doi: 10.3934/ipi.2020068

An adaptive total variational despeckling model based on gray level indicator frame

1. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

School of Management, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Zhichang Guo

Received  October 2019 Revised  July 2020 Published  November 2020

Fund Project: This work is partially supported by the National Natural Science Foundation of China (11971131, U1637208, 61873071, 51476047, 11871133, 71773024), the Natural Sciences Foundation of Heilongjiang Province of China (LC2018001, G2018006) and the Heilongjiang Postdoctoral Scientific Research Developmental Fund (LBH-Q18064)

For the characteristics of the degraded images with multiplicative noise, the gray level indicators for constructing adaptive total variation are proposed. Based on the new regularization term, we propose the new convex adaptive variational model. Then, considering the existence, uniqueness and comparison principle of the minimizer of the functional. The finite difference method with rescaling technique and the primal-dual method with adaptive step size are used to solve the minimization problem. The paper ends with a report on numerical tests for the denoising of images subject to multiplicative noise, the comparison with other methods is provided as well.

Citation: Yu Zhang, Songsong Li, Zhichang Guo, Boying Wu. An adaptive total variational despeckling model based on gray level indicator frame. Inverse Problems & Imaging, doi: 10.3934/ipi.2020068
References:
[1]

G. Aubert and J.-F. Aujol, A variational approach to remove multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946.  doi: 10.1137/060671814.  Google Scholar

[2]

A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing, 18 (2009), 2419-2434.  doi: 10.1109/TIP.2009.2028250.  Google Scholar

[3]

A. C. BovikT. S. Huang and D. C. Munson, A generalization of median filtering using linear combinations of order statistics, IEEE Trans Acoustics Speech Signal Processing, 31 (1983), 1342-1350.   Google Scholar

[4]

A. C. BovikT. S. Huang and D. C. Munson, The effect of median filtering on edge estimation and detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9 (1987), 181-194.  doi: 10.1109/TPAMI.1987.4767894.  Google Scholar

[5]

K. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[6]

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.   Google Scholar

[7]

A. Chambolle, An algorithm for mean curvature motion, Interfaces and Free Boundaries, 6 (2004), 195-218.  doi: 10.4171/IFB/97.  Google Scholar

[8]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[9]

C. ChauxJ.-C. Pesquet and N. Pustelnik, Nested iterative algorithms for convex constrained image recovery problems, SIAM Journal on Imaging Sciences, 2 (2009), 730-762.  doi: 10.1137/080727749.  Google Scholar

[10]

Y. Chen and M. Rao, Minimization problems and associated flows related to weighted p energy and total variation, SIAM Journal on mathematical Analysis, 34 (2003), 1084-1104.  doi: 10.1137/S0036141002404577.  Google Scholar

[11]

Y. Chen and T. Wunderli, Adaptive total variation for image restoration in BV space, Journal of Mathematical Analysis and Applications, 272 (2002), 117-137.  doi: 10.1016/S0022-247X(02)00141-5.  Google Scholar

[12]

P. L. Combettes and J.-C. Pesquet, A douglas-rachford splitting approach to nonsmooth convex variational signal recovery, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 564-574.  doi: 10.1109/JSTSP.2007.910264.  Google Scholar

[13]

A. Dauwe, B. Goossens, H. Luong and W. Philips, A fast non-local image denoising algorithm, Electronic Imaging, 6812 (2008), 681210. doi: 10.1117/12.765505.  Google Scholar

[14]

C.-A. Deledalle, L. Denis, S. Tabti and F. Tupin, MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?, IEEE Transactions on Image Processing, 26 (2017), 4389-4303. doi: 10.1109/TIP.2017.2713946.  Google Scholar

[15]

G. DongZ. Guo and B. Wu, A convex adaptive total variation model based on the gray level indicator for multiplicative noise removal, Abstract and Applied Analysis, 2013 (2013), 1-21.  doi: 10.1155/2013/912373.  Google Scholar

[16]

Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM Journal on Imaging Sciences, 6 (2013), 1598-1625.  doi: 10.1137/120870621.  Google Scholar

[17]

S. DurandJ. Fadili and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, Journal of Mathematical Imaging and Vision, 36 (2010), 201-226.  doi: 10.1007/s10851-009-0180-z.  Google Scholar

[18]

E. EsserX. Zhang and T. F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, Siam Journal on Imaging Sciences, 3 (2010), 1015-1046.  doi: 10.1137/09076934X.  Google Scholar

[19]

K. FlorianB. KristianP. Thomas and S. Rudolf, Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.  doi: 10.1002/mrm.22595.  Google Scholar

[20]

E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, Birkhäuser Verlag, Basel, 80 1984, 7-26. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[21]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[22]

H. NaM. KangM. Jung and M. Kang, Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters, Inverse Problems and Imaging, 13 (2019), 117-147.  doi: 10.3934/ipi.2019007.  Google Scholar

[23]

P. KornprobstR. Deriche and G. Aubert, Image sequence analysis via partial differential equations, Journal of Mathematical Imaging and Vision, 11 (1999), 5-26.  doi: 10.1023/A:1008318126505.  Google Scholar

[24]

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

[25]

T. LeR. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[26]

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

[27]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[28]

T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the mumford-Shah functional, IEEE International Conference on Computer Vision, (2009), 1133-1140. doi: 10.1109/ICCV.2009.5459348.  Google Scholar

[29]

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

[30]

C. Poynton, Digital Video and HD: Algorithms and Interfaces, 2$^nd$ edition, Elsevier, USA, 2012. doi: 10.1016/C2010-0-68987-5.  Google Scholar

[31]

X. ShanJ. Sun and Z. Guo, Multiplicative noise removal based on the smooth diffusion equation, Journal of Mathematical Imaging and Vision, 61 (2019), 763-779.  doi: 10.1007/s10851-018-00870-z.  Google Scholar

[32]

R. Soorajkumar, P. K. Kumar, D. Girish and J. Rajan, Fourth order PDE based ultrasound despeckling using ENI classification, IEEE International Conference on Signal Processing and Communications (SPCOM), (2016). doi: 10.1109/SPCOM.2016.7746633.  Google Scholar

[33]

D. M. Strong and T. F. Chan, Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing, Journal of Mathematical Imaging and Vision, (1996). Google Scholar

[34]

M. TurK. C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.  doi: 10.1364/AO.21.001157.  Google Scholar

[35]

L. Verdoliva, R. Gaetano, G. Ruello and G. Poggi, Optical-driven nonlocal SAR despeckling, IEEE Geoscience and Remote Sensing Letters, 12 (2015), 314-318. doi: 10.1109/LGRS.2014.2337515.  Google Scholar

[36]

Z. WangA. C. Bovik and H. R. Sheikh, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 53 (2015), 2765-2774.   Google Scholar

[37]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, ECMI Series, Teubner, Stuttgart, 1998.  Google Scholar

[38]

C. Wu and X.-C. Tai, Augmented Lagrangian method, dual Methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[39]

J.-H. YangX.-L. ZhaoT.-H. MaY. ChenT.-Z. Huang and M. Ding, Emote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization, Journal of Computational and Applied Mathematics, 363 (2020), 124-144.  doi: 10.1016/j.cam.2019.06.004.  Google Scholar

[40]

X.-L. ZhaoF. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imagingences, 7 (2014), 456-475.  doi: 10.1137/13092472X.  Google Scholar

[41]

Y. ZhaoJ. G. LiuB. ZhangW. Hong and Y.-R. Wu, Adaptive total variation regularization based SAR image despeckling and despeckling evaluation index, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 2765-2774.  doi: 10.1109/TGRS.2014.2364525.  Google Scholar

[42]

Z. ZhouZ. GuoG. DongJ. SunD. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Transactions on Image Processing, 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185.  Google Scholar

show all references

References:
[1]

G. Aubert and J.-F. Aujol, A variational approach to remove multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946.  doi: 10.1137/060671814.  Google Scholar

[2]

A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing, 18 (2009), 2419-2434.  doi: 10.1109/TIP.2009.2028250.  Google Scholar

[3]

A. C. BovikT. S. Huang and D. C. Munson, A generalization of median filtering using linear combinations of order statistics, IEEE Trans Acoustics Speech Signal Processing, 31 (1983), 1342-1350.   Google Scholar

[4]

A. C. BovikT. S. Huang and D. C. Munson, The effect of median filtering on edge estimation and detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9 (1987), 181-194.  doi: 10.1109/TPAMI.1987.4767894.  Google Scholar

[5]

K. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[6]

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.   Google Scholar

[7]

A. Chambolle, An algorithm for mean curvature motion, Interfaces and Free Boundaries, 6 (2004), 195-218.  doi: 10.4171/IFB/97.  Google Scholar

[8]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[9]

C. ChauxJ.-C. Pesquet and N. Pustelnik, Nested iterative algorithms for convex constrained image recovery problems, SIAM Journal on Imaging Sciences, 2 (2009), 730-762.  doi: 10.1137/080727749.  Google Scholar

[10]

Y. Chen and M. Rao, Minimization problems and associated flows related to weighted p energy and total variation, SIAM Journal on mathematical Analysis, 34 (2003), 1084-1104.  doi: 10.1137/S0036141002404577.  Google Scholar

[11]

Y. Chen and T. Wunderli, Adaptive total variation for image restoration in BV space, Journal of Mathematical Analysis and Applications, 272 (2002), 117-137.  doi: 10.1016/S0022-247X(02)00141-5.  Google Scholar

[12]

P. L. Combettes and J.-C. Pesquet, A douglas-rachford splitting approach to nonsmooth convex variational signal recovery, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 564-574.  doi: 10.1109/JSTSP.2007.910264.  Google Scholar

[13]

A. Dauwe, B. Goossens, H. Luong and W. Philips, A fast non-local image denoising algorithm, Electronic Imaging, 6812 (2008), 681210. doi: 10.1117/12.765505.  Google Scholar

[14]

C.-A. Deledalle, L. Denis, S. Tabti and F. Tupin, MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?, IEEE Transactions on Image Processing, 26 (2017), 4389-4303. doi: 10.1109/TIP.2017.2713946.  Google Scholar

[15]

G. DongZ. Guo and B. Wu, A convex adaptive total variation model based on the gray level indicator for multiplicative noise removal, Abstract and Applied Analysis, 2013 (2013), 1-21.  doi: 10.1155/2013/912373.  Google Scholar

[16]

Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM Journal on Imaging Sciences, 6 (2013), 1598-1625.  doi: 10.1137/120870621.  Google Scholar

[17]

S. DurandJ. Fadili and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, Journal of Mathematical Imaging and Vision, 36 (2010), 201-226.  doi: 10.1007/s10851-009-0180-z.  Google Scholar

[18]

E. EsserX. Zhang and T. F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, Siam Journal on Imaging Sciences, 3 (2010), 1015-1046.  doi: 10.1137/09076934X.  Google Scholar

[19]

K. FlorianB. KristianP. Thomas and S. Rudolf, Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.  doi: 10.1002/mrm.22595.  Google Scholar

[20]

E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, Birkhäuser Verlag, Basel, 80 1984, 7-26. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[21]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[22]

H. NaM. KangM. Jung and M. Kang, Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters, Inverse Problems and Imaging, 13 (2019), 117-147.  doi: 10.3934/ipi.2019007.  Google Scholar

[23]

P. KornprobstR. Deriche and G. Aubert, Image sequence analysis via partial differential equations, Journal of Mathematical Imaging and Vision, 11 (1999), 5-26.  doi: 10.1023/A:1008318126505.  Google Scholar

[24]

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

[25]

T. LeR. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[26]

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

[27]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[28]

T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the mumford-Shah functional, IEEE International Conference on Computer Vision, (2009), 1133-1140. doi: 10.1109/ICCV.2009.5459348.  Google Scholar

[29]

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

[30]

C. Poynton, Digital Video and HD: Algorithms and Interfaces, 2$^nd$ edition, Elsevier, USA, 2012. doi: 10.1016/C2010-0-68987-5.  Google Scholar

[31]

X. ShanJ. Sun and Z. Guo, Multiplicative noise removal based on the smooth diffusion equation, Journal of Mathematical Imaging and Vision, 61 (2019), 763-779.  doi: 10.1007/s10851-018-00870-z.  Google Scholar

[32]

R. Soorajkumar, P. K. Kumar, D. Girish and J. Rajan, Fourth order PDE based ultrasound despeckling using ENI classification, IEEE International Conference on Signal Processing and Communications (SPCOM), (2016). doi: 10.1109/SPCOM.2016.7746633.  Google Scholar

[33]

D. M. Strong and T. F. Chan, Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing, Journal of Mathematical Imaging and Vision, (1996). Google Scholar

[34]

M. TurK. C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.  doi: 10.1364/AO.21.001157.  Google Scholar

[35]

L. Verdoliva, R. Gaetano, G. Ruello and G. Poggi, Optical-driven nonlocal SAR despeckling, IEEE Geoscience and Remote Sensing Letters, 12 (2015), 314-318. doi: 10.1109/LGRS.2014.2337515.  Google Scholar

[36]

Z. WangA. C. Bovik and H. R. Sheikh, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 53 (2015), 2765-2774.   Google Scholar

[37]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, ECMI Series, Teubner, Stuttgart, 1998.  Google Scholar

[38]

C. Wu and X.-C. Tai, Augmented Lagrangian method, dual Methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[39]

J.-H. YangX.-L. ZhaoT.-H. MaY. ChenT.-Z. Huang and M. Ding, Emote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization, Journal of Computational and Applied Mathematics, 363 (2020), 124-144.  doi: 10.1016/j.cam.2019.06.004.  Google Scholar

[40]

X.-L. ZhaoF. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imagingences, 7 (2014), 456-475.  doi: 10.1137/13092472X.  Google Scholar

[41]

Y. ZhaoJ. G. LiuB. ZhangW. Hong and Y.-R. Wu, Adaptive total variation regularization based SAR image despeckling and despeckling evaluation index, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 2765-2774.  doi: 10.1109/TGRS.2014.2364525.  Google Scholar

[42]

Z. ZhouZ. GuoG. DongJ. SunD. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Transactions on Image Processing, 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185.  Google Scholar

Figure 1.  The relation between the influence of the noise and the gray levels. (a)1D original ladder-shaped signal; (b)1D speckle noise with mean of 1 and $ L = 4 $; (c) the comparison of original signal and speckle-corrupted signal; (d) The gray area shows values between the first and third quartiles, the dots represent the expectation and the blue line a single noisy realization. Expectation and quartiles are computed from 10000 noisy realizations
Figure 2.  Pretreatment effect map. The black curve is the original signal, and the green curve is the observed signal which has polluted by the Gamma noise with a standard deviation of $ 1/\sqrt{6} $ ($ L $ = 6). The red curve is result of preprocessing
Figure 3.  Adjustment for $ \alpha $. In the first row, the red curves are the observed signal which have polluted by the Gamma noise with a standard deviation of 1/2 (L = 4). In the second line, the blue and green curves represent the corresponding $ \alpha $ and $ \alpha^{'} $ respectively
Figure 4.  Comparison of AA, DZ and our model for multiplicative noise removal. The red curve is the denoising relization of the observed signals by each model. The random Gamma noise with a standard deviation of 1/$ \sqrt{6} $ ($ L $ = 6)
Figure 5.  Comparison of AA, DZ and our model for multiplicative noise removal. The random Gamma noise with a mean of 1 and a standard deviation of 1/2 (L = 4)
Figure 6.  Pie image (214 $ \times $ 216, $ L $ = 1.)
Figure 7.  Pie image (214 $ \times $ 216, $ L $=4.)
Figure 8.  Pie image (214 $ \times $ 216, $ L $ = 10.)
Figure 9.  Cameraman image (256 $ \times $ 256, $ L $ = 1)
Figure 10.  Cameraman image (256 $ \times $ 256, $ L $=4)
Figure 11.  Cameraman image (256 $ \times $ 256, $ L $ = 10)
Figure 12.  Parrot image (512 $ \times $ 512, $ L $ = 1.)
Figure 13.  Parrot image (512 $ \times $ 512, $ L $=4.)
Figure 14.  Parrot image (512 $ \times $ 512, $ L $ = 10.)
Figure 15.  Synthetic image (300 $ \times $ 300, $ L $ = 1.)
Figure 16.  Synthetic image (300 $ \times $ 300, $ L $=4.)
Figure 17.  Synthetic image (300 $ \times $ 300, $ L $ = 10.)
Figure 18.  Piglet image ($ 341 \times 199 $, $ L $ = 1.)
Figure 19.  Piglet image ($ 341 \times 199 $, $ L $=4.)
Figure 20.  Piglet image ($ 341 \times 199 $, $ L $ = 10.)
Figure 21.  Plots of Energy and PSNR of $ \tilde{u}^{k} $. The first column is about Pie(L=10) and the second column is about Cameraman(L=4)
Table 1.  Parameters used in the comparison study
Parameters
Algorithm $ L=1 $ $ L=4 $ $ L=10 $
The Pie image (214 $ \times $ 216)
AA ($ \lambda $) 5e-4 0.2 22
DZ ($ \alpha,\lambda $) 0.3, 0.2 0.5, 0.04 5, 0.03
$ E_{1} $-FDM ($ \lambda,\beta,p $) 3e-7, 3e-8, 1.3 2e-3, 1e-3, 1.3 2e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 1e-7, 1e-8, 1.39, 0.74 2e-3, 1e-3, 1.39, 0.74 2e-3, 3e-3, 1.39, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 4e-3, 6e-3, 1.3 4e-3, 5e-3, 1.3 3e-3, 6e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 4e-2, 6e-3, 1.39, 0.74 4e-3, 5e-3, 1.38, 0.74 4e-3, 4e-3, 1.34, 0.74
The Cameraman image($ 256\times256 $)
AA ($ \lambda $) 0.18 40 75
DZ ($ \alpha,\lambda $) 0.21, 0.25 1.6, 0.11 2.5, 0.03
$ E_{1} $-FDM ($ \lambda,\beta,p $) 1e-7, 5e-7, 1.3 2e-3, 2e-3, 1.3 3e-2, 2e-2, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 3e-7, 2e-7, 1.3, 0.8 3e-3, 2e-3, 1.33, 0.76 3e-2, 2e-2, 1.4, 0.78
$ E_{1} $-PDM ($ \lambda,\beta,p $) 6e-3, 8e-3, 1.3 8e-3, 6e-3, 1.3 8e-3, 3e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 1e-2, 6e-3, 1.3, 0.79 8e-3, 4e-3, 1.3, 0.75 6e-3, 6e-3, 1.3, 0.75
The Parrot image($ 512\times512 $)
AA ($ \lambda $) 0.6 1.5 8
DZ ($ \alpha,\lambda $) 0.11, 0.2 1, 0.2 15, 0.3
$ E_{1} $-FDM ($ \lambda,\beta,p $) 2e-6, 2e-6, 1.3 1e-4, 1e-4, 1.3 3e-3, 3e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 1e-6, 1e-6, 1.36, 0.74 1e-4, 1e-4, 1.35, 0.74 1e-3, 1e-3, 1.32, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 6e-3, 6e-3, 1.3 6e-3, 6e-3, 1.3 6e-3, 6e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 6e-3, 6e-3, 1.36, 0.74 6e-3, 6e-3, 1.34, 0.74 6e-3, 6e-3, 1.32, 0.74
The Synthetic image($ 300\times300 $)
AA ($ \lambda $) 7e-4 4e-3 1e-3
DZ ($ \alpha,\lambda $) 0.117, 0.125 0.155, 0.1 0.11, 0.2
$ E_{1} $-FDM ($ \lambda,\beta,p $) 1e-7, 1e-7, 1.3 2e-4, 2e-4, 1.3 1e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 1e-7, 1e-7, 1.35, 0.78 1e-4, 1e-4, 1.36, 0.74 1e-3, 1e-3, 1.35, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 4e-3, 5e-3, 1.3 4e-3, 4e-3, 1.3 4e-3, 4e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 4e-3, 5e-3, 1.4, 0.75 4e-3, 5e-3, 1.4, 0.74 4e-3, 4e-3, 1.32, 0.74
The Piglet image($ 341\times199 $)
AA ($ \lambda $) 5e-3 80 190
DZ ($ \alpha,\lambda $) 3e-3, 0.16 2e-3, 3e-2 0.1, 7e-3
$ E_{1} $-FDM ($ \lambda,\beta,p $) 1e-7, 9e-8, 1.3 1e-4, 1e-4, 1.3 1e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 2e-7, 9e-8, 1.38, 0.74 1e-4, 1e-4, 1.37, 0.74 1e-3, 1e-3, 1.39, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 4e-3, 4e-3, 1.3 4e-3, 7e-3, 1.3 6e-3, 8e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 4e-3, 7e-3, 1.35, 0.74 4e-3, 7e-3, 1.35, 0.74 6e-3, 8e-3, 1.32, 0.74
Parameters
Algorithm $ L=1 $ $ L=4 $ $ L=10 $
The Pie image (214 $ \times $ 216)
AA ($ \lambda $) 5e-4 0.2 22
DZ ($ \alpha,\lambda $) 0.3, 0.2 0.5, 0.04 5, 0.03
$ E_{1} $-FDM ($ \lambda,\beta,p $) 3e-7, 3e-8, 1.3 2e-3, 1e-3, 1.3 2e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 1e-7, 1e-8, 1.39, 0.74 2e-3, 1e-3, 1.39, 0.74 2e-3, 3e-3, 1.39, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 4e-3, 6e-3, 1.3 4e-3, 5e-3, 1.3 3e-3, 6e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 4e-2, 6e-3, 1.39, 0.74 4e-3, 5e-3, 1.38, 0.74 4e-3, 4e-3, 1.34, 0.74
The Cameraman image($ 256\times256 $)
AA ($ \lambda $) 0.18 40 75
DZ ($ \alpha,\lambda $) 0.21, 0.25 1.6, 0.11 2.5, 0.03
$ E_{1} $-FDM ($ \lambda,\beta,p $) 1e-7, 5e-7, 1.3 2e-3, 2e-3, 1.3 3e-2, 2e-2, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 3e-7, 2e-7, 1.3, 0.8 3e-3, 2e-3, 1.33, 0.76 3e-2, 2e-2, 1.4, 0.78
$ E_{1} $-PDM ($ \lambda,\beta,p $) 6e-3, 8e-3, 1.3 8e-3, 6e-3, 1.3 8e-3, 3e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 1e-2, 6e-3, 1.3, 0.79 8e-3, 4e-3, 1.3, 0.75 6e-3, 6e-3, 1.3, 0.75
The Parrot image($ 512\times512 $)
AA ($ \lambda $) 0.6 1.5 8
DZ ($ \alpha,\lambda $) 0.11, 0.2 1, 0.2 15, 0.3
$ E_{1} $-FDM ($ \lambda,\beta,p $) 2e-6, 2e-6, 1.3 1e-4, 1e-4, 1.3 3e-3, 3e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 1e-6, 1e-6, 1.36, 0.74 1e-4, 1e-4, 1.35, 0.74 1e-3, 1e-3, 1.32, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 6e-3, 6e-3, 1.3 6e-3, 6e-3, 1.3 6e-3, 6e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 6e-3, 6e-3, 1.36, 0.74 6e-3, 6e-3, 1.34, 0.74 6e-3, 6e-3, 1.32, 0.74
The Synthetic image($ 300\times300 $)
AA ($ \lambda $) 7e-4 4e-3 1e-3
DZ ($ \alpha,\lambda $) 0.117, 0.125 0.155, 0.1 0.11, 0.2
$ E_{1} $-FDM ($ \lambda,\beta,p $) 1e-7, 1e-7, 1.3 2e-4, 2e-4, 1.3 1e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 1e-7, 1e-7, 1.35, 0.78 1e-4, 1e-4, 1.36, 0.74 1e-3, 1e-3, 1.35, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 4e-3, 5e-3, 1.3 4e-3, 4e-3, 1.3 4e-3, 4e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 4e-3, 5e-3, 1.4, 0.75 4e-3, 5e-3, 1.4, 0.74 4e-3, 4e-3, 1.32, 0.74
The Piglet image($ 341\times199 $)
AA ($ \lambda $) 5e-3 80 190
DZ ($ \alpha,\lambda $) 3e-3, 0.16 2e-3, 3e-2 0.1, 7e-3
$ E_{1} $-FDM ($ \lambda,\beta,p $) 1e-7, 9e-8, 1.3 1e-4, 1e-4, 1.3 1e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $) 2e-7, 9e-8, 1.38, 0.74 1e-4, 1e-4, 1.37, 0.74 1e-3, 1e-3, 1.39, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $) 4e-3, 4e-3, 1.3 4e-3, 7e-3, 1.3 6e-3, 8e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $) 4e-3, 7e-3, 1.35, 0.74 4e-3, 7e-3, 1.35, 0.74 6e-3, 8e-3, 1.32, 0.74
Table 2.  Comparison of PSNR, MAE, SSIM and iterations
PSNR MAE SSIM #iter
$ L $ $ 1 $ $ 4 $ $ 10 $ $ 1 $ $ 4 $ $ 10 $ $ 1 $ $ 4 $ $ 10 $ $ 1 $ $ 4 $ $ 10 $
The Pie image($ 214\times216 $)
AA $ 13.12 $ $ 18.43 $ $ 22.20 $ $ 42.51 $ $ 22.66 $ $ 13.40 $ 0.57 0.77 0.88 1978 743 464
DZ $ 14.25 $ $ 19.94 $ $ 23.45 $ $ 32.77 $ $ 16.26 $ $ 10.14 $ 0.63 0.86 0.92 637 1162 1027
$ E_{1} $-FDM $ 17.26 $ $ 21.35 $ $ 24.00 $ $ 23.71 $ $ 14.33 $ 10.15 0.67 0.84 0.90 481 275 273
$ E_{2} $-FDM $ 17.45 $ $ 21.30 $ 24.08 $ 23.50 $ $ 14.44 $ 10.10 0.69 0.84 0.90 362 256 211
$ E_{1} $-PDM $ 17.22 $ $ 21.30 $ $ 23.86 $ $ 24.00 $ $ 14.29 $ $ 10.15 $ 0.71 0.84 0.90 1973 609 312
$ E_{2} $-PDM $ 17.46 $ $ 21.21 $ $ 23.87 $ $ 23.19 $ $ 14.53 $ $ 10.28 $ 0.71 0.83 0.90 1475 485 239
The cameraman image($ 256\times256 $)
AA 19.06 22.96 25.09 16.79 10.09 7.94 0.56 0.69 0.74 1800 672 351
DZ 19.34 22.81 25.22 16.64 10.45 7.94 0.53 0.70 0.75 387 291 338
$ E_{1} $-FDM 21.56 24.44 26.37 12.95 9.19 7.29 0.63 0.71 0.77 442 361 289
$ E_{2} $-FDM 21.56 24.50 26.42 12.98 9.18 7.27 0.63 0.71 0.76 395 315 238
$ E_{1} $-PDM 21.70 24.28 26.21 12.60 9.01 7.29 0.65 0.73 0.78 1469 455 243
$ E_{2} $-PDM 21.76 24.30 26.23 11.96 9.00 7.35 0.66 0.73 0.77 1167 382 203
The Parrot image($ 512\times512 $)
AA 20.52 24.46 26.89 14.73 10.15 7.86 0.60 0.68 0.72 1833 666 359
DZ 20.59 25.19 27.52 14.22 9.38 7.29 0.57 0.71 0.76 533 187 69
$ E_{1} $-FDM 23.84 26.73 28.57 10.88 7.95 6.49 0.66 0.74 0.78 445 391 342
$ E_{2} $-FDM 23.84 26.75 28.61 10.96 8.00 6.48 0.65 0.73 0.78 372 309 227
$ E_{1} $-PDM 23.88 26.73 28.55 10.93 7.94 6.50 0.65 0.73 0.78 1907 629 323
$ E_{2} $-PDM 23.81 26.71 28.60 11.96 9.00 6.48 0.65 0.73 0.78 1572 523 240
The Synthetic image($ 300\times300 $)
AA $ 22.31 $ $ 27.44 $ $ 30.43 $ $ 11.90 $ $ 6.32 $ $ 4.35 $ 0.81 0.88 0.92 2664 1010 643
DZ $ 22.43 $ $ 27.90 $ $ 30.54 $ $ 10.30 $ $ 5.42 $ $ 4.04 $ 0.77 0.94 0.96 1195 570 291
$ E_{1} $-FDM $ 24.71 $ $ 29.08 $ $ 31.68 $ $ 9.02 $ $ 4.88 $ 3.30 0.84 0.93 0.95 507 442 371
$ E_{2} $-FDM $ 24.77 $ $ 29.07 $ 31.71 $ 8.61 $ $ 4.94 $ 3.41 0.83 0.92 0.95 457 337 282
$ E_{1} $-PDM $ 25.44 $ $ 28.89 $ $ 31.46 $ $ 7.30 $ $ 4.29 $ $ 3.24 $ 0.89 0.93 0.96 2506 787 425
$ E_{2} $-PDM $ 25.36 $ $ 28.93 $ $ 31.42 $ $ 7.43 $ $ 4.43 $ $ 3.24 $ 0.89 0.92 0.96 1759 645 338
The Piglet image($ 341 \times 199 $)
AA $ 21.70 $ $ 26.96 $ $ 30.08 $ $ 12.27 $ $ 6.88 $ $ 4.72 $ 0.83 0.89 0.91 2363 1068 660
DZ $ 21.74 $ $ 27.17 $ $ 30.18 $ $ 11.94 $ $ 6.67 $ $ 4.66 $ 0.73 0.89 0.91 739 1540 4440
$ E_{1} $-FDM $ 24.92 $ $ 28.45 $ $ 30.89 $ $ 9.65 $ $ 6.08 $ 4.53 0.82 0.89 0.92 499 390 345
$ E_{2} $-FDM $ 24.82 $ $ 28.51 $ 30.75 $ 9.44 $ $ 5.88 $ 4.58 0.78 0.88 0.91 432 321 297
$ E_{1} $-PDM $ 24.96 $ $ 28.33 $ $ 30.68 $ $ 8.99 $ $ 5.95 $ $ 4.58 $ 0.84 0.88 0.91 1904 666 335
$ E_{2} $-PDM $ 24.89 $ $ 28.31 $ $ 30.68 $ $ 9.12 $ $ 5.96 $ $ 4.46 $ 0.80 0.88 0.92 1463 530 280
PSNR MAE SSIM #iter
$ L $ $ 1 $ $ 4 $ $ 10 $ $ 1 $ $ 4 $ $ 10 $ $ 1 $ $ 4 $ $ 10 $ $ 1 $ $ 4 $ $ 10 $
The Pie image($ 214\times216 $)
AA $ 13.12 $ $ 18.43 $ $ 22.20 $ $ 42.51 $ $ 22.66 $ $ 13.40 $ 0.57 0.77 0.88 1978 743 464
DZ $ 14.25 $ $ 19.94 $ $ 23.45 $ $ 32.77 $ $ 16.26 $ $ 10.14 $ 0.63 0.86 0.92 637 1162 1027
$ E_{1} $-FDM $ 17.26 $ $ 21.35 $ $ 24.00 $ $ 23.71 $ $ 14.33 $ 10.15 0.67 0.84 0.90 481 275 273
$ E_{2} $-FDM $ 17.45 $ $ 21.30 $ 24.08 $ 23.50 $ $ 14.44 $ 10.10 0.69 0.84 0.90 362 256 211
$ E_{1} $-PDM $ 17.22 $ $ 21.30 $ $ 23.86 $ $ 24.00 $ $ 14.29 $ $ 10.15 $ 0.71 0.84 0.90 1973 609 312
$ E_{2} $-PDM $ 17.46 $ $ 21.21 $ $ 23.87 $ $ 23.19 $ $ 14.53 $ $ 10.28 $ 0.71 0.83 0.90 1475 485 239
The cameraman image($ 256\times256 $)
AA 19.06 22.96 25.09 16.79 10.09 7.94 0.56 0.69 0.74 1800 672 351
DZ 19.34 22.81 25.22 16.64 10.45 7.94 0.53 0.70 0.75 387 291 338
$ E_{1} $-FDM 21.56 24.44 26.37 12.95 9.19 7.29 0.63 0.71 0.77 442 361 289
$ E_{2} $-FDM 21.56 24.50 26.42 12.98 9.18 7.27 0.63 0.71 0.76 395 315 238
$ E_{1} $-PDM 21.70 24.28 26.21 12.60 9.01 7.29 0.65 0.73 0.78 1469 455 243
$ E_{2} $-PDM 21.76 24.30 26.23 11.96 9.00 7.35 0.66 0.73 0.77 1167 382 203
The Parrot image($ 512\times512 $)
AA 20.52 24.46 26.89 14.73 10.15 7.86 0.60 0.68 0.72 1833 666 359
DZ 20.59 25.19 27.52 14.22 9.38 7.29 0.57 0.71 0.76 533 187 69
$ E_{1} $-FDM 23.84 26.73 28.57 10.88 7.95 6.49 0.66 0.74 0.78 445 391 342
$ E_{2} $-FDM 23.84 26.75 28.61 10.96 8.00 6.48 0.65 0.73 0.78 372 309 227
$ E_{1} $-PDM 23.88 26.73 28.55 10.93 7.94 6.50 0.65 0.73 0.78 1907 629 323
$ E_{2} $-PDM 23.81 26.71 28.60 11.96 9.00 6.48 0.65 0.73 0.78 1572 523 240
The Synthetic image($ 300\times300 $)
AA $ 22.31 $ $ 27.44 $ $ 30.43 $ $ 11.90 $ $ 6.32 $ $ 4.35 $ 0.81 0.88 0.92 2664 1010 643
DZ $ 22.43 $ $ 27.90 $ $ 30.54 $ $ 10.30 $ $ 5.42 $ $ 4.04 $ 0.77 0.94 0.96 1195 570 291
$ E_{1} $-FDM $ 24.71 $ $ 29.08 $ $ 31.68 $ $ 9.02 $ $ 4.88 $ 3.30 0.84 0.93 0.95 507 442 371
$ E_{2} $-FDM $ 24.77 $ $ 29.07 $ 31.71 $ 8.61 $ $ 4.94 $ 3.41 0.83 0.92 0.95 457 337 282
$ E_{1} $-PDM $ 25.44 $ $ 28.89 $ $ 31.46 $ $ 7.30 $ $ 4.29 $ $ 3.24 $ 0.89 0.93 0.96 2506 787 425
$ E_{2} $-PDM $ 25.36 $ $ 28.93 $ $ 31.42 $ $ 7.43 $ $ 4.43 $ $ 3.24 $ 0.89 0.92 0.96 1759 645 338
The Piglet image($ 341 \times 199 $)
AA $ 21.70 $ $ 26.96 $ $ 30.08 $ $ 12.27 $ $ 6.88 $ $ 4.72 $ 0.83 0.89 0.91 2363 1068 660
DZ $ 21.74 $ $ 27.17 $ $ 30.18 $ $ 11.94 $ $ 6.67 $ $ 4.66 $ 0.73 0.89 0.91 739 1540 4440
$ E_{1} $-FDM $ 24.92 $ $ 28.45 $ $ 30.89 $ $ 9.65 $ $ 6.08 $ 4.53 0.82 0.89 0.92 499 390 345
$ E_{2} $-FDM $ 24.82 $ $ 28.51 $ 30.75 $ 9.44 $ $ 5.88 $ 4.58 0.78 0.88 0.91 432 321 297
$ E_{1} $-PDM $ 24.96 $ $ 28.33 $ $ 30.68 $ $ 8.99 $ $ 5.95 $ $ 4.58 $ 0.84 0.88 0.91 1904 666 335
$ E_{2} $-PDM $ 24.89 $ $ 28.31 $ $ 30.68 $ $ 9.12 $ $ 5.96 $ $ 4.46 $ 0.80 0.88 0.92 1463 530 280
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