An introduction to the mathematical theory of inverse problems
An introduction to the mathematical theory of inverse problems
The mathematics of computerized tomography
The mathematics of computerized tomography
Parallel Optimization: Theory, Algorithms and Applications
Parallel Optimization: Theory, Algorithms and Applications
SIAM Journal on Matrix Analysis and Applications
On Diagonally Relaxed Orthogonal Projection Methods
SIAM Journal on Scientific Computing
Convexly constrained linear inverse problems: iterativeleast-squares and regularization
IEEE Transactions on Signal Processing
On the Convergence of Generalized Simultaneous Iterative Reconstruction Algorithms
IEEE Transactions on Image Processing
Sparse angular CT reconstruction using non-local means based iterative-correction POCS
Computers in Biology and Medicine
AIR Tools - A MATLAB package of algebraic iterative reconstruction methods
Journal of Computational and Applied Mathematics
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The Landweber scheme is an algebraic reconstruction method and includes several important algorithms as its special cases. The convergence of the Landweber scheme is of both theoretical and practical importance. Using the singular value decomposition (SVD), we derive an iterative representation formula for the Landweber scheme and consequently establish the necessary and sufficient conditions for its convergence. In addition to verifying the necessity and sufficiency of known convergent conditions, we find new convergence conditions allowing relaxation coefficients in an interval not covered by known results. Moreover, it is found that the Landweber scheme can converge within finite iterations when the relaxation coefficients are chosen to be the inverses of squares of the nonzero singular values. Furthermore, the limits of the Landweber scheme in all convergence cases are shown to be the sum of the minimum norm solution of a weighted least-squares problem and an oblique projection of the initial image onto the null space of the system matrix.