Anisotropic Wavelet-Based Image Denoising Using Multiscale Products

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Procedia

Engineering

Procedia Engineering 00 (2011) 000–000 www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

Anisotropic Wavelet-Based Image Denoising

Using Multiscale Products

Ming He

a

_{, Hao Wang}

a

_{, Hong-fu Xie}

a

_{,Ke-jun Yang}

a

_{, Qing-Wei Gao}

b*

a_{Anhui Jiyuan Electric Power System Tech Co.Ltd.}_，_{Hefei 230088, China} b_{School of Electrical Engineering and Automation of Anhui University, Hefei 230039, China}

Abstract

Images are often corrupted by noise in the process of acquisition and transmission. Thus, image denoising is a key issue in all image processing researches. The great challenge of image denoising is how to preserve the details of an image when reduces the noise. In this paper, a new denoising algorithm based on anisotropic wavelet transform (AWT) using multiscale products is proposed. Before denoising, the noisy images are first decomposed by the anisotropic wavelet. The multiscale product threshold is then applied to the multiscale products of the AWT coefficients instead of directly to the AWT coefficients. Since the multiplicating operation amplifies the significant features and dilute noise, the method reduces speckle effectively while preserving edge structures. Experimental results show that the proposed scheme can outperform standard wavelet-based denoising with soft and hard threshold.

1. Introduction

Image denoising is a fundamental problem in the field of image processing. Denoising refers to the filtering operation which removes noise from an noisy image while preserving the details and significant features. The common types of noise can be roughly categorized into two groups: the additive and multiplicative noise, and the typical representatives of the two groups are Gaussian noise and speckle

* Corresponding author. Tel.: +8613856922457.

E-mail address:[email protected].

Open access under CC BY-NC-ND license.

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noise respectively. For the multiplicative noise, it can be converted into white additive noise after applying the logarithmic transform. This process is related to homomorphic filtering. The early methods used linear or smoothing [1] operation to reduce the noise. These linear filters were used mainly due to the simplicity of theory and easy to application, but it must be noted that they often blurred or smoothed out the edges which is undesired phenomenon.

In the past two decades, with the advent of the multiresolution analysis, a significant breakthrough was made in the filed of denoising. Since their introduction, wavelets have been universally regarded as extremely powerful tool for analysis of nonstationary signals and images. All the wavelet-based methods mainly includes thresholding schemes which are based on Donoho’s pioneering work [2] and MAP estimators [3] which take into consideration the statistics of the signal and noise components. Although the discrete wavelet transform (DWT) can provides a better reduction of the noise as compared with that of the simple filters, a disadvantage relating to wavelet-based methods must be noted, that is, the DWT can’t capture edges and contours properly due to its isotropic property. To overcome the difficulties, some anisotropic transform were proposed to represent the signals, such as ridgelet transform [4], contourlet transform [5], shearlet transform [6], curvelet transform [7]. These transforms obtained better denoising effect compared with the general DWT schemes. However, we are still suffer from the complication of constructing the directional filters.

In this paper, a new image denoising algorithm based on anisotropic wavelet using multiscale products is proposed. The anisotropy is achieved by an unequal iteration of 1-D transform steps along two directions, that is, the transform is applied more along one than along the other direction. This operation avoid the difficulties of directional filter design. To capture the edges and contours of the images more effectively, we employ the undecimated AWT to decompose the noisy images. Two adjacent scale subbands are multiplied to amplify the significant features and dilute noise. Unlike other wavelet-based thresholding methods, this scheme apply threshold to the multiscale products instead of the wavelet coefficients directly. A new noise level estimator used to distinguish the edge structures from noise is also proposed in this paper.

2. Anisotropic Wavelet Transform and Multiscale Products

2.1. Anisotropic Wavelet Transform

The anisotropic wavelet transform have been explained in detail in [8]. Here, we briefly review the main theory.

In the anisotropic wavelet transform, the number of 1-D transform applied along the horizontal and vertical directions is unequal, that is, there are horizontal and vertical transform at each scale, where is not equal to . Then, the identical operation is iterated in the approximation subband until the expected scale, like the standard wavelet transform. The anisotropic transform is denoted as AWT . Obviously, the number of the subbands at each scale is .

1 n n₂ 1 n n₂ (n1,n2) (1 2) 2n+n 2.2. Multiscale Products

In [9], Mallat et al. pointed out that the signal and noise have totally different behaviors in the wavelet domain. According to Meyer [10], the wavelet transform satisfies

( )

_x _K

( )

jα f

W₂j ≤ 2 (1)

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( ) ∏ ( ) = + = 2 1 k k i j i jf x W f x P

The above theorem implies that if the uniform Lipshchitz regularity is positive, the amplitude of the wavelet transform should increase with increasing scales. On the contrary, wavelet transform magnitude should decrease for negative Lipshchitz regularity with increasing scales. Thus, multiplication of the DWT coefficients between the adjacent scales can lead to enhance edge structures while weakening noise. In this paper, the multiscale products are defined as

(2) where and 1 k2 are nonnegative integers.

To simplify the computation, the AWT multiscale products for the images are denoted as k ( )x y W f( )x y W f( )x y d LHLLHH HHLLLH HLLHLH HHH f

P_jd , = _jd , ⋅ _jd₊₁ , , = , , , , , , (3)

3. Denoising Algorithm and Threshold Estimation

In this section, we first propose the AWT-based thresholding denoising algorithm using multiscale products, and then discuss the estimation of threshold used in the denoising scheme. The complete algorithm is summarized as follows.

y Apply the AWT(n1,n2) to the noisy SAR image f( )x,y up to J scales.

y Multiply the adjacent scales coefficients to obtain the multiscale products ( )x y W f( )x y W f( )x y

f

Pdj , = dj , ⋅ _jd₊₁ , d=LHL,LHH,HHL,LLH,HLL,HLH,HHH, . j= ,12LJ

d

y Compute the hard thresholds tdjp, and apply them to Pjdf( )x,y to identify the W∧j f( )x,y by

(4) ( ) ( ) _{( )} , , , , 0 , , p t y x f P p t y x f P y x f W y x f W _d j d j j d j d j d j < ≥ ⎪⎩ ⎪ ⎨ ⎧ = ∧ ( ) d

y Reconstruct the denoised image from the thresholded coefficients W∧dj f( )x,y via the inverse AWT.

It can be seen that the performance of despeckling algorithm depend on the threshold value critically. So the main task of the rest is how to estimate properly.

p tdj p

tdj

In this paper, we adopt the adaptive threshold to shrink the AWT coefficients. It can be defined as

( )

( ) ( )

( )

d j d j d j j d j d j d j d j E P E W W k 5 2 2 ₊₁ 2⎥= 1+2 ₊₁_, 2⋅ ₊₁ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅ = ε ε ε ρ σ σ (5) where d is computed as j j+1,

ρ ρd_j₊₁_,_j=_∫∫ψd_j( )x,y ⋅ψ_jd₊₁( )x,ydxdy _∫∫

(

ψd_j( )x,y

)

2dxdy⋅_∫∫

(

ψd_j₊₁( )x,y

)

2dxdy and is

the noise standard deviation along direction at the

d j σ HHH HLH HLL LLH HHL LHH LHL d= , , , , , , j scale. It is often estimated as σ ϕ σ j d j= ∧

, where σ∧dj = ∫∫

(

ϕd_j( )x,y

)

2dxdy⋅

(

Median

(

HHH₁

)

0.6745

)

(6) where signifies the median operator. For a more detailed discussion on threshold estimation, please refer to [11]. Median(•)

4. Simulation Results

In this section, we demonstrate the effectiveness of the proposed method using grayscale image “Lena” with different Gaussian noise levels σ=5,10,20,30. To assess the performance of the 256×256

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proposed method, the obtained results were compared with wavelet-based soft thresholding ( WST ) and hard thresholding ( WHT ) schemes, and two quantitative measures Peak Signal to Noise Ratio ( PSNR )

β values of image Lenaby the three schemes Table 1. PSNR and Method σ=5 σ=10 σ=20 σ=30 PSNR β PSNR β PSNR β PSNR β WST 30.0393 0.5186 28.1658 0.4383 26.1603 0.3078 25.0435 0.2152 WHT 31.8829 0.5587 30.0926 0.4921 27.4413 0.3590 25.7702 0.2371 Proposed 36.8333 0.8430 33.0192 0.6833 29.4681 0.4619 27.5674 0.3426 (a) (b) (c) (d) (e) (f) (g) (h) (i)

Fig. 1. (a) Original image; (b) Noisy image degraded with σ=10; (c) Denoised image by WST; (d) Denoised image by WHT; (e) Denoised image by the proposed method; (f) Noisy image degraded with σ=20 ; (g) Denoised image by WST; (h) Denoised image by WHT; (c) Denoised image by the proposed method.

and edge preservation [11] denoted as β were used to quantify the achieved performance improvement. The PSNR is defined as

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⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⋅ = ∑ ∑ = = ∧ N i N j Sij Sij MN PSNR 1 1 2 , , 2 255 lg 10 (7)

where is the original image, is the denoised image and is the image size. S ∧S N

Considering the computational complexity, 1-D transform steps along horizontal and vertical directions were one and two respectively at each scale in the AWT. Daubechies’ symmlet wavelet with 4 vanishing moments was used for a 3-level decomposition of the images. Table 1. reports the values of PSNR obtained by the three filters. From the table, it can be seen that the proposed method achieves larger PSNR values over all noise levels. Table 1. also shows that the our scheme performs better edge preservation than other filters in terms of β. For a visual comparison, the original image, noisy image degraded by σ=10,20 and denoised images are presented in Fig.1 (a)-(i), respectively.

5. Conclusion

In this paper, a new method based on anisotropic wavelet transform using multiscale products for image denoising has been proposed. The AWT coefficients in the adjacent scales subbands were multiplied to amplify significant features of the images, and then applied the thresholding to multiscale products instead of directly to the directionlet coefficients. The experimental results confirm that the proposed algorithm allows efficient removal of noise, outperforming the previously proposed filters.

Acknowledgements

The authors want to express their sincere thanks to the editors and the anonymous reviewers for their constructive comments and suggestions which helped us improve the paper greatly.

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