Abstract

Positron emission tomography (PET) is becoming increasingly important in the fields of medicine and biology. Penalized iterative algorithms based on maximum a posteriori (MAP) estimation for image reconstruction in emission tomography place conditions on which types of images are accepted as solutions. The recently introduced median root prior (MRP) favors locally monotonic images. MRP can preserve sharp edges, but a steplike streaking effect and much noise are still observed in the reconstructed image, both of which are undesirable. An MRP tomography reconstruction combined with nonlinear anisotropic diffusion interfiltering is proposed for removing noise and preserving edges. Analysis shows that the proposed algorithm is capable of producing better reconstructed images compared with those reconstructed by conventional maximum-likelihood expectation maximization (MLEM), MAP, and MRP-based algorithms in PET image reconstruction.

© 2007 Optical Society of America

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  1. T. F. Budinger, G. T. Gullberg, and R. H. Huesman, "Emission computed tomography," in Image Reconstruction from Projections: Implementation and Applications, G.T.Herman, ed. (Springer-Verlag, 1979).
  2. S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
    [CrossRef]
  3. S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).
  4. J. Biemond, R. L. Lagendijk, and R. M. Mersereau, "Iterative methods for image deblurring," Proc. IEEE 78, 856-883 (1990).
    [CrossRef]
  5. G. Kontaxakis, L. G. Strauss, and G. van Kaick, "Optimized image reconstruction for emission tomography using ordered subsets, median root prior, successive substitutions and a web-based interface," in Proceedings of IEEE Conference on Medical Imaging (Institute of Electrical and Electronics Engineers, 1998), pp. 1347-1352.
  6. S. Alenius, "On noise reduction in iterative image reconstruction algorithms for emission tomography: median root prior," Ph.D. dissertation (Tampere University of Technology, Tampere, Finland, 1999).
  7. I. T. Hsiao, A. Rangarjan, and G. Gindi, "A new convex edge-preserving median prior with applications to tomography," IEEE Trans. Med. Imaging 22, 580-585 (2003).
    [CrossRef] [PubMed]
  8. P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
    [CrossRef]
  9. M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
    [CrossRef]
  10. J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
    [CrossRef]
  11. O. Demirkaya, "Anisotropic diffusion filtering of PET attenuation data to improve emission images," Phys. Med. Biol. 47, N271-N278 (2002).
    [CrossRef] [PubMed]
  12. Y. L. You and M. Kaveh, "Fourth-order partial differential equations for noise removal," IEEE Trans. Image Process. 9, 1723-1730 (2000).
    [CrossRef]
  13. J. Ling and A. C. Bovik, "Smoothing low-SNR molecular images via anisotropic median-diffusion," IEEE Trans. Med. Imaging 21, 377-384 (2002).
    [CrossRef] [PubMed]
  14. G. Gilboa, N. Sochen, and Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," IEEE Trans. Image Process. 11, 689-703 (2002).
    [CrossRef]
  15. L. A. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
    [CrossRef]
  16. P. J. Green, "Bayesian reconstruction from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
    [CrossRef] [PubMed]
  17. Z. Zhou, R. M. Leahy, and J. Qi, "Approximate maximum likelihood hyperparameter estimation for Gibbs prior," IEEE Trans. Image Process. 6, 844-861 (1997).
    [CrossRef] [PubMed]
  18. T. Herbert and R. Leahy, "Statistic based MAP image reconstruction from Poisson data using Gibbs priors," IEEE Trans. Signal Process. 40, 2290-2303 (1992).
    [CrossRef]
  19. J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
    [CrossRef]
  20. S. Alenius, U. Ruotsalainen, and J. Astola, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
    [CrossRef]
  21. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.
  22. J. W. Stayman and J. A. Fessler, "Compensation for nonuniform resolution using penalized likelihood reconstruction in space-variant imaging systems," IEEE Trans. Med. Imaging 23, 269-284 (2004).
    [CrossRef] [PubMed]
  23. J. A. Fessler, "Grouped coordinate descent algorithms for robust edge-preserving image restoration," in Proc. SPIE 3170, 184-194 (1997).
    [CrossRef]
  24. W. Chlewicki, F. Hermansen, and S. B. Hansen, "Noise reduction and convergence of Bayesian algorithms with blobs based on the Huber function and median root prior," Phys. Med. Biol. 49, 4717-4730 (2004).
    [CrossRef] [PubMed]
  25. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

2004

J. W. Stayman and J. A. Fessler, "Compensation for nonuniform resolution using penalized likelihood reconstruction in space-variant imaging systems," IEEE Trans. Med. Imaging 23, 269-284 (2004).
[CrossRef] [PubMed]

W. Chlewicki, F. Hermansen, and S. B. Hansen, "Noise reduction and convergence of Bayesian algorithms with blobs based on the Huber function and median root prior," Phys. Med. Biol. 49, 4717-4730 (2004).
[CrossRef] [PubMed]

2003

I. T. Hsiao, A. Rangarjan, and G. Gindi, "A new convex edge-preserving median prior with applications to tomography," IEEE Trans. Med. Imaging 22, 580-585 (2003).
[CrossRef] [PubMed]

2002

O. Demirkaya, "Anisotropic diffusion filtering of PET attenuation data to improve emission images," Phys. Med. Biol. 47, N271-N278 (2002).
[CrossRef] [PubMed]

J. Ling and A. C. Bovik, "Smoothing low-SNR molecular images via anisotropic median-diffusion," IEEE Trans. Med. Imaging 21, 377-384 (2002).
[CrossRef] [PubMed]

G. Gilboa, N. Sochen, and Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," IEEE Trans. Image Process. 11, 689-703 (2002).
[CrossRef]

J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
[CrossRef]

2000

Y. L. You and M. Kaveh, "Fourth-order partial differential equations for noise removal," IEEE Trans. Image Process. 9, 1723-1730 (2000).
[CrossRef]

1999

S. Alenius, "On noise reduction in iterative image reconstruction algorithms for emission tomography: median root prior," Ph.D. dissertation (Tampere University of Technology, Tampere, Finland, 1999).

1998

G. Kontaxakis, L. G. Strauss, and G. van Kaick, "Optimized image reconstruction for emission tomography using ordered subsets, median root prior, successive substitutions and a web-based interface," in Proceedings of IEEE Conference on Medical Imaging (Institute of Electrical and Electronics Engineers, 1998), pp. 1347-1352.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

S. Alenius, U. Ruotsalainen, and J. Astola, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[CrossRef]

1997

J. A. Fessler, "Grouped coordinate descent algorithms for robust edge-preserving image restoration," in Proc. SPIE 3170, 184-194 (1997).
[CrossRef]

M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
[CrossRef]

Z. Zhou, R. M. Leahy, and J. Qi, "Approximate maximum likelihood hyperparameter estimation for Gibbs prior," IEEE Trans. Image Process. 6, 844-861 (1997).
[CrossRef] [PubMed]

1993

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).

1992

T. Herbert and R. Leahy, "Statistic based MAP image reconstruction from Poisson data using Gibbs priors," IEEE Trans. Signal Process. 40, 2290-2303 (1992).
[CrossRef]

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.

1990

P. J. Green, "Bayesian reconstruction from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, "Iterative methods for image deblurring," Proc. IEEE 78, 856-883 (1990).
[CrossRef]

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

1984

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

1982

L. A. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

1980

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

1979

T. F. Budinger, G. T. Gullberg, and R. H. Huesman, "Emission computed tomography," in Image Reconstruction from Projections: Implementation and Applications, G.T.Herman, ed. (Springer-Verlag, 1979).

Alenius, S.

S. Alenius, "On noise reduction in iterative image reconstruction algorithms for emission tomography: median root prior," Ph.D. dissertation (Tampere University of Technology, Tampere, Finland, 1999).

S. Alenius, U. Ruotsalainen, and J. Astola, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[CrossRef]

Astola, J.

S. Alenius, U. Ruotsalainen, and J. Astola, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[CrossRef]

Bequ, D.

J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
[CrossRef]

Biemond, J.

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, "Iterative methods for image deblurring," Proc. IEEE 78, 856-883 (1990).
[CrossRef]

Bovik, A. C.

J. Ling and A. C. Bovik, "Smoothing low-SNR molecular images via anisotropic median-diffusion," IEEE Trans. Med. Imaging 21, 377-384 (2002).
[CrossRef] [PubMed]

Budinger, T. F.

T. F. Budinger, G. T. Gullberg, and R. H. Huesman, "Emission computed tomography," in Image Reconstruction from Projections: Implementation and Applications, G.T.Herman, ed. (Springer-Verlag, 1979).

Chlewicki, W.

W. Chlewicki, F. Hermansen, and S. B. Hansen, "Noise reduction and convergence of Bayesian algorithms with blobs based on the Huber function and median root prior," Phys. Med. Biol. 49, 4717-4730 (2004).
[CrossRef] [PubMed]

Demirkaya, O.

O. Demirkaya, "Anisotropic diffusion filtering of PET attenuation data to improve emission images," Phys. Med. Biol. 47, N271-N278 (2002).
[CrossRef] [PubMed]

Dupont, P.

J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
[CrossRef]

Fdez-Valdivia, J.

M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
[CrossRef]

Fessler, J. A.

J. W. Stayman and J. A. Fessler, "Compensation for nonuniform resolution using penalized likelihood reconstruction in space-variant imaging systems," IEEE Trans. Med. Imaging 23, 269-284 (2004).
[CrossRef] [PubMed]

J. A. Fessler, "Grouped coordinate descent algorithms for robust edge-preserving image restoration," in Proc. SPIE 3170, 184-194 (1997).
[CrossRef]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.

Garcia, J. A.

M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
[CrossRef]

Garcia-Silvente, M.

M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
[CrossRef]

Garrido, A.

M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
[CrossRef]

Geman, D.

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

Gilboa, G.

G. Gilboa, N. Sochen, and Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," IEEE Trans. Image Process. 11, 689-703 (2002).
[CrossRef]

Gindi, G.

I. T. Hsiao, A. Rangarjan, and G. Gindi, "A new convex edge-preserving median prior with applications to tomography," IEEE Trans. Med. Imaging 22, 580-585 (2003).
[CrossRef] [PubMed]

Green, P. J.

P. J. Green, "Bayesian reconstruction from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

Gullberg, G. T.

T. F. Budinger, G. T. Gullberg, and R. H. Huesman, "Emission computed tomography," in Image Reconstruction from Projections: Implementation and Applications, G.T.Herman, ed. (Springer-Verlag, 1979).

Hansen, S. B.

W. Chlewicki, F. Hermansen, and S. B. Hansen, "Noise reduction and convergence of Bayesian algorithms with blobs based on the Huber function and median root prior," Phys. Med. Biol. 49, 4717-4730 (2004).
[CrossRef] [PubMed]

Herbert, T.

T. Herbert and R. Leahy, "Statistic based MAP image reconstruction from Poisson data using Gibbs priors," IEEE Trans. Signal Process. 40, 2290-2303 (1992).
[CrossRef]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

Hermansen, F.

W. Chlewicki, F. Hermansen, and S. B. Hansen, "Noise reduction and convergence of Bayesian algorithms with blobs based on the Huber function and median root prior," Phys. Med. Biol. 49, 4717-4730 (2004).
[CrossRef] [PubMed]

Hsiao, I. T.

I. T. Hsiao, A. Rangarjan, and G. Gindi, "A new convex edge-preserving median prior with applications to tomography," IEEE Trans. Med. Imaging 22, 580-585 (2003).
[CrossRef] [PubMed]

Huesman, R. H.

T. F. Budinger, G. T. Gullberg, and R. H. Huesman, "Emission computed tomography," in Image Reconstruction from Projections: Implementation and Applications, G.T.Herman, ed. (Springer-Verlag, 1979).

Iizawa, T.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

Ishikawa, C.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

Ishizaka, T.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

Kaveh, M.

Y. L. You and M. Kaveh, "Fourth-order partial differential equations for noise removal," IEEE Trans. Image Process. 9, 1723-1730 (2000).
[CrossRef]

Kay, S.

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).

Kontaxakis, G.

G. Kontaxakis, L. G. Strauss, and G. van Kaick, "Optimized image reconstruction for emission tomography using ordered subsets, median root prior, successive substitutions and a web-based interface," in Proceedings of IEEE Conference on Medical Imaging (Institute of Electrical and Electronics Engineers, 1998), pp. 1347-1352.

Lagendijk, R. L.

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, "Iterative methods for image deblurring," Proc. IEEE 78, 856-883 (1990).
[CrossRef]

Leahy, R.

T. Herbert and R. Leahy, "Statistic based MAP image reconstruction from Poisson data using Gibbs priors," IEEE Trans. Signal Process. 40, 2290-2303 (1992).
[CrossRef]

Leahy, R. M.

Z. Zhou, R. M. Leahy, and J. Qi, "Approximate maximum likelihood hyperparameter estimation for Gibbs prior," IEEE Trans. Image Process. 6, 844-861 (1997).
[CrossRef] [PubMed]

Ling, J.

J. Ling and A. C. Bovik, "Smoothing low-SNR molecular images via anisotropic median-diffusion," IEEE Trans. Med. Imaging 21, 377-384 (2002).
[CrossRef] [PubMed]

Maeda, J.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

Malik, J.

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

Mersereau, R. M.

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, "Iterative methods for image deblurring," Proc. IEEE 78, 856-883 (1990).
[CrossRef]

Mortelmans, L.

J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
[CrossRef]

Nuyts, J.

J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
[CrossRef]

Perona, P.

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.

Qi, J.

Z. Zhou, R. M. Leahy, and J. Qi, "Approximate maximum likelihood hyperparameter estimation for Gibbs prior," IEEE Trans. Image Process. 6, 844-861 (1997).
[CrossRef] [PubMed]

Rangarjan, A.

I. T. Hsiao, A. Rangarjan, and G. Gindi, "A new convex edge-preserving median prior with applications to tomography," IEEE Trans. Med. Imaging 22, 580-585 (2003).
[CrossRef] [PubMed]

Ruotsalainen, U.

S. Alenius, U. Ruotsalainen, and J. Astola, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[CrossRef]

Shepp, L. A.

L. A. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

Sochen, N.

G. Gilboa, N. Sochen, and Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," IEEE Trans. Image Process. 11, 689-703 (2002).
[CrossRef]

Stayman, J. W.

J. W. Stayman and J. A. Fessler, "Compensation for nonuniform resolution using penalized likelihood reconstruction in space-variant imaging systems," IEEE Trans. Med. Imaging 23, 269-284 (2004).
[CrossRef] [PubMed]

Strauss, L. G.

G. Kontaxakis, L. G. Strauss, and G. van Kaick, "Optimized image reconstruction for emission tomography using ordered subsets, median root prior, successive substitutions and a web-based interface," in Proceedings of IEEE Conference on Medical Imaging (Institute of Electrical and Electronics Engineers, 1998), pp. 1347-1352.

Suzuki, Y.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.

van Kaick, G.

G. Kontaxakis, L. G. Strauss, and G. van Kaick, "Optimized image reconstruction for emission tomography using ordered subsets, median root prior, successive substitutions and a web-based interface," in Proceedings of IEEE Conference on Medical Imaging (Institute of Electrical and Electronics Engineers, 1998), pp. 1347-1352.

Vardi, Y.

L. A. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.

You, Y. L.

Y. L. You and M. Kaveh, "Fourth-order partial differential equations for noise removal," IEEE Trans. Image Process. 9, 1723-1730 (2000).
[CrossRef]

Zeevi, Y.

G. Gilboa, N. Sochen, and Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," IEEE Trans. Image Process. 11, 689-703 (2002).
[CrossRef]

Zhou, Z.

Z. Zhou, R. M. Leahy, and J. Qi, "Approximate maximum likelihood hyperparameter estimation for Gibbs prior," IEEE Trans. Image Process. 6, 844-861 (1997).
[CrossRef] [PubMed]

IEEE Trans. Image Process.

Y. L. You and M. Kaveh, "Fourth-order partial differential equations for noise removal," IEEE Trans. Image Process. 9, 1723-1730 (2000).
[CrossRef]

G. Gilboa, N. Sochen, and Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," IEEE Trans. Image Process. 11, 689-703 (2002).
[CrossRef]

Z. Zhou, R. M. Leahy, and J. Qi, "Approximate maximum likelihood hyperparameter estimation for Gibbs prior," IEEE Trans. Image Process. 6, 844-861 (1997).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

L. A. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

P. J. Green, "Bayesian reconstruction from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

J. Ling and A. C. Bovik, "Smoothing low-SNR molecular images via anisotropic median-diffusion," IEEE Trans. Med. Imaging 21, 377-384 (2002).
[CrossRef] [PubMed]

I. T. Hsiao, A. Rangarjan, and G. Gindi, "A new convex edge-preserving median prior with applications to tomography," IEEE Trans. Med. Imaging 22, 580-585 (2003).
[CrossRef] [PubMed]

J. W. Stayman and J. A. Fessler, "Compensation for nonuniform resolution using penalized likelihood reconstruction in space-variant imaging systems," IEEE Trans. Med. Imaging 23, 269-284 (2004).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci.

J. Nuyts, D. Bequ, P. Dupont, and L. Mortelmans, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 49, 56-60 (2002).
[CrossRef]

S. Alenius, U. Ruotsalainen, and J. Astola, "A concave prior penalizing relative differences for maximum a posteriori reconstruction in emission tomography," IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

IEEE Trans. Signal Process.

T. Herbert and R. Leahy, "Statistic based MAP image reconstruction from Poisson data using Gibbs priors," IEEE Trans. Signal Process. 40, 2290-2303 (1992).
[CrossRef]

Image Vis. Comput.

M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scale-spectrum information," Image Vis. Comput. 15, 913-923 (1997).
[CrossRef]

Pattern Recogn.

J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Suzuki, "Segmentation of natural images using anisotropic diffusion and linking of boundary edges," Pattern Recogn. 31, 1993-1999 (1998).
[CrossRef]

Phys. Med. Biol.

O. Demirkaya, "Anisotropic diffusion filtering of PET attenuation data to improve emission images," Phys. Med. Biol. 47, N271-N278 (2002).
[CrossRef] [PubMed]

W. Chlewicki, F. Hermansen, and S. B. Hansen, "Noise reduction and convergence of Bayesian algorithms with blobs based on the Huber function and median root prior," Phys. Med. Biol. 49, 4717-4730 (2004).
[CrossRef] [PubMed]

Proc. IEEE

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, "Iterative methods for image deblurring," Proc. IEEE 78, 856-883 (1990).
[CrossRef]

Proc. SPIE

J. A. Fessler, "Grouped coordinate descent algorithms for robust edge-preserving image restoration," in Proc. SPIE 3170, 184-194 (1997).
[CrossRef]

Other

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

T. F. Budinger, G. T. Gullberg, and R. H. Huesman, "Emission computed tomography," in Image Reconstruction from Projections: Implementation and Applications, G.T.Herman, ed. (Springer-Verlag, 1979).

G. Kontaxakis, L. G. Strauss, and G. van Kaick, "Optimized image reconstruction for emission tomography using ordered subsets, median root prior, successive substitutions and a web-based interface," in Proceedings of IEEE Conference on Medical Imaging (Institute of Electrical and Electronics Engineers, 1998), pp. 1347-1352.

S. Alenius, "On noise reduction in iterative image reconstruction algorithms for emission tomography: median root prior," Ph.D. dissertation (Tampere University of Technology, Tampere, Finland, 1999).

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), Chap. 15.

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Figures (6)

Fig. 1
Fig. 1

Modified Shepp–Logan phantom.

Fig. 2
Fig. 2

Images reconstructed by different algorithms after 30 iterations: (a) MLEM, (b) MAP with the Huber prior, (c) MRP, and (d) PDEmedian.

Fig. 3
Fig. 3

Profile plots along two lines and the corresponding absolute value error plots through the original and the reconstructed images produced by MLEM, MAP with the Huber prior, MRP, and PDEmedian algorithms: (a) profile plots along the center row line, (b) profile plots along the 135° diagonal line, (c) the absolute value error plots along the center row line, and (d) the absolute value error plots along the 135° diagonal line.

Fig. 4
Fig. 4

Images reconstructed by MLEM, MAP with the Huber prior, MRP, and PDEmedian at 1,000,000 emission counts [(a), (b), (c), (d), respectively] and at 400,000 emission counts [(e), (f), (g), (h), respectively] after 30 iterations.

Fig. 5
Fig. 5

(a) Real emission thorax phantom. (b)–(e) Images reconstructed by different algorithms after 30 iterations with real emission phantom data. (b) MLEM, (c) MAP with the Huber prior, (d) MRP, and (e) PDEmedian.

Fig. 6
Fig. 6

NRMSE versus iteration plots for PDEmedian, MLEM, MAP with the Huber prior, and MRP for three coincidence count levels (a) 4,000,000, (b) 1,000,000, (c) 400,000, and (d) real thorax phantom.

Equations (27)

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g i Poisson ( j = 1 M H i j f j ) , i = 1 , , N ,
P L ( g f ) = i = 1 N [ λ i g i g i ! exp ( l i ) ] ,
λ i = j = 1 M H i j f j .
L ( f ) = i = 1 N [ g i log λ i λ i ] .
f j k + 1 = f j k i = 1 N H i j [ g i ( l = 1 M H i l f l k ) ] i = 1 N H i j , j = 1 , , M ,
f MAP arg max f 0 L ( f ) + log P ( f ) .
P ( f ) = ( 1 Z ) exp [ β j j N j ω j j V ( f j , f j ) ] ,
f j k + 1 = f j k i = 1 N H i j [ g i ( l = 1 M H i l f l k ) ] i = 1 N H i j + β j N j ω j j ( V ( f j , f j ) f j f j k ) .
f ̂ = arg min f 0 Φ ( f ) = arg min f 0 { L ( f ) + β Φ p mrp ( f ; m k ) } ,
m j k + 1 = median ( f j k + 1 ; j N j ) ,
Φ j mrp ( f ; m k ) = j ( f j m j k ) 2 2 m j k .
f j ( k + 1 ) = f j k i = 1 N H i j [ g i ( l = 1 M H i l f l k ) ] i = 1 N H i j + β [ ( f j k M b ) M b ] ,
f t = div [ g ( f ) f ] ,
g 1 ( t ) = 1 1 + t 2 K 2 ,
g 2 ( t ) = exp ( t 2 K 2 ) ,
E 1 ( m ) = Ψ ( m ) d x ,
E 2 ( f ) = L ( f ) + β Φ p ( f ; m ) .
Φ p ( f ; m ) = j j N j ω j j ϕ ( f j m j ) .
Φ p abs ( f ; m ) = j j N j f j m j .
m t = div ( ψ ( m ) m m ) .
m j k , l + 1 = m j k , l + ω N j j N j ( g ( ( m k , l ) j , j ) m j , j k , l ) ,
q j , j = q j q j , j N j
f j k + 1 = f j k i = 1 N H i j [ g i ( l = 1 M H i l f l k ) ] i = 1 N H i j + β [ ( f j k M b AD ) M b AD ] ,
M b AD = median ( m j k , l + 1 , j N j ) .
H ( t ) = { t 2 2 , t δ δ t δ 2 2 , t > δ } .
V ( f j , f i ) = { ( f j f i ) 2 2 , f j f i δ δ f j f i δ 2 2 , f j f i > δ } .
d = { j = 1 M θ j final θ j 0 2 j = 1 M θ j 0 θ 0 ¯ 2 } 1 2 .

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