First International Conference on Emerging Trends in Engineering and Technology
Total Variation based Wavelet Domain Filter for Image Denoising Nilamani Bhoi Ph. D Scholar, Dept. of Electronics & Communication Engineering National Institute of Technology, Rourkela
[email protected]
Dr. Sukadev Meher Asst. Professor, Dept. of Electronics & Communication Engineering National Institute of Technology, Rourkela
[email protected] We prevent the blurring across edges but fail to smooth the homogenous regions. An iterative method known as Total Variation [5] is proposed by Rudin et al. to smoothen the homogenous region while retaining edges. It is a constrained optimization type of algorithm where the total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using gradient projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As time increases the solution converges to a steady state which is the denoised image. For a small variance Gaussian noise, large time i.e. more iteration leads to blurring effect. The method suppresses noise very well but introduces blurring with more iteration. Wavelet based methods are always a good choice for image denoising and has been discussed widely in literatures for past two decades [6]-[12]. The most straightforward way of distinguishing information from noise in the wavelet domain consists of thresholding the wavelet coefficients. Of the various thresholding strategies, soft-thresholding proposed by Donoho and Johnstone [13] is most popular. They have introduced a universal threshold T, given by:
Abstract In this paper the method Total Variation (TV) is applied on noisy image decomposed in wavelet domain for removal of Additive White Gaussian Noise (AWGN). LL subband of a single decomposed noisy image is used to find the horizontal, vertical and diagonal edges. Using the pixel position of horizontal edges, the corresponding wavelet coefficients in HL subband is retained threshoding others to zero. Adopting the same procedure the vertical and diagonal details of LH and HH subband is retained. The method TV is applied to LL subband for one iteration only. Applying inverse wavelet transform on modified wavelet coefficients we get back the image with little noise. This little noise can be removed using TV filter with single iteration. The method performs well in terms of Peak Signal to Noise Ratio (PSNR) over many well known spatial and wavelet domain methods. The method also retains the edges and other detailed information very well.
Keywords: Gaussian noise, Total variation, Discrete Wavelet Transform, Peak signal to noise ratio. 1. Introduction
T = 2 σ 2 logN
Image denoising is an import pre-processing task for further processing of image like segmentation, feature extraction, texture analysis etc. Denoising refers to suppressing the noise while retaining the edges and other important detailed structures as much as possible. The image gets corrupted by Additive White Gaussian Noise during the process of acquisition, transmission, storage and retrieval. Traditionally this noise can be removed using linear spatial domain filter such as wiener filter [1]. The disadvantage of using linear filter is that it introduces blurring effect to the filter image. Many efforts have been devoted to reducing this undesired effect [2]-[4].
where, σ2 is the noise variance and N is the number of samples in the signal. The use of the universal threshold to denoise images in wavelet domain is known as VisuShrink [14]. In addition, subband adaptive system have superior performance, such as SureShrink [15], which is a data driven system. Recently, another data driven subband adaptive technique is proposed by Chang et al., namely BayesShrink [16] which outperforms SureShrink and VisuShrink. They have also stated another two shrinkage methods known as OracleShrink and OracleThresh. In wavelet domain methods the noisy image is decomposed to around five level of decomposition for
978-0-7695-3267-7/08 $25.00 © 2008 IEEE DOI 10.1109/ICETET.2008.6
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(1)
efficient denoising. This leads to increase in computational complexity, extra hardware and cost. In the proposed method, the noisy image is decomposed to one level which results in LL, HL, HL and HH subband. LL subband contains less noise which can be easily eliminated using Total variation with single iteration. The horizontal, vertical and diagonal edges are found out from the LL subband. Using the pixel positions at these edges the corresponding horizontal, vertical and detail wavelet coefficients of HL, LH and HH subbands are retained, thresholding others to zero. From these modified coefficients the image that reconstructed using inverse transform is contaminated with less noise at the edges which can be suppressed using TV method for single iteration. The rest of the paper is organized as follows. Section 2 provides a brief introduction of Total Variation. In Section 3 the proposed method is discussed. Experimental results and discussion are given in Section 4. Finally the paper is concluded in Section 5.
3. Proposed Method In the proposed method the method Total Variation (TV) is applied in wavelet domain for suppressing Gaussian noise. In TV based denoising the noisy image undergoes several iterations for suppressing Gaussian noise. More number of iteration leads to blurring effect. Hence, to estimate the number of iteration for suppressing noise is a major disadvantage in TV based denoising. Less iteration with suitable parameter λ is required for filtration of an image corrupted with low variance Gaussian noise. When a noisy image is decomposed in wavelet domain LL subband contains the low frequency coefficients along with less noise. This little noise can be easily suppressed using TV based denoising with single iteration. Fig.1 shows the zoomed version of LL subband of original Pepper, noisy with σ = 15 and filter image filtered with TV. The resulted output looks almost same as the LL subband of original decomposed image. Now we have to estimate the HL, LH and HH subband of original decomposed image from the corresponding noisy decomposition. In order to estimate these coefficients first we have to find the horizontal, vertical and diagonal edges of LL subband. Looking at the pixel positions of these edges the corresponding wavelet coefficients of HL, LH and HH subbands are retained thresholding others to zero. Negative version of the vertical edges of LL subband of noisy Pepper image, the vertical details of original decomposed image and modified vertical details of noisy decomposed image are shown in Fig.2. Inverse discrete wavelet transform (IDWT) is applied on the modified wavelet coefficients to reconstruct an image which contains little noise at the edges. The reconstructed image is shown in Fig. 3(a) where the noise at the edges is prominent. The noise at these edges can be removed by applying TV filter to image obtained in Fig. 3(a) for single iteration. The effect is shown in Fig. 3(b).
2. Total Variation Rudlin, Osher and Fatemi [5] have proposed the method of total variation as: ut =
uy ux ∂ + ∂ ∂x u x 2 + u y 2 ∂y u x 2 + u y 2 − λ (u − u0 ),
(2) The equation (2) can be discretized four nearest neighbours and can be written as: for t > 0, x, y ∈ Ω
uijn +1 = uijn ∆ +x u ijn ∆ t x + ∆ − 2 h x n 2 y n y n (∆ + uij ) + (m (∆ + uij , ∆ − uij ))
(
∆+y uijn +∆−y 2 2 (∆+y uijn ) + (m (∆+x uijn , ∆−x uijn ))
(
)
)
1 2
1 2
4. Experimental Results And Discussion
−∆t λ n (uijn − u0 (ih, jh )) , for i, j =1, …, N (3)
The proposed method is implemented and tested on various test images and its performance is compared with that of many other existing techniques. Extensive simulation is conducted on variety of standard Gray scale images with distinct features, including Lena, Pepper, Boat and Flintstones. Simulation work is done through MATLAB 7.0 platform. The size of the images taken is all 512×512. Daubechies tap-8 filter is used for wavelet decomposition. The proposed method is compared in terms of peak signal to noise ratio (PSNR) with Mean filter, Wiener filter, Anisotropic Diffusion (AD), Total
where, ∆∓x uij = ∓(ui ∓1, j − uij ) and similarly for ∆∓y uij .
m( a, b ) = min mod( a, b ) sgn( a ) + sgn(b ) = min ( a , b ) 2
‘λ’ is the tuning parameter. Larger value of λ gives blurring effect to the image.
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Variation (TV), VisuShrink, SureShrink, OracleShrink and OracleThresh.
Table II (PSNR [dB] RESULTS OF IMAGES AT = 25)
Peak-Signal-to-Noise-Ratio (PSNR)
Image Method Mean filter Wiener filter AD filter TV filter Visu Shrink Sure Shrink Oracle Shrink Oracle Thresh Proposed filter
The PSNR is defined by:
255 PSNR = 20 × log10 ε
(4)
where, ε is the root mean-square error given by:
ε=
1 N
∑∑ ( f ( x, y ) − fˆ ( x, y ) ) x
2
(5)
y
where, N is the total number of pixels and f , fˆ are original and filtered image respectively. Table I and Table II shows the PSNR values of various test images at σ =20 and σ = 25 respectively. For subjective evaluation the output of various methods are compared through Fig. 4 - Fig.6.
Lena
Pepper
Goldhill
Flintstones
28.48
29.02
27.72
24.39
29.00
29.22
27.75
23.87
28.21 29.32
27.81 28.03
27.35 28.00
24.05 24.36
27.86
25.35
25.92
21.52
28.71
27.69
27.38
22.21
24.53
24.74
24.18
20.87
21.84
22.33
22.93
22.04
29.36
29.42
28.10
24.47
Table I (PSNR [dB] RESULTS OF IMAGES AT = 20) Image Lena Method Mean filter Wiener filter
Pepper
Goldhill
29.78
29.55
28.77
24.58
30.19
29.45
28.70
24.55
AD filter
29.30
28.80
28.25
24.54
TV filter
30.41
29.08
29.13
25.15
28.75
25.92
26.92
21.98
29.72
28.65
28.12
22.44
26.47
26.68
25.48
22.11
23.84
24.58
24.57
23.60
30.43
30.34
29.17
25.21
Visu Shrink Sure Shrink Oracle Shrink Oracle Thresh Proposed filter
(a)
Flintstones
(b)
(c)
Fig.1 Zoomed version of single decomposition of Pepper image in wavelet (db8) domain (a) LL subband of Original image (b) LL subband of Noisy image with = 15 (c) LL subband of Filter image of TV for one iteration with = 0.9.
(a)
(b)
(c)
Fig. 2 Negative version of (a) vertical edges of LL subband of noisy Pepper image (b) vertical details of original decomposed image (c) modified vertical details of noisy decomposed image.
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(a)
(b)
Fig.3 Reconstructed image after IDWT (a) before TV filter (b) after TV filter.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Fig.4 Comparing the performance of the various methods for Pepper with AWGN =20 (a) Original Image (b) Noisy image (c) Mean filter (d) Wiener filter (e) AD filter (f) TV filter (g) VisuShrink (h) SureShrink (i) OracleShrink (j) OracleThresh (k) Proposed method (l) Error between Original and Filtered Image by proposed method shown as Image with magnifying the pixel value by 10 times.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(k)
(l)
(i)
(j)
Fig.5 Comparing the performance of the various methods for Flintstones with AWGN =20 (a) Original Image (b) Noisy image (c) Mean filter (d) Wiener filter (e) AD filter (f) TV filter (g) VisuShrink (h) SureShrink (i) OracleShrink (j) OracleThresh (k) Proposed method (l) Error between Original and Filtered Image by proposed method shown as Image with magnifying the pixel value by 10 times.
(a)
(c)
(b)
(d)
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(e)
(i)
(g)
(f)
(h)
(k)
(j)
Fig.6 Comparing the performance of the various methods for zoomed Flintstones with AWGN =20 (a) Original Image (b) Noisy image (c) Mean filter (d) Wiener filter (e) AD filter (f) TV filter (g) VisuShrink (h) SureShrink (i) OracleShrink (j) OracleThresh (k) Proposed method.
5. Conclusion In this paper, a novel method for removal of Gaussian noise is presented. Experiments are conducted to assess the performance of proposed method in comparison with Mean filter, Wiener filter, AD filter, TV filter, VisuShrink, SureShrink, OracleShrink and OracleThresh. From the experimental results, the proposed denoising scheme is observed to perform better than the various standard denoising schemes. This filter outperforms even the wavelet based bench mark filtering schemes.
[7]
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