Chapter 88
Filter Parameter Estimation in Non-Local Means Algorithm Hong-jun Li, Wei Hu, Zheng-guang Xie and Yan Yan
Abstract In this paper, improvements to the Non-local Means (NL-Means) algorithm introduced by Buades et al. are presented. The filtering parameter is unclearly defined in the original NL-Means algorithm. In order to solve this problem, we calculated filtering parameter by the relation of noise variance, and then proposed a noise variance estimate method. In this paper, noisy image is transformed by wavelet. The wavelet coefficients in each sub-band can be well modelized by a Generalized Gaussian Distribution (GGD) whose parameters can be used to estimate noise variance. The simulation results show that the noise variance estimate method is not only exact but also makes the algorithm adaptive. The adaptive NL-Means algorithm can obtain approximately optimal value, and need less computing time. Keywords Non-local means algorithm Gaussian Distribution Wavelet Domain
Filtering Parameter
Generalized
88.1 Introduction In recent years, patch-based methods have drawn a lot of attention in the image processing community. Buades et al. have proposed the Non-local Means denoising algorithm [1]. That paper proposed a very elegant nonlocal denoising method H. Li (&) Z. Xie School of Electronic Information Engineering, Nantong University, 226019 Nantong, China e-mail:
[email protected] W. Hu Nantong University, 226019 Jiangsu, Nantong, China Y. Yan School of Computer Sciences and Technology, Nantong University, 226019 Nantong, Jiangsu, China
Z. Sun and Z. Deng (eds.), Proceedings of 2013 Chinese Intelligent Automation Conference, Lecture Notes in Electrical Engineering 256, DOI: 10.1007/978-3-642-38466-0_88, Ó Springer-Verlag Berlin Heidelberg 2013
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shown to propose a state-of-the-art result. The NL-Means restores the original image by considering Non-local neighborhoods of a given pixel. The aim of this paper is to discuss the choice of parameters of the standard NL-means filter. To discuss the tuning of parameters of the NL-means algorithm, we interpret this choice as a bias-variance dilemma. Contrary to the approach of [2], which also relies on a bias variance analysis, we mainly focus on the choice of the smoothing parameter rather than on the search window. This choice should be made depending on the regularity of the image, a notion that is to be defined. The study of [3] discussed the influence of two important parameters on this algorithm: the size of the searching window and the weight given to the central patch. The Filter parameter in [1] is searched in oracle, and set an optimal value as the filtering parameter in NL-Means algorithm. The author mentioned that the value of Filter parameter is correlated to image noise variance, but the definition of Filter parameter is unclearly. In this paper, we first present an overview of the NL-Means algorithm introduced in [1]; also analyze the relation between Filter parameter and noise variance. Then we estimate the noise variance by the parameters of GGD model. Finally, we use the estimated Filter parameters to solved traditional NL-Means algorithm drawbacks.
88.2 NL-Means Image Denoising Method In this section, a brief overview of NL-Means algorithm is introduced. Given a discrete noisy image v ¼ fvðiÞji 2 I g, the estimated value NL½vðiÞ is computed as a weighted average of all the pixels in the image, X NL½vðiÞ ¼ wði; jÞvð jÞ ð88:1Þ j2I
where the family of weight fwði; jÞgj depend on the similarity between the pixels i P and j, and satisfy the usual conditions 0 wði; jÞ 1 and j wði; jÞ ¼ 1. The similarity between two pixels i and j depends on the similarity of intensity gray level vectors vðNi Þ and v Nj , where Nk denotes a square neighborhood of fixed size and centered at a pixel k. This similarity is measured as a decreasing 2 function of the weighted Euclidean distance, vðNi Þ v Nj , where a is the 2;a
standard deviation of the Gaussian kernel. The application of Euclidean distance to noisy neighborhoods raises the following equality. 2 2 EvðNi Þ v Nj 2;a ¼ uðNi Þ u Nj 2;a þ2r2 ð88:2Þ This equality shows the robustness of the algorithm since in expectation the Euclidean distance conserves the order of similarity between pixels. So the most
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similar pixels to i in v also are expected to the most similar pixels i in u. The weights express the amount of similarity between the neighborhoods of each pair of pixels involved in the computation.
88.3 Obtain the Value of Filter Parameter In [1] Buades obtained the filtering parameter h by experiments, and found that the value of the filtering parameter h is between 10 r and 15 r. Buades made an error, he wanted to say h2 ¼ 10 r but he said h ¼ 10 r. In [4] the author discussed the relation between filtering parameter h and noise variance, and gave a relation between them. However, too many parameters are considered in estimation and new parameters lead the relation uncertainly. The [4] proposed that h parameter and noise variance are linear correlative. In Eq. (88.2), the weight function is Gaussian kernel with standard deviation pffiffiffi h= 2 acts as a filtering parameter. On the other hand, under the circumstances that the noise presents as a Gaussian distribution, the pixel and neighborhood pixels have the same value of gray. But the added Gaussian white noise will make them have different value of gray. Gaussian white noise will cause the difference between current pixels which can be calculated by Euclidean difference. So the pffiffiffi value of filtering parameter is approximate 2 times of noise sigma. It is difficult to obtain the value of filtering parameter accurately, so we chose the approximate one to substitute. The filtering parameter can be calculated by noise variance estimation. We show how to estimate in next section.
88.4 Noises Estimate Many image denoising algorithms assume that the noise level is known prior; and the algorithms are adapted to the amount of noise instead of using fixed parameters. In 1993, Olsen [5] gave a complete description and comparison of six earlier estimation algorithms. They are classified into two different approaches: filterbased and block-based. In filter-based methods, the noisy image is firstly filtered by a low-pass filter to suppress the image structures. Then the noise variance is computed from the difference between the noisy image and the filtered image. The main difficulty of filter-based methods is that the difference image is assumed to be the noise but this assumption is not true as images have structures or details. In block-based methods, images are tessellated into a number of blocks. The noise variance is then computed from a set of homogeneous blocks. The main issue of block-based methods is how to identify the homogeneous blocks. In this section we used the image block method in the wavelet domain, and proposed a novel image noise estimate method. In the wavelet domain, the most simple method about noise variance estimation is calculated by diagonal sub-bands
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coefficients. The noise variance estimate method proposed by Donoho and Johnstone [6] is expressed as follows: rn ¼ MAD=0:6745; Where MAD is middle value of diagonal sub-bands coefficients. In methods of [6], it is inappropriate to take the entire coefficients in HH sub-band to estimate noise variance without considering the coefficients also contain the edge information in direction 45 and 135°. So these methods are no suitable to estimate noise variance. The threshold proposed by Chang [7] is derived in a Bayesian framework, and the prior used on the wavelet coefficients is the GGD widely used in image processing applications. The proposed Bayesian risk minimization is sub-banded dependent. Given the signal being generalized Gaussian distributed and the noise being Gaussian, via numerical calculation a nearly optimal threshold defined as: TB ðrs Þ ¼ r2 rs ; where r2 is the noise variance, and r2s is the signal variance. This threshold only on the standard deviation and not on the shape parameter, it may not yield a good approximation for values of shape parameter, and the threshold may need to be modified to incorporate shape parameter. Our method is based on the Bayes-shrink threshold and takes the two parameters of GGD model into account.
88.4.1 Wavelet Coefficients Modelized by GGD Model Mallat [8] proved that the histogram of wavelet coefficients can be well fitted by GGD model. Define GGD, pðx; a; bÞ, b expðj xj=aÞb 2aCð1=bÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cð1=bÞ b[0 a ¼ rx Cð3=bÞ
pðx; a; bÞ ¼
ð88:3Þ
ð88:4Þ
where, rx is the standard deviation of the signal and Cð:Þ is the Gamma function; a is the scale parameter and b is the shape parameter. The value of b determines the decay rate of the probability density function. Then our purpose is to obtain the scale and shape parameters.
88.4.2 Analyze the Parameter of GGD Model Zhang [9] find that many well-known types of image distortions lead to significant changes in wavelet coefficient histograms, and measurement is based on the parameters of generalized Gaussian model. The paper introduce a new way to calculate the distortion of image, so there are some relation on the image distortion and the parameters of generalized Gaussian model indeed. So to find out a proper relationship of them is most important. In the previous work, Do and Vetterli [10]
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used GGD model to fit distribution of wavelet coefficients, and proved shape parameter is among [0.7 2.0]. When noise variance increases, shape parameter changes smoothly. When shape parameter is among [0.7 2.0], the relation of scale parameter and the standard deviation of the signal is approximately linear. On the experiment, when noise variance increases, shape parameter changes smoothly. So when add noise increase, and research the point, shape parameter can not well calculate by the standard deviation of the signal. So we need to analysis the relationship of shape parameter and noise variance. We used several images for test under different noise variance, and obtained the sequence value of a on different types of images. In order to obvious the relationship of scale parameter and noise variance deeply; we used experiments to find out the relationship of them. It is an interesting observation that the reciprocals of scale parameter changes smoothly when noise variance research the point, it is similarly the character of shape parameter. The value of 1=a is reduced dramatically when the value of noise variance is small, and then keeps a fixed value when noise variance reach a certain value. As the noise variance increase the value of 1=a changes small. When r ¼ 20; it keeps among [0.035 0.040]. When the noise sigma is bigger than 20, the value of b is bigger than 1 in most images. We find that in the situation of low density noise; if the value of 1=a is smaller than a certain value, the image has the character of abundant information and detail structure.
88.4.3 Noise Estimate Method We found that the scale parameter and shape parameter are closely related to the standard deviation of noise, and have some superior characters. Our estimation method is consider both the scale parameter and shape parameter. We found the shape parameter increases a little, when standard deviation of noise increases. We observed that a GGD with the shape parameter ranging from 0.5 to 1 can adequately describe the wavelet coefficients of a large set of natural images. The detail of estimate method is show in the follow steps. 1. Wavelet transformation in noisy image; 2. Used GGD to model wavelet coefficients probability histogram in each scales and directions; 3. Donoho noise variance estimate method; 4. Image noise _ estimate, using the scale and shape parameter: (a) When b 1, r ¼ rn _ ðrn =rx Þð2bÞ=b ; (b) when b 1 and a1 0:2, r ¼ rn ðrn =rx Þð1bÞ=b ; (c) when _ ð 1þb Þ=b . rx is considering all coefficients in b 1 and a1 [ 0:2, r ¼ rn ðrn =rx Þ image, so that the value is bigger than rn in most condition. The value of rn =rx is a little smaller than 1. In high density noise, rn is approximate to the noise value. So we set rn as a reference value. In low density noise, we use scale parameter to sort different kind of images mentioned above. If scale parameter smaller than 0.2, the image has complex structure and abundant details. In this situation, rn is bigger than the added standard deviation of noise.
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88.5 Adaptive Non-local Means Algorithm 88.5.1 Filter Parameter h Estimate We can calculate the filter parameter h by the noise variance method proposed above. In Table 88.1, we compare key parameters in original NL-Means algorithm and the method we proposed. Our method can estimate noise variance in good condition, and compute the h filtering parameter in low computational complexity. The complexity is 10 % of the original method. The proposed value of h is approximate to the optimal value, and the Peak Signal to Noise Ratio (PSNR) is acceptable.
88.5.2 Adaptive Non-local Means Algorithm We used parameter of GGD to obtain the noise variance, and then used the noise variance to estimate parameter h. It makes the method adaptive to preprocessing image instead of searching an optimal filtering parameter h. When image is decomposed by the wavelet, the size of image is reduced half. If the decomposition scale increases, the size will be reduced. Also the computational complexity is reduced quickly. So our algorithm not only solved the set of filtering parameter h, but also reduced the computation complexity. Figure 88.1 gives the denoising results about different algorithms; Visual quality of our method outperforms the Non-local means method [1], and better the method [11] with different value of filter parameter. Table 88.1 Compare with different method r 10 20
30
40
50
Method in [1] using oracle Sigma([1]) 10 Time(S) 480 h(0ptimal) 14.1 PSNR(0ptimal) 31.5
20.4 462 28.6 28.7
30.9 473 43.2 26.8
41.2 479 57.7 25.2
51.0 461 71.4 24
Method in [1] using h2 ¼ 10 r h([1]) 10 Time(S) 47 PSNR 31.4
14.3 48 27.8
17.6 47 24.0
20.3 45 20.8
22.6 44 18.3
Method proposed in this paper Sigma(proposed) 10.5 h(proposed) 14.7 Time(S) 42 PSNR 31.4
21.1 29.5 46 28.5
30.5 42.7 45 26.7
41 57.4 48 25.2
50.3 70.4 43 23.9
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Fig. 88.1 Image denoising results based on different algorithms: a Original image; b noisy image; c method in [11] with h ¼ 1:2r; d method in [11] with h ¼ 1:4r; e method in [1] h2 ¼ 10r, 2007; f our method
88.6 Conclusions In this paper, we proposed a noise estimate method to solve the problem about the definition of parameter h and the estimation of noise variance. We used the parameters of GGD model to estimate the noise variance, also took the noise variance in whole area into account. The classical noise estimate method is the fundamental of our method and support it in theoretic. Noise estimate method proposed in this paper, not only solved the problem about parameter h, and reduced the computation complexity of traditional Non-local Means algorithm when searching an optimal filtering parameter h. So the Adaptive NL-Means Algorithm can succeed automatically. Acknowledgments This work was supported by the National Natural Science Foundation of China (NO. 61171077), University Science Research Project of Jiangsu Province(NO. 12KJB510025, NO. 12KJB510026), the Traffic Department of applied basic research project of China (NO. 2011-319-813-510), the science and technology supporting plan (social development) of Jiangsu Province (NO. BE2010686), joint tackle hard-nut problems in science and technology on traffic and transportation industry of state ministry of communications (NO. 2010-353-332-110).
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