- Email: [email protected]
RHFMs similarity based nonlocal means image denoising in PDTDFB domain
G Model
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IJLEO 56514 1–3
Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Short note
1
RHFMs similarity based nonlocal means image denoising in PDTDFB domain
2
3
4 5 6
Q1 Q2
Hong-Ying Yang a,b,∗ , Na Zhang a , Xiang-Yang Wang a,∗∗ , Yu Zhang a a b
School of Computer and Information Technology, Liaoning Normal University, Dalian 116029, PR China State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing 100190, PR China
7
8 21
a r t i c l e
i n f o
a b s t r a c t
9 10 11 12 13
Article history: Received 22 December 2014 Accepted 10 October 2015 Available online xxx
14
20
Keywords: Image denoising Nonlocal means Radial harmonic Fourier moments (RHFMs) Shiftable complex directional pyramid (PDTDFB)
22
1. Introduction
15 16 17 18 19
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
It is a challenging work to design an edge-preserving image denoising scheme. Nonlocal means methods have been widely adopted as simple algorithms for image denoising, but they tend to overblur the image and sharpen the boundary with many texture details lost. In this paper, we propose a new nonlocal means image denoising approach in shiftable complex directional pyramid (PDTDFB) domain, in which the robust radial harmonic Fourier moments (RHFMs) are utilized. The novelty of the proposed approach includes: (1) PDTDFB based multi-scale structural similarity is explored, which has good robustness to noise interference, (2) Geometric invariant radial harmonic Fourier moments (RHFMs) are introduced into similarity measure function, which can improve the denoising ability. Experimental results demonstrate that our image denoising method can obtain better performances in terms of both subjective and objective evaluations than some other state-of-the-art denoising techniques. Especially, it can preserve edges very well while removing noise. © 2015 Published by Elsevier GmbH.
Denoising has become an essential step in image processing and analysis. Indeed, due to sensors imperfections, transmission channels defects, as well as physical constraints, noise deteriorates the quality of almost every acquired image [1]. For image denoising, the main challenge is how to preserve the information-bearing structures such as edges and textures to get satisfactory visual quality when improving the signal-to-noise-ratio (SNR). During the past three decades, a variety of denoising methods have been developed in the image processing and computer vision communities. Although seemingly very different, they all share the same property: to keep the meaningful edges and remove less meaningful ones. Generally, the existing image denoising work can be roughly divided into Nonlocal Methods, Random Fields, Bilateral Filtering, Anisotropic Diffusion, and Statistical Model [2]. Among them, nonlocal denoising methods has attracted significant interests in recent years, largely due to its success in preserving edges in denoised images [3]. Since the nonlocal means (NLM) was first proposed by Buades et al. [4], numerous methods have been developed for better
∗ Corresponding author at: Liaoning Normal University, School of Computer and
Q3 Information Technology, Dalian, China. Q4 ∗∗ Corresponding author. Tel.: +86 411 85992415; fax: +86 411 85992005. E-mail addresses: yhy [email protected] (H.-Y. Yang), [email protected] (X.-Y. Wang).
denoising performance, among which similarity measure for patches is one of most studied problems. Kervrann et al. [5] proposed a Bayesian NLM framework that measures the noisy patches similarities based on the statistical distribution of noise. Chen et al. [6] presented a homogeneity similarity based denoising method, in which the homogeneity similarity is defined in adaptive weighted neighborhoods. The homogeneity similarity can effectively find more similar points, especially for points with less repetitive patterns, such as corner and end points. The NLM method based on reprojection (NLM-REP) [7] projected the patches space to the original (image) pixel space to reduce the halo of noise. The principle neighborhood dictionary (PND) proposed by Tasdizen et al. [8], as a transformation domain based method, measures the patch similarity in the domain of principle component analysis (PCA). Tracey et al. [9] used a variational approach to combine multiple NLM estimates, seeking a solution that balances positivity constraints and gradient penalties against Stein’s Unbiased Risk Estimate. Zhong et al. [10] proposed a two stage scheme which improved the weight calculation method by incorporating the role of both the predenoised result and corresponding method noise. Despite conventional NLM and above-mentioned improvements have obtained better denoising effect, they always tend to overblur the image and sharpen the boundary with many texture details lost. The reason for this is that: First, the adopted similarity measure function is often fragile to noise. Second, the similarity measure of patches is only translation invariants. In other words, we can only match patches that
http://dx.doi.org/10.1016/j.ijleo.2015.10.054 0030-4026/© 2015 Published by Elsevier GmbH.
Please cite this article in press as: H.-Y. Yang, et al., RHFMs similarity based nonlocal means image denoising in PDTDFB domain, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.054
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
G Model
ARTICLE IN PRESS
IJLEO 56514 1–3
H.-Y. Yang et al. / Optik xxx (2015) xxx–xxx
2
75
are simply in different locations, but the orientation and scale unchanged. In this paper, we propose a RHFMs similarity based nonlocal means image denoising in PDTDFB domain. The proposed scheme has the following advantages: (1) PDTDFB based multi-scale structural similarity is explored, (2) Geometric invariant RHFMs are introduced into similarity measure function.
76
2. RHFMs similarity for nonlocal means image denoising
69 70 71 72 73 74
77 78 79 80 81
82
Our image denoising method makes a simple modification to the classical nonlocal means filter. Assuming the additive white Gaussian noise model f = fˆ + n, where f denotes the noisy observation polluted by the noise n, and fˆ is the noise-free image, the nonlocal means filter is defined as [4]
1 w (i, j) f (j) fˆ (i) = C(i)
(1)
j ∈ ˝i
83 84 85 86
where i is the pixel index, the restored intensity fˆ (i) is computed as a weighted average of the noisy observation f (i) in the search domain ˝i (a larger neighbourhood of pixel i, for example, 7 × 7 in this paper), C(i) = w(i, j) is a normalization constant, and weight j∈˝ i
87 88
89
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
w(i, j) is determined by the similarity of the Gaussian neighborhood between pixels i and j, which can be expressed as
⎛ ⎞ Ni − Nj 2 2,a ⎠ w(i, j) = exp ⎝−
and where N i denotes a square neighborhood centered at pixel i, 22,a is a Gaussian weighted Euclidean distance function, a is the standard deviation of the Gaussian kernel. The smoothing factor h acts as a degree of filtering. It controls the decay of the exponential function and therefore the decay of the weights as a function of the Euclidean distances. Moment descriptors have been studied for image recognition and computer vision since the 1960s. In Ren et al. [11] suggested to choose a triangular function as the radial function and introduced new orthogonal moments named radial harmonic Fourier moments (RHFMs). Compared with other orthogonal moments, RHFMs has a better image reconstruction, lower noise sensitivity, lower computational complexity, and magnitude invariance. Besides, the RHFMs are free of numerical instability issues so that high order moments can be obtained accurately. In this paper, a RHFMs similarity is proposed, based on which the weight w(i, j) can be written
106
w(i, j) = exp 107
108 109 110 111 112
(2)
h2
v(i) − v(j)2 −
113
The shiftable complex directional pyramid (PDTDFB) is a new multiresolution and multidirectional image decomposition, which has several desirable properties for image analysis applications [12]. It is energy shift-invariant, which makes the representation stable with respect to image translation. It also has a very low redundant ratio compared to other energy shift invariant image decompositions. The multiscale and multidirectional representation creates favorable conditions for efficient subsequent processing algorithms. Unlike other shiftable directional transforms whose angular resolutions are limited, the number of directional subbands in the PDTDFB can be increased adaptively depending on the processed image without increasing the redundant ratio of the representation. Furthermore, the decomposition provides phase information on the image feature, which can be very useful in several image processing tasks such as edge detection. Based on RHFMs similarity, we can straightforwardly extend the nonlocal means image denoising into the PDTDFB domain. The nonlocal means filtering in the PDTDFB domain includes three main steps as follow: Step 1. Perform the PDTDFB decomposition on the noisy image, and obtain a low-pass subband and a series of high-pass subbands. Step 2. The high-pass PDTDFB coefficients of the noisy image are estimated by the nonlocal means filtering with RHFMs similarity (Eq. (3)). See Section 2. Step 3. The smoothing factor h acts as a degree of filtering. It controls the decay of the exponential function and therefore the decay of the weights as a function of the Euclidean distances. In this paper, an adaptive smoothing factor hs,d is defined in PDTDFB domain as follow √ 2˛02 hs,d = 4s,d (4) Median(abs(Vs,d (·))) s,d = 0.675 where 0 is the noise standard deviation of the input image, V (·) denotes the high-pass PDTDFB coefficients at scale s and direction d, Median denotes the computation of the median, and abs denotes the function that calculates the absolute value. Step 4. Conduct the PDTDFB reconstruction to obtain the denoised image. 4. Experimental results
h2
v(i) = M 0,0 (i) , M0,1 (i) , M0,2 (i) , M1,1 (i) , M1,2 (i) , M2,2 (i)
3. Proposed image denoising algorithm
(3)
where Mm,l (i) denotes the magnitude of RHFMs M n,l (i) for image pixels in the search domain +. n, l denotes the order and repetition of RHFMs.
20
30
Input PSNR Method ProbShrink (2006) BLS-GSM (2003) SUREbivariate (2007) NL-means (2005) MCNLM (2014) Our method
22.11 18.59 Lena (512 × 512) 31.24 29.36 31.32 29.47 31.37 29.56 31.16 28.30 32.72 30.49 32.23 31.12
40
20
16.09
22.11 18.59 Barbara (512 × 512) 28.40 26.27 28.28 25.92 27.98 25.83 27.42 24.16 29.24 26.86 30.08 28.89
28.01 28.21 28.31 26.98 28.76 29.27
30
We tested our algorithm on a variety of standard images, namely, ‘Lena’, ‘Barbara’, ‘Peppers’, ‘House’ to make a comparison with other image denoising algorithms (including ProbShrink (2006), BLS-GSM (2003), SUREbivariate (2007), and NL-means (2005). We have applied three decomposition stages of a PDTDFB for our denoising procedure. Square-shape patch size for structure similarity is selected as 7 × 7. Because of space limitation, the
40
20
16.09
22.11 18.59 Peppers (256 × 256) 28.85 26.70 29.07 26.97 29.33 27.13 27.96 25.52 30.42 28.03 30.15 28.20
24.89 24.44 24.54 23.69 25.27 25.26
30
115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
143
144 145 146 147 148 149
150
Table 1 The PSNR results for various image denoising and noise intensities (dB).
114
40
20
16.09
22.11 18.59 House (256 × 256) 30.29 28.35 30.79 28.72 30.93 28.96 30.56 27.79 32.48 30.03 32.52 31.63
25.40 25.53 25.62 23.82 26.21 27.32
30
40 16.09 27.14 27.44 27.61 26.09 28.29 29.47
Please cite this article in press as: H.-Y. Yang, et al., RHFMs similarity based nonlocal means image denoising in PDTDFB domain, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.054
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Fig. 1. Comparison of denoising performance on noisy images corrupted by AWGN of = 20 (standard image Lena): (a) SUREbivariate, (b) ProbShrink, (c) NL-means, (h) our scheme.
Fig. 2. Comparison of denoising performance on noisy images corrupted by AWGN of = 20 (standard image Barbara): (a) SUREbivariate, (b) ProbShrink, (c) NL-means, (h) our scheme.
158 159 160 161 162 163 164
165
results of only four standard images are shown in Table 1 ( = 20, 30, 40). Figs. 1–2 show the visual metric of different denoising methods. From Table 1 and Figs. 1–2, it can be seen that our algorithm achieves the maximal PSNR output among other image denoising algorithms, especially for large noise variance. Also, we can see that the subjective quality of the proposed approach is better than other related methods in sharp edges and flat area. 5. Conclusions
179
We have presented an improved nonlocal means image denoising approach. The proposed approach has the following advantages: 1) PDTDFB based multi-scale structural similarity is explored, which has good robustness to noise interference. 2) Geometric invariant RHFMs are introduced into similarity measure function. The RHFMs based similarity measure can get more similar pixels or patches than traditional NL-means algorithm, which means that our approach has stronger denoising ability. The comparison of the denoising results obtained with our approach, and with the state-of-the-art denoising techniques, demonstrate the effectiveness of our new image denoising which gave better output PSNRs for most of the images, especially for large noise variance. The visual quality of our denoised images is moreover characterized by fewer artifacts than the other methods.
180
Acknowledgments
166 167 168 169 170 171 172 173 174 175 176 177 178
Q5 181
This work was supported by the National Natural Science Foundation of China under grant nos. 61472171 & 61272416, the Open
Foundation of State KeyLaboratory of Information Security of China under Grant No. 04-06-1, and Liaoning Research Project for Institutions of Higher Education of China under grant no. L2013407. Q6 References [1] C.Y. Lien, C.C. Huang, P.Y. Chenl, An efficient denoising architecture for removal of impulse noise in images, IEEE Trans. Comput. 62 (4) (2013) 631–643. [2] C. Liu, R. Szeliski, S.B. Kang, Automatic estimation and removal of noise from a single image, IEEE Trans. Pattern Anal. Mach. Intell. 30 (2) (2008) 299–3142. [3] S.H. Chan, T. Zickler, Y.M. Lu, Monte Carlo non-local means: random sampling for large-scale image filtering, IEEE Trans. Image Process. 23 (8) (2014) 3711–3725. [4] A. Buades, B. Coll, J.M. Morel, A non-local algorithm for image denoising, in: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), vol. 2, 2005, pp. 60–65. [5] C. Kervrann, J. Boulanger, P. Coupe, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal, in: Proceedings of Conference on Scale-Space and Variational Methods (SSVM 07), vol. 4485, 2007, pp. 520–532. [6] Q. Chen, Q. Sun, D. Xia, Homogeneity similarity based image denoising, Pattern Recognit. 43 (12) (2010) 4089–4100. [7] J. Salmon, Y. Strozecki, Patch reprojections for non-local methods, Signal Process. 92 (2) (2012) 477–489. [8] T. Tasdizen, Principal neighborhood dictionaries for nonlocal means image denoising, IEEE Trans. Image Process. 18 (12) (2009) 2649–2660. [9] B.H. Tracey, E.L. Miller, Y. Wu, A constrained optimization approach to combining multiple non-local means denoising estimates, Signal Process. 103 (2014) 60–68. [10] H. Zhong, C. Yang, X. Zhang, A new weight for nonlocal means denoising using method noise, IEEE Signal Process Lett. 19 (8) (2012) 535–538. [11] H. Ren, Z. Ping, W. Bo, W. Wu, Multidistortion-invariant image recognition with radial harmonic Fourier moments, J. Opt. Soc. Am. A 20 (4) (2003) 631–637. [12] T.T. Nguyen, S. Oraintara, The shiftable complex directional pyramid—part I: Theoretical aspects, IEEE Trans. Signal Process. 56 (10) (2008) 4651–4660.
Please cite this article in press as: H.-Y. Yang, et al., RHFMs similarity based nonlocal means image denoising in PDTDFB domain, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.054
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ARTICLE IN PRESS
IJLEO 56514 1–3
Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Short note
1
RHFMs similarity based nonlocal means image denoising in PDTDFB domain
2
3
4 5 6
Q1 Q2
Hong-Ying Yang a,b,∗ , Na Zhang a , Xiang-Yang Wang a,∗∗ , Yu Zhang a a b
School of Computer and Information Technology, Liaoning Normal University, Dalian 116029, PR China State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing 100190, PR China
7
8 21
a r t i c l e
i n f o
a b s t r a c t
9 10 11 12 13
Article history: Received 22 December 2014 Accepted 10 October 2015 Available online xxx
14
20
Keywords: Image denoising Nonlocal means Radial harmonic Fourier moments (RHFMs) Shiftable complex directional pyramid (PDTDFB)
22
1. Introduction
15 16 17 18 19
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
It is a challenging work to design an edge-preserving image denoising scheme. Nonlocal means methods have been widely adopted as simple algorithms for image denoising, but they tend to overblur the image and sharpen the boundary with many texture details lost. In this paper, we propose a new nonlocal means image denoising approach in shiftable complex directional pyramid (PDTDFB) domain, in which the robust radial harmonic Fourier moments (RHFMs) are utilized. The novelty of the proposed approach includes: (1) PDTDFB based multi-scale structural similarity is explored, which has good robustness to noise interference, (2) Geometric invariant radial harmonic Fourier moments (RHFMs) are introduced into similarity measure function, which can improve the denoising ability. Experimental results demonstrate that our image denoising method can obtain better performances in terms of both subjective and objective evaluations than some other state-of-the-art denoising techniques. Especially, it can preserve edges very well while removing noise. © 2015 Published by Elsevier GmbH.
Denoising has become an essential step in image processing and analysis. Indeed, due to sensors imperfections, transmission channels defects, as well as physical constraints, noise deteriorates the quality of almost every acquired image [1]. For image denoising, the main challenge is how to preserve the information-bearing structures such as edges and textures to get satisfactory visual quality when improving the signal-to-noise-ratio (SNR). During the past three decades, a variety of denoising methods have been developed in the image processing and computer vision communities. Although seemingly very different, they all share the same property: to keep the meaningful edges and remove less meaningful ones. Generally, the existing image denoising work can be roughly divided into Nonlocal Methods, Random Fields, Bilateral Filtering, Anisotropic Diffusion, and Statistical Model [2]. Among them, nonlocal denoising methods has attracted significant interests in recent years, largely due to its success in preserving edges in denoised images [3]. Since the nonlocal means (NLM) was first proposed by Buades et al. [4], numerous methods have been developed for better
∗ Corresponding author at: Liaoning Normal University, School of Computer and
Q3 Information Technology, Dalian, China. Q4 ∗∗ Corresponding author. Tel.: +86 411 85992415; fax: +86 411 85992005. E-mail addresses: yhy [email protected] (H.-Y. Yang), [email protected] (X.-Y. Wang).
denoising performance, among which similarity measure for patches is one of most studied problems. Kervrann et al. [5] proposed a Bayesian NLM framework that measures the noisy patches similarities based on the statistical distribution of noise. Chen et al. [6] presented a homogeneity similarity based denoising method, in which the homogeneity similarity is defined in adaptive weighted neighborhoods. The homogeneity similarity can effectively find more similar points, especially for points with less repetitive patterns, such as corner and end points. The NLM method based on reprojection (NLM-REP) [7] projected the patches space to the original (image) pixel space to reduce the halo of noise. The principle neighborhood dictionary (PND) proposed by Tasdizen et al. [8], as a transformation domain based method, measures the patch similarity in the domain of principle component analysis (PCA). Tracey et al. [9] used a variational approach to combine multiple NLM estimates, seeking a solution that balances positivity constraints and gradient penalties against Stein’s Unbiased Risk Estimate. Zhong et al. [10] proposed a two stage scheme which improved the weight calculation method by incorporating the role of both the predenoised result and corresponding method noise. Despite conventional NLM and above-mentioned improvements have obtained better denoising effect, they always tend to overblur the image and sharpen the boundary with many texture details lost. The reason for this is that: First, the adopted similarity measure function is often fragile to noise. Second, the similarity measure of patches is only translation invariants. In other words, we can only match patches that
http://dx.doi.org/10.1016/j.ijleo.2015.10.054 0030-4026/© 2015 Published by Elsevier GmbH.
Please cite this article in press as: H.-Y. Yang, et al., RHFMs similarity based nonlocal means image denoising in PDTDFB domain, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.054
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
G Model
ARTICLE IN PRESS
IJLEO 56514 1–3
H.-Y. Yang et al. / Optik xxx (2015) xxx–xxx
2
75
are simply in different locations, but the orientation and scale unchanged. In this paper, we propose a RHFMs similarity based nonlocal means image denoising in PDTDFB domain. The proposed scheme has the following advantages: (1) PDTDFB based multi-scale structural similarity is explored, (2) Geometric invariant RHFMs are introduced into similarity measure function.
76
2. RHFMs similarity for nonlocal means image denoising
69 70 71 72 73 74
77 78 79 80 81
82
Our image denoising method makes a simple modification to the classical nonlocal means filter. Assuming the additive white Gaussian noise model f = fˆ + n, where f denotes the noisy observation polluted by the noise n, and fˆ is the noise-free image, the nonlocal means filter is defined as [4]
1 w (i, j) f (j) fˆ (i) = C(i)
(1)
j ∈ ˝i
83 84 85 86
where i is the pixel index, the restored intensity fˆ (i) is computed as a weighted average of the noisy observation f (i) in the search domain ˝i (a larger neighbourhood of pixel i, for example, 7 × 7 in this paper), C(i) = w(i, j) is a normalization constant, and weight j∈˝ i
87 88
89
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
w(i, j) is determined by the similarity of the Gaussian neighborhood between pixels i and j, which can be expressed as
⎛ ⎞ Ni − Nj 2 2,a ⎠ w(i, j) = exp ⎝−
and where N i denotes a square neighborhood centered at pixel i, 22,a is a Gaussian weighted Euclidean distance function, a is the standard deviation of the Gaussian kernel. The smoothing factor h acts as a degree of filtering. It controls the decay of the exponential function and therefore the decay of the weights as a function of the Euclidean distances. Moment descriptors have been studied for image recognition and computer vision since the 1960s. In Ren et al. [11] suggested to choose a triangular function as the radial function and introduced new orthogonal moments named radial harmonic Fourier moments (RHFMs). Compared with other orthogonal moments, RHFMs has a better image reconstruction, lower noise sensitivity, lower computational complexity, and magnitude invariance. Besides, the RHFMs are free of numerical instability issues so that high order moments can be obtained accurately. In this paper, a RHFMs similarity is proposed, based on which the weight w(i, j) can be written
106
w(i, j) = exp 107
108 109 110 111 112
(2)
h2
v(i) − v(j)2 −
113
The shiftable complex directional pyramid (PDTDFB) is a new multiresolution and multidirectional image decomposition, which has several desirable properties for image analysis applications [12]. It is energy shift-invariant, which makes the representation stable with respect to image translation. It also has a very low redundant ratio compared to other energy shift invariant image decompositions. The multiscale and multidirectional representation creates favorable conditions for efficient subsequent processing algorithms. Unlike other shiftable directional transforms whose angular resolutions are limited, the number of directional subbands in the PDTDFB can be increased adaptively depending on the processed image without increasing the redundant ratio of the representation. Furthermore, the decomposition provides phase information on the image feature, which can be very useful in several image processing tasks such as edge detection. Based on RHFMs similarity, we can straightforwardly extend the nonlocal means image denoising into the PDTDFB domain. The nonlocal means filtering in the PDTDFB domain includes three main steps as follow: Step 1. Perform the PDTDFB decomposition on the noisy image, and obtain a low-pass subband and a series of high-pass subbands. Step 2. The high-pass PDTDFB coefficients of the noisy image are estimated by the nonlocal means filtering with RHFMs similarity (Eq. (3)). See Section 2. Step 3. The smoothing factor h acts as a degree of filtering. It controls the decay of the exponential function and therefore the decay of the weights as a function of the Euclidean distances. In this paper, an adaptive smoothing factor hs,d is defined in PDTDFB domain as follow √ 2˛02 hs,d = 4s,d (4) Median(abs(Vs,d (·))) s,d = 0.675 where 0 is the noise standard deviation of the input image, V (·) denotes the high-pass PDTDFB coefficients at scale s and direction d, Median denotes the computation of the median, and abs denotes the function that calculates the absolute value. Step 4. Conduct the PDTDFB reconstruction to obtain the denoised image. 4. Experimental results
h2
v(i) = M 0,0 (i) , M0,1 (i) , M0,2 (i) , M1,1 (i) , M1,2 (i) , M2,2 (i)
3. Proposed image denoising algorithm
(3)
where Mm,l (i) denotes the magnitude of RHFMs M n,l (i) for image pixels in the search domain +. n, l denotes the order and repetition of RHFMs.
20
30
Input PSNR Method ProbShrink (2006) BLS-GSM (2003) SUREbivariate (2007) NL-means (2005) MCNLM (2014) Our method
22.11 18.59 Lena (512 × 512) 31.24 29.36 31.32 29.47 31.37 29.56 31.16 28.30 32.72 30.49 32.23 31.12
40
20
16.09
22.11 18.59 Barbara (512 × 512) 28.40 26.27 28.28 25.92 27.98 25.83 27.42 24.16 29.24 26.86 30.08 28.89
28.01 28.21 28.31 26.98 28.76 29.27
30
We tested our algorithm on a variety of standard images, namely, ‘Lena’, ‘Barbara’, ‘Peppers’, ‘House’ to make a comparison with other image denoising algorithms (including ProbShrink (2006), BLS-GSM (2003), SUREbivariate (2007), and NL-means (2005). We have applied three decomposition stages of a PDTDFB for our denoising procedure. Square-shape patch size for structure similarity is selected as 7 × 7. Because of space limitation, the
40
20
16.09
22.11 18.59 Peppers (256 × 256) 28.85 26.70 29.07 26.97 29.33 27.13 27.96 25.52 30.42 28.03 30.15 28.20
24.89 24.44 24.54 23.69 25.27 25.26
30
115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
143
144 145 146 147 148 149
150
Table 1 The PSNR results for various image denoising and noise intensities (dB).
114
40
20
16.09
22.11 18.59 House (256 × 256) 30.29 28.35 30.79 28.72 30.93 28.96 30.56 27.79 32.48 30.03 32.52 31.63
25.40 25.53 25.62 23.82 26.21 27.32
30
40 16.09 27.14 27.44 27.61 26.09 28.29 29.47
Please cite this article in press as: H.-Y. Yang, et al., RHFMs similarity based nonlocal means image denoising in PDTDFB domain, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.054
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3
Fig. 1. Comparison of denoising performance on noisy images corrupted by AWGN of = 20 (standard image Lena): (a) SUREbivariate, (b) ProbShrink, (c) NL-means, (h) our scheme.
Fig. 2. Comparison of denoising performance on noisy images corrupted by AWGN of = 20 (standard image Barbara): (a) SUREbivariate, (b) ProbShrink, (c) NL-means, (h) our scheme.
158 159 160 161 162 163 164
165
results of only four standard images are shown in Table 1 ( = 20, 30, 40). Figs. 1–2 show the visual metric of different denoising methods. From Table 1 and Figs. 1–2, it can be seen that our algorithm achieves the maximal PSNR output among other image denoising algorithms, especially for large noise variance. Also, we can see that the subjective quality of the proposed approach is better than other related methods in sharp edges and flat area. 5. Conclusions
179
We have presented an improved nonlocal means image denoising approach. The proposed approach has the following advantages: 1) PDTDFB based multi-scale structural similarity is explored, which has good robustness to noise interference. 2) Geometric invariant RHFMs are introduced into similarity measure function. The RHFMs based similarity measure can get more similar pixels or patches than traditional NL-means algorithm, which means that our approach has stronger denoising ability. The comparison of the denoising results obtained with our approach, and with the state-of-the-art denoising techniques, demonstrate the effectiveness of our new image denoising which gave better output PSNRs for most of the images, especially for large noise variance. The visual quality of our denoised images is moreover characterized by fewer artifacts than the other methods.
180
Acknowledgments
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This work was supported by the National Natural Science Foundation of China under grant nos. 61472171 & 61272416, the Open
Foundation of State KeyLaboratory of Information Security of China under Grant No. 04-06-1, and Liaoning Research Project for Institutions of Higher Education of China under grant no. L2013407. Q6 References [1] C.Y. Lien, C.C. Huang, P.Y. Chenl, An efficient denoising architecture for removal of impulse noise in images, IEEE Trans. Comput. 62 (4) (2013) 631–643. [2] C. Liu, R. Szeliski, S.B. Kang, Automatic estimation and removal of noise from a single image, IEEE Trans. Pattern Anal. Mach. Intell. 30 (2) (2008) 299–3142. [3] S.H. Chan, T. Zickler, Y.M. Lu, Monte Carlo non-local means: random sampling for large-scale image filtering, IEEE Trans. Image Process. 23 (8) (2014) 3711–3725. [4] A. Buades, B. Coll, J.M. Morel, A non-local algorithm for image denoising, in: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), vol. 2, 2005, pp. 60–65. [5] C. Kervrann, J. Boulanger, P. Coupe, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal, in: Proceedings of Conference on Scale-Space and Variational Methods (SSVM 07), vol. 4485, 2007, pp. 520–532. [6] Q. Chen, Q. Sun, D. Xia, Homogeneity similarity based image denoising, Pattern Recognit. 43 (12) (2010) 4089–4100. [7] J. Salmon, Y. Strozecki, Patch reprojections for non-local methods, Signal Process. 92 (2) (2012) 477–489. [8] T. Tasdizen, Principal neighborhood dictionaries for nonlocal means image denoising, IEEE Trans. Image Process. 18 (12) (2009) 2649–2660. [9] B.H. Tracey, E.L. Miller, Y. Wu, A constrained optimization approach to combining multiple non-local means denoising estimates, Signal Process. 103 (2014) 60–68. [10] H. Zhong, C. Yang, X. Zhang, A new weight for nonlocal means denoising using method noise, IEEE Signal Process Lett. 19 (8) (2012) 535–538. [11] H. Ren, Z. Ping, W. Bo, W. Wu, Multidistortion-invariant image recognition with radial harmonic Fourier moments, J. Opt. Soc. Am. A 20 (4) (2003) 631–637. [12] T.T. Nguyen, S. Oraintara, The shiftable complex directional pyramid—part I: Theoretical aspects, IEEE Trans. Signal Process. 56 (10) (2008) 4651–4660.
Please cite this article in press as: H.-Y. Yang, et al., RHFMs similarity based nonlocal means image denoising in PDTDFB domain, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.054
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