The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Representation of High Resolution Images from Low Sampled Fourier Data: Applications to Dynamic MRI
Journal of Mathematical Imaging and Vision
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In this work we consider an inverse ill-posed problem coming from the area of dynamic magnetic resonance imaging (MRI), where high resolution images must be reconstructed from incomplete data sets collected in the Fourier domain. The reduced-encoding imaging by generalized-series reconstruction (RIGR) method used leads to ill-conditioned linear systems with Hermitian Toeplitz matrix and noisy right-hand side. We analyze the behavior of some regularization methods such as the truncated singular value decomposition (TSVD), the Lavrent'yev regularization method and conjugate gradients (CG) type iterative methods. For what concerns the choice of the regularization parameter, we use some known methods and we propose new heuristic criteria for iterative regularization methods. The simulations are carried on test problems obtained from real data acquired on a 1.5 T Phillips system.