A hybrid Kaczmarz-Conjugate Gradient algorithm for image reconstruction

The present paper is a theoretical contribution to the field of iterative methods for solving inconsistent linear least squares problems arising in image reconstruction from projections in computerized tomography. It consists on a hybrid algorithm which includes in each iteration a CG-like step for modifying the right-hand side and a Kaczmarz-like step for producing the approximate solution. We prove convergence of the hybrid algorithm for general inconsistent and rank-deficient least-squares problems. Although the new algorithm has potential for more applied experiments and comparisons, we restrict them in this paper to a regularized image reconstruction problem involving a 2D medical data set.

[1]  R. Zdunek,et al.  Electromagnetic geotomography-selection of measuring frequency , 2005, IEEE Sensors Journal.

[2]  G. B. Smith,et al.  Preface to S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images” , 1987 .

[3]  Constantin Popa,et al.  Least-squares solution of overdetermined inconsistent linear systems using kaczmarz's relaxation , 1995, Int. J. Comput. Math..

[4]  Tommy Elfving,et al.  A stationary iterative pseudoinverse algorithm , 1998 .

[5]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[6]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[8]  Rafal Zdunek,et al.  Kaczmarz extended algorithm for tomographic image reconstruction from limited-data , 2004, Math. Comput. Simul..

[9]  Gene H. Golub,et al.  Matrix computations , 1983 .

[10]  M. Z. Nashed,et al.  On the Convergence of the Conjugate Gradient Method for Singular Linear Operator Equations , 1972 .

[11]  C. Popa.  Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deficient least-squares problems , 1998 .

[12]  Andrzej Stachurski,et al.  Parallel Optimization: Theory, Algorithms and Applications , 2000, Parallel Distributed Comput. Pract..

[13]  Y. Censor.  Row-Action Methods for Huge and Sparse Systems and Their Applications , 1981 .

[14]  Elena Bautu,et al.  Tikhonov Regularization in Image Reconstruction with Kaczmarz Extended Algorithm , 2005 .

[15]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[16]  K. Tanabe.  Projection method for solving a singular system of linear equations and its applications , 1971 .

[17]  Yair Censor,et al.  Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems , 2001, Parallel Comput..

[18]  Jean-Baptiste Thibault,et al.  A three-dimensional statistical approach to improved image quality for multislice helical CT. , 2007, Medical physics.

[19]  Yoram Bresler,et al.  Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography , 1998, IEEE Trans. Image Process..

[20]  Y. Censor,et al.  Parallel Optimization: Theory, Algorithms, and Applications , 1997 .

[21]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .