October  2019, 13(5): 903-930. doi: 10.3934/ipi.2019041

Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors

1. 

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China

2. 

School of Mathematical Sciences, Nankai University, Tianjin 300071, China

3. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

* Corresponding author: Chunlin Wu

Received  March 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by HDU grant KYS075618129. The second author is supported by NSFC grant 11871035 and 11531013 and Recruitment Program of Global Young Expert. The third author is supported by NSFC grant 11771420

Piecewise constant signals and images are an important kind of data. Typical examples include bar code signals, logos, cartoons, QR codes (Quick Response codes), and text images, which are widely used in both general commercial and automotive industry use. One previous work called a general selective averaging method (GSAM) was introduced to remove noise from them. It chooses homogeneous neighbors from the two closest pixels (one pixel at each side) to update the current pixel. One limitation is that it suffered from appearing sparse noisy pixels in the denoised result when the noise level is high. In this paper, we try to solve this problem by proposing a selective averaging method with multiple neighbors. To update the intensity value at each pixel, the proposed algorithm averages more homogeneous neighbors selected from a large domain, which is based on the property of the local geometry of signals and images. This greatly reduces sparse noisy pixels left in the final result by GSAM. Similarly, our method adopts the Neumann boundary condition at edges, and thus preserves edges well. In 1D case, some theoretical results are given to guarantee the convergence of our algorithm. In 2D case, except eliminating additive Gaussian noise, this algorithm can be used for restoring noisy images corrupted by speckle noise. Intensive experiments on both gray and color image denoising demonstrate that the proposed method is quite effective for piecewise constant image denoising and achieves superior performance visually and quantitatively.

Citation: Weina Wang, Chunlin Wu, Jiansong Deng. Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors. Inverse Problems & Imaging, 2019, 13 (5) : 903-930. doi: 10.3934/ipi.2019041
References:
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P. Blomgren and T. F. Chan, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. Image Process., 7 (1998), 304-309.   Google Scholar

[2]

J. E. Boyd and J. Meloche, Binary restoration of thin objects in multidimensional imagery, IEEE Trans. Pattern Anal. Mach. Intell., 20 (1998), 647-651.  doi: 10.1109/34.683781.  Google Scholar

[3]

C. Brito-LoezaK. Chen and V. Uc-Cetina, Image denoising using the Gaussian curvature of the image surface, Numer. Math. Part. D. E., 32 (2016), 1066-1089.  doi: 10.1002/num.22042.  Google Scholar

[4]

A. BuadesB. Coll and J. M. Morel, A review of image denoising algorithms with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar

[5]

J. CaiS. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8 (2009), 337-369.  doi: 10.1137/090753504.  Google Scholar

[6]

J. CaiB. Dong and Z. Shen, Image restoration: A wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41 (2015), 94-138.  doi: 10.1016/j.acha.2015.06.009.  Google Scholar

[7]

J. CaiH. JiZ. Shen and G. Ye, Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105.  doi: 10.1016/j.acha.2013.10.001.  Google Scholar

[8]

R. H. ChanT. F. ChanL. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction, SIAM J. Sci. Comput., 24 (2003), 1408-1432.  doi: 10.1137/S1064827500383123.  Google Scholar

[9]

T. F. Chan, S. Esedoglu and M. Nikolova, Finding the global minimum for binary image restoration, in IEEE International Conference on Image Processing 2005, (2005), 75-89. doi: 10.1109/ICIP.2005.1529702.  Google Scholar

[10]

C. S. Cho and S. Lee, Effective Five Directional Partial Derivatives-based Image Smoothing and a Parallel Structure Design, IEEE Trans. Image Process., 25 (2016), 1617-1625.  doi: 10.1109/TIP.2016.2526785.  Google Scholar

[11]

R. ChoksiY. van Gennip and A. Oberman, Anisotropic total variation regularized L1 approximation and denoising/deblurring of 2D bar codes, Inverse Probl. Imaging, 5 (2011), 591-617.  doi: 10.3934/ipi.2011.5.591.  Google Scholar

[12]

N. ChumchobK. Chen and C. Brito-Loeza, A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation, Int. J. Comput. Math., 90 (2013), 140-161.  doi: 10.1080/00207160.2012.709625.  Google Scholar

[13]

P. CoupéP. HellierC. Kervrann and C. Barillot, Nonlocal means-based speckle filtering for ultrasound images, IEEE Trans. Image Process., 18 (2009), 2221-2229.  doi: 10.1109/TIP.2009.2024064.  Google Scholar

[14]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.  Google Scholar

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C. A. DeledalleV. Duval and J. Salmon, Anisotropic non-local means with spatially adaptive patch shapes, SSVM: Scale Space and Variational Methods in Computer Vision, 6667 (2011), 231-242.  doi: 10.1007/978-3-642-24785-9_20.  Google Scholar

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C. A. DeledalleA. CharlesV. Duval and J. Salmon, Non-local Methods with Shape-Adaptive Patches (NLM-SAP), J. Math. Imaging Vis., 43 (2012), 103-120.  doi: 10.1007/s10851-011-0294-y.  Google Scholar

[17]

F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imag., 10 (2016), 1066-1089.  doi: 10.3934/ipi.2016.10.27.  Google Scholar

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[19]

D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inf. Theory, 41 (1995), 613-627.  doi: 10.1109/18.382009.  Google Scholar

[20]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736-3745.  doi: 10.1109/TIP.2006.881969.  Google Scholar

[21]

M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar

[22] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2 edition, Prentice Hall, 2002.   Google Scholar
[23]

X. Gu, H. Wang and D. Yu, Binary Image Restoration Using Pulse Coupled Neural Network, in ICNIP, (2001), 922-927. Google Scholar

[24]

W. GuoJ. Qin and V. A. Luminita, A geometry guided image denoising scheme, Inverse Probl. Imag., 7 (2013), 1066-1089.   Google Scholar

[25] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.   Google Scholar
[26]

Y. HuangM. K. Ng and Y. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593.  Google Scholar

[27]

H. JiY. Luo and Z. Shen, Image recovery via geometrically structured approximation, Appl. Comput. Harmon. Anal., 41 (2016), 75-93.  doi: 10.1016/j.acha.2015.08.012.  Google Scholar

[28]

Z. Jin and X. Yang, A variational model to remove the multiplicative noise in ultrasound images, J. Math. Imaging Vis., 39 (2011), 62-74.  doi: 10.1007/s10851-010-0225-3.  Google Scholar

[29]

M. KaramanM. A. Kutay and G. Bozdagi, An adaptive speckle suppression filter for medical ultrasonic imaging, IEEE Trans. Med. Imag., 14 (1995), 283-292.  doi: 10.1109/42.387710.  Google Scholar

[30]

D. T. KuanA. SawchukC. Timothy and P. Chavel, Adaptive noise smoothing filter for images with signal-dependent noise, IEEE Trans. Pattern Anal. Mach. Intell., 7 (1985), 165-177.  doi: 10.1109/TPAMI.1985.4767641.  Google Scholar

[31]

J. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980), 165-168.  doi: 10.1109/TPAMI.1980.4766994.  Google Scholar

[32]

F. LiM. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM J. Imaging Sci., 3 (2010), 1-20.  doi: 10.1137/090748421.  Google Scholar

[33]

J. E. Odegard, H. Guo, M. Lang, C. Burrus, R. Wells, L. Novak and M. Margarita, Wavelet based SAR speckle reduction and image compression, in SPIE, (1995), 17-21. Google Scholar

[34]

S. Ono, K. Morinaga and S. Nakayama, Two-dimensional barcode decoration based on real-coded genetic algorithm, in IEEE CEC, (2008), 1068-1073. Google Scholar

[35]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[36]

S. V. Richard, Matrix Iterative Analysis, 2nd edition, Springer-Verlag, New York, 2009. Google Scholar

[37]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[38]

Y. ShenB. Han and E. Braverman, Adaptive frame-based color image denoising, Appl. Comput. Harmon. Anal., 41 (2015), 54-74.  doi: 10.1016/j.acha.2015.04.001.  Google Scholar

[39]

Y. ShenE. Y. Lam and N. Wong, A signomial programming approach for binary image restoration by penalized least squares, IEEE Trans. Circuits Sys. Ⅱ: Exp. Briefs, 55 (2008), 41-45.  doi: 10.1109/TCSII.2007.907751.  Google Scholar

[40]

C. Sheng, Y. Xin, L. Yao and S. Kun, Total variation-based speckle reduction using multi-grid algorithm for ultrasound images, in ICIAP, (2005), 245-252. doi: 10.1007/11553595_30.  Google Scholar

[41]

M. E. Taylor, Partial Differential Equations Ⅰ: Basic Theory, 2nd edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7055-8.  Google Scholar

[42]

C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in ICCV, (1998), 839-846. doi: 10.1109/ICCV.1998.710815.  Google Scholar

[43]

S. WangT. HuangX. ZhaoJ. Mei and J. Huang, Speckle noise removal in ultrasound images by first- and second-order total variation, Numer. Algor., 78 (2018), 513-533.  doi: 10.1007/s11075-017-0386-x.  Google Scholar

[44]

W. WangC. Wu and J. Deng, A general selective averaging method for piecewise constant signal and image processing, J. Sci. Comput., 76 (2018), 1078-1104.  doi: 10.1007/s10915-018-0650-9.  Google Scholar

[45]

W. WangS. WenC. Wu and J. Deng, Denoising piecewise constant images with selective averaging and outlier removal, Numer. Math. Theor. Meth. Appl., 12 (2019), 467-491.  doi: 10.4208/nmtma.OA-2017-0130.  Google Scholar

[46]

C. Wu and X. Tai, Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[47]

L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Trans. Graph., 30 (2011), 174: 1-174: 12. Google Scholar

[48]

J. YuJ. Tan and Y. Wang, Ultrasound speckle reduction by a SUSAN-controlled anisotropic diffusion method, Pattern Recogn., 43 (2010), 3083-3092.  doi: 10.1016/j.patcog.2010.04.006.  Google Scholar

[49]

J. Zhang and K. Chen, A total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.  Google Scholar

[50]

X. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imaging Sci., 7 (2014), 456-475. arXiv: 1809.09783v2 arXiv: 1809.03948v1 doi: 10.1137/13092472X.  Google Scholar

show all references

References:
[1]

P. Blomgren and T. F. Chan, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. Image Process., 7 (1998), 304-309.   Google Scholar

[2]

J. E. Boyd and J. Meloche, Binary restoration of thin objects in multidimensional imagery, IEEE Trans. Pattern Anal. Mach. Intell., 20 (1998), 647-651.  doi: 10.1109/34.683781.  Google Scholar

[3]

C. Brito-LoezaK. Chen and V. Uc-Cetina, Image denoising using the Gaussian curvature of the image surface, Numer. Math. Part. D. E., 32 (2016), 1066-1089.  doi: 10.1002/num.22042.  Google Scholar

[4]

A. BuadesB. Coll and J. M. Morel, A review of image denoising algorithms with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar

[5]

J. CaiS. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8 (2009), 337-369.  doi: 10.1137/090753504.  Google Scholar

[6]

J. CaiB. Dong and Z. Shen, Image restoration: A wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41 (2015), 94-138.  doi: 10.1016/j.acha.2015.06.009.  Google Scholar

[7]

J. CaiH. JiZ. Shen and G. Ye, Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105.  doi: 10.1016/j.acha.2013.10.001.  Google Scholar

[8]

R. H. ChanT. F. ChanL. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction, SIAM J. Sci. Comput., 24 (2003), 1408-1432.  doi: 10.1137/S1064827500383123.  Google Scholar

[9]

T. F. Chan, S. Esedoglu and M. Nikolova, Finding the global minimum for binary image restoration, in IEEE International Conference on Image Processing 2005, (2005), 75-89. doi: 10.1109/ICIP.2005.1529702.  Google Scholar

[10]

C. S. Cho and S. Lee, Effective Five Directional Partial Derivatives-based Image Smoothing and a Parallel Structure Design, IEEE Trans. Image Process., 25 (2016), 1617-1625.  doi: 10.1109/TIP.2016.2526785.  Google Scholar

[11]

R. ChoksiY. van Gennip and A. Oberman, Anisotropic total variation regularized L1 approximation and denoising/deblurring of 2D bar codes, Inverse Probl. Imaging, 5 (2011), 591-617.  doi: 10.3934/ipi.2011.5.591.  Google Scholar

[12]

N. ChumchobK. Chen and C. Brito-Loeza, A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation, Int. J. Comput. Math., 90 (2013), 140-161.  doi: 10.1080/00207160.2012.709625.  Google Scholar

[13]

P. CoupéP. HellierC. Kervrann and C. Barillot, Nonlocal means-based speckle filtering for ultrasound images, IEEE Trans. Image Process., 18 (2009), 2221-2229.  doi: 10.1109/TIP.2009.2024064.  Google Scholar

[14]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[15]

C. A. DeledalleV. Duval and J. Salmon, Anisotropic non-local means with spatially adaptive patch shapes, SSVM: Scale Space and Variational Methods in Computer Vision, 6667 (2011), 231-242.  doi: 10.1007/978-3-642-24785-9_20.  Google Scholar

[16]

C. A. DeledalleA. CharlesV. Duval and J. Salmon, Non-local Methods with Shape-Adaptive Patches (NLM-SAP), J. Math. Imaging Vis., 43 (2012), 103-120.  doi: 10.1007/s10851-011-0294-y.  Google Scholar

[17]

F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imag., 10 (2016), 1066-1089.  doi: 10.3934/ipi.2016.10.27.  Google Scholar

[18]

D. L. Donoho and J. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425-455.  doi: 10.1093/biomet/81.3.425.  Google Scholar

[19]

D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inf. Theory, 41 (1995), 613-627.  doi: 10.1109/18.382009.  Google Scholar

[20]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736-3745.  doi: 10.1109/TIP.2006.881969.  Google Scholar

[21]

M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar

[22] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2 edition, Prentice Hall, 2002.   Google Scholar
[23]

X. Gu, H. Wang and D. Yu, Binary Image Restoration Using Pulse Coupled Neural Network, in ICNIP, (2001), 922-927. Google Scholar

[24]

W. GuoJ. Qin and V. A. Luminita, A geometry guided image denoising scheme, Inverse Probl. Imag., 7 (2013), 1066-1089.   Google Scholar

[25] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.   Google Scholar
[26]

Y. HuangM. K. Ng and Y. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593.  Google Scholar

[27]

H. JiY. Luo and Z. Shen, Image recovery via geometrically structured approximation, Appl. Comput. Harmon. Anal., 41 (2016), 75-93.  doi: 10.1016/j.acha.2015.08.012.  Google Scholar

[28]

Z. Jin and X. Yang, A variational model to remove the multiplicative noise in ultrasound images, J. Math. Imaging Vis., 39 (2011), 62-74.  doi: 10.1007/s10851-010-0225-3.  Google Scholar

[29]

M. KaramanM. A. Kutay and G. Bozdagi, An adaptive speckle suppression filter for medical ultrasonic imaging, IEEE Trans. Med. Imag., 14 (1995), 283-292.  doi: 10.1109/42.387710.  Google Scholar

[30]

D. T. KuanA. SawchukC. Timothy and P. Chavel, Adaptive noise smoothing filter for images with signal-dependent noise, IEEE Trans. Pattern Anal. Mach. Intell., 7 (1985), 165-177.  doi: 10.1109/TPAMI.1985.4767641.  Google Scholar

[31]

J. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980), 165-168.  doi: 10.1109/TPAMI.1980.4766994.  Google Scholar

[32]

F. LiM. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM J. Imaging Sci., 3 (2010), 1-20.  doi: 10.1137/090748421.  Google Scholar

[33]

J. E. Odegard, H. Guo, M. Lang, C. Burrus, R. Wells, L. Novak and M. Margarita, Wavelet based SAR speckle reduction and image compression, in SPIE, (1995), 17-21. Google Scholar

[34]

S. Ono, K. Morinaga and S. Nakayama, Two-dimensional barcode decoration based on real-coded genetic algorithm, in IEEE CEC, (2008), 1068-1073. Google Scholar

[35]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[36]

S. V. Richard, Matrix Iterative Analysis, 2nd edition, Springer-Verlag, New York, 2009. Google Scholar

[37]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[38]

Y. ShenB. Han and E. Braverman, Adaptive frame-based color image denoising, Appl. Comput. Harmon. Anal., 41 (2015), 54-74.  doi: 10.1016/j.acha.2015.04.001.  Google Scholar

[39]

Y. ShenE. Y. Lam and N. Wong, A signomial programming approach for binary image restoration by penalized least squares, IEEE Trans. Circuits Sys. Ⅱ: Exp. Briefs, 55 (2008), 41-45.  doi: 10.1109/TCSII.2007.907751.  Google Scholar

[40]

C. Sheng, Y. Xin, L. Yao and S. Kun, Total variation-based speckle reduction using multi-grid algorithm for ultrasound images, in ICIAP, (2005), 245-252. doi: 10.1007/11553595_30.  Google Scholar

[41]

M. E. Taylor, Partial Differential Equations Ⅰ: Basic Theory, 2nd edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7055-8.  Google Scholar

[42]

C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in ICCV, (1998), 839-846. doi: 10.1109/ICCV.1998.710815.  Google Scholar

[43]

S. WangT. HuangX. ZhaoJ. Mei and J. Huang, Speckle noise removal in ultrasound images by first- and second-order total variation, Numer. Algor., 78 (2018), 513-533.  doi: 10.1007/s11075-017-0386-x.  Google Scholar

[44]

W. WangC. Wu and J. Deng, A general selective averaging method for piecewise constant signal and image processing, J. Sci. Comput., 76 (2018), 1078-1104.  doi: 10.1007/s10915-018-0650-9.  Google Scholar

[45]

W. WangS. WenC. Wu and J. Deng, Denoising piecewise constant images with selective averaging and outlier removal, Numer. Math. Theor. Meth. Appl., 12 (2019), 467-491.  doi: 10.4208/nmtma.OA-2017-0130.  Google Scholar

[46]

C. Wu and X. Tai, Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[47]

L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Trans. Graph., 30 (2011), 174: 1-174: 12. Google Scholar

[48]

J. YuJ. Tan and Y. Wang, Ultrasound speckle reduction by a SUSAN-controlled anisotropic diffusion method, Pattern Recogn., 43 (2010), 3083-3092.  doi: 10.1016/j.patcog.2010.04.006.  Google Scholar

[49]

J. Zhang and K. Chen, A total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.  Google Scholar

[50]

X. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imaging Sci., 7 (2014), 456-475. arXiv: 1809.09783v2 arXiv: 1809.03948v1 doi: 10.1137/13092472X.  Google Scholar

Figure 1.  An illustration of the updating order from $ i = 1 $ to $ i = m $
Figure 3.  The first two rows show the results by GSAM [44] ($ T = 0.44, p = 0.49 $), Algorithm 1 ($ T = 0.44, k = 2, p = 0.25 $), TV[37] ($ \alpha = 1.3, r_p = 10 $) and $ L_{0} $[47] ($ \lambda = 0.03, \kappa = 1.01 $), when $ \sigma = 0.12 $. The last two rows show the results by GSAM [44] ($ T = 0.43, p = 0.49 $), Algorithm 1 ($ T = 0.43, k = 3, p = 0.11 $), TV[37] ($ \alpha = 1.4, r_p = 10 $) and $ L_{0} $[47] ($ \lambda = 0.05, \kappa = 1.01 $), when $ \sigma = 0.16 $. The corresponding SNR values are listed in brackets
Figure 4.  Test images. (a) Logo. (b) Cartoon1. (c) QRcode1. (d) Text. (e) QArtcode. (f) Blobs. (g) QRcode2. (h) QRcode3. (i) Cartoon2
Figure 5.  From top to down and left to right: the noisy Logo, the results by TV[37], $ L_{0} $[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
Figure 6.  From top to down and left to right: the noisy Cartoon1, the results by TV[37], $ L_{0} $[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little line in each image are displayed for further comparison. The corresponding SNR values are listed in brackets
Figure 7.  From top to down and left to right: the noisy QRcode1, the results by TV[37], $ L_{0} $[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets
Figure 8.  From top to down and left to right: the noisy Text, the results by TV[37], $ L_{0} $[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets
Figure 9.  From top to down and left to right: the noisy QArtcode, the results by TV[37], $ L_{0} $[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
Figure 10.  From left to right: the noisy QRcode2, the results by TV[1], $ L_{0} $[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the first image are displayed for further comparison. The corresponding SNR values are listed in brackets
Figure 11.  From up to down: Denoising results by different methods for two noisy QRcode3 and Cartoon2 images. From left to right: the noisy images, the results by TV[1], $ L_{0} $[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the second row-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
Figure 12.  From top to down and left to right: the clean and noisy "Blobs", the results by Wang et.al [43], $ L_{0} $ [47], AGSAM [44] and our method. The zoom-in views of the region indicited by the little line in the each image are displayed for further comparison. The corresponding SNR values are listed in brackets
Figure 13.  From up to down: Denoising results by different methods for Cartoon1 corrupted by two levels of noise. From left to right: the noisy images, the results by Wang et.al [43], $ L_{0} $ [47], AGSAM [44] and our method. The corresponding SNR values are listed in brackets
Figure 2.  Different types to compute $ u^{(n+1)}_{i} $ based on five cases of $ K^{l}_{i-1} $ and $ K^{r}_{i-1} $ from (2) to (6). (a1-a2), (b1-b2), (c1-c2), (d) and (e) correspond to Case 1, Case 2, Case 3, Case 4 and Case 5, respectively
Table 1.  Iterative number and time comparison(in seconds) by GSAM [44] and Algorithm 1 in Fig. 3
Signal $ \sigma $ GSAM [44] Proposed
iter t iter t
Fig. 3 $ 0.12 $ 6231 1.61 3695 1.07
$ 0.16 $ 3322 0.98 2181 0.79
Signal $ \sigma $ GSAM [44] Proposed
iter t iter t
Fig. 3 $ 0.12 $ 6231 1.61 3695 1.07
$ 0.16 $ 3322 0.98 2181 0.79
Table 2.  The parameter settings of different methods for gray images
Image $ \sigma $ TV[37] $ L_{0} $[47] NLM-SAP[16] AGSAM[44] Proposed
$ \alpha $ $ \lambda $ $ h $/$ T $ $ T $ $ T $/$ p $
Logo $ 10 $ 30 0.01 1.5/0.6 5 5/0.13
$ 15 $ 20 0.02 0.9/1.12 4 4/0.13
$ 20 $ 14 0.03 0.7/2 2.4 3.3/0.05
Cartoon1 $ 15 $ 20 0.02 1.6/1.35 5 5/0.13
$ 20 $ 15 0.03 1.1/2.8 4.5 4.5/0.13
$ 25 $ 12 0.04 0.8/5 4 4/0.06
QRcode1 $ 15 $ 20 0.02 1.8/1.12 5 5/0.13
$ 25 $ 12 0.04 1.1/3.12 4.5 4.5/0.13
$ 35 $ 8 0.1 0.9/11.03 3.2 3.2/0.07
Text $ 15 $ 24 0.02 1.7/1.8 4.5 4.5/0.13
$ 25 $ 14 0.05 1.1/3.75 4.5 4.5/0.13
$ 35 $ 10 0.06 1/9.8 3.2 3.5/0.13
QArtcode $ 15 $ 23 0.02 1.7/1.8 5 5/0.13
$ 25 $ 14 0.04 0.9/4.38 4.5 4.5/0.13
$ 35 $ 10 0.07 0.6/9.8 3.4 3.4/0.07
Image $ \sigma $ TV[37] $ L_{0} $[47] NLM-SAP[16] AGSAM[44] Proposed
$ \alpha $ $ \lambda $ $ h $/$ T $ $ T $ $ T $/$ p $
Logo $ 10 $ 30 0.01 1.5/0.6 5 5/0.13
$ 15 $ 20 0.02 0.9/1.12 4 4/0.13
$ 20 $ 14 0.03 0.7/2 2.4 3.3/0.05
Cartoon1 $ 15 $ 20 0.02 1.6/1.35 5 5/0.13
$ 20 $ 15 0.03 1.1/2.8 4.5 4.5/0.13
$ 25 $ 12 0.04 0.8/5 4 4/0.06
QRcode1 $ 15 $ 20 0.02 1.8/1.12 5 5/0.13
$ 25 $ 12 0.04 1.1/3.12 4.5 4.5/0.13
$ 35 $ 8 0.1 0.9/11.03 3.2 3.2/0.07
Text $ 15 $ 24 0.02 1.7/1.8 4.5 4.5/0.13
$ 25 $ 14 0.05 1.1/3.75 4.5 4.5/0.13
$ 35 $ 10 0.06 1/9.8 3.2 3.5/0.13
QArtcode $ 15 $ 23 0.02 1.7/1.8 5 5/0.13
$ 25 $ 14 0.04 0.9/4.38 4.5 4.5/0.13
$ 35 $ 10 0.07 0.6/9.8 3.4 3.4/0.07
Table 5.  The parameter settings of different methods for color images
Image $ \sigma $ TV[1] $ L_{0} $[47] AGSAM[44] Proposed
$ \alpha $ $ \lambda $ $ T $ $ T $/$ p $
QRcode2 $ 10 $ 16 0.01 5 5/0.13
$ 20 $ 8 0.04 4 4/0.13
$ 30 $ 5 0.08 3.5 3.5/0.13
$ 40 $ 4 1.6 2.3 2.6/0.1
QRcode3 $ 10 $ 17 0.02 5 5/0.13
$ 20 $ 9 0.04 4 4/0.13
$ 30 $ 6 0.12 2.5 2.6/0.11
$ 40 $ 4 0.5 2 2/0.06
Cartoon2 $ 10 $ 19 0.01 5 5/0.13
$ 20 $ 10 0.05 4 4.2/0.13
$ 30 $ 6 0.09 2.5 2.8/0.13
$ 40 $ 5 1.16 1.9 2.3/0.1
Image $ \sigma $ TV[1] $ L_{0} $[47] AGSAM[44] Proposed
$ \alpha $ $ \lambda $ $ T $ $ T $/$ p $
QRcode2 $ 10 $ 16 0.01 5 5/0.13
$ 20 $ 8 0.04 4 4/0.13
$ 30 $ 5 0.08 3.5 3.5/0.13
$ 40 $ 4 1.6 2.3 2.6/0.1
QRcode3 $ 10 $ 17 0.02 5 5/0.13
$ 20 $ 9 0.04 4 4/0.13
$ 30 $ 6 0.12 2.5 2.6/0.11
$ 40 $ 4 0.5 2 2/0.06
Cartoon2 $ 10 $ 19 0.01 5 5/0.13
$ 20 $ 10 0.05 4 4.2/0.13
$ 30 $ 6 0.09 2.5 2.8/0.13
$ 40 $ 5 1.16 1.9 2.3/0.1
Table 3.  The SNR values by different methods for gray images
Image $ \sigma $ TV[37] $ L_{0} $[47] NLM-SAP[16] AGSAM[44] Proposed
Logo $ 10 $ 30.07 46.80 46.41 48.05 $ \bf{49.26} $
$ 20 $ 24.41 $ \bf{39.15} $ 34.96 29.05 34.77
Cartoon1 $ 15 $ 26.86 44.60 43.22 44.95 $ \bf{46.87} $
$ 25 $ 22.64 40.17 36.01 41.20 $ \bf{42.35} $
QRcode1 $ 15 $ 27.20 41.98 42.00 43.78 $ \bf{44.97} $
$ 25 $ 22.93 37.54 35.77 39.85 $ \bf{40.78} $
Text $ 15 $ 23.55 39.19 39.75 42.57 $ \bf{43.06} $
$ 35 $ 16.48 $ \bf{30.46} $ 25.00 29.79 29.97
QArtcode $ 15 $ 24.45 36.14 $ \bf{40.01} $ 39.37 39.51
$ 25 $ 20.16 31.70 35.08 35.20 $ \bf{35.25} $
Image $ \sigma $ TV[37] $ L_{0} $[47] NLM-SAP[16] AGSAM[44] Proposed
Logo $ 10 $ 30.07 46.80 46.41 48.05 $ \bf{49.26} $
$ 20 $ 24.41 $ \bf{39.15} $ 34.96 29.05 34.77
Cartoon1 $ 15 $ 26.86 44.60 43.22 44.95 $ \bf{46.87} $
$ 25 $ 22.64 40.17 36.01 41.20 $ \bf{42.35} $
QRcode1 $ 15 $ 27.20 41.98 42.00 43.78 $ \bf{44.97} $
$ 25 $ 22.93 37.54 35.77 39.85 $ \bf{40.78} $
Text $ 15 $ 23.55 39.19 39.75 42.57 $ \bf{43.06} $
$ 35 $ 16.48 $ \bf{30.46} $ 25.00 29.79 29.97
QArtcode $ 15 $ 24.45 36.14 $ \bf{40.01} $ 39.37 39.51
$ 25 $ 20.16 31.70 35.08 35.20 $ \bf{35.25} $
Table 4.  The iterative number and time comparison(in seconds) by different methods for gray images
Image $ \sigma $ TV[37] $ L_{0} $[47] NLM-SAP[16] AGSAM[44] Proposed
iter/t iter/t iter/t iter/t iter/t
Logo $ 10 $ 90/0.7 1551/5.5 1/141.4 139/0.5 $ \bf{66} $/$ \bf{0.2} $
$ 20 $ $ \bf{91} $/0.6 1481/5.0 1/145.7 182/0.6 138/$ \bf{0.4} $
Cartoon1 $ 15 $ $ \bf{96} $/2.5 1481/19.6 1/698.3 170/3.5 102/$ \bf{2.2} $
$ 25 $ $ \bf{98} $/$ \bf{2.6} $ 1355/18.2 1/720.5 235/5.0 177/3.8
QRcode1 $ 15 $ $ \bf{83} $/$ \bf{2.2} $ 1479/18.0 1/994.1 177/3.6 103/2.3
$ 25 $ $ \bf{84} $/$ \bf{2.1} $ 1411/17.2 1/998.0 254/5.2 150/3.5
Text $ 15 $ $ \bf{112} $/$ \bf{0.7} $ 1482/3.9 1/165.9 295/1.6 147/0.8
$ 35 $ $ \bf{109} $/$ \bf{0.6} $ 1355/3.5 1/166.4 502/2.7 324/1.8
QArtcode $ 15 $ $ \bf{67} $/0.3 1478/2.9 1/77.6 175/0.3 109/$ \bf{0.2} $
$ 25 $ $ \bf{69} $/0.3 1389/2.3 1/78.3 254/0.4 155/$ \bf{0.2} $
Image $ \sigma $ TV[37] $ L_{0} $[47] NLM-SAP[16] AGSAM[44] Proposed
iter/t iter/t iter/t iter/t iter/t
Logo $ 10 $ 90/0.7 1551/5.5 1/141.4 139/0.5 $ \bf{66} $/$ \bf{0.2} $
$ 20 $ $ \bf{91} $/0.6 1481/5.0 1/145.7 182/0.6 138/$ \bf{0.4} $
Cartoon1 $ 15 $ $ \bf{96} $/2.5 1481/19.6 1/698.3 170/3.5 102/$ \bf{2.2} $
$ 25 $ $ \bf{98} $/$ \bf{2.6} $ 1355/18.2 1/720.5 235/5.0 177/3.8
QRcode1 $ 15 $ $ \bf{83} $/$ \bf{2.2} $ 1479/18.0 1/994.1 177/3.6 103/2.3
$ 25 $ $ \bf{84} $/$ \bf{2.1} $ 1411/17.2 1/998.0 254/5.2 150/3.5
Text $ 15 $ $ \bf{112} $/$ \bf{0.7} $ 1482/3.9 1/165.9 295/1.6 147/0.8
$ 35 $ $ \bf{109} $/$ \bf{0.6} $ 1355/3.5 1/166.4 502/2.7 324/1.8
QArtcode $ 15 $ $ \bf{67} $/0.3 1478/2.9 1/77.6 175/0.3 109/$ \bf{0.2} $
$ 25 $ $ \bf{69} $/0.3 1389/2.3 1/78.3 254/0.4 155/$ \bf{0.2} $
Table 6.  The SNR values by different methods for color images
Image $ \sigma $ TV[1] $ L_{0} $[47] AGSAM[44] Proposed
QRcode2 $ 10 $ 31.29 46.49 46.02 $ \bf{47.62} $
$ 30 $ 22.23 36.37 37.53 $ \bf{37.81} $
$ 40 $ 20.01 29.14 33.39 $ \bf{33.85} $
QRcode3 $ 10 $ 29.68 44.10 44.44 $ \bf{46.11} $
$ 20 $ 23.97 38.08 38.91 $ \bf{40.77} $
$ 40 $ 18.48 $ \bf{25.57} $ 21.10 25.30
Cartoon2 $ 10 $ 29.39 40.31 40.93 $ \bf{41.05} $
$ 20 $ 23.65 38.36 38.46 $ \bf{39.53} $
$ 30 $ 20.37 36.09 35.80 $ \bf{36.99} $
Image $ \sigma $ TV[1] $ L_{0} $[47] AGSAM[44] Proposed
QRcode2 $ 10 $ 31.29 46.49 46.02 $ \bf{47.62} $
$ 30 $ 22.23 36.37 37.53 $ \bf{37.81} $
$ 40 $ 20.01 29.14 33.39 $ \bf{33.85} $
QRcode3 $ 10 $ 29.68 44.10 44.44 $ \bf{46.11} $
$ 20 $ 23.97 38.08 38.91 $ \bf{40.77} $
$ 40 $ 18.48 $ \bf{25.57} $ 21.10 25.30
Cartoon2 $ 10 $ 29.39 40.31 40.93 $ \bf{41.05} $
$ 20 $ 23.65 38.36 38.46 $ \bf{39.53} $
$ 30 $ 20.37 36.09 35.80 $ \bf{36.99} $
Table 7.  The iterative number and time comparison(in seconds) by different methods for color images
Image $ \sigma $ TV[1] $ L_{0} $[47] AGSAM[44] Proposed
iter/t iter/t iter/t iter/t
QRcode2 $ 10 $ 76/3.0 1551/33.3 144/2.7 $ \bf{68} $/$ \bf{1.4} $
$ 30 $ $ \bf{82} $/3.3 1342/29.6 253/3.4 172/$ \bf{2.9} $
$ 40 $ $ \bf{75} $/$ \bf{3.0} $ 1041/23.9 311/5.8 215/4.4
QRcode3 $ 10 $ $ \bf{66} $/2.6 1411/31.3 139/2.7 68/$ \bf{1.5} $
$ 20 $ $ \bf{63} $/2.8 1340/32.3 183/3.5 118/$ \bf{2.7} $
$ 40 $ $ \bf{72} $/$ \bf{2.9} $ 1158/25.9 308/6.0 283/6.1
Cartoon2 $ 10 $ $ \bf{70} $/3.4 1551/46.1 146/3.4 79/$ \bf{1.9} $
$ 20 $ $ \bf{54} $/3.2 1355/44.1 189/4.2 116/$ \bf{2.8} $
$ 30 $ $ \bf{78} $/3.8 1240/39.5 245/5.4 152/$ \bf{3.7} $
Image $ \sigma $ TV[1] $ L_{0} $[47] AGSAM[44] Proposed
iter/t iter/t iter/t iter/t
QRcode2 $ 10 $ 76/3.0 1551/33.3 144/2.7 $ \bf{68} $/$ \bf{1.4} $
$ 30 $ $ \bf{82} $/3.3 1342/29.6 253/3.4 172/$ \bf{2.9} $
$ 40 $ $ \bf{75} $/$ \bf{3.0} $ 1041/23.9 311/5.8 215/4.4
QRcode3 $ 10 $ $ \bf{66} $/2.6 1411/31.3 139/2.7 68/$ \bf{1.5} $
$ 20 $ $ \bf{63} $/2.8 1340/32.3 183/3.5 118/$ \bf{2.7} $
$ 40 $ $ \bf{72} $/$ \bf{2.9} $ 1158/25.9 308/6.0 283/6.1
Cartoon2 $ 10 $ $ \bf{70} $/3.4 1551/46.1 146/3.4 79/$ \bf{1.9} $
$ 20 $ $ \bf{54} $/3.2 1355/44.1 189/4.2 116/$ \bf{2.8} $
$ 30 $ $ \bf{78} $/3.8 1240/39.5 245/5.4 152/$ \bf{3.7} $
Table 8.  The parameter settings of different methods for gray images with speckle noise
Image $ \sigma $ Wang et.al [43] $ L_{0} $[47] AGSAM[44] Proposed
$ \lambda $/$ \mu $ $ \lambda $ $ T $ $ T $/$ p $/$ k $
Blobs $ 2 $ 1.2/11 0.04 60 60/0.04/12
Cartoon1 $ 2 $ 0.8/2 0.05 58 58/0.13/4
$ 2.5 $ 0.6/2 0.06 45 45/0.03/5
Image $ \sigma $ Wang et.al [43] $ L_{0} $[47] AGSAM[44] Proposed
$ \lambda $/$ \mu $ $ \lambda $ $ T $ $ T $/$ p $/$ k $
Blobs $ 2 $ 1.2/11 0.04 60 60/0.04/12
Cartoon1 $ 2 $ 0.8/2 0.05 58 58/0.13/4
$ 2.5 $ 0.6/2 0.06 45 45/0.03/5
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