Regularization in tomographic reconstruction using thresholding estimators

In tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras, image reconstruction is an unstable inverse problem, due to the presence of additive noise. A new family of regularization methods for reconstruction, based on a thresholding procedure in wavelet and wavelet packet (WP) decompositions, is studied. This approach is based on the fact that the decompositions provide a near-diagonalization of the inverse Radon transform and of prior information in medical images. A WP decomposition is adaptively chosen for the specific image to be restored. Corresponding algorithms have been developed for both two-dimensional and full three-dimensional reconstruction. These procedures are fast, noniterative, and flexible. Numerical results suggest that they outperform filtered back-projection and iterative procedures such as ordered-subset-expectation-maximization.

[1]  Minh N. Do,et al.  The finite ridgelet transform for image representation , 2003, IEEE Trans. Image Process..

[2]  A. Medl,et al.  Time Frequency and Wavelets in Biomedical Signal Processing , 1998, IEEE Engineering in Medicine and Biology Magazine.

[3]  Shiying Zhao.  Wavelet Filtering for Filtered Backprojection in Computed Tomography , 1999 .

[4]  Ronald R. Coifman,et al.  Entropy-based algorithms for best basis selection , 1992, IEEE Trans. Inf. Theory.

[5]  P. Toft.  The Radon Transform - Theory and Implementation , 1996 .

[6]  P. Fry.  Poisson Intensity Estimation Using Wavelets and the Fisz Transformation , 2001 .

[7]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[8]  Michel Barlaud,et al.  A fast tomographic reconstruction algorithm in the 2-D wavelet transform domain , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[9]  W. Clem Karl,et al.  Tomographic reconstruction and estimation based on multiscale natural-pixel bases , 1997, IEEE Trans. Image Process..

[10]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[11]  Nam-Yong Lee,et al.  Wavelet methods for inverting the Radon transform with noisy data , 2001, IEEE Trans. Image Process..

[12]  M. Lautsch,et al.  A spline inversion formula for the Radon transform , 1989 .

[13]  Ge Wang,et al.  Wavelet Sampling and Localization Schemes for the Radon Transform in Two Dimensions , 1997, SIAM J. Appl. Math..

[14]  Jean-Pierre V. Guédon,et al.  Bandlimited and Haar filtered back-projection reconstructions , 1994, IEEE Trans. Medical Imaging.

[15]  References , 1971 .

[16]  Yong Choi,et al.  Image reconstruction using the wavelet transform for positron emission tomography , 2001, IEEE Transactions on Medical Imaging.

[17]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[18]  E. Kolaczyk.  A Wavelet Shrinkage Approach to Tomographic Image Reconstruction , 1996 .

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

[20]  S. Deans.  The Radon Transform and Some of Its Applications , 1983 .

[21]  W. Clem Karl,et al.  A wavelet-based method for multiscale tomographic reconstruction , 1996, IEEE Trans. Medical Imaging.

[22]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[23]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[24]  Michel Defrise,et al.  Exact and approximate rebinning algorithms for 3-D PET data , 1997, IEEE Transactions on Medical Imaging.

[25]  C. Stein.  Estimation of the Mean of a Multivariate Normal Distribution , 1981 .

[26]  A. Aldroubi,et al.  Wavelets in Medicine and Biology , 1997 .

[27]  K. J. Ray Liu,et al.  Wavelet-based multiresolution local tomography , 1997, IEEE Trans. Image Process..

[28]  William Moran,et al.  An Application of Wavelets in Tomography , 1998, Digit. Signal Process..

[29]  H. Malcolm Hudson,et al.  Accelerated image reconstruction using ordered subsets of projection data , 1994, IEEE Trans. Medical Imaging.

[30]  Andrew F. Laine,et al.  Improving PET-based physiological quantification through methods of wavelet denoising , 2001, IEEE Transactions on Biomedical Engineering.

[31]  S. Mallat,et al.  Thresholding estimators for linear inverse problems and deconvolutions , 2003 .

[32]  E. Candès,et al.  Recovering edges in ill-posed inverse problems: optimality of curvelet frames , 2002 .

[33]  Yoram Bresler,et al.  Multiresolution tomographic reconstruction using wavelets , 1994, Proceedings of 1st International Conference on Image Processing.

[34]  Tim Olson,et al.  Wavelet localization of the Radon transform , 1994, IEEE Trans. Signal Process..

[35]  Metin Akay,et al.  Time frequency and wavelets in biomedical signal processing , 1998 .

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

[37]  E.E. Pissaloux,et al.  Image Processing , 1994, Proceedings. Second Euromicro Workshop on Parallel and Distributed Processing.

[38]  N. Katz,et al.  WAVELETS: CALDERÓN-ZYGMUND AND MULTILINEAR OPERATORS (Cambridge Studies in Advanced Mathematics 48) , 1999 .

[39]  W. R. Madych,et al.  Tomography, Approximate Reconstruction, and Continuous Wavelet Transforms , 1999 .

[40]  Y. Meyer,et al.  Wavelets: Calderón-Zygmund and Multilinear Operators , 1997 .