Wavelet shrinkage,

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Figure 1 graphically shows the difference between the practice of soft and hard thresholding.

Fig. 2. Qian Thresholding

Fig. 1. Soft and Hard Thresholding

We can expect that the technique of soft thresholding would introduce more error or bias than hard thresholding does. But on the other hand, soft thresholding is more efficient in denoising. We will see examples later. 3. QIAN THRESHOLDING


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5. REFERENCES [1] D.L. Donoho, I.M. Johnstone, “Ideal Spatial Adaptation by Wavelet Shrinkage,” Biometrika vol 81, 1994.

Motivated by finding a more general case that incorperates the soft and hard thresholding, I propose the following thresh& olding rule,
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clearly seen in Figure 4, which shows the error image associated the three methods. An example on the “cameraman” image is given in Figure 5.



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Where  is chosen as an estimate of noise level. When Q = 1, it is equivalent to soft thresholding; when Q = A , it is equivalent to hard thresholding. We can also see that when Q = 2 and  approaches zero, the method becomes a Wiener filter. Figure 2 graphically shows its relation with soft and hard thresholding. It can be clearly seen that Qian thresholding is something between hard and soft thresholding. With careful tuning of parameter Q for a particular signal, one can achieve best denoising effect within thresholding framework. 4. RESULTS Figure 3 shows the effects of soft, hard, and Qian thresholding upon the simulation image. Note that soft thresholding is best in reducing noise but worst in preserving edges, and hard thresholding is best in preserving edges but worst in denoising. Qian thresholding, with Q = 2, achieves a compromise between the two extremes. It significantly reduces noise and well preserves edges at the same time. This can

[2] D.L. Donoho, “De-noising by soft-thresholding”, IEEE Trans. on Inf. Theory, 41, 3, 1995. [3] C. Taswell, “The Why, How, and Why of Wavelet Shrinkage Denoising”, Computing in Science and Engineering, June 2000

Noisy Signal

Hard Thresholding

Soft Thresholding

Qian Thresholding with Q = 2

Fig. 3. Simulation results

Original Signal

Error of Hard Thresholding

Error of Soft Thresholding

Error of Qian Thresholding

Fig. 4. Error of three thresholding methods

Noisy Signal

Hard Thresholding

Soft Thresholding

Qian Thresholding with Q = 2

Fig. 5. Result on real image