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UW-Madison ECE 533 - Image Denoising In The Wavelet Domain Using Wiener Filtering

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Image Denoising In The Wavelet Domain Using Wiener FilteringNevine Jacob and Aline MartinDecember 17, 2004Abstract: Wavelet transforms have b ecome a very powerful tool in the area of image denoising. One ofthe most popular method consists of thresholding the wavelet coefficients (using the Hard threshold or theSoft threshold) as introduced by Donoho [4,5]. In this paper, we perform Wiener filtering on the waveletcoefficients to denoise an image degraded by an Additive White Gaussian Noise (AWGN). This method isthen compared to other well known denoising methods.1 IntroductionImage Denoising has become a very essential exercise in Image Restoration. Today several techniques existsuch as Wiener filtering which accomplish image denoising. They have been successfully used in areas suchas medical imaging and astronomy. The wavelet transform has recently entered the arena of image denoisingand it has firmly established its stand as a powerful denoising tool. In [4,5] Donoho presents a method forimage denoising by thresholding the wavelet coefficients. He shows that this method is nearly minimax. Another denoising method in the wavelet domain consists of Wiener filtering the wavelet coefficients. In thispaper, we investigate the performance of this method on a degraded image X such that X = S + N whereS is the original image and N is an AWGN. The performance of this method is then compared to otherwell-known techniques both visually and in the mean square sense. The rest of the pap er is organized asfollows. In the second section we review the wavelet transform. In the third section we present the methodof Wiener filtering in the Fourier and Wavelet domain. The fourth section introduces the Soft and Hardthresholding techniques. The simulation results are discussed in part five. We conclude in part six.2 Wavelet Transform2.1 Introduction to the wavelet transformIn several applications, it might be essential to analyze a given signal. The structure and features of thegiven signal may be better understood by transforming the data into another domain. There are severaltransforms available like the Fourier transform, Hilbert transform, wavelet transform, etc. The Fouriertransform is probably the most popular transform. However the Fourier transform gives only the frequency-amplitude representation of the raw signal. The time information is lost. So we cannot use the Fouriertransform in applications which require both time as well as frequency information at the same time. TheShort Time Fourier Transform (STFT) was developed to overcome this drawback. However the STFT gives afixed resolution at all times and this shortcoming was overcome by the development of the wavelet transform.The frequency component of a signal at a particular time instant cannot be exactly determined. This followsdirectly from the Heisenberg’s Uncertainty Principle which states that the momentum and position of amoving particle cannot be exactly determined. Thus the best we can do is to investigate which frequencycomponents exist in any given interval of time. The high frequency components are better resolved in timeand low frequency components are better resolved in frequency. This is the reason why the wavelet transformhas overtaken the STFT.1110xFigure 1: 1D Haar scaling function.2.2 Haar wavelet transformIn our method we use the Haar wavelet to perform the Wavelet transform. So before we proceed, we give ashort introduction to the Haar wavelet. The Haar Wavelet is the first known wavelet and was proposed byAlfred Haar in 1909. It is the simplest of all wavelets and its operation is easy to understand. Haar waveletshave their limitations too. They are piecewise constant and hence produce irregular, blocky approximations.There are several other wavelets available like the Daubechies wavelet, Donoho’s wavelet, Meyer wavelet,etc. However these wavelets are not easy to comprehend and are also computationally intensive. Most ofthe equations used in this section are from the textbook “Digital Image Processing” by R. C. Gonzalez andR. E. Woods.2.2.1 1-D Haar Wavelet TransformThe 1-D Haar wavelet (Figure 1) is expressed as shown below:For φ : < → < :φ(t) =½1 t ∈ [0, 1)0 t /∈ [0, 1)Scaling FunctionThe Haar scaling function is given by:φj,k(t) = 2j/2φ(2jt − k)Here j refers to the dilation and k refers to the translation.Also do note that φ0,0(t) corresponds to φ(t). φ0,0(t) belongs to the V0subspace which represents the lowestresolution and φj,k(t) belongs to the Vjsubspace. The higher resolution subspaces contain the lower resolu-tion subspaces. HenceV0⊂ V1⊂ V2⊂ ....... ⊂ L2(R)where L2(R) is the set of all measurable, square-integrable functions.The scaling function can be expressed as:φj,k(t) =Xnhφ(n)2(j+1)/2φ(2j+1t − n)2110x0.5Figure 2: 1D Haar wavelet function.Since φ(t) = φ0,0(t), both j and k can be set to 0 to obtain the simpler expressionφ(t) =Xnhφ(n)√2φ(2t − n)where hφ(n) represents the scaling coefficients.This expression conveys that a low resolution scaling function can b e expressed as the sum of scaled andshifted versions of the double-resolution copies of themselves.Wavelet FunctionThe Haar wavelet function (Figure 2) is expressed as shown below:For ψ : < → < :ψ(t) =1 t ∈ [0,12)−1 t ∈ [12, 1)0 t /∈ [0, 1)The wavelet function in general can be expressed as:ψj,k(t) = 2j/2ψ(2jt − k)where ψj,kbelongs to the Wjsubspace and ψ belongs to the W0subspace.The scaling and wavelet function subspaces are related byVj+1= VjMWjThat is, Wjis the orthogonal complement of Vjin Vj+1. The wavelet function can also be expressed as thesum of scaled and shifted versions of the double resolution scaling functions. That is,ψ(t) =Xnhψ(n)√2φ(2t − n)where hψ(n) represents the wavelet coefficients.3Wavelet Series ExpansionsThe wavelet series expansion of a function f (t) ∈ L2(R) relative to wavelet ψ(t) and scaling function φ(t) isexpressed asf(t) =Xkcj0(k)φj0,k(t) +∞Xj=j0Xkdj(k)ψj,k(t)where cj0(k)’s and dj(k)’s are called the approximation or scaling coefficients and detail or wavelet coeffi-cients respectively. They can be calculated as follows:cj0(k) = hf(t), φj0,k(t)i =Zf(t)φj0,k(t)dtanddj(k) = hf(t), ψj,k(t)i =Zf(t)ψj,k(t)dtThe Discrete Wavelet TransformThe 1D-DWT transform pairs are expressed as shown below:Wφ(j0, k) =1√MXxf(t)φj0,k(t)Wψ(j, k) =1√MXxf(t)ψj,k(t)for j ≥ j0andf(t) =1√MXkWφ(j0, k)φj0,k(t)


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UW-Madison ECE 533 - Image Denoising In The Wavelet Domain Using Wiener Filtering

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