Document Type : Original Paper

**Authors**

Department of Applied Mathematics, Faculty of Mathematics, Iran University of Science & Technology, Tehtan, Iran

**Abstract**

In this study, the problem of image reconstruction (Inverse Problem) and its semantic relationship

with a large range of applied and engineering problems are explained. The mechanism of tomographic

which is an example of the image reconstruction problem is described. The existence and uniqueness of

solutions and also regularization techniques are discussed. In order to better imagine abstract concepts, the process of digital X-ray imaging is presented using MATLAB. Image reconstruction techniques

are described by Radon and Fourier transforms and algebraic methods. The effects of selecting the number of angles in tomography and additive noise in the data, on the quality of the reconstructed image are investigated. The numerical results and the comparison of images reconstructed by different methods confirm the special ability of algebraic techniques in image reconstruction. In addition, it is shown that in algebraic methods, desirable features such as positivity and smoothness can be considered for the reconstructed image, which has a significant effect on reducing the approximation error.

with a large range of applied and engineering problems are explained. The mechanism of tomographic

which is an example of the image reconstruction problem is described. The existence and uniqueness of

solutions and also regularization techniques are discussed. In order to better imagine abstract concepts, the process of digital X-ray imaging is presented using MATLAB. Image reconstruction techniques

are described by Radon and Fourier transforms and algebraic methods. The effects of selecting the number of angles in tomography and additive noise in the data, on the quality of the reconstructed image are investigated. The numerical results and the comparison of images reconstructed by different methods confirm the special ability of algebraic techniques in image reconstruction. In addition, it is shown that in algebraic methods, desirable features such as positivity and smoothness can be considered for the reconstructed image, which has a significant effect on reducing the approximation error.

**Keywords**

- Image Reconstruction
- Regularization
- Algebraic Image Reconstruction Technique
- Radon Inverse transform

**Main Subjects**

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Algorithms. 7(1) (2014) 121–144.

[2] K. E. Atkinson, An introduction to numerical analysis, 2sd edn, John Wiley & Sons, 1987.

[3] A.B. Bakushinskii, Remarks on choosing a regularization parameter using the quasioptimality and

ratio criterion, USSR Computational Mathematics and Mathematical Physics. 24(4) (1984) 181–182.

[4] Å. Björck, Numericla Methods in Matrix Computations, Springer, 2015.

[5] E. Boy, Regularization of inverse problems by the landweber iteration, Masters’ thesis, 2019.

[6] R. N. Bracewell, Two-Dimensional Imaging, Prentice Hall, 1995.

[7] W. L. Briggs and V. E. Henson, The DFT - An Owner’s Manual for the Discrete Fourier Transform,

SIAM, 1995.

[8] Y. Censor, and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the

linear feasibility problem, SIAM Journal on Matrix Analysis and Applications. 24(1) (2002) 40–58.

[9] M. Defrise, C. De Mol and P. C. Sabatier, A note on stopping rules for iterative regularization methods

and filtered SVD, Inverse Problems : An interdisciplinary Study ed P C Sabatier. (1987) 261–268.

[10] T. Elfving, T. Nikazad, Stopping rules for Landweber-type iteration, Inverse Problem. 23(4) (2007)

1417–1432.

[11] T. Elfving, T. Nikazad, Properties of a class of block-iterative methods, Inverse Problem. 25(11)

(2009), 115011.

[12] T. Elfving, T. Nikazad, and C. Popa. A class of iterative methods: semi-convergence, stopping rules,

inconsistency, and constraining. In Y. Censor, M. Jiang, and G. Wang, editors, Biomed-ical Mathematics:

Promising Directions in Imaging, Therapy Planning, and Inverse Problems.Medical Physics

Publishing, Madison, WI. (2010) 157–184.

Algorithms. 7(1) (2014) 121–144.

[2] K. E. Atkinson, An introduction to numerical analysis, 2sd edn, John Wiley & Sons, 1987.

[3] A.B. Bakushinskii, Remarks on choosing a regularization parameter using the quasioptimality and

ratio criterion, USSR Computational Mathematics and Mathematical Physics. 24(4) (1984) 181–182.

[4] Å. Björck, Numericla Methods in Matrix Computations, Springer, 2015.

[5] E. Boy, Regularization of inverse problems by the landweber iteration, Masters’ thesis, 2019.

[6] R. N. Bracewell, Two-Dimensional Imaging, Prentice Hall, 1995.

[7] W. L. Briggs and V. E. Henson, The DFT - An Owner’s Manual for the Discrete Fourier Transform,

SIAM, 1995.

[8] Y. Censor, and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the

linear feasibility problem, SIAM Journal on Matrix Analysis and Applications. 24(1) (2002) 40–58.

[9] M. Defrise, C. De Mol and P. C. Sabatier, A note on stopping rules for iterative regularization methods

and filtered SVD, Inverse Problems : An interdisciplinary Study ed P C Sabatier. (1987) 261–268.

[10] T. Elfving, T. Nikazad, Stopping rules for Landweber-type iteration, Inverse Problem. 23(4) (2007)

1417–1432.

[11] T. Elfving, T. Nikazad, Properties of a class of block-iterative methods, Inverse Problem. 25(11)

(2009), 115011.

[12] T. Elfving, T. Nikazad, and C. Popa. A class of iterative methods: semi-convergence, stopping rules,

inconsistency, and constraining. In Y. Censor, M. Jiang, and G. Wang, editors, Biomed-ical Mathematics:

Promising Directions in Imaging, Therapy Planning, and Inverse Problems.Medical Physics

Publishing, Madison, WI. (2010) 157–184.

[13] T. Elfving,P. C. Hansen and T. Nikazad , Semi-convergence properties of Kaczmarz’s method, Inverse

Problems. 30(5) (2014) 055007.

[14] T. Elfving, T. Nikazad And P. C. Hansen, Semi-convergence and relaxation paremeters for a class of

SIRT algorithms, ETNA. 37(274) (2010) 321–336.

[15] T. Elfving, P. C. Hansen,T. Nikazad, Semi-convergence and relaxation paremeters for Projected SIRT

algorithms, SIAM Journal on Scientific Computing. 34(4) (2017) A2000–A2017.

[16] H. W. Engl, M. Hanke, and .A Neubauer, Regularization of inverse problems, Springer Science &

Business Media, 2000.

[17] S. Gazzola,Y. Wiaux , Fast nonnegative least squares through flexible Krylov subspaces. SIAM Journal

on Scientific Computing. 39(2) (2017) A655–A679.

[18] J. D. Gibson, A. Bovik, Handbook of Image and Video Processing,Academic press, 2000.

[19] M. S. Gockenbach, Linear Inverse Problems and Tikhonov Regularization, 32, American Mathematical

Soc, 2016.

[20] J. Hadamard, Lectures on Cauch’s Problem in Linear Partial Differenctal Equations, Yale University

Press, 1923.

[21] P. C. Hansen , Analysis of discrete ill-posed problems by means of the L-curve, SIAM review. 34(4)

(1992) 561–580.

[22] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problem, numerical aspects of linear inversion,

SIAM. Philadelphia, 1998.

[23] P. C. Hansen, M. Saxild-Hansen, AIRTools- a MATLAB package of algebraic iterative reconstruction

methods Journal of Computational and Applied Mathematics. 236(8) (2012) 2167–78.

[24] G. T. Herman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections,

Springer, 2009.

[25] M. R. Hestenes and E. Stiefel, Methods of Conjugate Gradient for Solving Linear Systemsw, J. Res.

Nat. Bur. Standards, 49(1) Washington, DC: NBS, (1952).

[26] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, SIAM, 2001.

[27] J. S. Lim, Two-Dimensional Signal and Image Processing, Englewood Cliffs, 1990.

[28] V. A. Morozov and M. Stessin, Regularization methods for ill-posed problems, CRC press Boca

Raton, FL, 1993.

[29] F. Natterer, The Mathematics of Computerized Tomography,John Wiley & Sons Ltd, 2001.

[30] D. Needell, J. A. Tropp, Paved with good intentions: analysis of randomized block Kaczmarz method

, Linear Algebra Appl. 441 (2014) 199–221.

[31] T. Nikazad, M. Karimpour, Controlling noise error in block iterative methods, Numerical Algorithms.

73(4) (2016) 907–925.

[32] T. Nikazad, M. Abbasi and T. Elfving, Error minimizing relaxation strategies in Landweber and

Kaczmarz type iterations, Journal of Inverse and Ill-posed Problems. 25(1) (2017) 35–56.

Problems. 30(5) (2014) 055007.

[14] T. Elfving, T. Nikazad And P. C. Hansen, Semi-convergence and relaxation paremeters for a class of

SIRT algorithms, ETNA. 37(274) (2010) 321–336.

[15] T. Elfving, P. C. Hansen,T. Nikazad, Semi-convergence and relaxation paremeters for Projected SIRT

algorithms, SIAM Journal on Scientific Computing. 34(4) (2017) A2000–A2017.

[16] H. W. Engl, M. Hanke, and .A Neubauer, Regularization of inverse problems, Springer Science &

Business Media, 2000.

[17] S. Gazzola,Y. Wiaux , Fast nonnegative least squares through flexible Krylov subspaces. SIAM Journal

on Scientific Computing. 39(2) (2017) A655–A679.

[18] J. D. Gibson, A. Bovik, Handbook of Image and Video Processing,Academic press, 2000.

[19] M. S. Gockenbach, Linear Inverse Problems and Tikhonov Regularization, 32, American Mathematical

Soc, 2016.

[20] J. Hadamard, Lectures on Cauch’s Problem in Linear Partial Differenctal Equations, Yale University

Press, 1923.

[21] P. C. Hansen , Analysis of discrete ill-posed problems by means of the L-curve, SIAM review. 34(4)

(1992) 561–580.

[22] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problem, numerical aspects of linear inversion,

SIAM. Philadelphia, 1998.

[23] P. C. Hansen, M. Saxild-Hansen, AIRTools- a MATLAB package of algebraic iterative reconstruction

methods Journal of Computational and Applied Mathematics. 236(8) (2012) 2167–78.

[24] G. T. Herman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections,

Springer, 2009.

[25] M. R. Hestenes and E. Stiefel, Methods of Conjugate Gradient for Solving Linear Systemsw, J. Res.

Nat. Bur. Standards, 49(1) Washington, DC: NBS, (1952).

[26] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, SIAM, 2001.

[27] J. S. Lim, Two-Dimensional Signal and Image Processing, Englewood Cliffs, 1990.

[28] V. A. Morozov and M. Stessin, Regularization methods for ill-posed problems, CRC press Boca

Raton, FL, 1993.

[29] F. Natterer, The Mathematics of Computerized Tomography,John Wiley & Sons Ltd, 2001.

[30] D. Needell, J. A. Tropp, Paved with good intentions: analysis of randomized block Kaczmarz method

, Linear Algebra Appl. 441 (2014) 199–221.

[31] T. Nikazad, M. Karimpour, Controlling noise error in block iterative methods, Numerical Algorithms.

73(4) (2016) 907–925.

[32] T. Nikazad, M. Abbasi and T. Elfving, Error minimizing relaxation strategies in Landweber and

Kaczmarz type iterations, Journal of Inverse and Ill-posed Problems. 25(1) (2017) 35–56.

[33] T. Nikazad, M. Karimpour, and M. Abbasi, Notes on flexible sequential block iterative methods,

Computers & Mathematics with Applications. 76(6) (2018) 1321–1332.

[34] T. Nikazad, R. Davidi, G. T. Herman, Accelerated perturbation-resilient block-iterative projection

methods with application to image reconstruction, Inverse Problems. 28(3) (2012) 035005.

[35] T. Nikazad, M. Abbasi, Perturbation-resilient iterative methods with an infinite pool of mappings,

SIAM J. Numerical Analysis. 53 (1) (2015) 390–404.

[36] T. Nikazad, M. Abbasi, L. Afzalipour,T. Elfving, A new step size rule for the superiorization method

and its application in computerized tomography, Numerical Algorithms. (2021) 1–25.

[37] Y. Saad, Iterative Methods for Sparse Linear System, PWS Publishing Company, 1996.

[38] J. Semmlow, Circuits, signals and systems for bioengineers: A MATLAB-based introduction Academic

Press, 2017.

[39] I. Tomba, Iterative regularization methods for ill-posed problems, PHD thesis, 2013.

[40] M.V. W. Zebetti, C. Lin and G. T. Herman, Total variation superiorized conjugate gradient method

for image reconstruction, Inverse Problems. 34(3) (2018) 034001.

Computers & Mathematics with Applications. 76(6) (2018) 1321–1332.

[34] T. Nikazad, R. Davidi, G. T. Herman, Accelerated perturbation-resilient block-iterative projection

methods with application to image reconstruction, Inverse Problems. 28(3) (2012) 035005.

[35] T. Nikazad, M. Abbasi, Perturbation-resilient iterative methods with an infinite pool of mappings,

SIAM J. Numerical Analysis. 53 (1) (2015) 390–404.

[36] T. Nikazad, M. Abbasi, L. Afzalipour,T. Elfving, A new step size rule for the superiorization method

and its application in computerized tomography, Numerical Algorithms. (2021) 1–25.

[37] Y. Saad, Iterative Methods for Sparse Linear System, PWS Publishing Company, 1996.

[38] J. Semmlow, Circuits, signals and systems for bioengineers: A MATLAB-based introduction Academic

Press, 2017.

[39] I. Tomba, Iterative regularization methods for ill-posed problems, PHD thesis, 2013.

[40] M.V. W. Zebetti, C. Lin and G. T. Herman, Total variation superiorized conjugate gradient method

for image reconstruction, Inverse Problems. 34(3) (2018) 034001.

April 2022

Pages 118-137

**Receive Date:**24 January 2022**Revise Date:**05 March 2022**Accept Date:**16 March 2022**First Publish Date:**21 March 2022