A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method
Abstract
:1. Introduction
2. Background and Related Algorithms
3. Preliminaries
 (i)
 ${\parallel tu+(1t)v\parallel}^{2}={t\parallel u\parallel}^{2}+{(1t)\parallel v\parallel}^{2}t(1t){\parallel uv\parallel}^{2}$$\forall t\in [0,1],$$\forall u,v\in H;$
 (ii)
 ${\parallel u\pm v\parallel}^{2}={\parallel u\parallel}^{2}\pm 2\langle u,v\rangle +{\parallel v\parallel}^{2}$$\forall u,v\in H.$
 (i)
 For every$p\in \mathsf{\Gamma},$${lim}_{n\to +\infty}\parallel {x}_{n}p\parallel $exists;
 (ii)
 Each weakcluster point of the sequence$\left\{{x}_{n}\right\}$is in$\mathsf{\Gamma}.$
4. Main Results
 H is a real Hilbert space;
 $\{{T}_{n}:H\to H\}$ is a family of nonexpansive operators;
 $\left\{{T}_{n}\right\}$ satisfies the NST*condition;
 $\mathsf{\Gamma}:={\cap}_{n=1}^{\infty}Fix\left({T}_{n}\right)\ne \varnothing .$
Algorithm 1: (MWA): A modified Walgorithm 

 (i)
 $\parallel {x}_{n+1}{x}^{*}\parallel \le K\xb7{\prod}_{j=1}^{n}(1+2{\theta}_{j}),$where$K=max\{\parallel {x}_{1}{x}^{*}\parallel ,\parallel {x}_{2}{x}^{*}\parallel \}$and${x}^{*}\in \mathsf{\Gamma}.$
 (ii)
 $\left\{{x}_{n}\right\}$converges weakly to a point in$\mathsf{\Gamma}.$
Algorithm 2: (FBMWA): A forwardbackward modified Walgorithm. 

 (i)
 $\parallel {x}_{n+1}{x}^{*}\parallel \le K\xb7{\prod}_{j=1}^{n}(1+2{\theta}_{j}),$where$K=max\{\parallel {x}_{1}{x}^{*}\parallel ,\parallel {x}_{2}{x}^{*}\parallel \}$and${x}^{*}\in Argmin(f+h).$
 (ii)
 $\left\{{x}_{n}\right\}$converges weakly to a point in$Argmin(f+h).$
5. Simulated Results for the Image Restoration Problem
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Cameraman  Lenna  

Algorithms  PSNR  Tol.  PSNR  Tol. 
FBS  27.1953  2.32 × ${10}^{5}$  29.4907  1.73 × ${10}^{5}$ 
IFBS  27.1953  2.32 × ${10}^{5}$  29.4907  1.73 × ${10}^{5}$ 
FISTA  34.6659  4.13 × ${10}^{5}$  36.9324  3.34 × ${10}^{5}$ 
NAGA  35.6670  4.15 × ${10}^{5}$  37.8088  3.32 × ${10}^{5}$ 
FBMWA  36.2783  4.21 × ${10}^{5}$  38.2989  3.31 × ${10}^{5}$ 
Cameraman  Lenna  

Case  Parameters  PSNR  Tol.  PSNR  Tol. 
1  ${\mu}_{n}=\frac{1}{{2}^{n}}$  27.8911  2.13 × ${10}^{5}$  30.1603  1.66 × ${10}^{5}$ 
2  ${\mu}_{n}=\frac{10}{{n}^{2}}$  27.9003  2.12 × ${10}^{5}$  30.1693  1.65 × ${10}^{5}$ 
3  ${\mu}_{n}=0.5$  28.7146  2.00 × ${10}^{5}$  30.9771  1.60 × ${10}^{5}$ 
4  ${\mu}_{n}=0.9$  30.9920  1.81 × ${10}^{5}$  33.2838  1.47 × ${10}^{5}$ 
5  ${\mu}_{n}=\frac{{t}_{n}1}{{t}_{n+1}},{t}_{1}=1,$ ${t}_{n+1}=\frac{1+\sqrt{1+4{t}_{n}^{2}}}{2},$  36.2783  4.21 × ${10}^{5}$  38.2989  3.31 × ${10}^{5}$ 
6  ${\mu}_{n}=\frac{n}{n+1}$  37.0979  1.63 × ${10}^{4}$  38.8562  1.30 × ${10}^{4}$ 
7  ${\mu}_{n}=1$  30.6832  9.13 × ${10}^{4}$  32.7996  7.07 × ${10}^{4}$ 
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Hanjing, A.; Suantai, S. A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method. Mathematics 2020, 8, 378. https://doi.org/10.3390/math8030378
Hanjing A, Suantai S. A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method. Mathematics. 2020; 8(3):378. https://doi.org/10.3390/math8030378
Chicago/Turabian StyleHanjing, Adisak, and Suthep Suantai. 2020. "A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method" Mathematics 8, no. 3: 378. https://doi.org/10.3390/math8030378