Image Denoising In The Wavelet Domain Using Wiener Filtering

Nevine Jacob and Aline Martin

December 17, 2004

Abstract: Wavelet transforms have become 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 Introduction

Image 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 paper 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 Transform

2.1 Introduction to the wavelet transform

In 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.

1

1

10

x

Figure 1: 1D Haar scaling function.

2.2 Haar wavelet transform

In 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 Transform

The 1-D Haar wavelet (Figure 1) is expressed as shown below:

For φ : < → < :

φ(t) ={

1 t ∈ [0, 1)0 t /∈ [0, 1)

Scaling Function

The 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 V0 subspace which represents the lowestresolution and φj,k(t) belongs to the Vj subspace. The higher resolution subspaces contain the lower resolu-tion subspaces. Hence

V0 ⊂ 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) =∑

n

hφ(n)2(j+1)/2φ(2j+1t− n)

2

1

10

x

0.5

Figure 2: 1D Haar wavelet function.

Since φ(t) = φ0,0(t), both j and k can be set to 0 to obtain the simpler expression

φ(t) =∑

n

hφ(n)√

2φ(2t− n)

where hφ(n) represents the scaling coefficients.This expression conveys that a low resolution scaling function can be expressed as the sum of scaled andshifted versions of the double-resolution copies of themselves.

Wavelet Function

The 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,k belongs to the Wj subspace and ψ belongs to the W0 subspace.The scaling and wavelet function subspaces are related by

Vj+1 = Vj

⊕Wj

That is, Wj is the orthogonal complement of Vj in 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) =∑

n

hψ(n)√

2φ(2t− n)

where hψ(n) represents the wavelet coefficients.

3

Wavelet Series Expansions

The wavelet series expansion of a function f(t) ∈ L2(R) relative to wavelet ψ(t) and scaling function φ(t) isexpressed as

f(t) =∑

k

cj0(k)φj0,k(t) +∞∑

j=j0

∑

k

dj(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) = 〈f(t), φj0,k(t)〉 =∫

f(t)φj0,k(t)dt

and

dj(k) = 〈f(t), ψj,k(t)〉 =∫

f(t)ψj,k(t)dt

The Discrete Wavelet Transform

The 1D-DWT transform pairs are expressed as shown below:

Wφ(j0, k) =1√M

∑x

f(t)φj0,k(t)

Wψ(j, k) =1√M

∑x

f(t)ψj,k(t)

for j ≥ j0 and

f(t) =1√M

∑

k

Wφ(j0, k)φj0,k(t) +1√M

∞∑

j=j0

∑

k

Wψ(j, k)ψj,k(t)

2.2.2 2D Haar Wavelet Transform

Scaling and Wavelet Functions

The 2D scaling and wavelet functions are expressed as follows:

φj,m,n(x, y) = 2j/2φ(2jx−m, 2jy − n)

ψij,m,n(x, y) = 2j/2ψi(2jx−m, 2jy − n), i = {H, V, D}

where H represents the Horizontal components, V the vertical components and D the Diagonal components.

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original image

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200

250

Figure 3: cameraman image.

Decomposition Level 1

0

10

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90

100

Figure 4: Haar Wavelet Transform - First decomposition level.

Discrete Wavelet Transform

The Discrete Wavelet Transform of a function f(x,y) of size M ×N is:

Wφ(j0,m, n) =1√MN

M−1∑x=0

N−1∑y=0

f(x, y)φj0,m,n(x, y)

W iψ(j, m, n) =

1√MN

M−1∑x=0

N−1∑y=0

f(x, y)ψij,m,n(x, y), i = {H, V, D}

The inverse discrete wavelet transform is given as:

f(x, y) =1√MN

∑m

∑n

Wφ(j0,m, n)φj0,m,n(x, y) +1√MN

∑

i=H,V,D

∞∑

j=j0

∑m

∑n

W iψ(j, m, n)ψi

j,m,n(x, y)

In the special case of a Haar Wavelet, the scaling and wavelet functions are as shown below:

3 Wiener Filter

A Wiener filter is a minimum mean square error filter. It has capabilities of handling both the degradationfunction as well as the noise, unlike the inverse filter.

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Decomposition Level 2

0

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30

40

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90

100

Figure 5: Haar Wavelet Transform - Second decomposition level.

Decomposition Level 3

0

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20

30

40

50

60

70

80

90

100

Figure 6: Haar Wavelet Transform - Third decomposition level.

3.1 Wiener Filter in the Fourier Domain

The Wiener Filter in the Fourier domain is given by the expression

F (u, v) =

H∗(u, v)

|H(u, v)|2 + Sη(u,v)Sf (u,v)

G(u, v)

whereH(u,v) is the degraded functionH∗(u, v) is the complex conjugate of H(u,v)|H(u, v)|2 = H∗(u, v)H(u, v)Sη(u, v) = |N(u, v)|2 is the power spectrum of the noiseSf (u, v) = |F (u, v)|2 is the power spectral density (PSD) of the undegraded image

The above expression is based on the following assumptions:

1. The noise and the image are uncorrelated

2. The noise or the image has zero mean

3. The gray levels in the estimate are a linear function of the levels in the degraded image.

The Wiener filter handles the situation in which the degradation function is zero, unless both the degradationfunction as well as the noise power spectrum is zero.

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3.1.1 Global Wiener Filter

In Global Wiener filtering, the above expression for a Wiener filter is applied over the whole image. Thismethod does a good job at deblurring; however, it behaves very poorly in the presence of large noise. TheWiener filter would work well for an image which has similar local statistics throughout the entire image.However in most natural images, the first order statistics vary from one part of an image to another andhence its poor performance in the presence of large noise.The advantages of this method are that it is not computationally intensive and that it works well for smoothimages (whose local statistics do not vary much from one part of image to another)

In our project, we study only the denoising characteristics of the Wiener filter and hence we assume theabsence of blurring. So in our computational analysis, we take the value of H(u,v) to be 1 in the expressionfor a Wiener filter.

3.1.2 Local Wiener Filter

When Wiener filtering is performed on small blocks of an image at a time, the method is called Local WienerFiltering. In this method, the PSD of the undegraded image is estimated for each block. This calculatedPSD is then used in the expression of the Wiener filter. Thus, the local statistics are also accounted for inthe calculation of the Wiener filtered image. Images with many edges are handled much better by the localWiener filter than the global Wiener filter. We used a window of size 3 × 3 in the calculation of the localWiener filtered image. The drawback of this method is that it is more computationally demanding.

3.2 Wiener Filter in the wavelet domain

In this method, the Wiener filtering is performed on the wavelet coefficients. This is with reference to [1,2,3]by Marian Kazubek. In this model we assume that the wavelet coefficients are conditionally independentGaussian random variables. The noise is also modeled as stationary independent zero-mean Gaussian vari-ables.

Let us consider an image corrupted by a zero-mean Gaussian noise. The coefficients of the noisy imagein the wavelet domain are given by

yi,j = si,j + ni,j

where yi,j represent the coefficients of the noisy image in the wavelet domain, si,j represent the coefficientsof the undegraded image, ni,j represent the coefficients of the noise.

The ni,js are normal with zero mean and variance σ2n, if the noise sequence is modelled by independent

normal random variables with zero mean and variance σ2. This follows directly from the orthogonality ofthe Haar wavelet transform. Due to the decorrelation property of the orthogonal wavelet transform, weassume that the signal components si,js are uncorrelated.Based on these assumptions, the Wiener filter expression is given by:

ai,j =E{s2

i,j}E{y2

i,j}si,j = aiyi,j

where si,j is the best linear estimate of the signal component si,j

On the assumption that the signal and noise are independent, we can write

E{s2i,j} = E{y2

i,j} − σ2n

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Without loss of generality, we can assume that the E{y2i,j}’s can be determined by averaging the squared

values of yi,j in a window centered at (i,j). This information can be expressed as

Qi,j =R∑

k=−R

R∑

l=−R

y2i−k,j−l, M = (2R + 1)2

qi,j =Qi,j

M

Thus E{y2i,j} ≈ qi,j and hence

E{s2i,j} = qi,j − σ2

n

As a result, the coefficients of the Wiener filter can be expressed as

ai,j =qi,j − σ2

n

qi,j

Restricting the values of E{s2i,j} to only positive values, the numerator of the above equation takes the form

max(qi,j − σ2n, 0) and so

ˆsi,j = max(ai,j , 0)yi,j

We perform this wiener filtering under two conditions - one in which noise variance is assumed to be knownand the other in which the noise variance is estimated using the Mean Absolute Deviation (MAD) methodas proposed by David L. Donoho in [5]. The MAD method for estimating the noise is given by

variance =(

MAD

0.6745

)2

whereMAD((wi)i) = Median((|wi|)i)

and wi represents the wavelet coefficients.

The method is implemented using the following steps:1) {qi,j} is computed by convolving {y2

i,j} with a kernel of size 9.2) The wiener filter is then applied using the formula

s =max(qi,j − σ2

n, 0)qi,j

yi,j = ˆai,jyi,j

4 Thresholding

4.1 Soft Thresholding

In [4,5] Donoho proposes a denoising method for images affected by an AWGN. This method consists ofthresholding the wavelet coefficients. He shows that Soft Thresholding the wavelet coefficients is nearlyminimax in the sense that the ratio of the worst case risk to minimax risk is not large. When using the Haarwavelet transform the steps for implementing denoising using the soft thresholding technique is as follows:

8

-T

T

x

y

Figure 7: Soft Thresholding.

(i) Apply the Haar wavelet transform to the original image(ii) Apply the Soft thresholding (Figure 7) on the wavelet coefficients y(i,j):

s(i, j) =

y − T if y(i,j) ≥ Ty + T if y(i,j) ≤ -T0 otherwise

Where σ is the standard deviation of the noise, n the number of wavelet coefficients, s(i, j) are the de-noised wavelet coefficients, and T is the Donoho threshold (also called Universal threshold) given by:T = σ

√2 log(n). In practical applications, the variance of the noise is estimated by the MAD method

as mentioned in section 3.2.

(iii) Apply the inverse Haar wavelet transform

4.2 Hard Thresholding

In [4] Donoho states that the denoising method described in the previous subsection (Soft Thresolding) canbe carried out using the Hard Threshold instead of the Soft Threshold on the wavelet coefficients in (ii).The hard thresholding formula is given to be:

s(i, j) ={

y(i, j) if |y(i, j)| ≥ T0 if y(i, j) < T

Where T is the Donoho threshold mentioned in the previous subsection. See Figure 8 for a representationof the Hard threshold.

5 Simulation Results

In this section we present some simulation results performed on the Cameraman image and the Lena image.We added white Gaussian noise to these images and denoised them with the methods described previouslyin the paper: Wiener Filtering in the Wavelet Domain (by MAD method), Wiener Filtering in the Waveletdomain with known noise variance, Hard and Soft Thresholding, Global Wiener Filtering and Local WienerFiltering. Figures (1 to ?) show the results of the different denoising methods presented in this paper for anoise variance of 400.

9

-T

T

x

y

Figure 8: Hard Thresholding.

5.1 Wiener filter in the Wavelet Domain

The Wiener filtering method in the wavelet domain that used the MAD method to estimate the noise vari-ance gave slightly better results both visually and in terms of MSE than the method for which the varianceof the noise is known. This can be clearly seen from the background of the cameraman image (Figures 11)which appears smoother than that seen on the cameraman image with known noise (Figures 11). The MSEwas found to be 107.5 as compared to the 114.2 for the latter method. As the noise variance is not the samein all the subbands, it is expected that the method that estimates the noise in the subbands would performbetter. Note that this method is better from a practical point of view as in practice one does not know thenoise variance.

The Wiener filter in the wavelet domain removes the noise pretty well in the smooth regions but performspoorly along the edges. That is why it performs better on smooth images like Lena (see Figures 19 and 20)than on images with edges like the cameraman. For a noise variance of 400, the MSE was found to be 107.5for the cameraman image and 80.5 for the Lena image. This has been the object of further research; In [2]the authors propose a denoising via block Wiener filtering in the wavelet domain to further remove the noisealong the edges.

On running the simulations, we noticed the impact of the value of R on the results. For large values of R,squared patterns appeared on the denoised image and it spread the noise along the edges. It was found thatfor a value of R equal to 1, the best results were obtained.

5.2 Wiener Filter in the wavelet domain vs Thresholding

The Wiener Filtering in the wavelet domain was compared to the Thresholding methods and we saw that theWiener Filter (Figure 11 and 19) outperforms both thresholding (see Figures 13 and 14 for the cameramanimage and Figures 21 and 22 for Lena) methods visually and in terms of MSE. More details were lost withthe thresholding methods especially for the hard Threshold wherein the background was not well denoised.If the Wiener could be thought as an other thresholding function, you get the intuition that it will performbetter as its shape is smoother than the Hard and Soft thresholds.

5.3 Wiener Filter in the wavelet domain vs Wiener Filter in the Fourier Domain

From Figures 15, 16, 23 and 24 it is clear that the method in the wavelet domain visually outperforms theglobal Wiener Filter and Local Wiener Filter. However, the Wiener filter has a smaller MSE (see figure 25and 26) than the Wiener Filter in the wavelet domain and this is true with all values of noise variances we

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150

200

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Figure 9: Cameraman image.

experimented with. This may be because the MSE is not always the best quantitative criterion.

5.4 MSE of the different denoising method for different noise variances

Figures 25 and 26 shows the plot of the MSE of the different methods we talked about in this paper versusthe noise variance (10 100 200 and 400 ). We see that in most cases the MSE for the Lena image wassmaller than that of the Cameraman image. Therefore we conclude that it seems easier to denoise a smoothimage than an image that has edges. Also, as mentioned earlier, the Wiener filter in the wavelet domainoutperforms the Thresholding method visually as well as in terms of MSE but does not outperform themethods in the Fourier Domain in terms of MSE. However the wavelet domain method had a better visualquality. This questions the MSE as the best quantitative criterion.

6 Conclusion

We thus conclude from the results obtained that the performance of a Wiener filter in the wavelet domainsurpass those of the other methods which were discussed. There are a couple of areas which we would like toimprove on. One area is in improving the denoising along the edges as the method we used did not performso well along the edges. Another area of improvement would be to develop a better optimality criterion asthe MSE is not always the best optimality criterion.

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Noisy Image − Noise variance = 400

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200

250−50

0

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Figure 10: Cameraman image with Additive White Gaussian Noise of variance 400.

Wavelet domain − Wiener Filtered image with estimated noise − variance = 400

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Figure 11: Denoised image using the Wiener Filter in the Wavelet Domain. The noise variance was estimatedusing the MAD method.

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Wavelet domain − Wiener Filtered image with known noise − variance = 400

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Figure 12: Denoised Image using the Wiener Filter in the Wavelet Domain with known noise variance.

Wavelet Domain − Soft thresholded image − variance = 400

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Figure 13: Denoised image obtained after thresholding the noisy wavelet coefficients with the Soft Threshold.The variance of the noise was estimated using the MAD method.

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Wavelet Domain − Hard thresholded image − variance = 400

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250 −50

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Figure 14: Denoised image obtained after thresholding the noisy wavelet coefficients with the Hard Threshold.The variance of the noise was estimated using the MAD method.

Fourier Domain − Global Wiener Filtered image − variance = 400

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Figure 15: Denoised image obtained after applying the Wiener Filter in the Fourier Domain.

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Fourier Domain − Local (4 by 4) Wiener Filtered image − variance = 400

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Figure 16: Denoised image obtained after applying a local Wiener Filter of size 4 by 4 in the Fourier Domain.

Original Image

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Figure 17: Lena image.

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Noisy Image − Noise variance = 400

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Figure 18: Lena image with Additive White Gaussian Noise of variance 400.

Wavelet domain − Wiener Filtered image with estimated noise − variance = 400

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Figure 19: Denoised Image using the Wiener Filter in the Wavelet Domain. The noise variance was estimatedusing the MAD method.

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Wavelet domain − Wiener Filtered image with known noise − variance = 400

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Figure 20: Denoised Image using the Wiener Filter in the Wavelet Domain with known noise variance.

Wavelet Domain − Soft thresholded image − variance = 400

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Figure 21: Denoised image obtained after thresholding the noisy wavelet coefficients with the Soft Threshold.The variance of the noise was estimated using the MAD method.

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Wavelet Domain − Hard thresholded image − variance = 400

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Figure 22: Denoised image obtained after thresholding the noisy wavelet coefficients with the Hard Threshold.The variance of the noise was estimated using the MAD method.

Fourier Domain − Global Wiener Filtered image − variance = 400

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Figure 23: Denoised image obtained after applying the Wiener Filter in the Fourier Domain.

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Fourier Domain − Local (4 by 4) Wiener Filtered image − variance = 400

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Figure 24: Denoised image obtained after applying a local Wiener Filter of size 4 by 4 in the Fourier Domain.

0 50 100 150 200 250 300 350 4000

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300MSE for different denoising methods

noise variance

MS

E

WF Wavelet D. − MADWF Wavelet D.HTSTWF Fourier D.Local WF Fourier D.

Figure 25: Cameramam image: MSE for different methods versus noise variance.

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300MSE for different denoising methods

noise variance

MS

E

WF Wavelet D. − MADWF Wavelet D.HTSTWF Fourier D.Local WF Fourier D.

Figure 26: Lena image: MSE for different methods versus noise variance.

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7 References

[1]Image Denoising via Wavelet-Domain Spatially Adaptive FIR Wiener Filtering Zhang, H.; Nosratinia,A.; Wells, R.O., Jr.; Acoustics, Speech, and Signal Processing, 2000. ICASSP ’00. Proceedings. 2000 IEEEInternational Conference on , Volume: 6 , 5-9 June 2000 Pages:2179 - 2182 vol.4

[2]Denoising via block Wiener filtering in wavelet domain, Strela, V.; Proc. Third European Congress ofMathematics, Progress in Mathematics series, Birkhauser Verlag, Barcelona (2000).

[3]Wavelet domain image denoising by thresholding and Wiener filtering Kazubek, M.; Signal ProcessingLetters, IEEE , Volume: 10 , Issue: 11 , Nov. 2003

[4]De-noising by soft-thresholding Donoho, D.L.; Information Theory, IEEE Transactions on , Volume: 41 ,Issue: 3 , May 1995 Pages: 613 - 627

[5] Wavelet Shrinkage and W.V.D.: A 10-minute Tour Donoho, D.L; (David L. Donoho’s website)

[6] Image Denoising using Wavelet Thresholding and Model Selection Zhong, S. ; Cherkassky, V.; ImageProcessing 2000, Proceedings. 2000 International Conference on, Volume: 3, 10-13 Sept. 2000 Pages: 262 -265 vol. 3.

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