Abstract

Image filtering techniques have numerous potential applications in biomedical imaging and image processing. The design of filters largely depends on the a priori, knowledge about the type of noise corrupting the image. This makes the standard filters application specific. Widely used filters such as average, Gaussian, and Wiener reduce noisy artifacts by smoothing. However, this operation normally results in smoothing of the edges as well. On the other hand, sharpening filters enhance the high-frequency details, making the image nonsmooth. An integrated general approach to design a finite impulse response filter based on Hebbian learning is proposed for optimal image filtering. This algorithm exploits the interpixel correlation by updating the filter coefficients using Hebbian learning. The algorithm is made iterative for achieving efficient learning from the neighborhood pixels. This algorithm performs optimal smoothing of the noisy image by preserving high-frequency as well as low-frequency features. Evaluation results show that the proposed finite impulse response filter is robust under various noise distributions such as Gaussian noise, salt-and-pepper noise, and speckle noise. Furthermore, the proposed approach does not require any a priori knowledge about the type of noise. The number of unknown parameters is few, and most of these parameters are adaptively obtained from the processed image. The proposed filter is successfully applied for image reconstruction in a positron emission tomography imaging modality. The images reconstructed by the proposed algorithm are found to be superior in quality compared with those reconstructed by existing PET image reconstruction methodologies.

© 2006 Optical Society of America

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2005 (3)

M. Izzetoglu, A. Devaraj, S. Bunce, and B. Onaral, 'Motion artifact cancellation in NIR spectroscopy using Wiener filtering,' IEEE Trans. Biomed. Eng. 52, 934-938 (2005).
[Crossref] [PubMed]

P. P. Mondal and K. Rajan, 'Fuzzy-rule-based image reconstruction for positron emission tomography,' J. Opt. Soc. Am. A 22, 1763-1771 (2005).
[Crossref]

P. P. Mondal and K. Rajan, 'Neural network based image reconstruction for positron emission tomography,' Appl. Opt. 44, 6245-6352 (2005).
[Crossref]

2004 (2)

S. Marshall, 'New direct, design method for weighted order statistic filters,' IEE Proc. Vision Image Signal Process. 151, 1-8 (2004).
[Crossref]

A. M. Wink and J. B. T. M. Roerdink, 'Denoising functional MR images: a comparison of wavelet denoising and gaussian smoothing,' IEEE Trans. Med. Imaging 23, 374-387 (2004).
[Crossref] [PubMed]

2003 (3)

M. Kazubek, 'Wavelet domain image denoising by thresholding and Wiener filtering,' IEEE Signal Process. Lett. 10, 324-326 (2003).
[Crossref]

P. Bao and L. Zhang, 'Noise reduction for magnetic resonance images via adaptive multiscale products thresholding,' IEEE Trans. Med. Imaging 22, 1089-1099 (2003).
[Crossref] [PubMed]

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, 'Versatile wavelet domain noise filtration technique for medical imaging,' IEEE Trans. Med. Imaging 22, 323-331 (2003).
[Crossref] [PubMed]

2002 (2)

H. G. Senel, R. A. Peters II, and B. Dawant, 'Topological median filters,' IEEE Trans. Image Process. 11, 89-104 (2002).
[Crossref]

S. Alenius and U. Ruotsalainen, 'Generalization of median root prior reconstruction,' IEEE Trans. Med. Imaging 21, 1413-1420 (2002).
[Crossref]

2001 (2)

T. Chen and H. Wu, 'Adaptive impulse detection using center-weighted median filters,' IEEE Signal Process. Lett. 8, 1-3 (2001).
[Crossref]

G. Fan and X. G. Xia, 'Image denoising using local contextual hidden Marcov model in the wavelet domain,' IEEE Signal Process. Lett. 8, 125-128 (2001).
[Crossref]

2000 (1)

S. A. G. Chang, B. Yu, and M. Vetterli, 'Adaptive wavelet thresholding for image denoising and compression,' IEEE Trans. Image Process. 9, 1532-l546 (2000).
[Crossref]

1999 (1)

M. K. Mihak, I. Kozinsev, K. Ramchandran, and P. Moulin, 'Low-complexity image denoising based on statistical modeling of wavelet coefficients,' IEEE Signal Process. Lett. 6, 300-303 (1999).
[Crossref]

1998 (3)

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

S. C. Pei, C L. Lai, and F. Y. Shih, 'Recursive order-statistic soft morphological filters,' IEE Proc. Vision Image Signal Process. 145, 333-342 (1998).
[Crossref]

S. Alenius and U. Ruotsalainen, 'Using local median as the location of prior distribution in iterative emission tomography reconstruction,' IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[Crossref]

1997 (1)

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]

1996 (1)

J. M. Links, J. L. Piince, and S. N. Gupta, 'A vector Wiener filter for dual-radionuclei imaging,' IEEE Trans. Med. Imaging 15, 700-709 (1996).
[Crossref] [PubMed]

1995 (3)

R. Yang, L. Lin, M. Gabbouj, J. Astola, and Y. Neuvo, 'Optimal weighted median filters under structural constraints,' IEEE Trans. Signal Process. 43, 591-604 (1995).
[Crossref]

H. T. M. Van der Voort and K. C. Strasters, 'Restoration of confocal images for quantitative image analysis,' J. Microsc. 178, 165-181 (1995).
[Crossref]

P. E. Hanninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, 'Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,' Appl. Phys. Lett. 66, 1698-1700 (1995).
[Crossref]

1994 (1)

T. Song, M. Gabbouj, and Y. Neuvo, 'Center weighted median filters: some properties and applications in image processing,' Signal Process. 35, 213-229 (1994).
[Crossref]

1992 (2)

T. Hebert and R. Leahy, 'Statistic based MAP image reconstruction from Poisson data using Gibbs priors,' IEEE Trans. Signal Process. 40, 2290-2303 (1992).
[Crossref]

F. Russo and G. Ramponi, 'Fuzzy operator for sharpening of noisy images,' IEE Electron. Lett. 28, 1715-1717 (1992).
[Crossref]

1990 (1)

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

1985 (3)

Y. Vardi, L. A. Shepp, and L. Kaufmann, 'A statistical model for positron emission tomography,' J. Am. Stat. Assoc. 80, 8-37 (1985).
[Crossref]

R. W. W. V. Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. K. H. Stelzer, and R. Stricker, 'Optical fluorescence microscopy in 3 dimensions: microtomoscopy,' J. Microsc. 138, 29-34 (1985).
[Crossref]

D. L. Snyder and M. I. Miller, 'The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,' IEEE Trans. Nucl. Sci. 32, 3864-3872 (1985).
[Crossref]

1984 (1)

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

1974 (1)

J. Besag, 'Spatial interaction and the statistical analysis of lattice systems,' J. R. Stat. Soc. Ser. B. Methodol. 36, 192-236 (1974).

Acheroy, M.

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, 'Versatile wavelet domain noise filtration technique for medical imaging,' IEEE Trans. Med. Imaging 22, 323-331 (2003).
[Crossref] [PubMed]

Alenius, S.

S. Alenius and U. Ruotsalainen, 'Generalization of median root prior reconstruction,' IEEE Trans. Med. Imaging 21, 1413-1420 (2002).
[Crossref]

S. Alenius and U. Ruotsalainen, 'Using local median as the location of prior distribution in iterative emission tomography reconstruction,' IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
[Crossref]

Andreasen, N. C.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Arndt, S.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Astola, J.

R. Yang, L. Lin, M. Gabbouj, J. Astola, and Y. Neuvo, 'Optimal weighted median filters under structural constraints,' IEEE Trans. Signal Process. 43, 591-604 (1995).
[Crossref]

Bao, P.

P. Bao and L. Zhang, 'Noise reduction for magnetic resonance images via adaptive multiscale products thresholding,' IEEE Trans. Med. Imaging 22, 1089-1099 (2003).
[Crossref] [PubMed]

Besag, J.

J. Besag, 'Spatial interaction and the statistical analysis of lattice systems,' J. R. Stat. Soc. Ser. B. Methodol. 36, 192-236 (1974).

Bunce, S.

M. Izzetoglu, A. Devaraj, S. Bunce, and B. Onaral, 'Motion artifact cancellation in NIR spectroscopy using Wiener filtering,' IEEE Trans. Biomed. Eng. 52, 934-938 (2005).
[Crossref] [PubMed]

Chang, S. A. G.

S. A. G. Chang, B. Yu, and M. Vetterli, 'Adaptive wavelet thresholding for image denoising and compression,' IEEE Trans. Image Process. 9, 1532-l546 (2000).
[Crossref]

Chen, T.

T. Chen and H. Wu, 'Adaptive impulse detection using center-weighted median filters,' IEEE Signal Process. Lett. 8, 1-3 (2001).
[Crossref]

Christian, B.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Cizadlo, T.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Cremer, C.

P. E. Hanninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, 'Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,' Appl. Phys. Lett. 66, 1698-1700 (1995).
[Crossref]

Davoust, J.

R. W. W. V. Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. K. H. Stelzer, and R. Stricker, 'Optical fluorescence microscopy in 3 dimensions: microtomoscopy,' J. Microsc. 138, 29-34 (1985).
[Crossref]

Dawant, B.

H. G. Senel, R. A. Peters II, and B. Dawant, 'Topological median filters,' IEEE Trans. Image Process. 11, 89-104 (2002).
[Crossref]

Devaraj, A.

M. Izzetoglu, A. Devaraj, S. Bunce, and B. Onaral, 'Motion artifact cancellation in NIR spectroscopy using Wiener filtering,' IEEE Trans. Biomed. Eng. 52, 934-938 (2005).
[Crossref] [PubMed]

Fan, G.

G. Fan and X. G. Xia, 'Image denoising using local contextual hidden Marcov model in the wavelet domain,' IEEE Signal Process. Lett. 8, 125-128 (2001).
[Crossref]

Flaum, M.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Gabbouj, M.

R. Yang, L. Lin, M. Gabbouj, J. Astola, and Y. Neuvo, 'Optimal weighted median filters under structural constraints,' IEEE Trans. Signal Process. 43, 591-604 (1995).
[Crossref]

T. Song, M. Gabbouj, and Y. Neuvo, 'Center weighted median filters: some properties and applications in image processing,' Signal Process. 35, 213-229 (1994).
[Crossref]

Geman, D.

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

Geman, S.

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

Gold, S.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Pearson Education, 2003).

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]

Gupta, S. N.

J. M. Links, J. L. Piince, and S. N. Gupta, 'A vector Wiener filter for dual-radionuclei imaging,' IEEE Trans. Med. Imaging 15, 700-709 (1996).
[Crossref] [PubMed]

Hanninen, P. E.

P. E. Hanninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, 'Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,' Appl. Phys. Lett. 66, 1698-1700 (1995).
[Crossref]

Haykin, S.

S. Haykin, Adaptive Filter Theory, 4th ed. (Prentice-Hall, 2002).

Hebb, D. O.

D. O. Hebb, The Organization of Behavior: A Neuropsychological Theory (Wiley, 1949).

Hebert, T.

T. Hebert and R. Leahy, 'Statistic based MAP image reconstruction from Poisson data using Gibbs priors,' IEEE Trans. Signal Process. 40, 2290-2303 (1992).
[Crossref]

Hell, S. W.

P. E. Hanninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, 'Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,' Appl. Phys. Lett. 66, 1698-1700 (1995).
[Crossref]

Izzetoglu, M.

M. Izzetoglu, A. Devaraj, S. Bunce, and B. Onaral, 'Motion artifact cancellation in NIR spectroscopy using Wiener filtering,' IEEE Trans. Biomed. Eng. 52, 934-938 (2005).
[Crossref] [PubMed]

Johnson, D. L.

S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
[Crossref] [PubMed]

Kaplan, R.

R. W. W. V. Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. K. H. Stelzer, and R. Stricker, 'Optical fluorescence microscopy in 3 dimensions: microtomoscopy,' J. Microsc. 138, 29-34 (1985).
[Crossref]

Kaufmann, L.

Y. Vardi, L. A. Shepp, and L. Kaufmann, 'A statistical model for positron emission tomography,' J. Am. Stat. Assoc. 80, 8-37 (1985).
[Crossref]

Kazubek, M.

M. Kazubek, 'Wavelet domain image denoising by thresholding and Wiener filtering,' IEEE Signal Process. Lett. 10, 324-326 (2003).
[Crossref]

Kozinsev, I.

M. K. Mihak, I. Kozinsev, K. Ramchandran, and P. Moulin, 'Low-complexity image denoising based on statistical modeling of wavelet coefficients,' IEEE Signal Process. Lett. 6, 300-303 (1999).
[Crossref]

Lai, C L.

S. C. Pei, C L. Lai, and F. Y. Shih, 'Recursive order-statistic soft morphological filters,' IEE Proc. Vision Image Signal Process. 145, 333-342 (1998).
[Crossref]

Leahy, R.

T. Hebert 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]

Lemahieu, I.

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, 'Versatile wavelet domain noise filtration technique for medical imaging,' IEEE Trans. Med. Imaging 22, 323-331 (2003).
[Crossref] [PubMed]

Lin, L.

R. Yang, L. Lin, M. Gabbouj, J. Astola, and Y. Neuvo, 'Optimal weighted median filters under structural constraints,' IEEE Trans. Signal Process. 43, 591-604 (1995).
[Crossref]

Links, J. M.

J. M. Links, J. L. Piince, and S. N. Gupta, 'A vector Wiener filter for dual-radionuclei imaging,' IEEE Trans. Med. Imaging 15, 700-709 (1996).
[Crossref] [PubMed]

Lucke, L.

S. Peng and L. Lucke, 'Fuzzy filtering for mixed noise removal during image processings,' in Proceedings of the Third IEEE International Conference on Fuzzy Systems (IEEE, 1994), pp. 89-93.
[Crossref]

Marshall, S.

S. Marshall, 'New direct, design method for weighted order statistic filters,' IEE Proc. Vision Image Signal Process. 151, 1-8 (2004).
[Crossref]

Marsman, H. J. B.

R. W. W. V. Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. K. H. Stelzer, and R. Stricker, 'Optical fluorescence microscopy in 3 dimensions: microtomoscopy,' J. Microsc. 138, 29-34 (1985).
[Crossref]

Mihak, M. K.

M. K. Mihak, I. Kozinsev, K. Ramchandran, and P. Moulin, 'Low-complexity image denoising based on statistical modeling of wavelet coefficients,' IEEE Signal Process. Lett. 6, 300-303 (1999).
[Crossref]

Miller, M. I.

D. L. Snyder and M. I. Miller, 'The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,' IEEE Trans. Nucl. Sci. 32, 3864-3872 (1985).
[Crossref]

Mondal, P. P.

P. P. Mondal and K. Rajan, 'Neural network based image reconstruction for positron emission tomography,' Appl. Opt. 44, 6245-6352 (2005).
[Crossref]

P. P. Mondal and K. Rajan, 'Fuzzy-rule-based image reconstruction for positron emission tomography,' J. Opt. Soc. Am. A 22, 1763-1771 (2005).
[Crossref]

Moulin, P.

M. K. Mihak, I. Kozinsev, K. Ramchandran, and P. Moulin, 'Low-complexity image denoising based on statistical modeling of wavelet coefficients,' IEEE Signal Process. Lett. 6, 300-303 (1999).
[Crossref]

Neuvo, Y.

R. Yang, L. Lin, M. Gabbouj, J. Astola, and Y. Neuvo, 'Optimal weighted median filters under structural constraints,' IEEE Trans. Signal Process. 43, 591-604 (1995).
[Crossref]

T. Song, M. Gabbouj, and Y. Neuvo, 'Center weighted median filters: some properties and applications in image processing,' Signal Process. 35, 213-229 (1994).
[Crossref]

Onaral, B.

M. Izzetoglu, A. Devaraj, S. Bunce, and B. Onaral, 'Motion artifact cancellation in NIR spectroscopy using Wiener filtering,' IEEE Trans. Biomed. Eng. 52, 934-938 (2005).
[Crossref] [PubMed]

Oppenheim, A.

A. Oppenheim and R. Schafer, Discrete Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

Pei, S. C.

S. C. Pei, C L. Lai, and F. Y. Shih, 'Recursive order-statistic soft morphological filters,' IEE Proc. Vision Image Signal Process. 145, 333-342 (1998).
[Crossref]

Peng, S.

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F. Russo and G. Ramponi, 'Fuzzy operator for sharpening of noisy images,' IEE Electron. Lett. 28, 1715-1717 (1992).
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F. Russo, 'A new class of fuzzy operators for image processing: design and implementation,' in Proceedings of the Second IEEE International Conference on Fuzzy Systems (IEEE, 1993), pp. 815-820.
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H. G. Senel, R. A. Peters II, and B. Dawant, 'Topological median filters,' IEEE Trans. Image Process. 11, 89-104 (2002).
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R. W. W. V. Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. K. H. Stelzer, and R. Stricker, 'Optical fluorescence microscopy in 3 dimensions: microtomoscopy,' J. Microsc. 138, 29-34 (1985).
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Appl. Opt. (1)

P. P. Mondal and K. Rajan, 'Neural network based image reconstruction for positron emission tomography,' Appl. Opt. 44, 6245-6352 (2005).
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Appl. Phys. Lett. (1)

P. E. Hanninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, 'Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,' Appl. Phys. Lett. 66, 1698-1700 (1995).
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S. Gold, B. Christian, S. Arndt, G. Zeien, T. Cizadlo, D. L. Johnson, M. Flaum, and N. C. Andreasen, 'Functional MRI statistical software packages: a comparative analysis,' Hum. Brain Mapp. 6, 73-84 (1998).
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F. Russo and G. Ramponi, 'Fuzzy operator for sharpening of noisy images,' IEE Electron. Lett. 28, 1715-1717 (1992).
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S. C. Pei, C L. Lai, and F. Y. Shih, 'Recursive order-statistic soft morphological filters,' IEE Proc. Vision Image Signal Process. 145, 333-342 (1998).
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[Crossref]

G. Fan and X. G. Xia, 'Image denoising using local contextual hidden Marcov model in the wavelet domain,' IEEE Signal Process. Lett. 8, 125-128 (2001).
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T. Chen and H. Wu, 'Adaptive impulse detection using center-weighted median filters,' IEEE Signal Process. Lett. 8, 1-3 (2001).
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M. Izzetoglu, A. Devaraj, S. Bunce, and B. Onaral, 'Motion artifact cancellation in NIR spectroscopy using Wiener filtering,' IEEE Trans. Biomed. Eng. 52, 934-938 (2005).
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Z. Zhou, R. M. Leahy, and J. Qi, 'Approximate maximum likelihood hyperparameter estimation for Gibbs prior,' IEEE Trans. Image Process. 6, 844-861 (1997).
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[Crossref]

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S. Alenius and U. Ruotsalainen, 'Generalization of median root prior reconstruction,' IEEE Trans. Med. Imaging 21, 1413-1420 (2002).
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J. M. Links, J. L. Piince, and S. N. Gupta, 'A vector Wiener filter for dual-radionuclei imaging,' IEEE Trans. Med. Imaging 15, 700-709 (1996).
[Crossref] [PubMed]

A. M. Wink and J. B. T. M. Roerdink, 'Denoising functional MR images: a comparison of wavelet denoising and gaussian smoothing,' IEEE Trans. Med. Imaging 23, 374-387 (2004).
[Crossref] [PubMed]

P. Bao and L. Zhang, 'Noise reduction for magnetic resonance images via adaptive multiscale products thresholding,' IEEE Trans. Med. Imaging 22, 1089-1099 (2003).
[Crossref] [PubMed]

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, 'Versatile wavelet domain noise filtration technique for medical imaging,' IEEE Trans. Med. Imaging 22, 323-331 (2003).
[Crossref] [PubMed]

IEEE Trans. Nucl. Sci. (2)

D. L. Snyder and M. I. Miller, 'The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,' IEEE Trans. Nucl. Sci. 32, 3864-3872 (1985).
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S. Alenius and U. Ruotsalainen, 'Using local median as the location of prior distribution in iterative emission tomography reconstruction,' IEEE Trans. Nucl. Sci. 45, 3097-3104 (1998).
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Y. Vardi, L. A. Shepp, and L. Kaufmann, 'A statistical model for positron emission tomography,' J. Am. Stat. Assoc. 80, 8-37 (1985).
[Crossref]

J. Microsc. (2)

R. W. W. V. Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. K. H. Stelzer, and R. Stricker, 'Optical fluorescence microscopy in 3 dimensions: microtomoscopy,' J. Microsc. 138, 29-34 (1985).
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R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Pearson Education, 2003).

J. Portilla, V. Strela, M. J. Wainnwright, and E. P. Simocelli, 'Adaptive Wiener denoising using a Gaussian scale mixture model in the wavelet domain,' in Proceedings of the Eighth International Conference on Image Processing (IEEE, 2001), pp. 37-40.

A. Oppenheim and R. Schafer, Discrete Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

S. Haykin, Adaptive Filter Theory, 4th ed. (Prentice-Hall, 2002).

I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, 1990).

F. Russo, 'A new class of fuzzy operators for image processing: design and implementation,' in Proceedings of the Second IEEE International Conference on Fuzzy Systems (IEEE, 1993), pp. 815-820.
[Crossref]

S. Peng and L. Lucke, 'Fuzzy filtering for mixed noise removal during image processings,' in Proceedings of the Third IEEE International Conference on Fuzzy Systems (IEEE, 1994), pp. 89-93.
[Crossref]

D. O. Hebb, The Organization of Behavior: A Neuropsychological Theory (Wiley, 1949).

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Figures (8)

Fig. 1
Fig. 1

Proposed spatial filtering approach.

Fig. 2
Fig. 2

The first row shows the noisy cameraman image, corrupted by Gaussian noise: (a) G ( 0 , 0.01 ) , (b) G ( 0 , 0.05 ) , (c) G ( 0 , 0.1 ) , (d) G ( 0 , 0.2 ) . The corresponding restored images using the average, Gaussian, Wiener, and the proposed filters are shown in the second, third, fourth, and fifth rows, respectively.

Fig. 3
Fig. 3

Zoomed section of images of Figs. 2p, 2t.

Fig. 4
Fig. 4

Noise variance versus NMSE plot for the noisy images filtered by average, Gaussian, Wiener, and the proposed filter. The images are corrupted by (a) Gaussian noise (b) salt-and-pepper noise, (c) speckle noise.

Fig. 5
Fig. 5

(a) Original brain image, (b) image corrupted by Gaussian noise G(0,0.1), (c)–(f) filtered images using the average, Gaussian, Wiener, and proposed filters, respectively.

Fig. 6
Fig. 6

Line plots through lines (red) passing through different regions of the filtered image shown in Fig. 3.

Fig. 7
Fig. 7

NMSE versus iteration plot for reconstructed images with learning parameter μ = 0.5 and 1.0.

Fig. 8
Fig. 8

(a) Original MRI test image. (b)–(d) The MRI reconstructed images after 50 iterations using ML, MAP, and the proposed algorithms. (e) Image corrupted by added Gaussian noise G ( 0.0.03 ) . (f)–(h) The reconstructed images after 50 iterations using ML, MAP, and the proposed algorithms.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

P ( λ ) > 0 λ Λ ( positivity ) , P ( λ i λ S { i } ) = P ( λ i λ N i ) ( Markovianity ) ,
P ( λ ) = 1 Z exp [ 1 β i j N i w i j V ( λ i , λ j ) ] ,
P ( λ ) = 1 Z exp [ 1 β i j N i w i j ( λ i λ j ) 2 ] ,
l ( λ ) = log 1 Z 1 β i j N i w i j ( λ i λ j ) 2 .
λ i log P ( λ ) = 2 β j N i w i j ( λ i λ j ) ,
2 λ i λ k ( log P ( λ ) ) = 2 β j N i w i j .
λ i log P ( λ ) λ = λ ̂ = 0 ,
j N i w i j ( λ i λ j ) λ = λ ̂ = 0 ,
λ i ̂ j N i w i j j N i w i j λ j λ = λ ̂ = 0 .
λ ̂ i = j N i w i j λ ̂ j .
λ ̂ i k + 1 = j N i w i j k λ ̂ j k .
w i j k + 1 = w i j k + μ ( λ i k λ j k ) j N i ,
NMSE = j = 1 M ( λ ̃ j λ j ) 2 j = 1 M λ j 2 ,

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