American Institute of Mathematical Sciences

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April  2018, 14(2): 597-611. doi: 10.3934/jimo.2017062

An optimized direction statistics for detecting and removing random-valued impulse noise

Hao Yang 1, , Hang Qiu 2, and  Leiting Chen 2,,

1. 

School of Computer Science, Chengdu University of Information Technology, No.24 Block 1, Xuefu Road, 610225, Chengdu, China
2. 

School of Computer Science and Engineering, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, 611731, Chengdu, China

* Corresponding author: Leiting Chen

Received  January 2016 Revised  September 2016 Published  June 2017

In this paper, we propose a robust local image statistic based on optimized direction, by which we can distinguish image details and edges from impulse noise effectively. Therefore it can identify noisy pixels more accurately. Meanwhile, we combine it with the edge-preserving regularization to remove random-valued impulse noise in the cause of precise estimated value. Simulation results show that our method can preserve edges and details efficiently even at high noise levels.

Keywords: Random-valued impulse noise, weight rank-order differences, detail-preserving regularization, image denoising.
Mathematics Subject Classification: Primary: 94A08; Secondary: 47A52.
Citation: Hao Yang, Hang Qiu, Leiting Chen. An optimized direction statistics for detecting and removing random-valued impulse noise. Journal of Industrial & Management Optimization, 2018, 14 (2) : 597-611. doi: 10.3934/jimo.2017062
References:
[1]

E. Abreu, M. Lightstone and S. K. Mitra, 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. Google Scholar

[2]

S. Akkoul, R. Lédée and R. Leconge, A new adaptive switching median filter, IEEE Signal Processing Letters, 17 (2010), 587-590. doi: 10.1109/LSP.2010.2048646. Google Scholar

[3]

G. Arce and J. Paredes, Recursive weighted median filters admitting negative weights and their optimization, IEEE Transactions on Image Processing, 48 (2000), 768-779. doi: 10.1109/78.824671. Google Scholar

[4]

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. Google Scholar

[5]

M. J. Black and A. Rangarajan, On the unification of line processes, outlier rejection, and robust statistics with applications in early vision, International Journal of Computer Vision, 19 (1996), 57-91. doi: 10.1007/BF00131148. Google Scholar

[6]

A. C. Bovik, Handbook of Image and Video Processing, 2nd edition, Academic press, 2010, New York, 2010.Google Scholar

[7]

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

[8]

J.-F. Cai, R. H. Chan and C. Fiore, Minimization of a detail-preserving regularization functional for impulse noise removal, IEEE Transactions on Image Processing, 29 (2007), 79-91. doi: 10.1007/s10851-007-0027-4. Google Scholar

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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. Google Scholar

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R. H. Chan, C.-W. Ho and C.-Y. Leung, Minimization of detail-preserving regularization functional by Newton's method with continuation, Proceedings -International Conference on Image Processing, ICIP, 1 (2005), 125-128. doi: 10.1109/ICIP.2005.1529703. Google Scholar

[11]

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. Google Scholar

[12]

P. Charbonnier, L. Blanc-Féraud and G. Aubert, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6 (1997), 298-311. doi: 10.1109/83.551699. Google Scholar

[13]

T. Chen and H. R. Wu, Adaptive impulse detection using center-weighted medial filters, IEEE Transactions on Image Processing Letters, 8 (2001), 1-3. doi: 10.1109/97.889633. Google Scholar

[14]

Y. Dong, H. R. Chan and S. Xu, Edge-preserving regularization, Image denoising, Noise detector, Random-valued impulse noise, IEEE Transactions on Image Processing, 16 (2007), 1112-1120. doi: 10.1109/TIP.2006.891348. Google Scholar

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Y. Dong and S. Xu, A new directional weighted median filter for removal of random-valued impulse noise, IEEE Transactions on Image Processing, 14 (2007), 193-196. doi: 10.1109/LSP.2006.884014. Google Scholar

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

[17]

U. Ghaneka, A. K. Singh and and R. Pandey, A contrast enhancement-based filter for removal of random valued impulse noise, IEEE Signal Processing Letters, 17 (2010), 47-50. doi: 10.1109/LSP.2009.2032479. Google Scholar

[18]

R. Gonzalez and R. Woods, Digital Image Processing, 2nd edition, Addision-Wesley Publishing Companyl, 2007.Google Scholar

[19]

P. J. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Transactions on Medical Imaging, 9 (1990), 84-93. doi: 10.1109/42.52985. Google Scholar

[20]

H. H. Dam, K. L. Teo and S. Nordebo, The dual parameterization approach to optimal least square FIR filter design subject to maximum error constraints, IEEE Transactions on Signal Processing, 48 (2000), 2314-2320. doi: 10.1109/78.852012. Google Scholar

[21]

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. Google Scholar

[22]

S. Z. Li, On discontinuity-adaptive smoothness priors in computer vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 17 (1995), 576-586. doi: 10.1109/34.387504. Google Scholar

[23]

L. Liu, C. P. Chen and Y. Zhou, A new weighted mean filter with a two-phase detector for removing impulse noise, Information Sciences, 315 (2015), 1-16. doi: 10.1016/j.ins.2015.03.067. Google Scholar

[24]

W. Luo, 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. doi: 10.1093/ietfec/e88-a.10.2579. Google Scholar

[25]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar

[26]

W. K. Pratt, Median Filtering, Image Proc Institute, University of Southern California, Los Angeles, Tech. Rep., 1975.Google Scholar

[27]

F. Russo, Hybrid neuro-fuzzy filter for impulse noise removal, Pattern Recognition, 32 (1999), 1843-1855. doi: 10.1016/S0031-3203(99)00009-6. Google Scholar

[28]

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. Google Scholar

[29]

K. Toh and N. Isa, Cluster-based adaptive fuzzy switching median filter for universal impulse noise reduction, IEEE Transactions on Consumer Electronics, 56 (2010), 2560-2568. doi: 10.1109/TCE.2010.5681141. Google Scholar

[30]

D. Van De Ville, M. Nachtegael and D. Van der Weken, Noise reduction by fuzzy image filtering, IEEE Transactions on Fuzzy Systems, 11 (2003), 429-436. doi: 10.1109/TFUZZ.2003.814830. Google Scholar

[31]

C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813-824. doi: 10.1109/83.679423. Google Scholar

[32]

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. Google Scholar

[33]

H. Xu, G. Zhu and H. Peng, Adaptive fuzzy switching filter for images corrupted by impulse noise, Pattern Recognition Letters, 25 (2004), 1657-1663. doi: 10.1016/j.patrec.2004.05.025. Google Scholar

[34]

M. E. Yüksel and A. Baştürk, A simple generalized neuro-fuzzy operator for efficient removal of impulse noise from highly corrupted digital images, AEU -International Journal of Electronics and Communications, 5 (1996), 1012-1025. doi: 10.1016/j.aeue.2004.10.002. Google Scholar

[35]

M. E. Yüksel, A hybrid neuro-fuzzy filter for edge preserving restoration of images corrupted by impulse noise, IEEE Transactions on Image Processing, 15 (2006), 928-936. doi: 10.1109/TIP.2005.863941. Google Scholar

[36]

X.-Y. Zeng and L.-H. Yang, Mixed impulse and gaussian noise removal using detail-preserving regularization, Optical Engineering, 49 (2010), 097002-097002. doi: 10.1117/1.3485756. Google Scholar

show all references

References:
[1]

E. Abreu, M. Lightstone and S. K. Mitra, 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. Google Scholar

[2]

S. Akkoul, R. Lédée and R. Leconge, A new adaptive switching median filter, IEEE Signal Processing Letters, 17 (2010), 587-590. doi: 10.1109/LSP.2010.2048646. Google Scholar

[3]

G. Arce and J. Paredes, Recursive weighted median filters admitting negative weights and their optimization, IEEE Transactions on Image Processing, 48 (2000), 768-779. doi: 10.1109/78.824671. Google Scholar

[4]

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. Google Scholar

[5]

M. J. Black and A. Rangarajan, On the unification of line processes, outlier rejection, and robust statistics with applications in early vision, International Journal of Computer Vision, 19 (1996), 57-91. doi: 10.1007/BF00131148. Google Scholar

[6]

A. C. Bovik, Handbook of Image and Video Processing, 2nd edition, Academic press, 2010, New York, 2010.Google Scholar

[7]

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

[8]

J.-F. Cai, R. H. Chan and C. Fiore, Minimization of a detail-preserving regularization functional for impulse noise removal, IEEE Transactions on Image Processing, 29 (2007), 79-91. doi: 10.1007/s10851-007-0027-4. Google Scholar

[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. Google Scholar

[10]

R. H. Chan, C.-W. Ho and C.-Y. Leung, Minimization of detail-preserving regularization functional by Newton's method with continuation, Proceedings -International Conference on Image Processing, ICIP, 1 (2005), 125-128. doi: 10.1109/ICIP.2005.1529703. Google Scholar

[11]

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. Google Scholar

[12]

P. Charbonnier, L. Blanc-Féraud and G. Aubert, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6 (1997), 298-311. doi: 10.1109/83.551699. Google Scholar

[13]

T. Chen and H. R. Wu, Adaptive impulse detection using center-weighted medial filters, IEEE Transactions on Image Processing Letters, 8 (2001), 1-3. doi: 10.1109/97.889633. Google Scholar

[14]

Y. Dong, H. R. Chan and S. Xu, Edge-preserving regularization, Image denoising, Noise detector, Random-valued impulse noise, IEEE Transactions on Image Processing, 16 (2007), 1112-1120. doi: 10.1109/TIP.2006.891348. Google Scholar

[15]

Y. Dong and S. Xu, A new directional weighted median filter for removal of random-valued impulse noise, IEEE Transactions on Image Processing, 14 (2007), 193-196. doi: 10.1109/LSP.2006.884014. Google Scholar

[16]

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

[17]

U. Ghaneka, A. K. Singh and and R. Pandey, A contrast enhancement-based filter for removal of random valued impulse noise, IEEE Signal Processing Letters, 17 (2010), 47-50. doi: 10.1109/LSP.2009.2032479. Google Scholar

[18]

R. Gonzalez and R. Woods, Digital Image Processing, 2nd edition, Addision-Wesley Publishing Companyl, 2007.Google Scholar

[19]

P. J. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Transactions on Medical Imaging, 9 (1990), 84-93. doi: 10.1109/42.52985. Google Scholar

[20]

H. H. Dam, K. L. Teo and S. Nordebo, The dual parameterization approach to optimal least square FIR filter design subject to maximum error constraints, IEEE Transactions on Signal Processing, 48 (2000), 2314-2320. doi: 10.1109/78.852012. Google Scholar

[21]

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. Google Scholar

[22]

S. Z. Li, On discontinuity-adaptive smoothness priors in computer vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 17 (1995), 576-586. doi: 10.1109/34.387504. Google Scholar

[23]

L. Liu, C. P. Chen and Y. Zhou, A new weighted mean filter with a two-phase detector for removing impulse noise, Information Sciences, 315 (2015), 1-16. doi: 10.1016/j.ins.2015.03.067. Google Scholar

[24]

W. Luo, 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. doi: 10.1093/ietfec/e88-a.10.2579. Google Scholar

[25]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar

[26]

W. K. Pratt, Median Filtering, Image Proc Institute, University of Southern California, Los Angeles, Tech. Rep., 1975.Google Scholar

[27]

F. Russo, Hybrid neuro-fuzzy filter for impulse noise removal, Pattern Recognition, 32 (1999), 1843-1855. doi: 10.1016/S0031-3203(99)00009-6. Google Scholar

[28]

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. Google Scholar

[29]

K. Toh and N. Isa, Cluster-based adaptive fuzzy switching median filter for universal impulse noise reduction, IEEE Transactions on Consumer Electronics, 56 (2010), 2560-2568. doi: 10.1109/TCE.2010.5681141. Google Scholar

[30]

D. Van De Ville, M. Nachtegael and D. Van der Weken, Noise reduction by fuzzy image filtering, IEEE Transactions on Fuzzy Systems, 11 (2003), 429-436. doi: 10.1109/TFUZZ.2003.814830. Google Scholar

[31]

C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813-824. doi: 10.1109/83.679423. Google Scholar

[32]

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. Google Scholar

[33]

H. Xu, G. Zhu and H. Peng, Adaptive fuzzy switching filter for images corrupted by impulse noise, Pattern Recognition Letters, 25 (2004), 1657-1663. doi: 10.1016/j.patrec.2004.05.025. Google Scholar

[34]

M. E. Yüksel and A. Baştürk, A simple generalized neuro-fuzzy operator for efficient removal of impulse noise from highly corrupted digital images, AEU -International Journal of Electronics and Communications, 5 (1996), 1012-1025. doi: 10.1016/j.aeue.2004.10.002. Google Scholar

[35]

M. E. Yüksel, A hybrid neuro-fuzzy filter for edge preserving restoration of images corrupted by impulse noise, IEEE Transactions on Image Processing, 15 (2006), 928-936. doi: 10.1109/TIP.2005.863941. Google Scholar

[36]

X.-Y. Zeng and L.-H. Yang, Mixed impulse and gaussian noise removal using detail-preserving regularization, Optical Engineering, 49 (2010), 097002-097002. doi: 10.1117/1.3485756. Google Scholar

Figure 1.  two kinds of edge contained in neighbor, (a) vertical edge, (b) slope edge
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Figure 2.  Directions and hops
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Figure 3.  The mean PSNR values associated with different $\alpha$ values
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Figure 4.  Total error detection
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Figure 5.  Results obtained by different algorithms for restoring the test lena image corrupted by random-valued impulse noise with 40 % noise density. (a) Noisy image, (b) ACWM, (c) Luo's method, (d) ASWM, (e) DWM, (f) ROAD-Trilateral, (g) ROR-NLM, (h) ROLD-EPR, (i) Proposed Method.
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Figure 6.  Run time of detection vs. removal noises with different density
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Figure 7.  Run time of detection vs. removal noises with different scale image
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Table 1.  sets along the $l^{th}$ direction and hop count $h$
$ S^{(1)}_{1}=\{ (-1,-1); (0, 0); (1, 1) \} $ $ S^{(1)}_{2}=\{ (-2,-2); (-1,-1); (0, 0); (1, 1); (2, 2) \} $
$ S^{(2)}_{1}=\{ (0,-1); (0, 0); (0, 1) \} $ $ S^{(2)}_{2}=\{(0,-2); (0,-1); (0, 0); (0, 1); (0, 2)\}$
$ S^{(3)}_{1}=\{ (1,-1); (0, 0); (-1, 1 \}$ $S^{(3)}_{2}=\{(2,-2); (1,-1); (0, 0); (-1, 1); (-2, 2)\}$
$ S^{(4)}_{1}=\{ (-1, 0); (0, 0); (1, 0) \} $ $ S^{(4)}_{2}=\{(-2, 0); (-1, 0); (0, 0); (1, 0); (2, 0) \} $
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$ S^{(1)}_{1}=\{ (-1,-1); (0, 0); (1, 1) \} $ $ S^{(1)}_{2}=\{ (-2,-2); (-1,-1); (0, 0); (1, 1); (2, 2) \} $
$ S^{(2)}_{1}=\{ (0,-1); (0, 0); (0, 1) \} $ $ S^{(2)}_{2}=\{(0,-2); (0,-1); (0, 0); (0, 1); (0, 2)\}$
$ S^{(3)}_{1}=\{ (1,-1); (0, 0); (-1, 1 \}$ $S^{(3)}_{2}=\{(2,-2); (1,-1); (0, 0); (-1, 1); (-2, 2)\}$
$ S^{(4)}_{1}=\{ (-1, 0); (0, 0); (1, 0) \} $ $ S^{(4)}_{2}=\{(-2, 0); (-1, 0); (0, 0); (1, 0); (2, 0) \} $
Table 2.  Comparison of noise detection results for image "Lena" with various ratios of random-valued impulse noise
Method40%50%60%
MissFalse-hitTotalMissFalse-hitTotalMissFalse-hitTotal
ACWM[13]142491928161772059636022419831165666837833
Luo[24]143651713160782059621352237133374288636260
CEF[17]147276141208681749077452523521314865729971
ASWM[2]73811104218423106141205022664195771684536422
DWM[15]1160079371953715035865223687153731421529588
ROR-NLM[32]124433056154991577836551943321601591727518
ROAD[16]1347680792155513771100552382617212933026542
ROLD[14]139877471214581633178752420617245922326468
Proposed101585234153921130265831788515234762322857
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Method40%50%60%
MissFalse-hitTotalMissFalse-hitTotalMissFalse-hitTotal
ACWM[13]142491928161772059636022419831165666837833
Luo[24]143651713160782059621352237133374288636260
CEF[17]147276141208681749077452523521314865729971
ASWM[2]73811104218423106141205022664195771684536422
DWM[15]1160079371953715035865223687153731421529588
ROR-NLM[32]124433056154991577836551943321601591727518
ROAD[16]1347680792155513771100552382617212933026542
ROLD[14]139877471214581633178752420617245922326468
Proposed101585234153921130265831788515234762322857
Table 3.  Comparison of restoration results in PSNR for images corrupted with random-valued impulse noise
Method"Lena" image"Bridge" image"Pentagon" image
40 %50 %60 %40 %50 %60 %40 %50 %60 %
ACWM[13]29.5824.6320.4023.5221.4119.1227.0925.4723.41
Luo[24]30.7727.1622.6223.5921.6219.1727.0025.3322.78
CEF[17]32.1129.7625.9023.8522.7921.4127.1626.2425.12
ASWM[2]32.2929.2325.0423.9722.5821.1127.2926.2024.98
DWM[15]32.3429.3225.4924.0722.5821.1327.2326.0725.03
ROR-NLM[32]32.9730.0225.6024.1822.8421.1927.6826.5625.36
ROAD[16]32.0730.2427.4223.7323.0921.8826.6125.9224.82
ROLD[14]32.7531.1228.9824.5123.5122.5227.5826.6525.61
Proposed33.6231.7329.5624.9823.8222.7927.9226.9825.93
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Method"Lena" image"Bridge" image"Pentagon" image
40 %50 %60 %40 %50 %60 %40 %50 %60 %
ACWM[13]29.5824.6320.4023.5221.4119.1227.0925.4723.41
Luo[24]30.7727.1622.6223.5921.6219.1727.0025.3322.78
CEF[17]32.1129.7625.9023.8522.7921.4127.1626.2425.12
ASWM[2]32.2929.2325.0423.9722.5821.1127.2926.2024.98
DWM[15]32.3429.3225.4924.0722.5821.1327.2326.0725.03
ROR-NLM[32]32.9730.0225.6024.1822.8421.1927.6826.5625.36
ROAD[16]32.0730.2427.4223.7323.0921.8826.6125.9224.82
ROLD[14]32.7531.1228.9824.5123.5122.5227.5826.6525.61
Proposed33.6231.7329.5624.9823.8222.7927.9226.9825.93
Table 4.  Run time of detection vs. removal noises with different density
Noise DensityRun Time(s)
DetectionRemovalTotal
30 % 4.7234.6939.41
40 % 4.8373.3078.13
50 %4.67163.53168.20
60 % 4.65239.58244.23
70 % 4.87271.64276.51
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Noise DensityRun Time(s)
DetectionRemovalTotal
30 % 4.7234.6939.41
40 % 4.8373.3078.13
50 %4.67163.53168.20
60 % 4.65239.58244.23
70 % 4.87271.64276.51
Table 5.  Run time of detection vs. removal noises with different scale image
Image ScaleRun Time(s)
DetectionRemoval
64×640.385.5
128× 1281.1314.28
256×2564.2734.69
512×51217.19111.64
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Image ScaleRun Time(s)
DetectionRemoval
64×640.385.5
128× 1281.1314.28
256×2564.2734.69
512×51217.19111.64
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