Abstract

We propose a new focus function Λ that, like many of the existing focus functions, consists of a convex function and an image enhancement filter. Λ is rather flexible because for any convex function and image enhancement filter, it is a focus function. We proved that Λ is a focus function using a model and Jensen’s inequality. Furthermore, we generated random Λs and experimentally applied them to simulated and real blurred images, finding that 98% and 99% of the random Λs, respectively, have a maximum value at the best-focused image and most of them decrease as the defocus increases. We also applied random Λs to motion-blurred images, blurred images in different-sized windows, and blurred images with different types of noise. We found that Λ can be applied to motion blur and is robust to different-sized windows and different noise types.

© 2014 Optical Society of America

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2012 (3)

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

P. Gao, B. Yao, R. Rupp, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, “Autofocusing based on wavelength dependence of diffraction in two-wavelength digital holographic microscopy,” Opt. Lett. 37(7), 1172–1174 (2012).
[CrossRef] [PubMed]

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (2)

H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. 29(3), 173–180 (2010).
[CrossRef]

M. J. Nasse and J. C. Woehl, “Realistic modeling of the illumination point spread function in confocal scanning optical microscopy,” J. Opt. Soc. Am. A 27(2), 295–302 (2010).
[CrossRef] [PubMed]

2009 (2)

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc. 236(3), 194–202 (2009).
[CrossRef] [PubMed]

2008 (2)

S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. 18(9), 1237–1246 (2008).
[CrossRef]

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17(10), 1737–1754 (2008).
[CrossRef] [PubMed]

2007 (1)

2004 (2)

Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech. 65(3), 139–149 (2004).
[CrossRef] [PubMed]

J. Daugman, “How iris recognition works,” IEEE Trans. Circuits Syst. Video Technol. 14(1), 21–30 (2004).
[CrossRef]

1997 (1)

A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
[CrossRef] [PubMed]

1995 (1)

A. P. Tzannes and J. M. Mooney, “Measurement of the modulation transfer function of infrared cameras,” Opt. Eng. 34(6), 1808–1817 (1995).
[CrossRef]

1994 (1)

S. K. Nayar and Y. Nakagawa, ““Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).

1993 (2)

M. Subbarao, T. S. Choi, and A. Nikzad, “Focusing techniques,” Opt. Eng. 32(11), 2824–2836 (1993).
[CrossRef]

G. V. Poropat, “Effect of system point spread function, apparent size, and detector instantaneous field of view on the infrared image contrast of small objects,” Opt. Eng. 32(10), 2598–2607 (1993).
[CrossRef]

1991 (2)

L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry 12(3), 195–206 (1991).
[CrossRef] [PubMed]

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30(2), 170–177 (1991).
[CrossRef]

1990 (2)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990).
[CrossRef]

F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys. 17(6), 962–966 (1990).
[CrossRef] [PubMed]

1987 (1)

E. Krotkov, “Focusing,” Int. J. Comput. Vis. 1(3), 223–237 (1987).
[CrossRef]

1985 (1)

F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6(2), 81–91 (1985).
[CrossRef] [PubMed]

1976 (1)

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
[CrossRef] [PubMed]

1972 (1)

M. L. Mendelsohn and B. H. Mayall, “Computer-oriented analysis of human chromosomes. 3. Focus,” Comput. Biol. Med. 2(2), 137–150 (1972).
[CrossRef] [PubMed]

1927 (1)

E. Artin, “Uber die Zerlegung definiter Funktionen in Quadrate,” Abh. Math. Seminar Univ. Hamburg 5, 85–99 (1927).

1906 (1)

J. L. W. V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Math. 30(1), 175–193 (1906).
[CrossRef]

Artin, E.

E. Artin, “Uber die Zerlegung definiter Funktionen in Quadrate,” Abh. Math. Seminar Univ. Hamburg 5, 85–99 (1927).

Bonam, O. P.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

Brázdilová, S. L.

S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc. 236(3), 194–202 (2009).
[CrossRef] [PubMed]

Brenner, J. F.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
[CrossRef] [PubMed]

Cen, Z.

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

Chaudhuri, B.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

Chen, Y.

Cho, J. M.

S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. 18(9), 1237–1246 (2008).
[CrossRef]

Choi, T. S.

M. Subbarao, T. S. Choi, and A. Nikzad, “Focusing techniques,” Opt. Eng. 32(11), 2824–2836 (1993).
[CrossRef]

Cook, K.

L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry 12(3), 195–206 (1991).
[CrossRef] [PubMed]

Culp, K.

L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry 12(3), 195–206 (1991).
[CrossRef] [PubMed]

Dan, D.

Daugman, J.

J. Daugman, “How iris recognition works,” IEEE Trans. Circuits Syst. Video Technol. 14(1), 21–30 (2004).
[CrossRef]

del Pozo, F.

A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
[CrossRef] [PubMed]

Dew, B. S.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
[CrossRef] [PubMed]

Distante, C.

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

Doi, K.

F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys. 17(6), 962–966 (1990).
[CrossRef] [PubMed]

Duthaler, S.

Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech. 65(3), 139–149 (2004).
[CrossRef] [PubMed]

Egiazarian, K.

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17(10), 1737–1754 (2008).
[CrossRef] [PubMed]

Elozory, D. T.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

Englmeier, K. H.

H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. 29(3), 173–180 (2010).
[CrossRef]

Feng, H.

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

Ferraro, P.

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

Finizio, A.

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

Firestone, L.

L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry 12(3), 195–206 (1991).
[CrossRef] [PubMed]

Foi, A.

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17(10), 1737–1754 (2008).
[CrossRef] [PubMed]

Gao, P.

Giger, M. L.

F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys. 17(6), 962–966 (1990).
[CrossRef] [PubMed]

Goldgof, D. B.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

Groen, F. C.

F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6(2), 81–91 (1985).
[CrossRef] [PubMed]

Guo, R.

Hall, L. O.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

Horton, J. B.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
[CrossRef] [PubMed]

Javidi, B.

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

Jensen, J. L. W. V.

J. L. W. V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Math. 30(1), 175–193 (1906).
[CrossRef]

Katkovnik, V.

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17(10), 1737–1754 (2008).
[CrossRef] [PubMed]

Kim, S. W.

S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. 18(9), 1237–1246 (2008).
[CrossRef]

King, T.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
[CrossRef] [PubMed]

Kozubek, M.

S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc. 236(3), 194–202 (2009).
[CrossRef] [PubMed]

Kramer, K. A.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

Krotkov, E.

E. Krotkov, “Focusing,” Int. J. Comput. Vis. 1(3), 223–237 (1987).
[CrossRef]

Kumar, Y.

S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. 18(9), 1237–1246 (2008).
[CrossRef]

Lee, S. W.

S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. 18(9), 1237–1246 (2008).
[CrossRef]

Lee, S. Y.

S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. 18(9), 1237–1246 (2008).
[CrossRef]

Lei, M.

Li, Q.

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

Li, T.

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

Li, X.

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

Ligthart, G.

F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6(2), 81–91 (1985).
[CrossRef] [PubMed]

Ma, B.

Malik, J.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990).
[CrossRef]

Malpica, N.

A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
[CrossRef] [PubMed]

Mater, M.

Mayall, B. H.

M. L. Mendelsohn and B. H. Mayall, “Computer-oriented analysis of human chromosomes. 3. Focus,” Comput. Biol. Med. 2(2), 137–150 (1972).
[CrossRef] [PubMed]

Memmolo, P.

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

Mendelsohn, M. L.

M. L. Mendelsohn and B. H. Mayall, “Computer-oriented analysis of human chromosomes. 3. Focus,” Comput. Biol. Med. 2(2), 137–150 (1972).
[CrossRef] [PubMed]

Min, J.

Mooney, J. M.

A. P. Tzannes and J. M. Mooney, “Measurement of the modulation transfer function of infrared cameras,” Opt. Eng. 34(6), 1808–1817 (1995).
[CrossRef]

Mouton, P. R.

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
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S. K. Nayar and Y. Nakagawa, ““Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).

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Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech. 65(3), 139–149 (2004).
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A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
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S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30(2), 170–177 (1991).
[CrossRef]

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P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

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A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
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H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. 29(3), 173–180 (2010).
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G. V. Poropat, “Effect of system point spread function, apparent size, and detector instantaneous field of view on the infrared image contrast of small objects,” Opt. Eng. 32(10), 2598–2607 (1993).
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L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry 12(3), 195–206 (1991).
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S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30(2), 170–177 (1991).
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A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
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H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. 29(3), 173–180 (2010).
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J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
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L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry 12(3), 195–206 (1991).
[CrossRef] [PubMed]

F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6(2), 81–91 (1985).
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IEEE Trans. Image Process. (1)

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17(10), 1737–1754 (2008).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990).
[CrossRef]

S. K. Nayar and Y. Nakagawa, ““Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).

Image Anal. Stereol. (1)

H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. 29(3), 173–180 (2010).
[CrossRef]

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E. Krotkov, “Focusing,” Int. J. Comput. Vis. 1(3), 223–237 (1987).
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J. Histochem. Cytochem. (1)

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24(1), 100–111 (1976).
[CrossRef] [PubMed]

J. Microsc. (3)

D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. 248(3), 245–259 (2012).
[CrossRef] [PubMed]

A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. 188(3), 264–272 (1997).
[CrossRef] [PubMed]

S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc. 236(3), 194–202 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Med. Phys. (1)

F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys. 17(6), 962–966 (1990).
[CrossRef] [PubMed]

Microsc. Res. Tech. (1)

Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech. 65(3), 139–149 (2004).
[CrossRef] [PubMed]

Opt. Eng. (4)

M. Subbarao, T. S. Choi, and A. Nikzad, “Focusing techniques,” Opt. Eng. 32(11), 2824–2836 (1993).
[CrossRef]

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30(2), 170–177 (1991).
[CrossRef]

A. P. Tzannes and J. M. Mooney, “Measurement of the modulation transfer function of infrared cameras,” Opt. Eng. 34(6), 1808–1817 (1995).
[CrossRef]

G. V. Poropat, “Effect of system point spread function, apparent size, and detector instantaneous field of view on the infrared image contrast of small objects,” Opt. Eng. 32(10), 2598–2607 (1993).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE 8384, 838408 (2012).

T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE 7498, 74981H (2009).
[CrossRef]

Other (7)

P. Favaro, “Shape from focus, and, defocus: convexity, quasiconvexity and defocus-invariant textures,” in ICCV (2007).

Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing algorithm selection in computer microscopy,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2005) (IEEE, 2005).

D. C. Tsai and H. H. Chen, “Effective autofocus decision using reciprocal focus profile,” in 18th IEEE International Conference on Image Processing (ICIP) (IEEE, 2011).
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F. Quan, K. Han, and X. C. Zhu, “A new auto-focusing method based on the center blocking DCT,” in Fourth International Conference on Image and Graphics ( ICIG 2007) (2007).

W. Jian and H. B. Chen, “A novel auto-focus function,” in 6th International Symposium on Advanced Optical Manufacturing and Testing Technologies (AOMATT 2012) (International Society for Optics and Photonics, 2012).

B. J. Kang and K. R. Park, “A study on iris image restoration,” in International Conference on Audio- and Video-Based Biometric Person Authentication (2005), pp. 31–40.
[CrossRef]

P. Langehanenberg, B. Kemper, and G. Bally, “Autofocus algorithms for digital-holographic microscopy,” in European Conference on Biomedical Optics, Optical Society of America (2007).

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Figures (19)

Fig. 1
Fig. 1

Curves of convex functions φ in Table 1. Functions a, b, c, d, and e are convex; f and g are approximately convex. Function h is reversed convex because defocusing an image always shrinks the histogram, which reverses the smoothing process.

Fig. 2
Fig. 2

PSF can be approximated by Gaussian function.

Fig. 3
Fig. 3

Brief illustration of the proof.

Fig. 4
Fig. 4

Shapes of 30 random convex functions generated by Eq. (9) in the range of [-1,1].

Fig. 5
Fig. 5

Meshes of four random image enhancement filters.

Fig. 6
Fig. 6

Original images: portrait (Lena), object, outdoor landscape, and microscopic image.

Fig. 7
Fig. 7

Examples of hot color maps of PSFs generated by PSF Lab.

Fig. 8
Fig. 8

Examples of blurred images. Blurred images were best focused at index 8, and the defocus increased as the index increased or decreased.

Fig. 9
Fig. 9

Examples of real out-of-focus blurred images captured by a microscope. The images were best focused at index 10, and the defocus increased as the indexes increased or decreased.

Fig. 10
Fig. 10

Examples of meshes of motion-blurred PSFs.

Fig. 11
Fig. 11

Examples of motion-blurred images. The images were the least motion blurred at index 1, and the blurring degree increased as the index increased.

Fig. 12
Fig. 12

Different-sized windows on Lena image.

Fig. 13
Fig. 13

Examples of blurred images in different-sized windows. Blurred images were best focused at index 8, and the defocus increased as the index increased or decreased.

Fig. 14
Fig. 14

Examples of blurred images with different types of noise. Blurred images were best focused at index 8, and the defocus increased as the index increased or decreased.

Fig. 15
Fig. 15

Experimental results for 196 random Λs applied to simulated out-of-focus blurred images. The blurred images were best focused at index 8, and the defocus increased as the index increased or decreased.

Fig. 16
Fig. 16

Experimental results for 196 constructed Λs applied to real out-of-focus blurred images. The images were best focused at index 10, and the defocus increased as the index increased or decreased.

Fig. 17
Fig. 17

Experimental results for 196 random Λs applied to motion-blurred images. The images were the least motion blurred at index 1, and the blurring degree increased as the index increased.

Fig. 18
Fig. 18

Experimental results for 196 constructed Λs applied to simulated out-of-focus blurred images in different-sized windows. The images were best focused at index 8, and the defocus increased as the index increased or decreased.

Fig. 19
Fig. 19

Experimental results for 196 constructed Λs applied to simulated out-of-focus blurred images with different types of noise. The images were best focused at index 8, and the defocus increased as the index increased or decreased.

Tables (1)

Tables Icon

Table 1 List of Known Focus Functions that Can Be Expressed by Eq. (1).

Equations (36)

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

Λ= φ(I(x,y)*g)dxdy ,
h(r)=4 [ J 1 (πr (λF) 1 ] 2 [πr (λF) 1 ] 2 ,
I 2 = I 0 *PS F 2 = I 0 *PS F 1 *G= I 1 *G,
φ(( I 2 *g)(x,y))dxdy φ(( I 1 *g)(x,y))dxdy .
φ(( I 2 *g)(x,y))dxdy = φ((( I 1 *G)*g)(x,y))dxdy = φ(( I 1 *(G*g))(x,y))dxdy ,
φ((I1*(G*g))(x,y))dxdy φ((I1*g)(x,y))dxdy ,
Ω φ(( I 2 *g)(x,y))dxdy Ω φ(( I 1 *g)(x,y))dxdy ,
p m 2 + p n 2 ,
( p m 2 + p n 2 ) .
I N =I+ n P + n G ,
1 a (I+ n P )~P( 1 a I),
n G ~N(0,b),
normΛ'( I i )= Λ'( I i ) max(abs(Λ'( I i ))) ,
σ(x)0,
σ(x)dx=1 .
φ((f*σ)(x))dx φ(f(x))dx .
φ((f*σ)(x))dx = φ( f(t)σ(xt)dt)dx .
p(x)0,
p(x)dx>0 .
φ( p(x)f(x)dx p(x)dx ) p(x)φ(f(x))dx p(x)dx .
φ( f(x)σ(x)dx) σ(x)φ(f(x))dx .
φ( f(t)σ(xt)dt)dx σ(xt)φ(f(t))dtdx .
σ(xt)φ(f(t))dtdx = σ(xt)φ(f(t))dxdt = φ(f(t)) σ(xt)dxdt = φ(f(t))dt . = φ(f(x))dx
φ((f*( σ 1 * σ 2 ))(x))dx φ((f* σ 1 )(x))dx .
φ(((f*( σ 1 * σ 2 ))*g)(x))dx φ(((f* σ 1 )*g)(x))dx .
i=1,j=1 m,n φ( I 2 (i,j)) i=1,j=1 m,n φ( I 1 (i,j)) .
i=1,j=1 m,n φ( I 2 (i,j)) i=1,j=1 m,n φ( r=1,s=1 p,q G(r,s)× I 1 (mod(i+r,m),mod(j+s,n)) ) = i=1,j=1 m,n φ( r=1,s=1 p,q G(r,s)× I 1 (i+r,j+s) ) ,
G(r,s)>0,1rp, 1sq,
r=1,s=1 p,q G(r,s)=1 .
i=1,j=1 m,n φ( I 1 (i,j)) =( r=1,s=1 p,q G(r,s) )× i=1,j=1 m,n φ( I 1 (i,j)) = i=1,j=1 m,n r=1,s=1 p,q G(r,s)×φ( I 1 (mod(i+r,m),mod(j+s,n))) , i=1,j=1 m,n r=1,s=1 p,q G(r,s)×φ( I 1 (i+r,j+s))
φ( a i x i a i ) a i φ( x i ) a i .
φ( r=1,s=1 p,q G(r,s)× I 1 (i+r,j+s) ) r=1,s=1 p,q G(r,s)×φ( I 1 (i+r,j+s)) .
i=1,j=1 m,n φ( r=1,s=1 p,q G(r,s)× I 1 (i+r,j+s) ) i=1,j=1 m,n r=1,s=1 p,q G(r,s)×φ( I 1 (i+r,j+s)) .
I 1 ' = I 1 *g,
I 2 ' = I 2 *g.
i=1,j=1 m,n φ( I 2 ' (i,j)) i=1,j=1 m,n φ( I 1 ' (i,j)) .

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