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December 2017, 11(6): 949-974. doi: 10.3934/ipi.2017044

Multiplicative noise removal with a sparsity-aware optimization model

1. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China

2. 

Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA

3. 

School of Data and Computer Science, Guangdong Provincial Key Lab of Computational Science, Sun Yat-sen University, Guangzhou 510275, China

4. 

Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA

* Corresponding author: Lixin Shen

Received  April 2016 Revised  August 2017 Published  September 2017

Restoration of images contaminated by multiplicative noise (also known as speckle noise) is a key issue in coherent image processing. Notice that images under consideration are often highly compressible in certain suitably chosen transform domains. By exploring this intrinsic feature embedded in images, this paper introduces a variational restoration model for multiplicative noise reduction that consists of a term reflecting the observed image and multiplicative noise, a quadratic term measuring the closeness of the underlying image in a transform domain to a sparse vector, and a sparse regularizer for removing multiplicative noise. Being different from popular existing models which focus on pursuing convexity, the proposed sparsity-aware model may be nonconvex depending on the conditions of the parameters of the model for achieving the optimal denoising performance. An algorithm for finding a critical point of the objective function of the model is developed based on coupled fixed-point equations expressed in terms of the proximity operator of functions that appear in the objective function. Convergence analysis of the algorithm is provided. Experimental results are shown to demonstrate that the proposed iterative algorithm is sensitive to some initializations for obtaining the best restoration results. We observe that the proposed method with SAR-BM3D filtering images as initial estimates can remarkably outperform several state-of-art methods in terms of the quality of the restored images.

Citation: Jian Lu, Lixin Shen, Chen Xu, Yuesheng Xu. Multiplicative noise removal with a sparsity-aware optimization model. Inverse Problems & Imaging, 2017, 11 (6) : 949-974. doi: 10.3934/ipi.2017044
References:
[1]

G. Aubert and J. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946. doi: 10.1137/060671814.

[2]

R. Bamler, Principles of synthetic aperture radar, Surv. Geophys., 21 (2000), 147-157.

[3]

H. L. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, AMS Books in Mathematics, Springer New York, 2011. doi: 10.1007/978-1-4419-9467-7.

[4]

J. M. Bioucas-Dias and M. A. T. Figueiredo, Multiplicative noise removal using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 1720-1730. doi: 10.1109/TIP.2010.2045029.

[5]

M. F. C. Chesneau and J. Starck, Stein block thresholding for image denoising, Appl. Computat. Harmon. Anal., 28 (2010), 67-88. doi: 10.1016/j.acha.2009.07.003.

[6]

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

[7]

D. DaiL. ShenY. Xu and N. Zhang, Noisy 1-bit compressive sensing: Models and algorithms, Appl. Computat. Harmon. Anal., 40 (2016), 1-32. doi: 10.1016/j.acha.2014.12.001.

[8]

I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS Conf. Series Appl. Math., SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970104.

[9]

I. DaubechiesB. HanA. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., 14 (2003), 1-46. doi: 10.1016/S1063-5203(02)00511-0.

[10]

L. DenisF. TupinJ. Darbon and M. Sigelle, SAR image regularization with fast approximation discrete minimization, IEEE Trans. Image Process., 18 (2009), 1588-1600. doi: 10.1109/TIP.2009.2019302.

[11]

Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imag. Sci., 6 (2013), 1598-1625. doi: 10.1137/120870621.

[12]

S. DurandJ. Fadili and M. Nikolova, Multiplicative noise removal using l1 fidelity on frame coefficients, J. Math. Imag. Vis., 36 (2010), 201-226.

[13]

J. W. Goodman, Some fundamental properties of speckle, J. Opt. Soc. of Amer., 66 (1976), 1145-1150. doi: 10.1364/JOSA.66.001145.

[14]

Y. HangL. MoisanM. K. Ng and T. Zeng, Multiplicative noise removal via a learned dictionary, IEEE Trans. Image Process., 21 (2012), 4534-4543. doi: 10.1109/TIP.2012.2205007.

[15]

Y.-H. HuangH. Yan and T. Zeng, Multiplicative noise removal based on unbiased box-cox transformation, Communications in Computational Physics, 22 (2017), 803-828. doi: 10.4208/cicp.OA-2016-0074.

[16]

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

[17]

M. KangS. Yun and H. Woo, Two-level convex relaxed variational model for multiplicative denoising, SIAM J. Imag. Sci., 6 (2013), 875-903. doi: 10.1137/11086077X.

[18]

D. Lazard, Quantifier elimination: Optimal solution for two classical examples, J. Symbol. Comput., 5 (1988), 261-266. doi: 10.1016/S0747-7171(88)80015-4.

[19]

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

[20]

J. LuL. ShenC. Xu and Y. Xu, Multiplicative noise removal in imaging: An exp-model and its fixed-point proximity algorithm, Appl. Comput. Harmon. Anal., 41 (2016), 518-539. doi: 10.1016/j.acha.2015.10.003.

[21]

C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising Inverse Probl., 27 (2011), 045009(30pp). doi: 10.1088/0266-5611/27/4/045009.

[22]

J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, C.R. Acad. Sci. Paris Sér. A Math., 255 (1962), 2897-2899.

[23] Y. Nesterov, Introductory Lectures on Convex Optimization, Kluwer, Boston, 2004. doi: 10.1007/978-1-4419-8853-9.
[24] C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Imaging, SciTech Publishing, Raleigh, NC, 2004.
[25]

S. ParrilliM. PodericoC. V. Angelino and L. Verdoliva, A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Trans. Geosci. Remote Sens., 50 (2012), 606-616. doi: 10.1109/TGRS.2011.2161586.

[26]

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

[27]

J. M. SchmittS. H. Xiang and K. M. Yung, Speckle in optical coherence tomography, J. Biomed. Opt., 4 (1999), 95-105. doi: 10.1117/1.429925.

[28]

J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise removal, SIAM J. Imag. Sci., 1 (2008), 294-321. doi: 10.1137/070689954.

[29]

G. Steidl and T. Teuber, Removing multiplicative noise by Douglad-Rachford splitting methods, J. Math. Imag. Vis., 36 (2010), 168-184. doi: 10.1007/s10851-009-0179-5.

[30]

T. Teuber and A. Lang, Nonlocal filters for removing multiplicative noise, Proc. of SSVM, LNCS, 6667 (2012), 50-61. doi: 10.1007/978-3-642-24785-9_5.

[31]

R. F. WagnerS. W. SmithJ. M. Sandrik and H. Lopez, Statistics of speckle in ultrasound B-scans, IEEE Trans. Sonics and Ultrason., 30 (1983), 156-163. doi: 10.1109/T-SU.1983.31404.

[32]

S. Yun and H. Woo, A new multiplicative denoising variational model based on m-th root transformation, IEEE Trans. Image Process., 21 (2012), 2523-2533. doi: 10.1109/TIP.2012.2185942.

show all references

References:
[1]

G. Aubert and J. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946. doi: 10.1137/060671814.

[2]

R. Bamler, Principles of synthetic aperture radar, Surv. Geophys., 21 (2000), 147-157.

[3]

H. L. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, AMS Books in Mathematics, Springer New York, 2011. doi: 10.1007/978-1-4419-9467-7.

[4]

J. M. Bioucas-Dias and M. A. T. Figueiredo, Multiplicative noise removal using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 1720-1730. doi: 10.1109/TIP.2010.2045029.

[5]

M. F. C. Chesneau and J. Starck, Stein block thresholding for image denoising, Appl. Computat. Harmon. Anal., 28 (2010), 67-88. doi: 10.1016/j.acha.2009.07.003.

[6]

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

[7]

D. DaiL. ShenY. Xu and N. Zhang, Noisy 1-bit compressive sensing: Models and algorithms, Appl. Computat. Harmon. Anal., 40 (2016), 1-32. doi: 10.1016/j.acha.2014.12.001.

[8]

I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS Conf. Series Appl. Math., SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970104.

[9]

I. DaubechiesB. HanA. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., 14 (2003), 1-46. doi: 10.1016/S1063-5203(02)00511-0.

[10]

L. DenisF. TupinJ. Darbon and M. Sigelle, SAR image regularization with fast approximation discrete minimization, IEEE Trans. Image Process., 18 (2009), 1588-1600. doi: 10.1109/TIP.2009.2019302.

[11]

Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imag. Sci., 6 (2013), 1598-1625. doi: 10.1137/120870621.

[12]

S. DurandJ. Fadili and M. Nikolova, Multiplicative noise removal using l1 fidelity on frame coefficients, J. Math. Imag. Vis., 36 (2010), 201-226.

[13]

J. W. Goodman, Some fundamental properties of speckle, J. Opt. Soc. of Amer., 66 (1976), 1145-1150. doi: 10.1364/JOSA.66.001145.

[14]

Y. HangL. MoisanM. K. Ng and T. Zeng, Multiplicative noise removal via a learned dictionary, IEEE Trans. Image Process., 21 (2012), 4534-4543. doi: 10.1109/TIP.2012.2205007.

[15]

Y.-H. HuangH. Yan and T. Zeng, Multiplicative noise removal based on unbiased box-cox transformation, Communications in Computational Physics, 22 (2017), 803-828. doi: 10.4208/cicp.OA-2016-0074.

[16]

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

[17]

M. KangS. Yun and H. Woo, Two-level convex relaxed variational model for multiplicative denoising, SIAM J. Imag. Sci., 6 (2013), 875-903. doi: 10.1137/11086077X.

[18]

D. Lazard, Quantifier elimination: Optimal solution for two classical examples, J. Symbol. Comput., 5 (1988), 261-266. doi: 10.1016/S0747-7171(88)80015-4.

[19]

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

[20]

J. LuL. ShenC. Xu and Y. Xu, Multiplicative noise removal in imaging: An exp-model and its fixed-point proximity algorithm, Appl. Comput. Harmon. Anal., 41 (2016), 518-539. doi: 10.1016/j.acha.2015.10.003.

[21]

C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising Inverse Probl., 27 (2011), 045009(30pp). doi: 10.1088/0266-5611/27/4/045009.

[22]

J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, C.R. Acad. Sci. Paris Sér. A Math., 255 (1962), 2897-2899.

[23] Y. Nesterov, Introductory Lectures on Convex Optimization, Kluwer, Boston, 2004. doi: 10.1007/978-1-4419-8853-9.
[24] C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Imaging, SciTech Publishing, Raleigh, NC, 2004.
[25]

S. ParrilliM. PodericoC. V. Angelino and L. Verdoliva, A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Trans. Geosci. Remote Sens., 50 (2012), 606-616. doi: 10.1109/TGRS.2011.2161586.

[26]

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

[27]

J. M. SchmittS. H. Xiang and K. M. Yung, Speckle in optical coherence tomography, J. Biomed. Opt., 4 (1999), 95-105. doi: 10.1117/1.429925.

[28]

J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise removal, SIAM J. Imag. Sci., 1 (2008), 294-321. doi: 10.1137/070689954.

[29]

G. Steidl and T. Teuber, Removing multiplicative noise by Douglad-Rachford splitting methods, J. Math. Imag. Vis., 36 (2010), 168-184. doi: 10.1007/s10851-009-0179-5.

[30]

T. Teuber and A. Lang, Nonlocal filters for removing multiplicative noise, Proc. of SSVM, LNCS, 6667 (2012), 50-61. doi: 10.1007/978-3-642-24785-9_5.

[31]

R. F. WagnerS. W. SmithJ. M. Sandrik and H. Lopez, Statistics of speckle in ultrasound B-scans, IEEE Trans. Sonics and Ultrason., 30 (1983), 156-163. doi: 10.1109/T-SU.1983.31404.

[32]

S. Yun and H. Woo, A new multiplicative denoising variational model based on m-th root transformation, IEEE Trans. Image Process., 21 (2012), 2523-2533. doi: 10.1109/TIP.2012.2185942.

Figure 1.  Five gray-level test images. (a) "Cameraman" ($512\times 512$). (b) "Lena" ($512\times 512$). (c) "Peppers" ($512\times 512$). (d) "Remote1" ($768\times 574$). (e) "Remote2" ($632\times 540$)
Figure 2.  (a) PSNR versus number of iterations. (b) Relative error versus number of iterations. Here, various $x^{(0)}$s are used as initial estimates in Algorithm 1. From top to bottom: the test images are the degraded "Cameraman" with multiplicative noise at levels $L=10$, $6$, $4$, and $2$
Figure 3.  (a) PSNR versus number of iterations for $L=4$. (b) PSNR versus number of iterations for $L=2$. Here, various $x^{(0)}$s are used as initializations in Algorithm 1 for the degraded "Cameraman" with multiplicative noise
Figure 4.  (a) PSNR versus number of iterations. (b) Relative error versus number of iterations. Here, two different $x^{(0)}$s are used as initial estimates in Algorithm 1. The solid lines are plotted firstly by selecting the parameters of Algorithm 1 to obtain (nearly) optimal PSNR values (marked by '$\circ$') under the prescribed tolerance $\text{TOL}=3\times 10^{-4}$; then the dashed lines are plotted by using the same parameters as those of corresponding solid lines. The test images are the degraded "Cameraman" with multiplicative noise at various noise levels ($L=10$, $6$, $4$, and $2$)
Figure 5.  Results of various denoising methods on "Cameraman" image corrupted by multiplicative noise with $L=2$ (the first column) and $L=4$ (the second column). From top to bottom: Noisy images (8.63 dB, 11.64 dB), DZ (24.27 dB, 25.85 dB), TwL-4V (25.28 dB, 26.72dB), Ⅰ-DIV (24.98 dB, 26.52 dB), HMNZ (25.30 dB, 27.33 dB), and Ours (25.47 dB, 27.38 dB)
Figure 6.  Results of various denoising methods on "Remote1" image corrupted by multiplicative noise with $L=2$ (the first column) and $L=4$ (the second column). From top to bottom: Noisy images (9.28 dB, 12.28 dB), DZ (21.80 dB, 23.01 dB), TwL-4V (22.57 dB, 23.68 dB), Ⅰ-DIV (22.39 dB, 23.48 dB), HMNZ (22.63 dB, 24.03 dB), and Ours (22.76 dB, 23.98 dB)
Algorithm 1 Fixed-point algorithm based on the proximity operators for model (3).
 Input: noisy image $f>0$ in $\mathbb{R}^{n}$; parameters $\lambda>0$, $\mu$, $\beta>1$; $\alpha>0$,
 Initialization: $x^{(0)}$ and $y^{(0)}=0$; positive numbers $\sigma$ and $\rho$ such $\mu<\sigma$ and $\frac{\rho}{\mu}>8\sin^2\frac{(\sqrt{n}-1)\pi}{2\sqrt{n}}$.
 repeat
   (a) $x^{(k+1)}\leftarrow\mathrm{prox}_{\frac{1}{\rho}\Phi}(x^{(k)}-\frac{\mu}{\rho}H^\top(Hx^{(k)}-y^{(k)}))$,
   (b) $y^{(k+1)}\leftarrow\mathrm{prox}_{\frac{\lambda}{\sigma}\psi}(y^{(k)}+\frac{\mu}{\sigma}(Hx^{(k+1)}-y^{(k)}))$,
 until converges or satisfies a stopping criteria.
 Write the output of $x^{(k+1)}$ from the above iteration as $\overline{x}$.
 The restored image is $u^\star=e^{\overline{x}}$.
Algorithm 1 Fixed-point algorithm based on the proximity operators for model (3).
 Input: noisy image $f>0$ in $\mathbb{R}^{n}$; parameters $\lambda>0$, $\mu$, $\beta>1$; $\alpha>0$,
 Initialization: $x^{(0)}$ and $y^{(0)}=0$; positive numbers $\sigma$ and $\rho$ such $\mu<\sigma$ and $\frac{\rho}{\mu}>8\sin^2\frac{(\sqrt{n}-1)\pi}{2\sqrt{n}}$.
 repeat
   (a) $x^{(k+1)}\leftarrow\mathrm{prox}_{\frac{1}{\rho}\Phi}(x^{(k)}-\frac{\mu}{\rho}H^\top(Hx^{(k)}-y^{(k)}))$,
   (b) $y^{(k+1)}\leftarrow\mathrm{prox}_{\frac{\lambda}{\sigma}\psi}(y^{(k)}+\frac{\mu}{\sigma}(Hx^{(k+1)}-y^{(k)}))$,
 until converges or satisfies a stopping criteria.
 Write the output of $x^{(k+1)}$ from the above iteration as $\overline{x}$.
 The restored image is $u^\star=e^{\overline{x}}$.
Table 1.  Parameter values in our algorithm (Algorithm 1) at various noise levels
$\lambda$ $\alpha$ $\beta$$\mu$ $\rho$ $\sigma$
$L=10$0.3060.00156.0630250150
$L=6$0.4060.0008515.0230250150
$L=4$0.5060.00010825030255.590.26
$L=2$0.80.000011655.0530290156.26
$\lambda$ $\alpha$ $\beta$$\mu$ $\rho$ $\sigma$
$L=10$0.3060.00156.0630250150
$L=6$0.4060.0008515.0230250150
$L=4$0.5060.00010825030255.590.26
$L=2$0.80.000011655.0530290156.26
Table 2.  Parameter values for all testing algorithms
$L$Method $\lambda$ $\alpha$ $\beta$$\mu$ $\rho$ $\sigma$
10Ours0.5691.00021.0005320.1425.520.140
TwL-4V3.6/$L$1.0$-$$-$0.3$-$
Ⅰ-DIV0.31378.0$-$$-$ $-$$-$
DZ0.07193.03.0 $-$$-$
HMNZ0.1$-$1017.5$-$$-$
6Ours0.5550.38851.00519.6285.519.667
TwL-4V2.9/$L$1.0$-$ $-$0.29$-$
Ⅰ-DIV0.45918.0 $-$$-$$-$$-$
DZ0.06$3.8$$3.0$$3.0$$-$$-$
HMNZ0.1$-$109$-$$-$
4Ours0.6590.185151.010529.658255.529.918
TwL-4V2.4/$L$1.0$-$$-$0.3
Ⅰ-DIV0.55658.0$-$$-$$-$$-$
DZ0.05$1.59$$3.0$ $3.0$$-$$-$
HMNZ0.1$-$106$-$$-$
2Ours0.80.000001115.021.016826.26
TwL-4V1.8/$L$1.0$-$$-$0.3$-$
Ⅰ-DIV0.841059.0$-$$-$$-$$-$
DZ0.065$0.45$$3.0$ $3.0$$-$$-$
HMNZ0.1$-$101.5$-$$-$
$L$Method $\lambda$ $\alpha$ $\beta$$\mu$ $\rho$ $\sigma$
10Ours0.5691.00021.0005320.1425.520.140
TwL-4V3.6/$L$1.0$-$$-$0.3$-$
Ⅰ-DIV0.31378.0$-$$-$ $-$$-$
DZ0.07193.03.0 $-$$-$
HMNZ0.1$-$1017.5$-$$-$
6Ours0.5550.38851.00519.6285.519.667
TwL-4V2.9/$L$1.0$-$ $-$0.29$-$
Ⅰ-DIV0.45918.0 $-$$-$$-$$-$
DZ0.06$3.8$$3.0$$3.0$$-$$-$
HMNZ0.1$-$109$-$$-$
4Ours0.6590.185151.010529.658255.529.918
TwL-4V2.4/$L$1.0$-$$-$0.3
Ⅰ-DIV0.55658.0$-$$-$$-$$-$
DZ0.05$1.59$$3.0$ $3.0$$-$$-$
HMNZ0.1$-$106$-$$-$
2Ours0.80.000001115.021.016826.26
TwL-4V1.8/$L$1.0$-$$-$0.3$-$
Ⅰ-DIV0.841059.0$-$$-$$-$$-$
DZ0.065$0.45$$3.0$ $3.0$$-$$-$
HMNZ0.1$-$101.5$-$$-$
Table 3.  PSNR (dB) and CPU time (s) for Ⅰ-DIV[29], DZ[11], TwL-4V[17], HMNZ[14], our algorithm (Algorithm 1 with $x^{(0)}=\log(\text{SAR-BM3D}(f))$), and SAR-BM3D[25] for test images of Fig. 1 corrupted by multiplicative noise with $L=10, 6, 4, 2$, respectively
Image $L$ Noisy Ⅰ-DIV DZ TwL-4V HMNZ Ours SAR-BM3D
Camer. 10 PSNR 15.61 28.69 28.30 28.83 29.96 30.15 28.89
Time $-$ 6.28 86.30 6.23 66.62 117.35+6.03 117.35
6 PSNR 13.39 27.35 26.93 27.59 28.46 28.53 26.35
Time $-$ 11.40 110.36 7.33 62.86 116.59+10.11 116.59
4 PSNR 11.64 26.52 25.85 26.72 27.33 27.38 23.67
Time $-$ 13.57 149.54 7.94 63.69 118.31+13.31 118.31
2 PSNR 8.63 24.98 24.27 25.28 25.30 25.47 16.45
Time $-$ 15.57 190.75 9.72 62.58 124.30+15.20 124.30
Lena 10 PSNR 15.64 28.47 27.51 28.60 29.41 29.65 28.48
Time $-$ 6.57 85.90 6.85 65.15 118.64+6.33 118.64
6 PSNR 13.42 27.34 26.24 27.48 28.07 28.14 25.92
Time $-$ 12.06 106.87 7.29 63.69 116.62+10.42 116.62
4 PSNR 11.68 26.64 25.45 26.72 27.01 27.37 23.63
Time $-$ 13.78 144.34 8.48 63.03 118.32+13.14 118.32
2 PSNR 8.71 25.07 24.01 25.17 25.21 25.50 16.46
Time $-$ 17.16 181.92 10.73 62.88 124.56+15.71 124.56
Pepp. 10 PSNR 15.93 28.83 27.20 28.86 29.13 29.53 28.09
Time $-$ 6.88 84.15 7.47 65.33 116.86+5.62 116.86
6 PSNR 13.70 27.95 26.15 27.92 28.12 28.51 25.78
Time $-$ 12.46 108.44 8.16 60.54 118.31+10.24 118.31
4 PSNR 11.98 27.10 25.10 27.05 27.14 27.76 23.63
Time $-$ 13.30 145.13 8.54 63.75 119.16+12.90 119.16
2 PSNR 8.93 25.57 23.72 25.54 25.52 25.97 16.45
Time $-$ 17.44 188.50 10.59 61.59 115.71+15.03 115.71
Rem.1 10 PSNR 16.27 25.22 25.15 25.33 26.16 26.21 25.43
Time $-$ 12.94 190.94 12.78 108.13 210.80+11.14 210.80
6 PSNR 14.00 24.17 24.01 24.41 25.02 25.07 23.64
Time $-$ 23.09 222.15 13.63 102.92 211.02+23.01 211.02
4 PSNR 12.28 23.48 23.01 23.68 24.03 23.98 22.20
Time $-$ 27.56 286.88 14.95 105.97 211.00+26.75 211.00
2 PSNR 9.28 22.39 21.80 22.57 22.63 22.76 16.45
Time $-$ 32.09 347.21 17.75 99.72 211.89+28.51 211.89
Rem.2 10 PSNR 16.23 25.56 25.59 25.63 26.76 26.69 25.92
Time $-$ 9.16 144.40 9.58 84.96 163.00+8.90 163.00
6 PSNR 14.02 24.55 24.30 24.45 25.25 25.43 24.07
Time $-$ 17.46 164.92 9.94 85.24 162.70+17.03 162.70
4 PSNR 12.27 23.45 23.38 23.63 23.93 24.26 22.26
Time $-$ 22.75 219.99 11.79 78.15 163.07+20.35 163.07
2 PSNR 9.22 22.01 21.86 22.18 22.12 22.72 16.08
Time $-$ 25.86 271.57 14.01 75.17 162.84+24.02 162.84
Image $L$ Noisy Ⅰ-DIV DZ TwL-4V HMNZ Ours SAR-BM3D
Camer. 10 PSNR 15.61 28.69 28.30 28.83 29.96 30.15 28.89
Time $-$ 6.28 86.30 6.23 66.62 117.35+6.03 117.35
6 PSNR 13.39 27.35 26.93 27.59 28.46 28.53 26.35
Time $-$ 11.40 110.36 7.33 62.86 116.59+10.11 116.59
4 PSNR 11.64 26.52 25.85 26.72 27.33 27.38 23.67
Time $-$ 13.57 149.54 7.94 63.69 118.31+13.31 118.31
2 PSNR 8.63 24.98 24.27 25.28 25.30 25.47 16.45
Time $-$ 15.57 190.75 9.72 62.58 124.30+15.20 124.30
Lena 10 PSNR 15.64 28.47 27.51 28.60 29.41 29.65 28.48
Time $-$ 6.57 85.90 6.85 65.15 118.64+6.33 118.64
6 PSNR 13.42 27.34 26.24 27.48 28.07 28.14 25.92
Time $-$ 12.06 106.87 7.29 63.69 116.62+10.42 116.62
4 PSNR 11.68 26.64 25.45 26.72 27.01 27.37 23.63
Time $-$ 13.78 144.34 8.48 63.03 118.32+13.14 118.32
2 PSNR 8.71 25.07 24.01 25.17 25.21 25.50 16.46
Time $-$ 17.16 181.92 10.73 62.88 124.56+15.71 124.56
Pepp. 10 PSNR 15.93 28.83 27.20 28.86 29.13 29.53 28.09
Time $-$ 6.88 84.15 7.47 65.33 116.86+5.62 116.86
6 PSNR 13.70 27.95 26.15 27.92 28.12 28.51 25.78
Time $-$ 12.46 108.44 8.16 60.54 118.31+10.24 118.31
4 PSNR 11.98 27.10 25.10 27.05 27.14 27.76 23.63
Time $-$ 13.30 145.13 8.54 63.75 119.16+12.90 119.16
2 PSNR 8.93 25.57 23.72 25.54 25.52 25.97 16.45
Time $-$ 17.44 188.50 10.59 61.59 115.71+15.03 115.71
Rem.1 10 PSNR 16.27 25.22 25.15 25.33 26.16 26.21 25.43
Time $-$ 12.94 190.94 12.78 108.13 210.80+11.14 210.80
6 PSNR 14.00 24.17 24.01 24.41 25.02 25.07 23.64
Time $-$ 23.09 222.15 13.63 102.92 211.02+23.01 211.02
4 PSNR 12.28 23.48 23.01 23.68 24.03 23.98 22.20
Time $-$ 27.56 286.88 14.95 105.97 211.00+26.75 211.00
2 PSNR 9.28 22.39 21.80 22.57 22.63 22.76 16.45
Time $-$ 32.09 347.21 17.75 99.72 211.89+28.51 211.89
Rem.2 10 PSNR 16.23 25.56 25.59 25.63 26.76 26.69 25.92
Time $-$ 9.16 144.40 9.58 84.96 163.00+8.90 163.00
6 PSNR 14.02 24.55 24.30 24.45 25.25 25.43 24.07
Time $-$ 17.46 164.92 9.94 85.24 162.70+17.03 162.70
4 PSNR 12.27 23.45 23.38 23.63 23.93 24.26 22.26
Time $-$ 22.75 219.99 11.79 78.15 163.07+20.35 163.07
2 PSNR 9.22 22.01 21.86 22.18 22.12 22.72 16.08
Time $-$ 25.86 271.57 14.01 75.17 162.84+24.02 162.84
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