A Cheaper Way to Compute Generalized Cross-Validation as a Stopping Rule for Linear Stationary Iterative Methods

We apply generalized cross-validation (GCV) as a stopping rule for general linear stationary iterative methods for solving very large-scale, ill-conditioned problems. We present a new general formula for the influence operator for these methods and, using this formula and a Monte Carlo approach, we show how to compute the GCV function at a cheaper cost. Then we apply our approach to a well known iterative method (ART) with simulated data in positron emission tomography (PET).

[1]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[2]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[3]  J. M. Ollinger,et al.  Positron Emission Tomography , 2018, Handbook of Small Animal Imaging.

[4]  Gabor T. Herman,et al.  Algebraic reconstruction techniques can be made computationally efficient [positron emission tomography application] , 1993, IEEE Trans. Medical Imaging.

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

[6]  L. Shepp,et al.  A Statistical Model for Positron Emission Tomography , 1985 .

[7]  G. Golub,et al.  Generalized cross-validation for large scale problems , 1997 .

[8]  G T Herman,et al.  Performance evaluation of an iterative image reconstruction algorithm for positron emission tomography. , 1991, IEEE transactions on medical imaging.

[9]  Albert Macovski,et al.  A Maximum Likelihood Approach to Emission Image Reconstruction from Projections , 1976, IEEE Transactions on Nuclear Science.

[10]  Stanley J. Reeves,et al.  A cross-validation framework for solving image restoration problems , 1992, J. Vis. Commun. Image Represent..

[11]  Stanley J. Reeves,et al.  Optimal space-varying regularization in iterative image restoration , 1994, IEEE Trans. Image Process..

[12]  Å. Björck,et al.  Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations , 1979 .

[13]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[14]  Reginaldo J. Santos.  Equivalence of regularization and truncated iteration for general ill-posed problems☆ , 1996 .

[15]  H. Fleming.  Equivalence of regularization and truncated iteration in the solution of III-posed image reconstruction problems , 1990 .

[16]  A. Girard.  A fast ‘Monte-Carlo cross-validation’ procedure for large least squares problems with noisy data , 1989 .

[17]  H. A. van der Vorst,et al.  Numerical solution of large, sparse linear algebraic systems arising from tomographic problems , 1987 .

[18]  Grace Wahba,et al.  THREE TOPICS IN ILL-POSED PROBLEMS , 1987 .

[19]  Stanley J. Reeves,et al.  A practical stopping rule for iterative signal restoration , 1994, IEEE Trans. Signal Process..

[20]  L. Shepp,et al.  Maximum Likelihood Reconstruction for Emission Tomography , 1983, IEEE Transactions on Medical Imaging.

[21]  H. V. D. Vorst,et al.  SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems , 1990 .

[22]  David M. Allen,et al.  The Relationship Between Variable Selection and Data Agumentation and a Method for Prediction , 1974 .

[23]  Linda Kaufman,et al.  Implementing and Accelerating the EM Algorithm for Positron Emission Tomography , 1987, IEEE Transactions on Medical Imaging.