Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images^{ †}
Abstract
:1. Introduction
2. Mathematical Description of Uniform Rectilinear Smearing of the Image
2.1. Direct Problem
Algorithm 1. The direct problem of uniform rectilinear smearing of an image. 
Input: exact (undistorted) image w

2.2. Inverse Problem
Algorithm 2. The inverse problem of the uniform rectilinear smearing (the first approach) 
Input: smeared image g(x, y), where x and y are directed horizontally and vertically, respectively.

Algorithm 3. The inverse problem of the uniform rectilinear smearing (the second approach) 
Input: smeared image g(x, y), where x and y are directed horizontally and vertically respectively

3. Mathematical Description of the Nonuniform Rectilinear Smearing
3.1. The First (Time) Approach [1,2,6]
3.2. The Second (Spatial) Approach [5,6]
3.2.1. The Direct Problem
3.2.2. The Inverse Problem
Algorithm 4. The inverse problem for illuminating the rowwise nonuniform rectilinear smear 
Input: image g(x, y) smeared nonuniformly along x 
(1) Presentation of a twodimensional image g(x, y) as a set of onedimensional images ${g}_{y}(x)$. 
(2) Determining the nonuniform smear Δ(x) from the spectrum (by the spectral method). 
(3) Calculating a PSF h(x,ξ) according to (11). 
(4) Writing matrix A and vectors ${g}_{y}$ of a SLAE. 
(5) Choosing the regularization parameter α in some approach. 
(6) Solving the regularized SLAE (13) in each yrow. 
(7) Obtaining the regularized solution ${w}_{y\mathsf{\alpha}}$ according to (14). 
(8) Forming image w_{α}(x, y) from a set of rowwise solutions ${w}_{y\mathsf{\alpha}}$. 
Output: w_{α}(x, y). 
4. Illustrative Example
4.1. Uniform Smearing of Image Using Boundary Conditions
4.2. Uniform Image Smearing with Truncation
4.3. Uniform Image Smearing with Diffusing the Image Edges
4.4. Error Estimation of Image Restoration
5. Direct and Inverse Problems of Nonuniform Image Smearing
The Approach of Divided Spectra
6. Noise Accounting
Algorithm 5. Noise eliminating by the modified median filter. 
Input: g – image noisy by bipolar impulse noise

7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
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Sizikov, V.; Dovgan, A.; Lavrov, A. Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images. Computers 2020, 9, 30. https://doi.org/10.3390/computers9020030
Sizikov V, Dovgan A, Lavrov A. Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images. Computers. 2020; 9(2):30. https://doi.org/10.3390/computers9020030
Chicago/Turabian StyleSizikov, Valery, Aleksandra Dovgan, and Aleksei Lavrov. 2020. "Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images" Computers 9, no. 2: 30. https://doi.org/10.3390/computers9020030