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NON-LOCAL DUAL IMAGE DENOISING

NON-LOCAL DUAL IMAGE DENOISING
NON-LOCAL DUAL IMAGE DENOISING

NON-LOCAL DUAL IMAGE DENOISING N. Pierazzo , M. Lebrun , M.E. Rais ? , J.M. Morel , and G. Facciolo CMLA, école Normale Supérieure de Cachan, France ? Dpt. Matemàtiques i Informàtica, UIB, Spain
ABSTRACT The current state-of-the-art non-local algorithms for image denoising have the tendency to remove many low contrast details. Frequency-based algorithms keep these details, but on the other hand many artifacts are introduced. Recently, the Dual Domain Image Denoising (DDID) method has been proposed to address this issue. While beating the state-of-the-art, this algorithm still causes strong frequency domain artifacts. This paper reviews DDID under a different light, allowing to understand their origin. The analysis leads to the development of NLDD, a new denoising algorithm that outperforms DDID, BM3D and other state-of-the-art algorithms. NLDD is also three times faster than DDID and easily parallelizable. Index Terms— Image denoising, Patch-Based methods, Fourier shrinkage, Dual Denoising, Non-Local Bayes 1. INTRODUCTION Image denoising is one of the fundamental image processing challenges [1]. Image denoising methods can be divided into two main categories: frequency-based or spatial-based. The frequency domain methods rely on an underlying image regularity assumption and work by compressing/thresholding coef?cients in some frequency domain [2, 3, 4, 5]. The Wiener ?lter [6] is one of the ?rst such methods. Donoho et al. [7] expand it to the wavelet domain. Among the spatial-based methods, non-linear variational methods such as Total Variation Minimization (Rudin et al. [8, 9]) were once the state-of-the-art. Nowadays, spatialbased methods achieve remarkable results by exploiting spatial self-similarity in the image itself. Non-Local Means (NL-Means) (Buades et al. [10, 11]) and UINTA (Awate et al. [12]) are among the ?rst methods of this kind. They denoise by averaging similar patches in the image. Patch-based denoising methods have developed into attempts to model the patch space of an image, or of a set of images. Recently, algorithms proposing sparse representations of patches using
Aknowledgements: work partially supported by Centre National d’Etudes Spatiales (MISS Project), European Research Council (Advanced Grant Twelve Labours), Of?ce of Naval Research (under Grant N0001497-1-0839), Direction Générale de l’Armement, Fondation Mathématique Jacques Hadamard and Agence Nationale de la Recherche (Stereo project).
(a) Original
(b) Noisy
(c) DDID
(d) NLDD
Fig. 1. A detail of the artifacts produced by DDID and the corresponding result of NLDD. In this example σ = 30.
dictionaries were introduced by Elad et al. [13], Mairal et al. [14, 15, 16] and Yu et al. [17]. Modeling image properties using a Gaussian Scale Mixture (GSM) model is the basic idea of a denoising algorithm proposed by Portilla et al. [18]. Rajaei recently proposed an improvement of the BLS-GSM method [19]. In 2011, Levin and Nadler [20] tried to model the patch space using a non-parametric approach by sampling from a huge database of patches. This method was later accelerated by Pierazzo and Rais [21]. A current state-of-the-art method that takes advantage of both space and frequency domain approaches is BM3D (Dabov et al. [22, 23]), which is one of the most ef?cient patch-based denoising methods to date. Finally NL-Bayes (Lebrun et al. [24]) is a spatial-based method that improves NL-means by considering a Gaussian probability model for each set of similar patches. Contrary to BM3D, NL-Bayes does not produce artifacts. In 2013 a new hybrid method called Dual-Domain Image Denoising (DDID) was published by Knaus and Zwicker [25]. It is remarkably simple to implement, and it provides results that are generally superior in terms of PSNR to stateof-the-art methods such as BM3D and NL-Bayes. Its main drawback is that it produces typical frequency domain artifacts, as shown in Fig. 1. This is unexpected, since the method itself was developed to avoid the artifacts of frequency-based methods. The present article explains the creation of those artifacts and presents Non-Local Dual Denoising (NLDD), a faster and better performing denoising algorithm. The organization of this paper is as follows. In section

2 the Dual-Domain Image Denoising algorithm is revisited from a different perspective that gives insight about the artifacts. In section 3 NLDD is presented, addressing DDID’s problems, and its results are shown in section 4. 2. DUAL DOMAIN IMAGE DENOISING This section presents an alternative interpretation of DDID that differs from the one originally proposed by Knaus and Zwicker [25]. The original description of DDID splits the image into a low- and a high-contrast layer, which are treated respectively with a spatial and a frequency domain method. In this work instead, the spatial domain ?ltering is seen as a pre-processing to improve the frequency domain denoising. DDID consists of three almost identical steps. The output of each step is used to guide the following one. Each step of the algorithm processes the noisy image y pixel-wise using the guide image g . Each pixel p is denoised using the d × d neighborhood (d = 31) of both the noisy and the guide image. Denoising in the frequency domain often results in the appearance of artifacts. To prevent it each patch is pre-processed to eliminate discontinuities corresponding to object’s edges and patch’s boundaries. To that end, a kernel k is created from g identifying the pixels belonging to the same object as its central pixel p. This kernel is the product of a spatial and range kernels, as used in the bilateral ?lter [26, 27] k (q ) = ks (q ) · kr (q ). (1) Fig. 2. Illustration of DDID’s preprocessing of a patch. The kernel k is computed using the guide g . In the modi?ed patch ym all object discontinuities have been removed, leaving only the texture information corresponding to the object selected by the kernel k . The removed pixels are replaced by s ?: the average of the meaningful portion of the patch. where the sums are computed over Np , the domain of the d × d patch centered at p. After that, the parts of the patch not taken into account by k are set to the respective average. The resulting modi?ed patch is ym (q ) = k (q )y (q ) + (1 ? k (q ))? s. (5)
As illustrated in Fig. 2 the patch ym is similar to y in the parts belonging to the same object as the central pixel (including the noise) and smooth in the rest. The same procedure is applied to the guide patch gm (q ) = k (q )g (q ) + (1 ? k (q ))? g. (6)
? The range kernel is used to identify the pixels belonging to the same object. The idea is that, in g , pixels belonging to the same object as the central pixel will have similar values. The kernel is kr (q ) = exp ? |g (q ) ? g (p)| γr σ 2
2
At this point, ym and gm are two patches, built in the same way, in which discontinuities have been strongly reduced and only information “relevant” to denoise the central pixel has been kept. It is therefore safe to apply the Fourier transform and to continue the process in the frequency domain G(f ) =
q ∈Np
,
(2)
exp ? exp ?
q ∈Np
2iπ (q ? p)f gm (q ), d 2iπ (q ? p)f ym (q ). d
(7) (8)
where γr is a parameter of the algorithm and σ is the standard deviation of the noise. ? The spatial kernel, identi?es the pixels close to the central one and smooths periodization discontinuities associated to the frequency domain processing. To achieve that a Gaussian kernel of standard deviation σs is used, where σs is a parameter of the algorithm: ks (q ) = exp ? |q ? p| 2 2σs
2
S (f ) =
.
(3)
Assuming that y contains an additive white Gaussian noise of variance σ 2 , the amount of noise present in ym depends on k . In particular, for a pixel q , ym (q ) contains a noise equal to σ 2 k (q ). An interesting property of the Fourier transform is that the noise in every pixel is evenly distributed over all frequencies. Thus every frequency of S has Gaussian noise with the same variance
2 σf = σ2 q ∈Np
Since denoising with Fourier coef?cients has problems in presence of edges (due to the Gibbs phenomenon), the goal is to make the parts of the patch not relevant to the denoising as regular as possible. k is used to compute the average of the “relevant” part of both the noisy and the guide patches: s ?= k (q )y (q ) , k (q ) g ?= k (q )g (q ) , k (q ) (4)
k (q )2 .
(9)
The patch is then denoised by shrinking its Fourier coef?cients S (f ) by the factor 1 K (f ) = exp
2 γf σ f ? |G(f ) |2
if f = 0, otherwise, (10)

3. NON-LOCAL DUAL DENOISING Since, as concluded in the previous section, most of the artifacts of DDID come from the guide image, a method to avoid them is to feed the algorithm with a cleaner image. Non-Local Dual Denoising uses the Non-Local Bayes [24] denoising algorithm to provide a clean guide, and then applies the last step of DDID to denoise the image (with parameters σs = 7, γr = 0.7 and γf = 0.8). NL-Bayes has been chosen over other state-of-the-art algorithms (such as BM3D) because it generally provides a smoother output (see [28]). BM3D has been tested too as the guide. However the results, while still improving the ones of both BM3D and DDID, were slightly worse than the ones of NLDD. A pseudo-code for NLDD is listed in Algorithm 1. Algorithm 1 Non-Local Dual Denoising function NLDD(y, σ ) g ← NL-BAYES(y, σ ) for all pixels p ∈ y do y ← E XTRACT PATCH(y, p) g ← E XTRACT PATCH(g, p) k ← C OMPUTE K(g, p) ym , gm ← M ODIFY PATCHES(y , g , k ) S ← FFT(ym ) G ← FFT(gm ) x(p) ← S HRINK(S , G, k , σ ) end for return x end function
Fig. 3. Artifacts in DDID. From left to right: the noisy image (with σ = 30), the result of the ?rst, second, and last iteration of the algorithm. where γf is a parameter of the algorithm. The denoised value of the central pixel is ?nally recovered by reversing the Fourier transform. Inverting equation (5) is unnecessary, since k (p) = 1. Since the inverse Fourier transform evaluated in the center of the patch is the average of the frequencies, the central pixel’s value is computed as x(p) = 1 d2 S (f )K (f ).
f
(11)
Eq. 1 Eq. 4-6
Equations (7-11) are slightly different from the ones presented in [25]. In fact, it can be easily proved that G(f ) and S (f ) differ from the ones presented in the original paper only at the zero frequency. This frequency is then restored after the shrinkage. In the presented version, the zero frequency is left untouched by the shrinkage, by imposing K (0) = 1. For color images, the kernel kr is computed by using the Euclidean distance in the color space, while the Fourier thresholding is done independently on each channel in the YUV color space. Artifacts in DDID The above description highlights that the denoising in DDID is accomplished in the frequency domain, while the spatial pre-processing is used to remove discontinuities from the image. The described procedure is applied three times with different parameters. Each time the result of the previous calculation is used as a guide, except in the ?rst iteration where the noisy image itself is used. It is worth noting that the image is denoised in the last iteration only. The other two are only used to obtain a suitable guide. Besides being slow to compute, the main drawback of DDID is that its results often present ringing artifacts (as seen in Fig. 1). This is surprising since removing the strong edges, as in equation (5), should prevent it. Since the guide image used in the ?rst iteration is noisy, and the kernel in (1) is computed from it, “parasite” information is retained and propagated in the following iterations (see Fig. 3). This yields a result that contains artifacts.
Eq. 10-11
This algorithm has several advantages over DDID. Since the guide image (provided by NL-Bayes) has less artifacts than the one computed in the ?rst two iterations of DDID, it generally provides better results, as shown in section 4. As expected, the results contain less artifacts. In addition, NLDD is faster than DDID as only one iteration is needed. Moreover, since both DDID and NL-Bayes are heavily parallelizable, NLDD could also be implemented on a GPU architecture [25]. Our multi-thread C++ implementation of DDID takes 69 seconds to denoise a 704 × 469 color image on a 8-core Intel Xeon 2.33 GHz. On the same machine, NLDD takes just 22 seconds, about one third of the time.1 4. EXPERIMENTAL RESULT NLDD has been compared against DDID, BM3D and NLBayes with different amounts of noise. For the tests an heterogeneous set of noise-free images was used. All the results are evaluated using the Peak Signal-to-Noise Ratio (PSNR) and SSIM [29], which isan alternative metric conceived to simulate the response of the Human Visual System.
1 The C++ code for NLDD along with a MATLAB wrapper and an online demo is available in the supplementary materials webpage [28].

Fig. 4. Crops from the results of the images Alley, Flowers and Computer. From left to right: the original image, the noisy image (σ = 30), the outputs of BM3D, NL-Bayes, DDID, and NLDD. Full results are available in the article’s website. Image Alley Computer Dice Flowers Girl Traf?c Trees Valldemossa Mean BM3D 29.32 30.66 38.02 33.76 36.95 28.83 24.62 27.24 31.18 NL-Bayes 29.12 30.68 37.97 33.85 36.62 29.00 25.02 27.37 31.20 DDID 29.33 31.00 38.45 34.36 37.26 29.20 24.85 27.27 31.46 NLDD 29.41 31.10 38.78 34.48 37.33 29.40 25.09 27.48 31.64 PSNR NL-Bayes 37.07 33.42 31.20 29.62 28.35 27.03 SSIM NL-Bayes 0.9680 0.9308 0.8928 0.8572 0.8084 0.7595
σ 10 20 30 40 60 80 σ 10 20 30 40 60 80
BM3D 36.84 33.22 31.18 29.70 27.45 26.53 BM3D 0.9684 0.9303 0.8938 0.8614 0.8064 0.7592
DDID 36.92 33.52 31.46 29.99 27.96 26.50 DDID 0.9699 0.9331 0.8957 0.8600 0.7994 0.7500
NLDD 36.92 33.64 31.64 30.21 28.38 26.89 NLDD 0.9691 0.9346 0.8995 0.8671 0.8100 0.7570
Table 1. Values of PSNR for σ = 30. The results for σ = 30, where DDID performs best, are summarized in Table 1. NLDD outperforms the other algorithms in terms of PSNR. The same holds for the SSIM comparison, which is available online, along with the complete database of results [28]. The results for other levels of noise are summarized in Table 2. NLDD provides the best results for values of σ between 20 and 60. These coincide with the values of noise for which DDID has the best performance. Looking closely at Fig. 4 fewer artifacts can be noticed for NLDD. However, the values of SSIM don’t re?ect the magnitude of this improvement, but the details in Fig. 1 suggest that the quality of the two reconstructions is signi?cantly different. It is worth noticing that when the guide image is inaccurate NLDD also performs relatively poorly. For example in the image “Flowers” NL-Bayes fails to recover the texture of the leaves. As a result, these areas of the image are not fully recovered by NLDD. 5. CONCLUSION In this paper the DDID denoising algorithm has been reviewed under a different light, allowing to understand the origin of most of its artifacts. This analysis has led to the
Table 2. Average values of PSNR and SSIM with different noise levels.
development of NLDD, a new denoising algorithm that addresses the creation of these artifacts and outperforms DDID and other state-of-the-art algorithms in almost every test. The results clearly show that NLDD is superior to the other methods in both PSNR and SSIM. However, a close inspection of the results leads to the conclusion that a robust metric for evaluating denoising algorithms is still needed. Indeed it is observed that even modern metrics such as SSIM, commonly applied in these cases, fall short in presence of isolated but blunt artifacts. Nonetheless the qualitative analysis of the results con?rms the superiority of NLDD. By construction the NLDD method is faster than DDID, but it is still slow compared to other methods. However, like DDID and NL-Bayes themselves it is heavily parallelizable and can be implemented in GPU hardware.

6. REFERENCES [1] M.C. Motwani, M.C. Gadiya, R.C. Motwani, and F.C. Harris Jr., “Survey of image denoising techniques,” in Proceedings of GSPX, 2004. [2] J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE TIP, vol. 11, no. 6, 2002. [3] H.Q. Li, S.Q. Wang, and C.Z. Deng, “New image denoising method based wavelet and curvelet transform,” in WASE ICIE, 2009, vol. 1. [4] D. Gnanadurai and V. Sadasivam, “Image denoising using double density wavelet transform based adaptive thresholding technique,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 03, no. 01, 2005. [5] G. Yu and G. Sapiro, “Dct image denoising: a simple and effective image denoising algorithm,” Image Processing On Line, 2011. [6] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, The MIT Press, 1964. [7] D.L. Donoho and J.M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, 1994. [8] L.I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D, vol. 60, 1992. [9] P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using split Bregman,” Image Processing On Line, vol. 2012, 2012. [10] A. Buades, B. Coll, and J.M. Morel, “A review of image denoising algorithms, with a new one,” SIAM Mult. Model. Simul., vol. 4, no. 2, 2006. [11] A. Buades, B. Coll, and J.M. Morel, “Non-local means denoising,” Image Processing On Line, 2011. [12] S.P. Awate and R.T. Whitaker, “Unsupervised, Information-Theoretic, adaptive image ?ltering for image restoration,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 28, 2006. [13] M. Elad and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE TIP, vol. 15, no. 12, 2006. [14] J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE TIP, vol. 17, no. 1, 2008.
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图像处理技术的应用论文

图像处理技术的应用先展示一下自己用Photoshop处理的图片(做的不好望见谅)

摘要:图像处理技术的研究和应用越来越收到社会发展的影响,并以自身的技术特点反过来影响整个社会技术的进步。本文主要简单概括了数字图像处理技术近期的发展及应用现状,列举了数字图像处理技术的主要优点和制约其发展的因素,同时设想了图像处理技术在未来的应用和发展。 关键字:图像处理发展技术应用 1.概述 1.1图像的概念 图像包含了它所表达的物体的描述信息。我们生活在一个信息时代,科学研究和统计表明,人类从外界获得的信息约有百分之七十来自视觉系统,也就是从图像中获得,即我们平常所熟知的照片,绘画,动画。视像等。 1.2图像处理技术 图像处理技术着重强调在图像之间进行的变换,主要目标是要对图像进行各种加工以改善图像的视觉效果并为其后的目标自动识别打基础,或对图像进行压缩编码以减少图像存储所需要的空间或图像传输所需的时间。图像处理是比较低层的操作,它主要在图像像素级上进行处理,处理的数据量非常大。 1.3优点分析 1.再现性好。数字图像处理与模拟图像处理的根本不同在于,它不会因图像的存储、传输或复制等一系列变换操作而导致图像质量的退化。 2.处理精度高。按目前的技术,几乎可将一幅模拟图像数字化为任意大小的二维数组,这主要取决于图像数字化设备的能力。现代扫描仪可以把每个像素的灰度等级量化为16位甚至更高,这意味着图像的数字化精度可以达到满足任一应用需求。 3.适用面宽。图像可以来自多种信息源,它们可以是可见光图像,也可以是不可见的波谱图像(例如X射线图像、射线图像、超声波图像或红外图像等)。从图像反映的客观实体尺度看,可以小到电子显微镜图像,大到航空照片、遥感图像甚至天文望远镜图像。即只要针对不同的图像信息源,采取相应的图像信息采集措施,图像的数字处理方法适用于任何一种图像。 4.灵活性高。图像处理大体上可分为图像的像质改善、图像分析和图像重建三大部分,每一部分均包含丰富的内容。而数字图像处理不仅能完成线性运算,而且能实现非线性处理,即凡是可以用数学公式或逻辑关系来表达的一切运算均可用数字图像处理实现。 2.应用领域 2.1图像技术应用领域

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