2011 18th IEEE International Conference on Image Processing

EFFECTIVE IMAGE NOISE REMOVAL

Haiying Tian, Hongmin Cai*, J.H. Lai

The Sun Yat-Sen University Department of Automation

Guangzhou, RRXhina

ABSTRACT Preservation of fine feature of an image is essential during the process of noise removal, especially via some types of smoothing such as using diffusion process-based methods to enhance images. In this paper, we present a new edge indi­cator called difference eigenvalue to measure image gradient magnitude in the diffusion process. Based on the eigenvalues of the Hessian matrix, the difference eigenvalue manifest it­self in terms of structural information of an image. We adapt the new edge indicator to a diffusion model to achieve a better balance between noise removal and detail preservation. Ex­periments on both synthetic and real images show that the new model can obtain good results and outperforms existing methods.

Index Terms— Hessian matrix, Edge indicator, Anisotropic diffusion, Noise reduction

1. INTRODUCTION

Digital images are often degraded by noises from various sources, such as imperfect illumination condition, limited ac­curacy in image quantization, compression and transmission. To effectively remove noise, many methods have been devel­oped over years.

Algorithms based on the theory of linear scale-space are shown to achieve a good balance between edge detection and noise removal by tracking the features across multiple scales [ 1, 2]. Among them, a classical anisotropic diffusion scheme was developed by Perona and Malik [3]. The method, however, has the shortcoming of generating a staircase-like effect in the processed image. Later on, mean curvature has been used to remove noises in a three-dimensional (3D) space [4]. A diffu­sion equation based on Gaussian curvature has also been pro­posed for removing noise in 3D [5]. In this type of approach, however, the two principal curvatures cannot be always cal­culated accurately and thus the performances of the mean cur­vature and Gaussian curvature-based algorithms may be less

*This work was supported in part by NSFC, NSF of Guangdong Province and the China Fundamental Research Funds for the Central Universities un­der award number 60902076, 9451027501002551 and lQykjctt, respectively.

trThe work was supported by NSF of USA under grant number 0958345.

BASED ON DIFFERENCE EIGENVALUE

Xiaoyin Xu T

Harvard Medical School Department of Radiology

Brigham and Women's Hospital, Boston, MA, USA,

than optimal. Chen et al. [6] developed a new edge indicator, difference curvature, to alleviate the staircase effect observed in the original Perona and Malik method. Yu et al. proposed to use kernel anisotropic diffusion to achieve smoothing by mapping spatial space into another space [7]. Most of the above methods could achieve a good trade-off between noise removal and edge preservation at relatively high signal-to-noise ratio (SNR), but they are prone to destroying the fine features or creating noise cluster in the image at low SNR.

In this paper, we propose an edge indicator, called differ­ence eigenvalue, to detect image details and apply it to control a diffusion function. Classical diffusion model is adapted by incorporating the difference eigenvalue to achieve balanced noise removal and preservation of fine details. We test the al­gorithm over various images and find it can preserve image details very well while effectively removing noise at the same time.

The remainder of this paper is organized as follows. In section 2, the backgrounds of anisotropic diffusion model are discussed. A new edge indicator is introduced in Section 3. The diffusion scheme based on the new edge indicator is de­rived in Section 4. Experimental results from testing and com­paring the new method with existing techniques are given in Section 5. Section 6 gives discussion and conclusion.

2. BACKGROUND OF DIFFUSION SMOOTHING

Diffusion model was derived from the process of heat dif­fusion as described in the work by Perona and Malik [3]. Mathematically, for an image w, the diffusion process is

ut = div(g(c)Vu) (1)

with a Neuman boundary condition where div is the diver­gence operator, Vw is the gradient of w, and g(-) is a diffusion function. Recently, Chen et al. proposed a new edge indicator called difference curvature to distinguish edges from ramps and then incorporated it into the classical diffusion model to tune the diffusion parameter [6]. An edge indicator is defined as

E = ||nw| - |«ff|| (2)

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2011 18th IEEE International Conference on Image Processing

where TJ and f are the direction of the gradient and the di­rection perpendicular to the gradient of image u, respectively. The second-order derivatives are given by

Utf

X XX ^"UyUyU^y I liyliyy

Ul+Uy '^y'^XX ^"UyUyU^y I U-^Uyy

"l+"y

(3)

(4)

Let Emax denote the maximal value across the entire differ­ence curvature^E and E be the normalized difference curva­ture given by E = E/Emax. Then the diffusion function used in [6] is

\* = t.exp(-f) where X and t are two positive constants. The difference cur­vature could accurately model important edge information in noise-free images. However, if image u is corrupted by noise, the performance of the method is affected, limiting its appli­cabilities in practice.

3. A NEW EDGE INDICATOR-DIFFERENCE EIGENVALUE

In order to accurately detect edges and boundaries in an image, we propose a new edge indicator to capture its fine details. The proposed edge indicator, named difference eigen­value, is calculated from the Hessian matrix of second-order derivatives [8] of an image. It has been shown image details can be better revealed by second-order operators [8, 9, 10].

Given an image u(x,y), its Hessian matrix is defined as

H = Axy

Matrix H is positive semidefinite with two eigenvalues A\ and A2 given by

1 * = -

1 A2 = -

(UXX + Uyy) + ^(UXX ~ Uyyf + 4M^

(UXX + Uyy) ~ ^(UXX ~ Uyy)2 + 4 ^

(5)

(6)

where A\ > A2. Here A\ corresponds to the maximum local variation at a pixel and the A2 corresponds to the minimum local variation. The difference eigenvalue edge indicator P(u) is defined as

P(u)=(Ai-A2)Aiw(u(x,y)) (7)

where w(u(x, y)) is a weighting factor. Behavioral analysis of the new edge indicator is as follows:

• On edges, A\ is large and A2 is small, thus (A\ - A2)A\ is large, giving a large P(u).

• In homogeneous areas, both A\ and A2 are small and thus (A\ -A2)A\ is small too, it follows that P(u) is small.

• For isolated noises, A\ - A2 is small, thus (A\ - A2)A\ is small and P(u) is small.

The role of weighting parameter w(u(x, y)) is to achieve a bal­ance between detail enhancement and noise suppression. Its value is estimated from the gray level variance, defined by

aj(x,y) -minfcrf} w(u(x,y)) = 6- (8)

maxfcrf} - minfcrf}

where minfof} and maxfof} are the minimum and maximum gray-level variances of u, respectively, and 6 is a constant. For a given pixel with coordinates (x,y), the gray-level variance is calculated from its 3 x 3 neighborhood

1 l l

oj(x,y) = — ^ YJ K X + W + J) ~ w(x^)] (9) i=-iy=-i

The behavior analysis aforementioned has shown that edge could be easily distinguished by the operator P(u).

4. DENOISING MODEL BASED ON DIFFERENCE EIGENVALUE

In this section, we incorporate P(u) into the classical dif­fusion model to remove noise. Given an image uo(x,y\ its smoothed versions comprise of a series of images ut with the variable t being time parameter. The continuous anisotropic diffusion is modeled as

ut = div(g(\Wu\)Wu)

Parameter k plays a key role in determine edge preservation as a k that is too large will cause edges of low contrast to be smoothed out while a small k has a diminished effect to remove noise in homogeneous regions. In this experiment, we estimate k by using normalized difference eigenvalue

A: = exp(-P(G*w)) (10)

where G denotes a Gaussian filter and P{G * u) is normal­ized difference eigenvalue operator defined by P(G * u) = P(G * u)/Pmax, and Pmax is the maximum value of\P(G * u). In homogeneous but noisy regions, the value of P(G * u) is very small, resulting in a high diffusion. In an edge region, the value of P(G * u) is large, leading to a low diffusion and promoting edge preservation.

5. EXPERIMENTAL RESULTS

We test and compare the new method with existing al­gorithms using a variety of images. Quantitative measure­ments in terms of peak SNR (PSNR) and Structure SIMilar-ity (SSIM) index [11] are used to evaluate their performance.

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2011 18th IEEE International Conference on Image Processing

Fig. 1(a) is a test image corrupted by additive Gaussian noise with a standard deviation of cr = 15. The gradient magni­tude of the noisy image is shown in Fig. 1(b). The result generated by difference curvature is shown in Fig. 1(c). The magnitude map after difference eigenvalue (P(u)) is shown in Fig. 1(d). We can observe that the gradient operator and difference curvature can preserve the edge information, but there are much remaining noise clusters within homogeneous regions. From the perspective of evaluating image quality, the difference eigenvalue method retains edge information and re­moves much noise clusters.

Fig. 2. Filtering results of the house image, (a) Original im­age; (b) The noisy image; (c) Mean curvature evolution; (d) Fourth-order PDEs; (e) Chen's model; (f) Proposed model.

Fig. 1. (a) The noisy image with Gaussian noise; (b) The gradient image; (c) The magnitude map given by the differ­ence curvature method; (d) The magnitude map given by the difference eigenvalue method (P(u)).

To test the performance of proposed diffusion model in noise removal, three classical anisotropic methods, the mean curvature evolution [4], the fourth-order PDEs [12] and Chen's model [6], were used for comparison. The PSNR and SSIM values of each method are reported in Table 1.

We use the "house" image to demonstrate the performance of the new method in noise removal. Fig. 2(a) is the origi­nal image and its noisy version, corrupted by Gaussian noise with a standard deviation of cr = 20 is shown in Fig. 2(b). The filtering result by mean curvature evolution is shown in Fig. 2(c). Fig. 2(d) and (e) are the results given by the fourth-order PDEs and Chen's model, respectively. Fig. 2(f) is the result after using the proposed model. Since the difference eigenvalue P(u) can indicate the image edge magnitude map correctly, the diffusion function g based on difference eigen­value is large in the homogeneous regions so that almost all of the noise has been removed. Fig. 3(a) shows the CT image of a human abdomen. The results from the four anisotropic diffusion methods are shown in Fig. 3(b)-(e), respectively.

On Fig. 3(f) and (g), two rectangular area are selected and zoomed-in. We can observe that the dark area seen on source image Fig. 3(f) is removed adequately, making it more smooth­ing and clearer in Fig. 3(g). For further evaluation, the pixel intensity along two dashed line (red, blue color) are plotted in Fig. 3(h) and (i), from which we can observe that the new method produces intensity values that more closely resemble a "Heaviside step function", indicating that the new method produces a better restoration.

The computational complexity of the proposed method is related with the iteration number. Since the Hessian matrix and weighting parameter can be calculated in advance, the computational complexity of proposed method is (mn)2(the size of image is m x n) in each iteration.

Table 1 summarizes the PSNR and SSIM values under different noise levels on various benchmark images. Quanti­tatively, the proposed model outperforms other three methods in regarding of PSNR and SSIM.

6. CONCLUSION

In this paper, we propose a novel edge indicator to ef­fectively characterize fine image structures. Based on this new indicator, the classical anisotropic diffusion method is adapted to obtain a new algorithm that can remove noise with­out destroying image details. We tested our algorithm on both synthetic and real images and compared its performance with similar algorithms. The experimental results demon­strate that the new method achieves better performance in terms of PSNR and SSIM than the mean curvature evolution, the fourth-order PDEs and difference curvature methods. The future directions is to further investigate the performance of proposed method in multi-scale cases.

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2011 18th IEEE International Conference on Image Processing

Table 1. The PSNR and SSIM results of the denoised images under different noise levels. The value in the parenthesis is the SSIM measurement. Abbreviations: FO for the fourth-order PDEs, MC for mean curvature evolution, DC for Chen's model and DE for difference eigenvalue method.

Methods geometry <x = 20 <x = 30 house <x = 20 <x = 30 tower <x = 20 <x = 30 Lenna <x = 20 <x = 30

FO

27.6(0.7) 24.6(0.6)

27.2(0.6) 25.0(0.5)

26.5(0.6) 24.5(0.5)

26.1(0.6) 24.0(0.5)

MC

29.4(0.7) 27.1(0.7)

28.4(0.7) 26.0(0.6)

28.3(0.7) 26.1(0.6)

27.2(0.7) 24.8(0.6)

DC

30.0(0.8) 26.0(0.6)

29.5(0.7) 26.5(0.6)

29.0(0.7) 26.6(0.6)

27.9(0.7) 25.2(0.6)

DE

31.1(0.8) 29.5(0.8)

30.0(0.8) 27.8(0.7)

29.7(0.8) 27.6(0.7)

28.4(0.8) 26.2(0.7)

Fig. 3. Experiments of on another CT image, (a) Original image; (b)-(e) are the filtering resultants by four anisotropic diffusion methods; The small rectangular areas in green color are zoomed-in, shown in (f, g); The pixel values along two dashed lines (red, blue) are plotted in (h) and (i).

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[11] Zhou Wang, A.C. Bovik, H.R. Sheikh, and E.P. Simon-celli, "Image quality assessment: from error visibility to structural similarity," IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600-612,2004.

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