American Institute of Mathematical Sciences

Advanced
April  2020, 14(2): 171-203. doi: 10.3934/ipi.2020009

Removing random-valued impulse noise with reliable weight

Qiyu Jin 1, , Li Bai 2, , Ion Grama 3, , Quansheng Liu 3,4,, and  Jie Yang 5,

1. 

School of mathematical science, Inner Mongolia University, No.235 Daxuexilu Road, 010021 Hohhot, Inner Mongolia, China
2. 

School of Computer Science, University of Nottingham, Nottingham NG8 1BB, UK
3. 

UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Université de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France
4. 

School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing 210044, China
5. 

Institute of Image Processing and Pattern Recognition, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Minhang District, Shanghai 200240, China

* Corresponding author: Quansheng Liu

Received  September 2018 Revised  November 2019 Published  February 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grants No. 61661039, No.61661040, No. 61661038, No. 11401590, No. 11571052 and No. 11731012), China Scholarship Council (Grant No. 201806810001) and Hunan Provincial Natural Science Foundation of China (Grant No.2017JJ2271). The work has benefited from the support of the Centre Henri Lebesgue (CHL, ANR-11-LABX-0020-01).

In this paper, we present a patch based weighted means filter for removing an impulse noise by adapting the fundamental idea of the non-local means filter to the random-valued impulse noise. Our approach is to give a weight to a pixel in order to evaluate the probability that the pixel is contaminated by the impulse noise, which we call Reliable Weight of the pixel. With the help of the Reliable Weights we introduce the similarity function to measure the similarity among patches of the image contaminated by a random impulse noise. It turns out that the similarity function has significant anti impulse noise interference ability. We then incorporate the Reliable Weights and the similarity function into a filter designed to remove the random impulse noise. Under suitable conditions, we establish two convergence theorems to demonstrate that our method is feasible. Simulation results confirm that our filter is competitive compared to recently proposed methods.

Keywords: Random-valued impulse noise, Reliable Weight, Denoising, Optimal Weights Filter, Non-Local Means, Rank Ordered Absolute Difference.
Mathematics Subject Classification: Primary: 62H35.
Citation: Qiyu Jin, Li Bai, Ion Grama, Quansheng Liu, Jie Yang. Removing random-valued impulse noise with reliable weight. Inverse Problems and Imaging, 2020, 14 (2) : 171-203. doi: 10.3934/ipi.2020009
References:
[1]

E. Abreu, M. Lightstone, S. K Mitra and K. Arakawa, A new efficient approach for the removal of impulse noise from highly corrupted images, IEEE Transactions on Image Processing, 5 (1996), 1012-1025.  doi: 10.1109/83.503916.

[2]

I. Aizenberg, C. Butakoff and D. Paliy, Impulsive noise removal using threshold boolean filtering based on the impulse detecting functions, IEEE Signal Processing Letters, 12 (2005), 63-66.  doi: 10.1109/LSP.2004.838198.

[3]

T. Y. Al-Naffouri, A. A. Quadeer and G. Caire, Impulse noise estimation and removal for ofdm systems, IEEE Transactions on communications, 62 (2014), 976-989.  doi: 10.1109/TCOMM.2014.012414.130244.

[4]

N. Alajlan, M. Kamel and E. Jernigan, Detail preserving impulsive noise removal, Signal Processing: Image Communication, 19 (2004), 993-1003.  doi: 10.1016/j.image.2004.08.003.

[5]

A. S. Awad, Standard deviation for obtaining the optimal direction in the removal of impulse noise, IEEE Signal Processing Letters, 18 (2011), 407-410.  doi: 10.1109/LSP.2011.2154330.

[6]

E. Beşdok and M. Emin Yüksel, Impulsive noise suppression from images with jarque-bera test based median filter, AEU-International Journal of Electronics and Communications, 59 (2005), 105-110. 

[7]

A. C. Bovik, Handbook of Image and Video Processing, Access Online via Elsevier, 2010.

[8]

D. R. K. Brownrigg, The weighted median filter, Communications of the ACM, 27 (1984), 807-818.  doi: 10.1145/358198.358222.

[9]

R. H. Chan, C.-W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Transactions on Image Processing, 14 (2005), 1479-1485.  doi: 10.1109/TIP.2005.852196.

[10]

R. H. Chan, C. Hu and M. Nikolova, An iterative procedure for removing random-valued impulse noise, IEEE Signal Processing Letters, 11 (2004), 921-924.  doi: 10.1109/LSP.2004.838190.

[11]

T. Chen, K.-K. Ma and L.-H. Chen, Tri-state median filter for image denoising, IEEE Transactions on Image Processing, 8 (1999), 1834-1838. 

[12]

T. Chen and H. R. Wu, Adaptive impulse detection using center-weighted median filters, IEEE Signal Processing Letters, 8 (2001), 1-3.  doi: 10.1109/97.889633.

[13]

___, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing 48 (2001), 784–789.

[14]

V. Crnojevic, V. Senk and Z. Trpovski, Advanced impulse detection based on pixel-wise mad, IEEE Signal Processing Letters, 11 (2004), 589-592.  doi: 10.1109/LSP.2004.830117.

[15]

J. Delon and A. Desolneux, A patch-based approach for removing impulse or mixed gaussian-impulse noise, SIAM Journal on Imaging Sciences, 6 (2013), 1140-1174.  doi: 10.1137/120885000.

[16]

J. Delon, A. Desolneux and T. Guillemot, Parigi: A patch-based approach to remove impulse-gaussian noise from images, Image Processing On Line, 6 (2016), 130-154.  doi: 10.5201/ipol.2016.161.

[17]

Y. Dong, R. H. Chan and S. Xu, A detection statistic for random-valued impulse noise, IEEE Transactions on Image Processing, 16 (2007), 1112-1120.  doi: 10.1109/TIP.2006.891348.

[18]

R. Garnett, T. Huegerich, C. Chui and W. He, A universal noise removal algorithm with an impulse detector, IEEE Transactions on Image Processing, 14 (2005), 1747-1754.  doi: 10.1109/TIP.2005.857261.

[19]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, Englewood Cliffs, NJ: Prentice-Hall, 2002.

[20]

M.-H. Hsieh, F.-C. Cheng, M.-C. Shie and S.-J. Ruan, Fast and efficient median filter for removing 1–99% levels of salt-and-pepper noise in images, Engineering Applications of Artificial Intelligence, 26 (2013), 1333-1338.  doi: 10.1016/j.engappai.2012.10.012.

[21]

H. Hu, B. Li and Q. Liu, Removing mixture of gaussian and impulse noise by patch-based weighted means, Journal of Scientific Computing, 67 (2016), 103-129.  doi: 10.1007/s10915-015-0073-9.

[22]

S. Huang and J. Zhu, Removal of salt-and-pepper noise based on compressed sensing, Electronics Letters, 46 (2010), 1198-1199.  doi: 10.1049/el.2010.0833.

[23]

I. F. Jafar, J. Amman, R. A. AlNa'mneh and K. A. Darabkh, Efficient improvements on the BDND filtering algorithm for the removal of high-density impulse noise, IEEE Transactions on Image Processing, 22 (2013), 1223-1232.  doi: 10.1109/TIP.2012.2228496.

[24]

Q. Jin, L. Bai, J. Yang, I. Grama and Q. Liu, A new method for removing random-valued impulse noise, International Conference on Neural Information Processing, Springer, 2014, 9–16. doi: 10.1007/978-3-319-12643-2_2.

[25]

Q. Jin, I. Grama, C. Kervrann and Q. Liu, Nonlocal means and optimal weights for noise removal, Siam Journal on Imaging Sciences, 10 (2017), 1878-1920.  doi: 10.1137/16M1080781.

[26]

Q. Jin, I. Grama and Q. Liu, A new poisson noise filter based on weights optimization, Journal of Scientific Computing, 58 (2014), 548-573.  doi: 10.1007/s10915-013-9743-7.

[27]

S.-J. Ko and Y. H. Lee, Center weighted median filters and their applications to image enhancement, IEEE Transactions on Circuits and Systems, 38 (1991), 984-993.  doi: 10.1109/31.83870.

[28]

B. Li, Q. Liu, J. Xu and X. Luo, A new method for removing mixed noises, Science China Information Sciences, 54 (2011), 51-59.  doi: 10.1007/s11432-010-4128-0.

[29]

C.-Y. Lien, P.-Y. Chen, L.-Y. Chang, Y.-M. Lin and P.-K. Chang, An efficient denoising chip for the removal of impulse noise, IEEE International Symposium on Circuits and Systems, 2010, 1169–1172. doi: 10.1109/ISCAS.2010.5537308.

[30]

C.-Y. Lien, C.-C. Huang, P.-Y. Chen and Y.-F. Lin, An efficient denoising architecture for removal of impulse noise in images, IEEE Transactions on Computers, 62 (2013), 631-643.  doi: 10.1109/TC.2011.256.

[31]

H.-M. Lin and A. N. Willson Jr, Median filters with adaptive length, IEEE Transactions on Circuits and Systems, 35 (1988), 675-690.  doi: 10.1109/31.1805.

[32]

T. Melange, M. Nachtegael and E. E. Kerre, Fuzzy Random Impulse Noise Removal From Color Image Sequences, IEEE Transactions on Image Processing, 20 (2011), 959-970.  doi: 10.1109/TIP.2010.2077305.

[33]

M. S. Nair and P. M. Ameera Mol, Noise adaptive weighted switching median filter for removing high density impulse noise, Advances in Computing and Communications, Springer, 2011, 193–204. doi: 10.1007/978-3-642-22720-2_19.

[34]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. 

[35]

G. Pok, J.-C. Liu and A. S. Nair, Selective removal of impulse noise based on homogeneity level information, IEEE Transactions on Image Processing, 12 (2003), 85-92. 

[36]

K. Prathiba, R. Rathi and C. S. Christopher, Random valued impulse denoising using robust direction based detector, Information & Communication Technologies (ICT), 2013 IEEE Conference on, IEEE, 2013, 1237–1242. doi: 10.1109/CICT.2013.6558290.

[37]

W. K. Pratt, Median Filtering, Image Process. Inst., Univ. Southern California, Los Angeles, 1975.

[38]

W. Pruitt, Summability of independent random variables, J. Math. Mech., 15 (1966), 769-776. 

[39]

F. Russo, Impulse noise cancellation in image data using a two-output nonlinear filter, Measurement, 36 (2004), 205-213.  doi: 10.1016/j.measurement.2004.09.002.

[40]

K. S. Srinivasan and D. Ebenezer, A new fast and efficient decision-based algorithm for removal of high-density impulse noises, IEEE Signal Processing Letters, 14 (2007), 189-192.  doi: 10.1109/LSP.2006.884018.

[41]

T. Sun and Y. Neuvo, Detail-preserving median based filters in image processing, Pattern Recognition Letters, 15 (1994), 341-347.  doi: 10.1016/0167-8655(94)90082-5.

[42]

C. Wang, T. Chen and Z. Qu, A novel improved median filter for salt-and-pepper noise from highly corrupted images, Systems and Control in Aeronautics and Astronautics (ISSCAA), 2010 3rd International Symposium on, IEEE, 2010, 718–722.

[43]

S.-S. Wang and C.-H. Wu, A new impulse detection and filtering method for removal of wide range impulse noises, Pattern Recognition, 42 (2009), 2194-2202.  doi: 10.1016/j.patcog.2009.01.022.

[44]

Z. Wang and D. Zhang, Progressive switching median filter for the removal of impulse noise from highly corrupted images, Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, 46 (1999), 78-80. 

[45]

M. Waqas, S. G. Javed and A. Khan, Random-valued impulse noise removal from images: K-means and luo-statistics based detector and nonlocal means based estimator, Applied Sciences and Technology (IBCAST), 2014 11th International Bhurban Conference on, IEEE, 2014, 130–135. doi: 10.1109/IBCAST.2014.6778135.

[46]

L. Wenbin, A new efficient impulse detection algorithm for the removal of impulse noise, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 88 (2005), 2579-2586. 

[47]

P. S. Windyga, Fast impulsive noise removal, IEEE Transactions on Image Processing, 10 (2001), 173-179.  doi: 10.1109/83.892455.

[48]

J. Wu and C. Tang, Pde-based random-valued impulse noise removal based on new class of controlling functions, IEEE Transactions on Image Processing, 20 (2011), 2428-2438.  doi: 10.1109/TIP.2011.2131664.

[49]

___, Random-valued impulse noise removal using fuzzy weighted non-local means, Signal, Image and Video Processing, 8 (2014), 349–355.

[50]

B. Xiong and Z. Yin, A universal denoising framework with a new impulse detector and nonlocal means, IEEE Transactions on Image Processing, 21 (2012), 1663-1675.  doi: 10.1109/TIP.2011.2172804.

[51]

Y. Y. Zhou, Z. F. Ye and J. J. Huang, Improved decision-based detail-preserving variational method for removal of random-valued impulse noise, IET Image Processing, 6 (2012), 976-985.  doi: 10.1049/iet-ipr.2011.0312.

[52]

Z. Zhou, Cognition and removal of impulse noise with uncertainty, IEEE Transactions on Image Processing, 21 (2012), 3157-3167.  doi: 10.1109/TIP.2012.2189577.

[53]

Z. Zhu, X. Zhang, X. Wan and Q. Wang, A random-valued impulse noise removal algorithm with local deviation index and edge-preserving regularization, Signal, Image and Video Processing, 9 (2015), 221-228.  doi: 10.1007/s11760-013-0426-5.

[54]

Z. Zhu, X. Zhang, Q. Wang, X. Wan and Y. Xiao, Edge-preserving regularized filter with spatial local outlier measure and q-estimate, Circuits, Systems, and Signal Processing, 33 (2014), 629-642.  doi: 10.1007/s00034-013-9644-x.

show all references

References:
[1]

E. Abreu, M. Lightstone, S. K Mitra and K. Arakawa, A new efficient approach for the removal of impulse noise from highly corrupted images, IEEE Transactions on Image Processing, 5 (1996), 1012-1025.  doi: 10.1109/83.503916.

[2]

I. Aizenberg, C. Butakoff and D. Paliy, Impulsive noise removal using threshold boolean filtering based on the impulse detecting functions, IEEE Signal Processing Letters, 12 (2005), 63-66.  doi: 10.1109/LSP.2004.838198.

[3]

T. Y. Al-Naffouri, A. A. Quadeer and G. Caire, Impulse noise estimation and removal for ofdm systems, IEEE Transactions on communications, 62 (2014), 976-989.  doi: 10.1109/TCOMM.2014.012414.130244.

[4]

N. Alajlan, M. Kamel and E. Jernigan, Detail preserving impulsive noise removal, Signal Processing: Image Communication, 19 (2004), 993-1003.  doi: 10.1016/j.image.2004.08.003.

[5]

A. S. Awad, Standard deviation for obtaining the optimal direction in the removal of impulse noise, IEEE Signal Processing Letters, 18 (2011), 407-410.  doi: 10.1109/LSP.2011.2154330.

[6]

E. Beşdok and M. Emin Yüksel, Impulsive noise suppression from images with jarque-bera test based median filter, AEU-International Journal of Electronics and Communications, 59 (2005), 105-110. 

[7]

A. C. Bovik, Handbook of Image and Video Processing, Access Online via Elsevier, 2010.

[8]

D. R. K. Brownrigg, The weighted median filter, Communications of the ACM, 27 (1984), 807-818.  doi: 10.1145/358198.358222.

[9]

R. H. Chan, C.-W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Transactions on Image Processing, 14 (2005), 1479-1485.  doi: 10.1109/TIP.2005.852196.

[10]

R. H. Chan, C. Hu and M. Nikolova, An iterative procedure for removing random-valued impulse noise, IEEE Signal Processing Letters, 11 (2004), 921-924.  doi: 10.1109/LSP.2004.838190.

[11]

T. Chen, K.-K. Ma and L.-H. Chen, Tri-state median filter for image denoising, IEEE Transactions on Image Processing, 8 (1999), 1834-1838. 

[12]

T. Chen and H. R. Wu, Adaptive impulse detection using center-weighted median filters, IEEE Signal Processing Letters, 8 (2001), 1-3.  doi: 10.1109/97.889633.

[13]

___, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing 48 (2001), 784–789.

[14]

V. Crnojevic, V. Senk and Z. Trpovski, Advanced impulse detection based on pixel-wise mad, IEEE Signal Processing Letters, 11 (2004), 589-592.  doi: 10.1109/LSP.2004.830117.

[15]

J. Delon and A. Desolneux, A patch-based approach for removing impulse or mixed gaussian-impulse noise, SIAM Journal on Imaging Sciences, 6 (2013), 1140-1174.  doi: 10.1137/120885000.

[16]

J. Delon, A. Desolneux and T. Guillemot, Parigi: A patch-based approach to remove impulse-gaussian noise from images, Image Processing On Line, 6 (2016), 130-154.  doi: 10.5201/ipol.2016.161.

[17]

Y. Dong, R. H. Chan and S. Xu, A detection statistic for random-valued impulse noise, IEEE Transactions on Image Processing, 16 (2007), 1112-1120.  doi: 10.1109/TIP.2006.891348.

[18]

R. Garnett, T. Huegerich, C. Chui and W. He, A universal noise removal algorithm with an impulse detector, IEEE Transactions on Image Processing, 14 (2005), 1747-1754.  doi: 10.1109/TIP.2005.857261.

[19]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, Englewood Cliffs, NJ: Prentice-Hall, 2002.

[20]

M.-H. Hsieh, F.-C. Cheng, M.-C. Shie and S.-J. Ruan, Fast and efficient median filter for removing 1–99% levels of salt-and-pepper noise in images, Engineering Applications of Artificial Intelligence, 26 (2013), 1333-1338.  doi: 10.1016/j.engappai.2012.10.012.

[21]

H. Hu, B. Li and Q. Liu, Removing mixture of gaussian and impulse noise by patch-based weighted means, Journal of Scientific Computing, 67 (2016), 103-129.  doi: 10.1007/s10915-015-0073-9.

[22]

S. Huang and J. Zhu, Removal of salt-and-pepper noise based on compressed sensing, Electronics Letters, 46 (2010), 1198-1199.  doi: 10.1049/el.2010.0833.

[23]

I. F. Jafar, J. Amman, R. A. AlNa'mneh and K. A. Darabkh, Efficient improvements on the BDND filtering algorithm for the removal of high-density impulse noise, IEEE Transactions on Image Processing, 22 (2013), 1223-1232.  doi: 10.1109/TIP.2012.2228496.

[24]

Q. Jin, L. Bai, J. Yang, I. Grama and Q. Liu, A new method for removing random-valued impulse noise, International Conference on Neural Information Processing, Springer, 2014, 9–16. doi: 10.1007/978-3-319-12643-2_2.

[25]

Q. Jin, I. Grama, C. Kervrann and Q. Liu, Nonlocal means and optimal weights for noise removal, Siam Journal on Imaging Sciences, 10 (2017), 1878-1920.  doi: 10.1137/16M1080781.

[26]

Q. Jin, I. Grama and Q. Liu, A new poisson noise filter based on weights optimization, Journal of Scientific Computing, 58 (2014), 548-573.  doi: 10.1007/s10915-013-9743-7.

[27]

S.-J. Ko and Y. H. Lee, Center weighted median filters and their applications to image enhancement, IEEE Transactions on Circuits and Systems, 38 (1991), 984-993.  doi: 10.1109/31.83870.

[28]

B. Li, Q. Liu, J. Xu and X. Luo, A new method for removing mixed noises, Science China Information Sciences, 54 (2011), 51-59.  doi: 10.1007/s11432-010-4128-0.

[29]

C.-Y. Lien, P.-Y. Chen, L.-Y. Chang, Y.-M. Lin and P.-K. Chang, An efficient denoising chip for the removal of impulse noise, IEEE International Symposium on Circuits and Systems, 2010, 1169–1172. doi: 10.1109/ISCAS.2010.5537308.

[30]

C.-Y. Lien, C.-C. Huang, P.-Y. Chen and Y.-F. Lin, An efficient denoising architecture for removal of impulse noise in images, IEEE Transactions on Computers, 62 (2013), 631-643.  doi: 10.1109/TC.2011.256.

[31]

H.-M. Lin and A. N. Willson Jr, Median filters with adaptive length, IEEE Transactions on Circuits and Systems, 35 (1988), 675-690.  doi: 10.1109/31.1805.

[32]

T. Melange, M. Nachtegael and E. E. Kerre, Fuzzy Random Impulse Noise Removal From Color Image Sequences, IEEE Transactions on Image Processing, 20 (2011), 959-970.  doi: 10.1109/TIP.2010.2077305.

[33]

M. S. Nair and P. M. Ameera Mol, Noise adaptive weighted switching median filter for removing high density impulse noise, Advances in Computing and Communications, Springer, 2011, 193–204. doi: 10.1007/978-3-642-22720-2_19.

[34]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. 

[35]

G. Pok, J.-C. Liu and A. S. Nair, Selective removal of impulse noise based on homogeneity level information, IEEE Transactions on Image Processing, 12 (2003), 85-92. 

[36]

K. Prathiba, R. Rathi and C. S. Christopher, Random valued impulse denoising using robust direction based detector, Information & Communication Technologies (ICT), 2013 IEEE Conference on, IEEE, 2013, 1237–1242. doi: 10.1109/CICT.2013.6558290.

[37]

W. K. Pratt, Median Filtering, Image Process. Inst., Univ. Southern California, Los Angeles, 1975.

[38]

W. Pruitt, Summability of independent random variables, J. Math. Mech., 15 (1966), 769-776. 

[39]

F. Russo, Impulse noise cancellation in image data using a two-output nonlinear filter, Measurement, 36 (2004), 205-213.  doi: 10.1016/j.measurement.2004.09.002.

[40]

K. S. Srinivasan and D. Ebenezer, A new fast and efficient decision-based algorithm for removal of high-density impulse noises, IEEE Signal Processing Letters, 14 (2007), 189-192.  doi: 10.1109/LSP.2006.884018.

[41]

T. Sun and Y. Neuvo, Detail-preserving median based filters in image processing, Pattern Recognition Letters, 15 (1994), 341-347.  doi: 10.1016/0167-8655(94)90082-5.

[42]

C. Wang, T. Chen and Z. Qu, A novel improved median filter for salt-and-pepper noise from highly corrupted images, Systems and Control in Aeronautics and Astronautics (ISSCAA), 2010 3rd International Symposium on, IEEE, 2010, 718–722.

[43]

S.-S. Wang and C.-H. Wu, A new impulse detection and filtering method for removal of wide range impulse noises, Pattern Recognition, 42 (2009), 2194-2202.  doi: 10.1016/j.patcog.2009.01.022.

[44]

Z. Wang and D. Zhang, Progressive switching median filter for the removal of impulse noise from highly corrupted images, Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, 46 (1999), 78-80. 

[45]

M. Waqas, S. G. Javed and A. Khan, Random-valued impulse noise removal from images: K-means and luo-statistics based detector and nonlocal means based estimator, Applied Sciences and Technology (IBCAST), 2014 11th International Bhurban Conference on, IEEE, 2014, 130–135. doi: 10.1109/IBCAST.2014.6778135.

[46]

L. Wenbin, A new efficient impulse detection algorithm for the removal of impulse noise, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 88 (2005), 2579-2586. 

[47]

P. S. Windyga, Fast impulsive noise removal, IEEE Transactions on Image Processing, 10 (2001), 173-179.  doi: 10.1109/83.892455.

[48]

J. Wu and C. Tang, Pde-based random-valued impulse noise removal based on new class of controlling functions, IEEE Transactions on Image Processing, 20 (2011), 2428-2438.  doi: 10.1109/TIP.2011.2131664.

[49]

___, Random-valued impulse noise removal using fuzzy weighted non-local means, Signal, Image and Video Processing, 8 (2014), 349–355.

[50]

B. Xiong and Z. Yin, A universal denoising framework with a new impulse detector and nonlocal means, IEEE Transactions on Image Processing, 21 (2012), 1663-1675.  doi: 10.1109/TIP.2011.2172804.

[51]

Y. Y. Zhou, Z. F. Ye and J. J. Huang, Improved decision-based detail-preserving variational method for removal of random-valued impulse noise, IET Image Processing, 6 (2012), 976-985.  doi: 10.1049/iet-ipr.2011.0312.

[52]

Z. Zhou, Cognition and removal of impulse noise with uncertainty, IEEE Transactions on Image Processing, 21 (2012), 3157-3167.  doi: 10.1109/TIP.2012.2189577.

[53]

Z. Zhu, X. Zhang, X. Wan and Q. Wang, A random-valued impulse noise removal algorithm with local deviation index and edge-preserving regularization, Signal, Image and Video Processing, 9 (2015), 221-228.  doi: 10.1007/s11760-013-0426-5.

[54]

Z. Zhu, X. Zhang, Q. Wang, X. Wan and Y. Xiao, Edge-preserving regularized filter with spatial local outlier measure and q-estimate, Circuits, Systems, and Signal Processing, 33 (2014), 629-642.  doi: 10.1007/s00034-013-9644-x.

Figure 1.  The mean of PSNR values of the denoised Lena image with impulse noise level $ p = 20\%, 40\% $ and $ 60\% $, as a function of the threshold b for SROAD values.
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Figure 2.  Mean values of ROAD of noisy pixels and uncorrupted pixels in the Lena image as a function of the impulse noise probability, with standard deviation error bars
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Figure 3.  Mean values of ROLD of noisy pixels and uncorrupted pixels in the Lena image as a function of the impulse noise probability, with standard deviation error bars
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Figure 4.  Mean values of Reliable Weight of noisy pixels and uncorrupted pixels in the Lena image as a function of the impulse noise probability, with standard deviation error bars
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Figure 5.  The evolution of PSNR value as a function of the size of image. Here we take the $ 1024\times 1024 $ test image Male as example, and zoom it out into $ 512\times 512 $, $ 256\times 256 $, $ 128\times128 $ and $ 64\times64 $ images
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Figure 6.  The evolution of PSNR value as a function of the size of similarity patch with level of the impulse noise $ p = 20\% $
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Figure 7.  The evolution of PSNR value as a function of the size of similarity patch with level of the impulse noise $ p = 40\% $
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Figure 8.  The evolution of PSNR value as a function of the size of similarity patch with level of the impulse noise $ p = 60\% $.
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Figure 9.  The set of 20 tested images. The commonly-used images (a)-(d) are provided by the authors of [17]; the images (e)-(m) are taken from USC-SIPI Image Database: http://sipi.usc.edu/database/; the images (n)-(t) are from: http://www.cs.tut.fi/ foi/GCF-BM3D/.
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Figure 10.  Results of different methods in restoring $ 40\% $ corrupted Baboon image
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Figure 11.  Results of different methods in restoring $ 40\% $ corrupted Lena image
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Figure 12.  Results of different methods in restoring $ 60\% $ corrupted Bridge image
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Figure 13.  Results of different methods in restoring $ 60\% $ corrupted Pentagon image
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Figure 14.  The Difference between the denoised image with impulse noise level $ p = 40\% $ and original image
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Table 1.  Comparison of restoration in PSNR(db) for images corrupted with random-valued impulse noise
Images Baboon Bridge Lena Pentagon Average
p $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
MF$^*$ [37] 22.52db 20.65db 19.36db 25.04db 22.17db 19.36db 32.37db 27.64db 21.58db 28.29db 25.16db 23.41db 27.06db 23.91db 20.93db
SS-II$^*$[41] 23.67db 20.85db 19.27db 26.26db 22.66db 19.13db 32.93db 27.90db 20.61db 29.34db 26.26db 23.90db 28.05db 24.42db 20.73db
SS-I$^*$ [41] 22.46db 21.35db 19.42db 25.90db 22.85db 19.04db 33.43db 27.75db 20.61db 28.28db 26.43db 23.85db 27.52db 24.60db 20.73db
SD-ROM$^*$[1] 23.81db 21.49db 19.45db 26.56db 23.80db 20.66db 35.71db 29.85db 23.41db 30.38db 27.27db 24.33db 29.12db 25.60db 21.96db
PSM$^*$[44] 23.43db 21.07db 19.56db 26.33db 22.75db 19.73db 35.09db 28.92db 22.06db 29.18db 26.19db 23.87db 28.51db 24.73db 21.31db
TSM$^*$[11] 23.73db 21.38db 19.44db 26.52db 22.89db 19.60db 34.21db 28.30db 21.67db 29.29db 26.29db 23.59db 28.44db 24.71db 21.08db
MSM$^*$[13] 24.02db 21.52db 19.63db 27.27db 23.55db 20.07db 35.44db 29.26db 22.14db 30.34db 27.04db 24.22db 29.27db 25.34db 21.52db
ACWM$^*$[12] 24.17db 21.58db 19.56db 27.08db 23.23db 19.27db 36.07db 28.79db 21.19db 30.23db 26.84db 23.50db 29.39db 25.11db 20.88db
PWMAD$^*$[14] 23.78db 21.56db 19.68db 26.90db 23.83db 20.83db 36.50db 31.41db 24.30db 30.11db 27.33db 24.46db 29.32db 26.03db 22.32db
Luo-IMF$^*$ [46] 24.18db 21.41db 19.08db 27.05db 23.88db 19.74db 36.90db 30.25db 22.96db 30.42db 26.93db 23.72db 29.64db 25.62db 21.38db
TriF$^*$[18] 24.18db 21.60db 19.52db 27.60db 24.01db 20.84db 36.70db 31.12db 26.08db 30.33db 27.14db 24.60db 29.70db 25.97db 22.76db
ACWM-EPR$^*$ [10] 23.97db 21.62db 19.87db 27.31db 24.60db 20.89db 36.57db 32.21db 24.62db 30.03db 27.35db 24.59db 29.47db 26.45db 22.49db
ROAD-EPR$^*$[17] 24.24db 21.53db 19.96db 27.42db 24.52db 22.04db 36.79db 32.32db 28.37db 30.35db 27.06db 25.00db 29.70db 26.36db 23.84db
ROLD-EPR$^*$ [17] 24.49db 21.92db 20.38db 27.86db 24.79db $\textbf{22.59}$db 37.45db 32.76db 29.03db 30.73db 27.73db 25.70db 30.13db 26.80db 24.43db
FWNLM [49] 23.45db 21.71db 20.45db 26.82db 24.23db 22.23db 34.95db 32.12db 28.03db 30.26db 27.48db 25.48db 28.87db 26.39db 24.05db
PWMF [21] 24.84db 22.00db 14.72db $\textbf{28.12}$db 24.71db 11.58db $\textbf{37.98}$db $\textbf{33.51}$db 10.41db $\textbf{31.53}$db 27.83db 11.39db $\textbf{30.62}$db 27.01db 12.03db
PARIGI [15,16] 24.46db 21.85db 19.79db 26.53db 24.06db 21.40db 36.62db 31.94db 27.61db 30.63db $\textbf{28.44db}$ 25.44db 29.56db 26.57db 23.56db
RRWF $\textbf{25.01}$db $\textbf{22.41db}$ $\textbf{20.46db}$ 27.86db $\textbf{24.91db}$ 22.49db 37.56db $\textbf{33.07db}$ $\textbf{29.05db}$ $\textbf{31.18db}$ 28.19db $\textbf{25.78db}$ 30.40db $\textbf{27.15}$db $ \textbf{24.45}$ db
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Images Baboon Bridge Lena Pentagon Average
p $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$ $20\%$ $40\%$ $60\%$
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
MF$^*$ [37] 22.52db 20.65db 19.36db 25.04db 22.17db 19.36db 32.37db 27.64db 21.58db 28.29db 25.16db 23.41db 27.06db 23.91db 20.93db
SS-II$^*$[41] 23.67db 20.85db 19.27db 26.26db 22.66db 19.13db 32.93db 27.90db 20.61db 29.34db 26.26db 23.90db 28.05db 24.42db 20.73db
SS-I$^*$ [41] 22.46db 21.35db 19.42db 25.90db 22.85db 19.04db 33.43db 27.75db 20.61db 28.28db 26.43db 23.85db 27.52db 24.60db 20.73db
SD-ROM$^*$[1] 23.81db 21.49db 19.45db 26.56db 23.80db 20.66db 35.71db 29.85db 23.41db 30.38db 27.27db 24.33db 29.12db 25.60db 21.96db
PSM$^*$[44] 23.43db 21.07db 19.56db 26.33db 22.75db 19.73db 35.09db 28.92db 22.06db 29.18db 26.19db 23.87db 28.51db 24.73db 21.31db
TSM$^*$[11] 23.73db 21.38db 19.44db 26.52db 22.89db 19.60db 34.21db 28.30db 21.67db 29.29db 26.29db 23.59db 28.44db 24.71db 21.08db
MSM$^*$[13] 24.02db 21.52db 19.63db 27.27db 23.55db 20.07db 35.44db 29.26db 22.14db 30.34db 27.04db 24.22db 29.27db 25.34db 21.52db
ACWM$^*$[12] 24.17db 21.58db 19.56db 27.08db 23.23db 19.27db 36.07db 28.79db 21.19db 30.23db 26.84db 23.50db 29.39db 25.11db 20.88db
PWMAD$^*$[14] 23.78db 21.56db 19.68db 26.90db 23.83db 20.83db 36.50db 31.41db 24.30db 30.11db 27.33db 24.46db 29.32db 26.03db 22.32db
Luo-IMF$^*$ [46] 24.18db 21.41db 19.08db 27.05db 23.88db 19.74db 36.90db 30.25db 22.96db 30.42db 26.93db 23.72db 29.64db 25.62db 21.38db
TriF$^*$[18] 24.18db 21.60db 19.52db 27.60db 24.01db 20.84db 36.70db 31.12db 26.08db 30.33db 27.14db 24.60db 29.70db 25.97db 22.76db
ACWM-EPR$^*$ [10] 23.97db 21.62db 19.87db 27.31db 24.60db 20.89db 36.57db 32.21db 24.62db 30.03db 27.35db 24.59db 29.47db 26.45db 22.49db
ROAD-EPR$^*$[17] 24.24db 21.53db 19.96db 27.42db 24.52db 22.04db 36.79db 32.32db 28.37db 30.35db 27.06db 25.00db 29.70db 26.36db 23.84db
ROLD-EPR$^*$ [17] 24.49db 21.92db 20.38db 27.86db 24.79db $\textbf{22.59}$db 37.45db 32.76db 29.03db 30.73db 27.73db 25.70db 30.13db 26.80db 24.43db
FWNLM [49] 23.45db 21.71db 20.45db 26.82db 24.23db 22.23db 34.95db 32.12db 28.03db 30.26db 27.48db 25.48db 28.87db 26.39db 24.05db
PWMF [21] 24.84db 22.00db 14.72db $\textbf{28.12}$db 24.71db 11.58db $\textbf{37.98}$db $\textbf{33.51}$db 10.41db $\textbf{31.53}$db 27.83db 11.39db $\textbf{30.62}$db 27.01db 12.03db
PARIGI [15,16] 24.46db 21.85db 19.79db 26.53db 24.06db 21.40db 36.62db 31.94db 27.61db 30.63db $\textbf{28.44db}$ 25.44db 29.56db 26.57db 23.56db
RRWF $\textbf{25.01}$db $\textbf{22.41db}$ $\textbf{20.46db}$ 27.86db $\textbf{24.91db}$ 22.49db 37.56db $\textbf{33.07db}$ $\textbf{29.05db}$ $\textbf{31.18db}$ 28.19db $\textbf{25.78db}$ 30.40db $\textbf{27.15}$db $ \textbf{24.45}$ db
Table 2.  Comparison of restoration in PSNR(db) for images corrupted with random-valued impulse noise with $ p = 20 $
Images Aerial Airplane B.wall Cam Clock Male512 Male1024 M.surface P.bubbles R.chart Barbara boat couple F.print hill house Average
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
FWNLM [49] 36.50db 40.78db 36.47db $\textbf{38.63db}$ 40.02db $\textbf{38.74db}$ $\textbf{39.59db}$ 37.08db 37.56db 37.51db $\textbf{38.56db}$ 38.51db 38.09db $\textbf{36.56db}$ 38.85db 40.51db 38.37db
PWMF[21] 36.04db 39.77db 36.15db 38.52db 39.52db 38.38db 38.76db 36.03db $\textbf{37.77db}$ 36.62db 37.51db 37.88db 37.55db 36.32db 38.54db 39.85db 37.83db
PARIGI [15,16] 33.13db 40.46db 32.64db 36.63db 38.95db 35.71db 37.09db 34.79db 34.85db $\textbf{38.35db}$ 37.06db 35.90db 35.67db 32.07db 36.53db 39.89db 36.23db
RRWF $\textbf{36.49db}$ $\textbf{41.73db}$ $\textbf{37.03db}$ 38.36db $\textbf{40.27db}$ 38.46db 39.56db $\textbf{37.61db}$ 37.57db 37.80db 37.88db $\textbf{38.87db}$ $\textbf{38.41db}$ 36.24db $\textbf{39.00db}$ $\textbf{41.20db}$ $\textbf{38.54db}$
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Images Aerial Airplane B.wall Cam Clock Male512 Male1024 M.surface P.bubbles R.chart Barbara boat couple F.print hill house Average
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
FWNLM [49] 36.50db 40.78db 36.47db $\textbf{38.63db}$ 40.02db $\textbf{38.74db}$ $\textbf{39.59db}$ 37.08db 37.56db 37.51db $\textbf{38.56db}$ 38.51db 38.09db $\textbf{36.56db}$ 38.85db 40.51db 38.37db
PWMF[21] 36.04db 39.77db 36.15db 38.52db 39.52db 38.38db 38.76db 36.03db $\textbf{37.77db}$ 36.62db 37.51db 37.88db 37.55db 36.32db 38.54db 39.85db 37.83db
PARIGI [15,16] 33.13db 40.46db 32.64db 36.63db 38.95db 35.71db 37.09db 34.79db 34.85db $\textbf{38.35db}$ 37.06db 35.90db 35.67db 32.07db 36.53db 39.89db 36.23db
RRWF $\textbf{36.49db}$ $\textbf{41.73db}$ $\textbf{37.03db}$ 38.36db $\textbf{40.27db}$ 38.46db 39.56db $\textbf{37.61db}$ 37.57db 37.80db 37.88db $\textbf{38.87db}$ $\textbf{38.41db}$ 36.24db $\textbf{39.00db}$ $\textbf{41.20db}$ $\textbf{38.54db}$
Table 3.  Comparison of restoration in PSNR(db) for images corrupted with random-valued impulse noise with $p = 40$
Images Aerial Airplane B.wall Cam Clock Male512 Male1024 M.surface P.bubbles R.chart Barbara boat couple F.print hill house Average
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
FWNLM [49] 33.17db 37.95db 33.32db 35.56db 36.91db 35.06db 36.28db 35.00db 34.47db 35.80db $\textbf{34.80db}$ 34.96db 34.40db 32.33db 35.56db 37.55db 35.20db
PWMF[21] $\textbf{33.74db}$ 38.14db 33.47db $\textbf{36.09db}$ $\textbf{37.50db}$ $\textbf{35.44db}$ 36.33db 34.77db 34.88db $\textbf{35.98db}$ 34.64db 35.28db 34.95db $\textbf{33.02db}$ 35.96db 37.93db 35.51db
PARIGI [15, 16] 30.80db 36.34db 30.93db 34.56db 34.72db 33.01db 34.15db 33.22db 32.27db 35.66db 33.08db 32.90db 32.45db 28.9594db 33.69db 36.29db 33.31db
RRWF 33.43db $\textbf{38.37db}$ $\textbf{34.21db}$ 35.79db 36.94db 35.41db $\textbf{36.70db}$ $\textbf{35.58db}$ $\textbf{35.04db}$ 35.74db 34.78db $\textbf{35.54db}$ $\textbf{35.19db}$ 32.55db $\textbf{36.25db}$ $\textbf{38.41db}$ $\textbf{35.62db} $
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Images Aerial Airplane B.wall Cam Clock Male512 Male1024 M.surface P.bubbles R.chart Barbara boat couple F.print hill house Average
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
FWNLM [49] 33.17db 37.95db 33.32db 35.56db 36.91db 35.06db 36.28db 35.00db 34.47db 35.80db $\textbf{34.80db}$ 34.96db 34.40db 32.33db 35.56db 37.55db 35.20db
PWMF[21] $\textbf{33.74db}$ 38.14db 33.47db $\textbf{36.09db}$ $\textbf{37.50db}$ $\textbf{35.44db}$ 36.33db 34.77db 34.88db $\textbf{35.98db}$ 34.64db 35.28db 34.95db $\textbf{33.02db}$ 35.96db 37.93db 35.51db
PARIGI [15, 16] 30.80db 36.34db 30.93db 34.56db 34.72db 33.01db 34.15db 33.22db 32.27db 35.66db 33.08db 32.90db 32.45db 28.9594db 33.69db 36.29db 33.31db
RRWF 33.43db $\textbf{38.37db}$ $\textbf{34.21db}$ 35.79db 36.94db 35.41db $\textbf{36.70db}$ $\textbf{35.58db}$ $\textbf{35.04db}$ 35.74db 34.78db $\textbf{35.54db}$ $\textbf{35.19db}$ 32.55db $\textbf{36.25db}$ $\textbf{38.41db}$ $\textbf{35.62db} $
Table 4.  Comparison of restoration in PSNR(db) for images corrupted with random-valued impulse noise with $p = 60$
Images Aerial Airplane B.wall Cam Clock Male512 Male1024 M.surface P.bubbles R.chart Barbara boat couple F.print hill house Average
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
FWNLM [49] 31.84db 37.39db 32.28db 33.54db $\textbf{35.43db}$ $\textbf{32.64db}$ 33.63db 33.64db 32.79db $\textbf{34.69db}$ 32.55db 33.18db 32.67db 30.12db 33.55db 35.60db 33.47db
PARIGI[15,16] 30.13db 35.83db 30.19db 33.44db 33.71db 31.50db 32.27db 32.14db 30.55db 34.56db 31.18db 31.62db 31.00db 28.21db 31.75db 34.54db 32.04db
RRWF $\textbf{32.07db}$ $\textbf{37.75db}$ $\textbf{32.73db}$ $\textbf{33.59db}$ 35.35db 32.61db $\textbf{33.70db}$ $\textbf{33.81db}$ $\textbf{33.21db}$ 34.10db $\textbf{33.06db}$ $\textbf{33.32db}$ $\textbf{33.00db}$ $\textbf{30.18db}$ $\textbf{33.74db}$ $\textbf{36.00db}$ $\textbf{3.64db}$
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Images Aerial Airplane B.wall Cam Clock Male512 Male1024 M.surface P.bubbles R.chart Barbara boat couple F.print hill house Average
Method PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR PSNR
FWNLM [49] 31.84db 37.39db 32.28db 33.54db $\textbf{35.43db}$ $\textbf{32.64db}$ 33.63db 33.64db 32.79db $\textbf{34.69db}$ 32.55db 33.18db 32.67db 30.12db 33.55db 35.60db 33.47db
PARIGI[15,16] 30.13db 35.83db 30.19db 33.44db 33.71db 31.50db 32.27db 32.14db 30.55db 34.56db 31.18db 31.62db 31.00db 28.21db 31.75db 34.54db 32.04db
RRWF $\textbf{32.07db}$ $\textbf{37.75db}$ $\textbf{32.73db}$ $\textbf{33.59db}$ 35.35db 32.61db $\textbf{33.70db}$ $\textbf{33.81db}$ $\textbf{33.21db}$ 34.10db $\textbf{33.06db}$ $\textbf{33.32db}$ $\textbf{33.00db}$ $\textbf{30.18db}$ $\textbf{33.74db}$ $\textbf{36.00db}$ $\textbf{3.64db}$
Table 5.  Comparison of the speed of different algorithms: average runtime in seconds (s) with a set of $ 512 \times 512 $ images using a Core(TM) i7-6820HQ CPU 2.70 GHz PC
Method TriF[18] FWNLM[49] PWMF[21] PARIGI[15,16] RRWF
Time 6.33s 1151.55s 2.16s 17.69s 1.51s
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Method TriF[18] FWNLM[49] PWMF[21] PARIGI[15,16] RRWF
Time 6.33s 1151.55s 2.16s 17.69s 1.51s
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