Towards Automatic Selection of the Regularization Parameters in Emission Tomgraphy by Fourier Synthesis

The problem of image reconstruction in emission tomography in an ill-posed inverse problem. The methodology FRECT (Fourier regularized computed tomography) allows not only for a priori analysis of the stability of the reconstruction process but also for an exact definition of the resolution in the slices. Its natural regularization parameter, namely the cutoff frequency ν of the filter underlying the definition of the FRECT solution, can be calibrated by estimating the condition number for a range of values of ν. We first outline the methodology FRECT. We then discuss the numerical strategies which can be implemented in order to estimate the condition numbers of large matrices. Finally, we present a few results obtained in the context of SPECT reconstructions, and discuss the possibility to determine automatically the best possible cutoff frequency from the analysis of the stability.

[1]  L. Blanc-Féraud,et al.  Conjugate gradient algorithm with edge- preserving regularization for microwave inverse scattering , 1996 .

[2]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[3]  R. F. Wagner,et al.  SNR and noise measurements for medical imaging: I. A practical approach based on statistical decision theory. , 1993, Physics in medicine and biology.

[4]  L. J. Thomas,et al.  Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography , 1987, IEEE Transactions on Medical Imaging.

[5]  P. Marchal,et al.  A new reconstruction methodology for computerized tomography: FRECT (Fourier regularized computed tomography) , 1999 .

[6]  D S Lalush,et al.  A fast and stable maximum a posteriori conjugate gradient reconstruction algorithm. , 1995, Medical physics.

[7]  Eric Anterrieu,et al.  Fourier interpolation and reconstruction via shannon-type techniques. I: Regularization principle , 1994 .

[8]  D Ros,et al.  Evaluation of a cross-validation stopping rule in MLE SPECT reconstruction. , 1998, Physics in medicine and biology.

[9]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[10]  P. Maréchal,et al.  A new reconstruction methodology for computerized tomography: FRECT (Fourier regularized computed tomography) , 1999, 1999 IEEE Nuclear Science Symposium. Conference Record. 1999 Nuclear Science Symposium and Medical Imaging Conference (Cat. No.99CH37019).

[11]  A. Lannes,et al.  Stabilized Reconstruction in Signal and Image Processing , 1987 .

[12]  T. J. Herbert.  Statistical stopping criteria for iterative maximum likelihood reconstruction of emission images , 1990 .

[13]  Eric Anterrieu,et al.  CLEAN and WIPE , 1997 .

[14]  Arnold Neumaier,et al.  PET regularization by envelope guided conjugate gradients , 1996, IEEE Trans. Medical Imaging.

[15]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[16]  B. Parlett,et al.  The Lanczos algorithm with selective orthogonalization , 1979 .

[17]  Eric C. Frey,et al.  Comparison between ML-EM and WLS-CG algorithms for SPECT image reconstruction , 1991 .

[18]  G. Gindi,et al.  Noise analysis of MAP-EM algorithms for emission tomography. , 1997, Physics in medicine and biology.

[19]  David S. Lalush,et al.  Simulation Evaluation Of Gibbs Prior Distributions For Use In Maximum A Posteriori SPECT Reconstructions , 1990, 1990 IEEE Nuclear Science Symposium Conference Record.

[20]  B R Zeeberg,et al.  Evaluation of reconstruction algorithms in SPECT neuroimaging: I. Comparison of statistical noise in SPECT neuroimages with 'naive' and 'realistic' predictions. , 1993, Physics in medicine and biology.

[21]  O. Nalcioglu,et al.  Constrained Iterative Reconstruction by the Conjugate Gradient Method , 1985, IEEE Transactions on Medical Imaging.

[22]  A. Formiconi,et al.  Physical modelling (geometrical system response, Compton scattering and attenuation) in brain SPECT using the conjugate gradients reconstruction method , 1993 .

[23]  Jeffrey A. Fessler.  Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography , 1996, IEEE Trans. Image Process..

[24]  S. Strother,et al.  Noise and signal decoupling in maximum-likelihood reconstructions and Metz filters for PET brain images. , 1994 .

[25]  Jorge Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[26]  Gabor T. Herman.  On the Noise in Images Produced by Computed Tomography , 1981 .

[27]  Donald W. Wilson,et al.  Noise properties of the EM algorithm. I. Theory , 1994 .

[28]  Soo Chin Liew,et al.  Noise propagation in SPECT images reconstructed using an iterative maximum-likelihood algorithm , 1993 .

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

[30]  Pierre Maréchal,et al.  Unification of some deterministic and probabilistic methods for the solution of linear inverse problems via the principle of maximum entropy on the mean , 1997 .

[31]  Ronald J. Jaszczak,et al.  Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data , 1997, IEEE Transactions on Medical Imaging.

[32]  J. Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[33]  A. Formiconi,et al.  Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique. , 1989, Physics in medicine and biology.

[34]  J. Cullum,et al.  A Lanczos Algorithm for Computing Singular Values and Vectors of Large Matrices , 1983 .

[35]  B. Tsui,et al.  Noise properties of filtered-backprojection and ML-EM reconstructed emission tomographic images , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

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

[37]  Jonathan M. Borwein,et al.  Computation and stability analysis for regularized tomographic reconstructions , 1999 .

[38]  R H Huesman,et al.  The effects of a finite number of projection angles and finite lateral sampling of projections on the propagation of statistical errors in transverse section reconstruction. , 1977, Physics in medicine and biology.

[39]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[40]  G. W. Stewart,et al.  Afternotes on numerical analysis , 1987 .

[41]  G. Gillen.  A simple method for the measurement of local statistical noise levels in SPECT , 1992 .