Abstract

In this paper we propose a robust method to suppress the noise components in digital holography (DH), called SPADEDH (SPArsity DEnoising of Digital Holograms), that does not consider any prior knowledge or estimation about the statistics of the noise. In the full digital holographic process we must mainly deal with two kinds of noise. The first one is an additive uncorrelated noise that corrupts the observed irradiance, the other one is the multiplicative phase noise called speckle noise. We consider both lensless and microscope configurations and we prove that the proposed algorithm works efficiently in all considered cases suppressing the aforementioned noise components. In addition, for digital holograms recorded in lensless configuration, we show the improvement in a display test by using a Spatial Light Modulator (SLM).

© 2012 OSA

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2012 (1)

2011 (2)

2010 (7)

K. Choi, R. Horisaki, J. Hahn, S. Lim, D.L. Marks, T.J. Schulz, and D.J. Brady, “Compressive holography of diffuse objects,” Appl. Opt.49, H1–H10 (2010).
[CrossRef] [PubMed]

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol.6(10), 506–509 (2010).
[CrossRef]

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process.4(2), 375–391 (2010).
[CrossRef]

M. Elad, M.A.T. Figueiredo, and M. Yi, “On the role of sparse and redundant representations in image processing,” Proc. IEEE98(6), 972–982 (2010).
[CrossRef]

M. Paturzo, P. Memmolo, A. Finizio, R. Nsnen, T.J. Naughton, and P. Ferraro, “Synthesis and display of dynamic holographic 3D scenes with real-world objects,” Opt. Express18, 8806–8815 (2010).
[CrossRef] [PubMed]

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

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

2008 (2)

C. Do and B. Javidi, “Three-dimensional computational holographic imaging and recognition using independent component analysis,” Proc. R. Soc. London464, 409–422 (2008).
[CrossRef]

J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag.25(2), 92–101 (2008).
[CrossRef]

2007 (1)

2006 (3)

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett.31, 1405–1407 (2006).
[CrossRef] [PubMed]

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 1289–1306 (2006).
[CrossRef]

2005 (5)

S. Mirza, R. Kumar, and C. Shakher, “Study of various preprocessing schemes and wavelet filters for speckle noise reduction in digital speckle pattern interferometric fringes,” Opt. Eng.44(4), 045603, (2005).
[CrossRef]

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik116, 44–48 (2005).
[CrossRef]

B. Javidi, P. Ferraro, S. Hong, and D. Alfieri, “Three-dimensional image fusion using multi-wavelengths digital holography,” Opt. Lett.30(2), 144–146 (2005).
[CrossRef] [PubMed]

T. Weissman, E. Ordentlich, G. Seroussi, S. Verdú, and M. Weinberger, “Universal discrete denoising: known channel,” IEEE Trans. Inf. Theory51(1), 5–28 (2005).
[CrossRef]

E. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory51(12), 4203–4215 (2005).
[CrossRef]

2004 (3)

2003 (1)

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

2000 (1)

1995 (1)

D. Donoho, “De-Noising by soft thresholding,” IEEE Trans. Inf. Theory38(2), 613–627 (1995).
[CrossRef]

1994 (1)

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D60, 259–268 (1992).
[CrossRef]

Alferi, D.

Alfieri, D.

Allaria, E.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

Angelini, E.

Atlan, M.

Bajwa, W.

J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag.25(2), 92–101 (2008).
[CrossRef]

Bertaux, N.

Brady, D.J.

Brugioni, S.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

Caire, G.

A.M. Tulino, G. Caire, S. Shamai, and S. Verdú, “Support recovery with sparsely sampled free random matrices,” The IEEE International Symposium on Information Theory (ISIT 2011), Saint-Petersburg, Russia, July, 31–August, 5, (2011).

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 1289–1306 (2006).
[CrossRef]

E. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory51(12), 4203–4215 (2005).
[CrossRef]

Castro, A.

Choi, K.

Coppola, G.

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phys. Lett.84, 2709–2711 (2004).
[CrossRef]

De Nicola, S.

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett.31, 1405–1407 (2006).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phys. Lett.84, 2709–2711 (2004).
[CrossRef]

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

De Petrocellis, L.

Distante, C.

Do, C.

C. Do and B. Javidi, “Three-dimensional computational holographic imaging and recognition using independent component analysis,” Proc. R. Soc. London464, 409–422 (2008).
[CrossRef]

Donoho, D.

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

D. Donoho, “De-Noising by soft thresholding,” IEEE Trans. Inf. Theory38(2), 613–627 (1995).
[CrossRef]

D. Donoho, Y. Tsaig, I. Drori, and J-L Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Stanford Technical Report 1–39 (2006).

M. Lustig, J. Santos, J. Lee, D. Donoho, and J. Pauly, “Application of compressed sensing for rapid MR imaging,” In Proc. Work. Struc. Parc. Rep. Adap. Signaux (SPARS), Rennes, France, Nov. (2005).

Drori, I.

D. Donoho, Y. Tsaig, I. Drori, and J-L Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Stanford Technical Report 1–39 (2006).

Elad, M.

M. Elad, M.A.T. Figueiredo, and M. Yi, “On the role of sparse and redundant representations in image processing,” Proc. IEEE98(6), 972–982 (2010).
[CrossRef]

Eldar, Y. C.

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process.4(2), 375–391 (2010).
[CrossRef]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D60, 259–268 (1992).
[CrossRef]

Ferraro, P.

P. Memmolo, C. Distante, M. Paturzo, A. Finizio, P. Ferraro, and B. Javidi, “Automatic focusing in digital holography and its application to stretched holograms,” Opt. Lett.36, 1945–1947 (2011).
[CrossRef] [PubMed]

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

M. Paturzo, P. Memmolo, A. Finizio, R. Nsnen, T.J. Naughton, and P. Ferraro, “Synthesis and display of dynamic holographic 3D scenes with real-world objects,” Opt. Express18, 8806–8815 (2010).
[CrossRef] [PubMed]

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett.31, 1405–1407 (2006).
[CrossRef] [PubMed]

B. Javidi, P. Ferraro, S. Hong, and D. Alfieri, “Three-dimensional image fusion using multi-wavelengths digital holography,” Opt. Lett.30(2), 144–146 (2005).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phys. Lett.84, 2709–2711 (2004).
[CrossRef]

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

Fessler, J. A.

Figueiredo, M.A.T.

M. Elad, M.A.T. Figueiredo, and M. Yi, “On the role of sparse and redundant representations in image processing,” Proc. IEEE98(6), 972–982 (2010).
[CrossRef]

Finizio, A.

Frauel, Y.

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik116, 44–48 (2005).
[CrossRef]

Geltrude, A.G.

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

Grilli, S.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

Hahn, J.

Harmany, Z.

R. Marcia, Z. Harmany, and R. Willett, “Compressive coded aperture imaging,” In Proc. IS&T/SPIE Symp. Elec. Imag.: Comp. Imag., San Jose, CA, (2009).

Haupt, J.

J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag.25(2), 92–101 (2008).
[CrossRef]

Hennelly, B. M.

Hong, S.

Horisaki, R.

Javidi, B.

Jptner, W.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2004).
[CrossRef]

Kumar, R.

S. Mirza, R. Kumar, and C. Shakher, “Study of various preprocessing schemes and wavelet filters for speckle noise reduction in digital speckle pattern interferometric fringes,” Opt. Eng.44(4), 045603, (2005).
[CrossRef]

Lee, J.

M. Lustig, J. Santos, J. Lee, D. Donoho, and J. Pauly, “Application of compressed sensing for rapid MR imaging,” In Proc. Work. Struc. Parc. Rep. Adap. Signaux (SPARS), Rennes, France, Nov. (2005).

Lim, S.

Locatelli, M.

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

Lustig, M.

M. Lustig, J. Santos, J. Lee, D. Donoho, and J. Pauly, “Application of compressed sensing for rapid MR imaging,” In Proc. Work. Struc. Parc. Rep. Adap. Signaux (SPARS), Rennes, France, Nov. (2005).

Majumdar, A.

A. Majumdar and R.K. Ward, “Sparsity promoting speckle denoising,” International Conference on Image Processing (2009).

Marcia, R.

R. Marcia, Z. Harmany, and R. Willett, “Compressive coded aperture imaging,” In Proc. IS&T/SPIE Symp. Elec. Imag.: Comp. Imag., San Jose, CA, (2009).

Marim, M. M.

Marks, D.L.

Maycock, J.

McDonald, J. B.

Memmolo, P.

Meucci, R.

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
[CrossRef]

Miccio, L.

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

Mirza, S.

S. Mirza, R. Kumar, and C. Shakher, “Study of various preprocessing schemes and wavelet filters for speckle noise reduction in digital speckle pattern interferometric fringes,” Opt. Eng.44(4), 045603, (2005).
[CrossRef]

Mishali, M.

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process.4(2), 375–391 (2010).
[CrossRef]

Naughton, T. J.

Naughton, T.J.

Nowak, R.

J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag.25(2), 92–101 (2008).
[CrossRef]

Nsnen, R.

Olivo-Marin, J-C

Ordentlich, E.

T. Weissman, E. Ordentlich, G. Seroussi, S. Verdú, and M. Weinberger, “Universal discrete denoising: known channel,” IEEE Trans. Inf. Theory51(1), 5–28 (2005).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D60, 259–268 (1992).
[CrossRef]

Paturzo, M.

Pauly, J.

M. Lustig, J. Santos, J. Lee, D. Donoho, and J. Pauly, “Application of compressed sensing for rapid MR imaging,” In Proc. Work. Struc. Parc. Rep. Adap. Signaux (SPARS), Rennes, France, Nov. (2005).

Pelagotti, A.

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

Pierattini, G.

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett.31, 1405–1407 (2006).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phys. Lett.84, 2709–2711 (2004).
[CrossRef]

Poggi, P.

A. Pelagotti, M. Locatelli, A.G. Geltrude, P. Poggi, R. Meucci, M. Paturzo, L. Miccio, and P. Ferraro, “Reliability of 3D imaging by digital holography at long IR wavelength,” J. Disp. Technol.6, 465–471 (2010).
[CrossRef]

Prieto, D. V.

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik116, 44–48 (2005).
[CrossRef]

Rabbat, M.

J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag.25(2), 92–101 (2008).
[CrossRef]

Ramirez, J. A. H.

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik116, 44–48 (2005).
[CrossRef]

Rfrgier, P.

Rivenson, Y.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol.6(10), 506–509 (2010).
[CrossRef]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 1289–1306 (2006).
[CrossRef]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D60, 259–268 (1992).
[CrossRef]

Santos, J.

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A.M. Tulino, G. Caire, S. Shamai, and S. Verdú, “Support recovery with sparsely sampled free random matrices,” The IEEE International Symposium on Information Theory (ISIT 2011), Saint-Petersburg, Russia, July, 31–August, 5, (2011).

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D. Donoho, Y. Tsaig, I. Drori, and J-L Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Stanford Technical Report 1–39 (2006).

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A.M. Tulino, G. Caire, S. Shamai, and S. Verdú, “Support recovery with sparsely sampled free random matrices,” The IEEE International Symposium on Information Theory (ISIT 2011), Saint-Petersburg, Russia, July, 31–August, 5, (2011).

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A. Majumdar and R.K. Ward, “Sparsity promoting speckle denoising,” International Conference on Image Processing (2009).

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Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phys. Lett.84, 2709–2711 (2004).
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J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag.25(2), 92–101 (2008).
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Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol.6(10), 506–509 (2010).
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E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, and R. Meucci, “Digital holography at 10.6 μm,” Opt. Commun.215, 257–262 (2003).
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Opt. Eng. (1)

S. Mirza, R. Kumar, and C. Shakher, “Study of various preprocessing schemes and wavelet filters for speckle noise reduction in digital speckle pattern interferometric fringes,” Opt. Eng.44(4), 045603, (2005).
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Opt. Express (1)

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Optik (1)

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik116, 44–48 (2005).
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M. Elad, M.A.T. Figueiredo, and M. Yi, “On the role of sparse and redundant representations in image processing,” Proc. IEEE98(6), 972–982 (2010).
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[CrossRef]

M. Lustig, J. Santos, J. Lee, D. Donoho, and J. Pauly, “Application of compressed sensing for rapid MR imaging,” In Proc. Work. Struc. Parc. Rep. Adap. Signaux (SPARS), Rennes, France, Nov. (2005).

A.M. Tulino, G. Caire, S. Shamai, and S. Verdú, “Support recovery with sparsely sampled free random matrices,” The IEEE International Symposium on Information Theory (ISIT 2011), Saint-Petersburg, Russia, July, 31–August, 5, (2011).

Y. Wu, “Shannon theory for compressed sensing,” Ph. D. dissertation, Princeton University, Sep. (2011).

D. Donoho, Y. Tsaig, I. Drori, and J-L Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Stanford Technical Report 1–39 (2006).

R. Marcia, Z. Harmany, and R. Willett, “Compressive coded aperture imaging,” In Proc. IS&T/SPIE Symp. Elec. Imag.: Comp. Imag., San Jose, CA, (2009).

A. Majumdar and R.K. Ward, “Sparsity promoting speckle denoising,” International Conference on Image Processing (2009).

S. Rangan, A.K. Fletcher, and V.K. Goyal, “Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing,” http://www.citebase.org/abstract?id=oai:arXiv.org:0906.3234 , (2009).

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Figures (2)

Fig. 1
Fig. 1

In-focus digital reconstructions of lensless test cases (a) and (c), and digital reconstruction in the BFP and in the in-focus plane (red boxes) of the MEMS hologram (e). (b), (d) and (f) are the denoised reconstructions using SPADEDH method.

Fig. 2
Fig. 2

Projections of lensless holograms using SLM. (a) and (c) are the projections of the original holograms, while (b) and (d) are the projections of the denoised holograms obtained by the SPADEDH algorithm.

Tables (2)

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Table 1 Calculation of SDR

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Table 2 Calculation of contrast C

Equations (11)

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

H = | O + R | 2 = | O | 2 + | R | 2 + O R * + O * R
H = q + b cos ( ϕ )
{ q = | O | 2 + | R | 2 b = 2 | O R | ϕ = arg ( O ) arg ( R )
H ˜ = q + b cos ( ϕ + n s ) + n a
H ˜ b cos ( ϕ + n s ) + n a .
v ^ = argmin v 1 subject to y = Φ v
v ^ = argmin v 1 subject to y Φ v 2 ε
v den = argmin v 1 subject to F δ h ˜ = v
SDR = I ˜ 2 I ˜ I ^ 2
C = σ ( I ) I
G = ( x , y ) S R ( I ^ I ˜ ) ( x , y ) S R I ˜

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