Mixed Fractional-Order and High-Order Adaptive Image Denoising Algorithm Based on Weight Selection Function
Abstract
:1. Introduction
2. Related Works
2.1. Fractional-Order Variation Model
2.2. High-Order Variation Model
2.3. Structural Tensor
- flat regions: ;
- edge regions: and ;
- corner regions: and
2.4. Primal–Dual Algorithm with Guaranteed Convergence
- Initialization: Choose with , and .
- Iteration : update as follows
3. Model Proposal and Solution
3.1. Proposed Model
3.2. Numerical Algorithm
Algorithm 1: Primal–Dual Algorithm for MOTV Image Restoration |
Require: f, H, .
|
3.3. Adaptive Regularization Parameter
Algorithm 2: Primal–Dual Algorithm for New MOTV Image Restoration |
Require: f, H, , .
|
3.4. Convergence Analysis
4. Simulation Results
4.1. Quantitative Evaluation Index Description
4.2. The Influence of Fractional-Order
4.3. The Influence of on Algorithm Performance
4.4. Comparison with Other Methods
5. Conclusions
- *
- The proposed MOTV model plays a significant role in improving the staircase effect and eliminating residual noise.
- *
- The appropriate fractional-order and the selection of based on the deviation principle have a certain effect on the performance of the algorithm.
- *
- According to the results of two groups of comparative experiments, it can be concluded that the algorithm proposed outperforms other algorithms and is effective.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Li, X.; Meng, X.; Xiong, B. A fractional variational image denoising model with two-component regularization terms. Appl. Math. Comput. 2022, 427, 127178. [Google Scholar] [CrossRef]
- Donoho, D.L. De-noising by soft-thresholding. IEEE Trans. Inf. Theory 1995, 41, 613–627. [Google Scholar] [CrossRef] [Green Version]
- Miclea, A.V.; Terebes, R.M.; Meza, S.; Cislariu, M. On spectral-spatial classification of hyperspectral images using image denoising and enhancement techniques, wavelet transforms and controlled data set partitioning. Remote Sens. 2022, 14, 1475. [Google Scholar] [CrossRef]
- Smitha, A.; Febin, I.P.; Jidesh, P. A retinex based non-local total generalized variation framework for OCT image restoration. Biomed. Signal Process. Control 2022, 71, 103234. [Google Scholar] [CrossRef]
- Bera, S.; Biswas, P.K. Noise conscious training of non local neural network powered by self attentive spectral normalized Markovian patch GAN for low dose CT denoising. IEEE Trans. Med. Imaging 2021, 40, 3663–3673. [Google Scholar] [CrossRef]
- Zhao, T.; Hoffman, J.; McNitt-Gray, M.; Ruan, D. Ultra-low-dose CT image denoising using modified BM3D scheme tailored to data statistics. Med. Phys. 2019, 46, 190–198. [Google Scholar] [CrossRef] [Green Version]
- Wen, Y.; Guo, Z.; Yao, W.; Yan, D.; Sun, J. Hybrid BM3D and PDE filtering for non-parametric single image denoising. Signal Process. 2021, 184, 108049. [Google Scholar] [CrossRef]
- Zhong, T.; Cheng, M.; Dong, X.; Wu, N. Seismic random noise attenuation by applying multiscale denoising convolutional neural network. IEEE Trans. Geosci. Remote Sens. 2021, 60, 5905013. [Google Scholar] [CrossRef]
- Dutta, S.; Basarab, A.; Georgeot, B.; Kouamé, D. DIVA: Deep unfolded network from quantum interactive patches for image restoration. arXiv 2022, arXiv:2301.00247. [Google Scholar] [CrossRef]
- Yang, D.; Sun, J. BM3D-Net: A convolutional neural network for transform-domain collaborative filtering. IEEE Signal Process. Lett. 2017, 25, 55–59. [Google Scholar] [CrossRef]
- Dutta, S.; Basarab, A.; Georgeot, B.; Kouamé, D. A novel image denoising algorithm using concepts of quantum many-body theory. Signal Process. 2022, 201, 108690. [Google Scholar] [CrossRef]
- Dutta, S.; Basarab, A.; Georgeot, B.; Kouamé, D. Quantum mechanics-based signal and image representation: Application to denoising. IEEE Open J. Signal Process. 2021, 2, 190–206. [Google Scholar] [CrossRef]
- Rudin, L.I.; Osher, S.; Fatemi, E. Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 1992, 60, 259–268. [Google Scholar] [CrossRef]
- Chan, T.; Marquina, A.; Mulet, P. High-order total variation-based image restoration. SIAM J. Sci. Comput. 2000, 22, 503–516. [Google Scholar] [CrossRef]
- Liu, X.; Li, Q.; Yuan, C.; Li, J.; Chen, X.; Chen, Y. High-order directional total variation for seismic noise attenuation. IEEE Trans. Geosci. Remote Sens. 2021, 60, 5903013. [Google Scholar] [CrossRef]
- Shi, B.; Gu, F.; Pang, Z.F.; Zeng, Y. Remove the salt and pepper noise based on the high order total variation and the nuclear norm regularization. Appl. Math. Comput. 2022, 421, 126925. [Google Scholar] [CrossRef]
- Tian, D.; Xue, D.; Wang, D. A fractional-order adaptive regularization primal–dual algorithm for image denoising. Inf. Sci. 2015, 296, 147–159. [Google Scholar] [CrossRef]
- Liu, Q.; Sun, L.; Gao, S. Non-convex fractional-order derivative for single image blind restoration. Appl. Math. Model. 2022, 102, 207–227. [Google Scholar] [CrossRef]
- Tian, C.; Zheng, M.; Zuo, W.; Zhang, B.; Zhang, Y.; Zhang, D. Multi-stage image denoising with the wavelet transform. Pattern Recognit. 2023, 134, 109050. [Google Scholar] [CrossRef]
- Chen, Y.; Cao, W.; Pang, L.; Cao, X. Hyperspectral image denoising with weighted nonlocal low-rank model and adaptive total variation regularization. IEEE Trans. Geosci. Remote Sens. 2022, 60, 5544115. [Google Scholar] [CrossRef]
- Kulathilake, K.A.S.H.; Abdullah, N.A.; Sabri, A.Q.M.; Bandara, A.M.R.R.; Lai, K.W. A review on self-adaptation approaches and techniques in medical image denoising algorithms. Multimed. Tools Appl. 2022, 81, 37591–37626. [Google Scholar] [CrossRef]
- Gu, X.M.; Huang, T.Z.; Ji, C.C.; Carpentieri, B.; Alikhanov, A.A. Fast iterative method with a second-order implicit difference scheme for time-space fractional convection–diffusion equation. J. Sci. Comput. 2017, 72, 957–985. [Google Scholar] [CrossRef]
- Li, M.; Gu, X.M.; Huang, C.; Fei, M.; Zhang, G. A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations. J. Comput. Phys. 2018, 358, 256–282. [Google Scholar] [CrossRef]
- Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: Hoboken, NJ, USA, 1993. [Google Scholar]
- Lian, W.; Liu, X. Non-convex fractional-order TV model for impulse noise removal. J. Comput. Appl. Math. 2023, 417, 114615. [Google Scholar] [CrossRef]
- Li, M.; Han, C.; Wang, R.; Guo, T. Shrinking gradient descent algorithms for total variation regularized image denoising. Comput. Optim. Appl. 2017, 68, 643–660. [Google Scholar] [CrossRef]
- Estellers, V.; Soatto, S.; Bresson, X. Adaptive regularization with the structure tensor. IEEE Trans. Image Process. 2015, 24, 1777–1790. [Google Scholar] [CrossRef]
- Prasath, V.B.S.; Pelapur, R.; Seetharaman, G.; Palaniappan, K. Multiscale structure tensor for improved feature extraction and image regularization. IEEE Trans. Image Process. 2019, 28, 6198–6210. [Google Scholar] [CrossRef]
- Hsieh, P.W.; Shao, P.C.; Yang, S.Y. A regularization model with adaptive diffusivity for variational image denoising. Signal Process. 2018, 149, 214–228. [Google Scholar] [CrossRef]
- Chambolle, A.; Pock, T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 2011, 40, 120–145. [Google Scholar] [CrossRef] [Green Version]
- Rockafellar, R.T. Convex Analysis; Princeton University Press: Princeton, NJ, USA, 1997. [Google Scholar]
- Duan, J.; Qiu, Z.; Lu, W.; Wang, G.; Pan, Z.; Bai, L. An edge-weighted second order variational model for image decomposition. Digit. Signal Process. 2016, 49, 162–181. [Google Scholar] [CrossRef]
- Deng, H.; Tao, J.; Song, X.; Zhang, C. Estimation of the parameters of a weighted nuclear norm model and its application in image denoising. Inf. Sci. 2020, 528, 246–264. [Google Scholar] [CrossRef]
- Harris, C.; Stephens, M. A combined corner and edge detector. Alvey Vis. Conf. 1988, 15, 10–5244. [Google Scholar] [CrossRef]
- Phan, T.D.K.; Tran, T.H.Y. Edge coherence-weighted second-order variational model for image denoising. Signal Image Video Process. 2022, 16, 2313–2320. [Google Scholar] [CrossRef]
- Chan, T.F.; Golub, G.H.; Mulet, P. A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 1999, 20, 1964–1977. [Google Scholar] [CrossRef] [Green Version]
- Wen, Y.W.; Chan, R.H.; Zeng, T.Y. Primal-dual algorithms for total variation based image restoration under poisson noise. Sci. China Math. 2016, 59, 141–160. [Google Scholar] [CrossRef]
- Boutaayamou, I.; Hadri, A.; Laghrib, A. An optimal bilevel optimization model for the generalized total variation and anisotropic tensor parameters selection. Appl. Math. Comput. 2023, 438, 127510. [Google Scholar] [CrossRef]
- Bertsekas, D.; Nedic, A.; Ozdaglar, A. Convex Analysis and Optimization; Athena Scientific: Nashua, NH, USA, 2003. [Google Scholar]
- Wen, Y.W.; Chan, R.H. Parameter selection for total-variation-based image restoration using discrepancy principle. IEEE Trans. Image Process. 2011, 21, 1770–1781. [Google Scholar] [CrossRef] [Green Version]
- Jiang, F.; Zhang, Z.; He, H. Solving saddle point problems: A landscape of primal-dual algorithm with larger stepsizes. J. Glob. Optim. 2022, 85, 821–846. [Google Scholar] [CrossRef]
- Jiang, F.; Wu, Z.; Cai, X.; Zhang, H. A first-order inexact primal-dual algorithm for a class of convex-concave saddle point problems. Numer. Algorithms 2021, 88, 1109–1136. [Google Scholar] [CrossRef]
- Chen, G.; Teboulle, M. A proximal-based decomposition method for convex minimization problems. Math. Program. 1994, 64, 81–101. [Google Scholar] [CrossRef]
- Nedić, A.; Ozdaglar, A. Subgradient methods for saddle-point problems. J. Optim. Theory Appl. 2009, 142, 205–228. [Google Scholar] [CrossRef]
- Pang, Z.F.; Zhang, H.L.; Luo, S.; Zeng, T. Image denoising based on the adaptive weighted TVp regularization. Signal Process. 2020, 167, 107325. [Google Scholar] [CrossRef]
- Sun, K.; Simon, S. Bilateral spectrum weighted total variation for noisy-image super-resolution and image denoising. IEEE Trans. Signal Process. 2021, 69, 6329–6341. [Google Scholar] [CrossRef]
- Galatsanos, N.P.; Katsaggelos, A.K. Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process. 1992, 1, 322–336. [Google Scholar] [CrossRef] [PubMed]
- Guo, J.; Chen, Q. Image denoising based on nonconvex anisotropic total-variation regularization. Signal Process. 2021, 186, 108124. [Google Scholar] [CrossRef]
- Li, M.; Cai, G.; Bi, S.; Zhang, X. Improved TV image denoising over inverse gradient. Symmetry 2023, 15, 678. [Google Scholar] [CrossRef]
- Phan, T.D.K. A weighted total variation based image denoising model using mean curvature. Optik 2020, 217, 164940. [Google Scholar] [CrossRef]
- Nguyen, M.P.; Chun, S.Y. Bounded self-weights estimation method for non-local means image denoising using minimax estimators. IEEE Trans. Image Process. 2017, 26, 1637–1649. [Google Scholar] [CrossRef] [Green Version]
- Zhang, X. Center pixel weight based on wiener filter for non-local means image denoising. Optik 2021, 244, 167557. [Google Scholar] [CrossRef]
- Zhang, B.; Zhu, G.; Zhu, Z.; Kwong, S. Alternating direction method of multipliers for nonconvex log total variation image restoration. Appl. Math. Model. 2023, 114, 338–359. [Google Scholar] [CrossRef]
Image | PSNR/SSIM [48] | PSNR/SSIM [49] | PSNR/SSIM [50] | PSNR/SSIM Proposed | |
---|---|---|---|---|---|
Parrots | 5 | 32.8362/0.9071 | 32.3292/0.9050 | 34.3292/0.9058 | 34.8288/0.9080 |
10 | 27.7650/0.9085 | 31.4979/0.9005 | 33.4346/0.9089 | 33.6375/0.9101 | |
20 | 23.9519/0.9146 | 28.2574/0.9162 | 28.5716/0.9218 | 28.5996/0.9241 | |
30 | 22.7536/0.92018 | 23.1230/0.9204 | 24.2050/0.9277 | 26.3916/0.9279 | |
Lena | 5 | 33.1229/0.9023 | 32.2596/0.8938 | 33.1639/0.8992 | 33.4520/0.9000 |
10 | 27.7962/0.8963 | 31.4268/0.8974 | 32.4873/0.9023 | 32.5379/0.9011 | |
20 | 24.0047/0.9144 | 28.0919/0.9050 | 28.1660/0.9141 | 28.4511/0.9153 | |
30 | 23.0918/0.8942 | 23.9741/0.8973 | 24.2123/0.9075 | 25.8134/0.8896 | |
Barbara | 5 | 33.1016/0.8908 | 29.2736/0.8904 | 31.0013/0.8909 | 31.0157/0.8924 |
10 | 27.7793/0.8954 | 28.8140/0.8973 | 30.0985/0.8974 | 30.1247/0.8978 | |
20 | 23.9799/0.9024 | 26.7437/0.9033 | 26.5292/0.9051 | 26.5692/0.9133 | |
30 | 22.1423/0.8937 | 22.9751/0.9102 | 22.6868/0.9188 | 22.9827/0.9124 | |
Cameraman | 5 | 33.1716/0.9021 | 29.8097/0.8991 | 31.8976/0.9000 | 32.2383/0.9006 |
10 | 27.8417/0.9056 | 29.3846/0.9015 | 30.3292/0.9034 | 31.4454/0.9059 | |
20 | 23.9418/0.9140 | 27.0952/0.9084 | 27.6781/0.9082 | 27.7247/0.9144 | |
30 | 22.8149/0.8902 | 23.1143/0.8907 | 23.5803/0.8905 | 24.6137/0.8911 | |
Pepper | 5 | 34.6851/0.8698 | 32.6693/0.8620 | 34.5614/0.8713 | 34.7247/0.8729 |
10 | 28.8834/0.8764 | 32.0109/0.8689 | 32.9939/0.8786 | 33.8284/0.8767 | |
20 | 24.2617/0.8853 | 28.6892/0.8808 | 29.1745/0.8902 | 29.3960/0.8913 | |
30 | 22.5738/0.8937 | 24.4121/0.8938 | 25.4702/0.8950 | 29.0906/0.9016 | |
Foot | 5 | 37.5861/0.8536 | 37.4102/0.8524 | 35.4676/0.8713 | 35.7180/0.8561 |
10 | 33.8384/0.8340 | 33.9815/0.8312 | 34.0148/0.8311 | 34.6360/0.8350 | |
20 | 27.1051/0.8286 | 28.2964/0.8279 | 28.7614/0.8253 | 29.8339/0.8272 | |
30 | 25.4720/0.8050 | 25.6541/0.8144 | 25.9459/0.8193 | 27.5594/0.8074 | |
Head | 5 | 33.5699/0.8953 | 33.4973/0.8874 | 34.6343/0.9030 | 33.6142/0.8423 |
10 | 32.3203/0.8470 | 31.0355/0.8309 | 32.1153/0.8576 | 32.3258/0.8310 | |
20 | 22.9786/0.7933 | 22.8761/0.78237 | 27.6259/0.8117 | 28.1583/0.7446 | |
30 | 19.4785/0.7403 | 19.4509/0.7371 | 23.3897/0.7649 | 24.5390/0.6468 |
Image | PSNR/SSIM [51] | PSNR/SSIM [52] | PSNR/SSIM [53] | PSNR/SSIM Proposed | |
---|---|---|---|---|---|
Cameraman | 10 | 31.3925/0.8948 | 33.1347/0.9049 | 25.0623/0.9089 | 33.6375/0.9101 |
20 | 28.2944/0.8272 | 28.5723/0.8309 | 24.9803/0.9218 | 28.5996/0.9241 | |
30 | 27.1039/0.7700 | 27.2705/0.7315 | 24.8771/0.9277 | 26.3916/0.9279 | |
Boats | 10 | 32.3014/0.9017 | 32.3438/0.9065 | 25.6609/0.9032 | 32.4058/0.8889 |
20 | 28.6036/0.8802 | 28.7042/0.8840 | 25.5432/0.9019 | 28.3939/0.9011 | |
30 | 27.5687/0.8204 | 27.5816/0.8253 | 25.3779/0.9087 | 26.0102/0.9112 | |
Barbara | 10 | 30.0487/0.9186 | 31.8501/0.9340 | 22.0302/0.8974 | 30.1247/0.8978 |
20 | 26.1845/0.9259 | 26.4952/0.9247 | 21.9840/0.9051 | 26.5692/0.9133 | |
30 | 25.7061/0.8790 | 25.7384/0.8790 | 21.9057/0.9188 | 22.9827/0.9124 | |
Pepper | 10 | 33.3949/0.8950 | 33.9108/0.9032 | 28.8618/0.9034 | 34.7247/0.8729 |
20 | 29.2765/0.8593 | 29.5572/0.8897 | 28.6174/0.8596 | 29.3960/0.8913 | |
30 | 28.2187/0.8746 | 28.2854/0.8731 | 28.4424/0.8905 | 29.0906/0.9016 |
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Bi, S.; Li, M.; Cai, G. Mixed Fractional-Order and High-Order Adaptive Image Denoising Algorithm Based on Weight Selection Function. Fractal Fract. 2023, 7, 566. https://doi.org/10.3390/fractalfract7070566
Bi S, Li M, Cai G. Mixed Fractional-Order and High-Order Adaptive Image Denoising Algorithm Based on Weight Selection Function. Fractal and Fractional. 2023; 7(7):566. https://doi.org/10.3390/fractalfract7070566
Chicago/Turabian StyleBi, Shaojiu, Minmin Li, and Guangcheng Cai. 2023. "Mixed Fractional-Order and High-Order Adaptive Image Denoising Algorithm Based on Weight Selection Function" Fractal and Fractional 7, no. 7: 566. https://doi.org/10.3390/fractalfract7070566