Abstract

Image denoising is important for high-quality imaging in adaptive optics. Richardson-Lucy deconvolution with total variation(TV) regularization is commonly used in image denoising. The selection of TV regularization parameter is an essential issue, yet no systematic approach has been proposed. A construction model for TV regularization parameter is proposed in this paper. It consists of four fundamental elements, the properties of which are analyzed in details. The proposed model bears generality, making it apply to different image recovery scenarios. It can achieve effective spatially adaptive image recovery, which is reflected in both noise suppression and edge preservation. Simulations are provided as validation of recovery and demonstration of convergence speed and relative mean-square error.

© 2014 Optical Society of America

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2014

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

2013

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013).
[CrossRef]

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013).
[CrossRef]

A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013).
[CrossRef]

2011

2010

M. Marim, M. Atlan, E. Angelini, J. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35(6), 871–873 (2010).
[CrossRef] [PubMed]

M. Freiberger, C. Clason, H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. 49(19), 3741–3747 (2010).
[CrossRef] [PubMed]

E. Y. Sidky, M. A. Anastasio, X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. 18(10), 10,404–10,422 (2010).
[CrossRef]

G. Krishnamurthi, C. Y. Wang, G. Steyer, D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. 18(21), 22,324–22,338 (2010).
[CrossRef]

J. Dahl, P. C. Hansen, S. H. Jensen, T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. 53(1), 67–92 (2010).
[CrossRef]

2009

D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009).
[CrossRef]

2008

E. Y. Sidky, X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53(17), 4777–4807 (2008).
[CrossRef] [PubMed]

2006

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006).
[CrossRef] [PubMed]

E. Candes, J. Romberg, T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[CrossRef]

2004

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20(1–2), 89–97 (2004).
[CrossRef]

1997

1995

1992

L. I. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992).
[CrossRef]

1988

1974

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745 (1974).
[CrossRef]

1972

1968

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on, 16(3), 437–466 (1968).
[CrossRef]

Anastasio, M. A.

E. Y. Sidky, M. A. Anastasio, X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. 18(10), 10,404–10,422 (2010).
[CrossRef]

Andrews, M.

Angelini, E.

Atlan, M.

Ayers, G. R.

Batenburg, K. J.

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013).
[CrossRef]

A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013).
[CrossRef]

Bequé, D.

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

Biggs, D. S. C.

Blanc-Feraud, L.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006).
[CrossRef] [PubMed]

Bostan, E.

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013).
[CrossRef]

Bourquard, A.

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013).
[CrossRef]

Brady, D.

D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009).
[CrossRef]

Brinicombe, A. M.

Cai, H.

H. Tian, H. Cai, J. Lai, X. Xu, “Effective image noise removal based on difference eigenvalue,” in Proceedings of ICIP 2011 International Conference on Image Processing (Brussels, 2011).
[CrossRef]

Candes, E.

E. Candes, J. Romberg, T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[CrossRef]

Cetin, M.

M. Cetin, W. Karl, A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, 2002).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20(1–2), 89–97 (2004).
[CrossRef]

Choi, K.

D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009).
[CrossRef]

Clason, C.

Conan, J.-M.

L. M. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence-degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Cozzini, C.

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

Dahl, J.

J. Dahl, P. C. Hansen, S. H. Jensen, T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. 53(1), 67–92 (2010).
[CrossRef]

Dainty, J. C.

Depeursinge, C.

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013).
[CrossRef]

Dey, N.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006).
[CrossRef] [PubMed]

Edic, P. M.

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

Fatemi, E.

L. I. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992).
[CrossRef]

Fish, D. A.

Freiberger, M.

Fusco, T.

L. M. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence-degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Hansen, P. C.

J. Dahl, P. C. Hansen, S. H. Jensen, T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. 53(1), 67–92 (2010).
[CrossRef]

Hope, D. A.

Horisaki, R.

D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009).
[CrossRef]

Jefferies, S. M.

Jensen, S. H.

J. Dahl, P. C. Hansen, S. H. Jensen, T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. 53(1), 67–92 (2010).
[CrossRef]

Jensen, T. L.

J. Dahl, P. C. Hansen, S. H. Jensen, T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. 53(1), 67–92 (2010).
[CrossRef]

Kam, Z.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006).
[CrossRef] [PubMed]

Karl, W.

M. Cetin, W. Karl, A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, 2002).
[CrossRef]

King, A.

A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013).
[CrossRef]

Kostenko, A.

A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013).
[CrossRef]

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013).
[CrossRef]

Krishnamurthi, G.

G. Krishnamurthi, C. Y. Wang, G. Steyer, D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. 18(21), 22,324–22,338 (2010).
[CrossRef]

Kudielka, G. P.

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

Lai, J.

H. Tian, H. Cai, J. Lai, X. Xu, “Effective image noise removal based on difference eigenvalue,” in Proceedings of ICIP 2011 International Conference on Image Processing (Brussels, 2011).
[CrossRef]

Lim, S.

D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745 (1974).
[CrossRef]

Mahdi, K.

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

Marim, M.

Marks, D.

D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009).
[CrossRef]

Meza, P.

Michau, V.

L. M. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence-degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Mugnier, L. M.

L. M. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence-degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Offerman, S. E.

A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013).
[CrossRef]

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013).
[CrossRef]

Olivo-Marin, J.

Olivo-Marin, J. C.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006).
[CrossRef] [PubMed]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on, 16(3), 437–466 (1968).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992).
[CrossRef]

Pan, X.

E. Y. Sidky, M. A. Anastasio, X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. 18(10), 10,404–10,422 (2010).
[CrossRef]

E. Y. Sidky, X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53(17), 4777–4807 (2008).
[CrossRef] [PubMed]

Pavillon, N.

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013).
[CrossRef]

Pike, E. R.

Richardson, W. H.

Romberg, J.

E. Candes, J. Romberg, T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[CrossRef]

Roux, P.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006).
[CrossRef] [PubMed]

Rudin, L. I.

L. I. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on, 16(3), 437–466 (1968).
[CrossRef]

Scharfetter, H.

Sidky, E. Y.

E. Y. Sidky, M. A. Anastasio, X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. 18(10), 10,404–10,422 (2010).
[CrossRef]

E. Y. Sidky, X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53(17), 4777–4807 (2008).
[CrossRef] [PubMed]

Sperl, J. I.

J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014).
[CrossRef]

Steyer, G.

G. Krishnamurthi, C. Y. Wang, G. Steyer, D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. 18(21), 22,324–22,338 (2010).
[CrossRef]

Stockham, T. G.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on, 16(3), 437–466 (1968).
[CrossRef]

Suhonen, H.

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013).
[CrossRef]

Tao, T.

E. Candes, J. Romberg, T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[CrossRef]

Tian, H.

H. Tian, H. Cai, J. Lai, X. Xu, “Effective image noise removal based on difference eigenvalue,” in Proceedings of ICIP 2011 International Conference on Image Processing (Brussels, 2011).
[CrossRef]

Torres, S.

Unser, M.

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013).
[CrossRef]

van Vliet, L. J.

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013).
[CrossRef]

A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013).
[CrossRef]

Vera, E.

Walker, J. G.

Wang, C. Y.

G. Krishnamurthi, C. Y. Wang, G. Steyer, D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. 18(21), 22,324–22,338 (2010).
[CrossRef]

Willsky, A.

M. Cetin, W. Karl, A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, 2002).
[CrossRef]

Wilson, D. L.

G. Krishnamurthi, C. Y. Wang, G. Steyer, D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. 18(21), 22,324–22,338 (2010).
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Figures (5)

Fig. 1:
Fig. 1:

Graph of different functions f1f4 as constructor function.

Fig. 2:
Fig. 2:

Recovery of real moon image. (a) Blurred moon image; (b)∼(f) Image recovered by f0f4, respectively.

Fig. 3:
Fig. 3:

Convergence results of constructor functions f0f4 in Fig. 2.

Fig. 4:
Fig. 4:

Recovery of simulated noisy image. (a) Original image; (b) Image blurred by simulated noise; (c)∼(g) Image recovered by f0f4, respectively.

Fig. 5:
Fig. 5:

Convergence results and relative mean-square error of f0f4 for recovery in Fig. 4. (a) Convergence results and (b) Relative mean-square error.

Equations (11)

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

i ( x ) = ( h * o ) ( x ) + n ( x )
P ( i | o , h ) = x [ ( h * o ) ( x ) ] i ( x ) exp [ ( h * o ) ( x ) ] i ( x ) !
J ( o ) = x { i ( x ) log [ ( h * o ) ( x ) ] + ( h * o ) ( x ) }
𝒥 ( o ) = J ( o ) + λ x o ( x ) = x { i ( x ) log [ ( h * o ) ( x ) ] + ( h * o ) ( x ) } + λ x o ( x )
o ( n + 1 ) = i o ( n ) * h * h * o ( n ) 1 λ div ( o ( n ) o ( n ) )
h ( n + 1 ) = h ( n ) ( i h ( n ) * o * o * )
h ( n + 1 ) = h ( n + 1 ) x h ( n + 1 ) ( x )
i h * o
o ( n + 1 ) o ( n ) 1 λ ( x ) div ( o ( n ) o ( n ) )
div ( o ( n ) o ( n ) ) = 2 o x 2 ( o y ) 2 + 2 o y 2 ( o x ) 2 2 o x o y 2 o x y [ ( o x ) 2 + ( o y ) 2 ] 3 / 2
λ ( x ) = λ 0 f ( β EI ( x ) )

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