A class of Landweber-type iterative methods based on the Radon transform for incomplete view tomography

Abstract

Background

We study the reconstruction problem for incomplete view tomography, including sparse view tomography and limited angle tomography, by the Landweber iteration and its accelerated version. Traditional implementations of these Landweber-type iterative methods necessitate multiple large-scale matrix-vector multiplications, which in turn require substantial time and storage resources.

Objective

This paper aims to develop and test a novel and efficient discretization approach for a class of Landweber-type methods that minimizes storage requirements by incorporating the specific structure of the incomplete view Radon transform.

Methods

We prove that the normal operator of incomplete view Radon transform in these methods is a compact convolution operator, and derive the explicit representation of its convolution kernel. Discretized by the pixel basis, these Landweber-type iterative methods can be implemented quickly and accurately by introducing a discretized convolution operation between two small-scale matrices with minimal storage requirements.

Results

For the simulated complete and limited angle data, the reconstruction results using various Landweber-type methods with our proposed discretization scheme achieve a 1-5dB improvement in PSNR and require one-third of computation time compared to the traditional approach. For the simulated sparse view data, our discretization scheme yields a valid image with the highest PSNR.

Conclusions

The Landweber-type iterative methods, when combined with our proposed discretization approach based on the Radon transform, are effective for addressing the incomplete view tomography problem.

Keywords

incomplete view tomography Radon transform Landweber-type iteration convolution representation discretization

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References

Natterer

. The mathematics of computerized tomography. Philadelphia: SIAM, 2001.

Herman

. Fundamentals of computerized tomography image reconstruction from projections. London: Springer, 2009.

Natterer

Wübbeling

. Mathematical methods in image reconstruction. Philadelphia: SIAM, 2001.

Shepp

Logan

. The Fourier reconstruction of a head section. IEEE Trans Nucl Sci 1974; 21: 21–43.

Niklason

Christian

, et al. Digital tomosynthesis in breast imaging. Radiology 1997; 205: 399–406.

Gao

Zhang

Chen

, et al. An extrapolation method for image reconstruction from a straight-line trajectory. In: 2006 IEEE nuclear science symposium conference record, 2006, pp.2304–2308. IEEE.

Sidky

Zou

Pan

. Volume image reconstruction from a straight-line source trajectory. In: IEEE nuclear science symposium conference record, 2005, pp.2441–2444. IEEE.

Zhao

, et al. An image reconstruction model regularized by edge-preserving diffusion and smoothing for limited-angle computed tomography. Inverse Probl 2019; 35: 085004.

Grünbaum

. A study of fourier space methods for “limited angle” image reconstruction. Numer Func Anal Optim 1980; 2: 31–42.

10.

Ramm

. Inversion of limited-angle tomographic data. Comput Math with Appl 1991; 22: 101–111.

11.

Davison

. The ill-conditioned nature of the limited angle tomography problem. SIAM J Appl Math 1983; 43: 428–448.

12.

Louis

. Incomplete data problems in X-ray computerized tomography: I. Singular value decomposition of the limited angle transform. Numer Math 1986; 48: 251–262.

13.

Quinto

. Singularities of the X-ray transform and limited data tomography in

R^{2}

and

R^{3}

. SIAM J Appl Math 1993; 24: 1215–1225.

14.

Quinto

. Artifacts and visible singularities in limited data X-ray tomography. Sens Imaging 2017; 18: 9.

15.

Borg

Frikel

Jørgensen

, et al. Analyzing reconstruction artifacts from arbitrary incomplete X-ray CT data. SIAM J Imaging Sci 2018; 11: 2786–2814.

16.

Frikel

Quinto

. Characterization and reduction of artifacts in limited angle tomography. Inverse Probl 2013; 29: 125007.

17.

Stefanov

. The Radon transform with finitely many angles. Inverse Probl 2023; 39: 105003.

18.

Jiang

. Landweber iterative methods for angle-limited image reconstruction. Acta Math Sin Engl Ser 2009; 25: 327–334.

19.

Ochs

. Local convergence of the heavy-ball method and iPiano for non-convex optimization. J Optim Theory Appl 2018; 177: 153–180.

20.

Polyak

. Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math Phys 1964; 4: 1–17.

21.

Brakhage

. On ill-posed problems and the method of conjugate gradients. In: Inverse and Ill-posed problems (Sankt Wolfgang, 1986) (Notes Rep. Math. Sci. Engrg. vol 4), 1987, pp.165–175. New York: Academic.

22.

Engl

Hanke

Neubauer

. Regularization of inverse problems. Dordrecht: Kluwer, 1996.

23.

Hanke

. Accelerated Landweber iterations for the solution of ill-posed equations. Numer Math 1991; 60: 341–373.

24.

Kindermann

. Optimal-order convergence of Nesterov acceleration for linear ill-posed problems. Inverse Probl 2021; 37: 065002.

25.

Neubauer

. On Nesterov acceleration for Landweber iteration of linear ill-posed problems. J Inverse Ill-Posed Probl 2017; 25: 381–390.

26.

Nesterov

. A method of solving a convex programming problem with convergence rate

O (1 / k^{2})

. Sov Math Dokl 1983; 27: 372–376.

27.

Beck

Teboulle

. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2009; 2: 183–202.

28.

Jin

. Landweber-Kaczmarz method in Banach spaces with inexact inner solvers. Inverse Probl 2016; 32: 104005.

29.

Klann

. A Mumford-Shah-like method for limited data tomography with an application to electron tomography. SIAM J Imaging Sci 2011; 4: 1029–1048.

30.

. Singularities of the Radon transform of a class of piecewise smooth functions in

R^{2}

. Acta Math Sin Engl Ser 2011; 27: 191–208.

31.

Tuy

. Reconstruction of a three-dimensional object from a limited range of views. J Math Anal Appl 1981; 80: 598–616.

32.

Hansen

Saxild-Hansen

. IR tools—a MATLAB package of algebraic iterative reconstruction methods. J Comput Appl Math 2012; 236: 2167–2178.

33.

Chen

. The reweighted Landweber scheme for the extrapolation of band-limited signals-II. Signal Process 2020; 177: 107754.

34.

Haltmeier

Leitão

Scherzer

. Kaczmarz methods for regularizing nonlinear ill-posed equations: I. Convergence analysis. Inverse Probl Imaging 2007; 1: 289–298.

35.

Jin

Wang

. Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals. Inverse Probl 2013; 29: 085011.

36.

Kirsch

. An introduction to the mathematical theory of inverse problems. 2nd ed New York: Springer, 2001.