Abstract

We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal’s expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability.

© 2004 Optical Society of America

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2004

2003

2002

M. Zibulevsky, Y. Y. Zeevi, “Extraction of a source from multichannel data using sparse decomposition,” Neuro- computing 49, 163–173 (2002).

P. R. Hobson, J. Watson, “The principles and practice of holographic recording of plankton,” J. Opt. A Pure Appl. Opt. 4, 34–49 (2002).
[CrossRef]

R. Yin, P. J. Flynn, S. L. Broschat, “Position-dependent defocus processing for acoustic holography images,” Int. J. Imaging Syst. Technol. 12, 101–111 (2002).
[CrossRef]

2001

2000

1999

1998

J. Widjaja, S. Jutamulia, “Use of wavelet analysis forimproving autofocusing capability,” Opt. Commun. 151, 12–14 (1998).
[CrossRef]

D. L. Donoho, M. Vetterli, R. A. DeVore, I. Daubechies, “Data compression and harmonic analysis,” IEEE Trans. Inf. Theory 44, 2435–2476 (1998).
[CrossRef]

1997

1996

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607–609 (1996).
[CrossRef] [PubMed]

1995

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Image Process. 43, 1568–1578 (1995).
[CrossRef]

1994

U. Schnars, W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

D. L. Donoho, I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

1992

D. L. Donoho, “Superresolution via sparsity constraints,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 23, 1309–1331 (1992).
[CrossRef]

1991

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magn. Reson. Med. 21, 288–295 (1991).
[CrossRef] [PubMed]

H. Lichte, “Optimum focus for taking electron holograms,” Ultramicroscopy 38, 13–22 (1991).
[CrossRef]

1989

S. G. Mallat, “A theory for multiresolution signal decomposition—the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

J. Gillespie, R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[CrossRef]

1988

P. G. Lemarié-Rieusset, “Ondelettes à localisation exponentielle,” J. Math. Pures Appl. 67, 227–236 (1988).

1987

G. Battle, “A block spin construction of ondelettes. 1. Lemarié functions,” Commun. Math. Phys. 110, 601–615 (1987).
[CrossRef]

P. Grossman, “Depth from focus,” Pattern Recogn. Lett. 5, 63–69 (1987).
[CrossRef]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

1986

1985

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

1982

S. J. Erasmus, K. C. A. Smith, “An automatic focusing and astigmatism correction system for the SEM and CTEM,” J. Microsc. 127, 185–199 (1982).
[CrossRef]

1981

1976

R. A. Jarvis, “Focus optimization criteria for computer image-processing,” Microscope 24, 163–180 (1976).

A. Erteza, “Sharpness index and its application to focus control,” Appl. Opt. 15, 877–881 (1976).
[CrossRef] [PubMed]

1974

1972

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

1967

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

1956

1948

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Battle, G.

G. Battle, “A block spin construction of ondelettes. 1. Lemarié functions,” Commun. Math. Phys. 110, 601–615 (1987).
[CrossRef]

Bauer, K. W.

Beresnev, L. A.

Bevilacqua, F.

Blu, Th.

M. Liebling, Th. Blu, M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004).
[CrossRef]

M. Liebling, Th. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

M. Unser, Th. Blu, “Wavelet theory demystified,” IEEE Trans. Signal Process. 51, 470–483 (2003).
[CrossRef]

Brent, R. P.

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice Hall, Englewood Cliffs, N.J., 1973).

Brigantic, R. T.

Broschat, S. L.

R. Yin, P. J. Flynn, S. L. Broschat, “Position-dependent defocus processing for acoustic holography images,” Int. J. Imaging Syst. Technol. 12, 101–111 (2002).
[CrossRef]

Buffington, A.

Carhart, G. W.

Coppola, G.

Cromwell, L. D.

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magn. Reson. Med. 21, 288–295 (1991).
[CrossRef] [PubMed]

Cuche, E.

Daubechies, I.

D. L. Donoho, M. Vetterli, R. A. DeVore, I. Daubechies, “Data compression and harmonic analysis,” IEEE Trans. Inf. Theory 44, 2435–2476 (1998).
[CrossRef]

De Nicola, S.

Depeursinge, Ch.

DeVore, R. A.

D. L. Donoho, M. Vetterli, R. A. DeVore, I. Daubechies, “Data compression and harmonic analysis,” IEEE Trans. Inf. Theory 44, 2435–2476 (1998).
[CrossRef]

Donoho, D. L.

D. L. Donoho, M. Vetterli, R. A. DeVore, I. Daubechies, “Data compression and harmonic analysis,” IEEE Trans. Inf. Theory 44, 2435–2476 (1998).
[CrossRef]

D. L. Donoho, I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

D. L. Donoho, “Superresolution via sparsity constraints,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 23, 1309–1331 (1992).
[CrossRef]

Erasmus, S. J.

S. J. Erasmus, K. C. A. Smith, “An automatic focusing and astigmatism correction system for the SEM and CTEM,” J. Microsc. 127, 185–199 (1982).
[CrossRef]

Erteza, A.

Ferraro, P.

Field, D. J.

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607–609 (1996).
[CrossRef] [PubMed]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

Fienup, J. R.

Finizio, A.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Flynn, P. J.

R. Yin, P. J. Flynn, S. L. Broschat, “Position-dependent defocus processing for acoustic holography images,” Int. J. Imaging Syst. Technol. 12, 101–111 (2002).
[CrossRef]

Forsythe, G. E.

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice Hall, Englewood Cliffs, N.J., 1977).

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Gillespie, J.

J. Gillespie, R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Groen, F. C. A.

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

Grossman, P.

P. Grossman, “Depth from focus,” Pattern Recogn. Lett. 5, 63–69 (1987).
[CrossRef]

Healy, D. M.

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magn. Reson. Med. 21, 288–295 (1991).
[CrossRef] [PubMed]

Hobson, P. R.

P. R. Hobson, J. Watson, “The principles and practice of holographic recording of plankton,” J. Opt. A Pure Appl. Opt. 4, 34–49 (2002).
[CrossRef]

Horn, B.

B. Horn, “Project MAC: Focusing,” , Massachusetts Institute of Technology, Cambridge, Mass. (1968), ftp://publications.ai.mit.edu/ai-publications/pdf/AIM-160.pdf.

Jarvis, R. A.

R. A. Jarvis, “Focus optimization criteria for computer image-processing,” Microscope 24, 163–180 (1976).

Johnstone, I. M.

D. L. Donoho, I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

Jüptner, W.

Justh, E. W.

Jutamulia, S.

J. Widjaja, S. Jutamulia, “Use of wavelet analysis forimproving autofocusing capability,” Opt. Commun. 151, 12–14 (1998).
[CrossRef]

King, R. A.

J. Gillespie, R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[CrossRef]

Kocatepe, M.

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Image Process. 43, 1568–1578 (1995).
[CrossRef]

Kreis, Th.

Krishnaprasad, P. S.

Kronrod, M. A.

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Lawrence, R. W.

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Lemarié-Rieusset, P. G.

P. G. Lemarié-Rieusset, “Ondelettes à localisation exponentielle,” J. Math. Pures Appl. 67, 227–236 (1988).

Lichte, H.

H. Lichte, “Optimum focus for taking electron holograms,” Ultramicroscopy 38, 13–22 (1991).
[CrossRef]

Liebling, M.

M. Liebling, Th. Blu, M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004).
[CrossRef]

M. Liebling, Th. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Ligthart, G.

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

Linfoot, E. H.

Lohmann, A. W.

Malcolm, M. A.

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice Hall, Englewood Cliffs, N.J., 1977).

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition—the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

S. G. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

Marquet, P.

Mendlovic, D.

Merzlyakov, N. S.

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, New York, 1980).

Miller, J. J.

Moler, C. B.

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice Hall, Englewood Cliffs, N.J., 1977).

Muller, R. A.

Olshausen, B. A.

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607–609 (1996).
[CrossRef] [PubMed]

Onural, L.

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Image Process. 43, 1568–1578 (1995).
[CrossRef]

Philips, W.

F. Rooms, A. Pizurica, W. Philips, “Estimating image blur in the wavelet domain,” in Proceedings of the Fifth Asian Conference on Computer Vision (ACCV) [Asian Federation of Computer Vision Societies (AFCV), Victoria, Australia, 2002], pp. 210–215.

Pierattini, G.

Pizurica, A.

F. Rooms, A. Pizurica, W. Philips, “Estimating image blur in the wavelet domain,” in Proceedings of the Fifth Asian Conference on Computer Vision (ACCV) [Asian Federation of Computer Vision Societies (AFCV), Victoria, Australia, 2002], pp. 210–215.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Roggemann, M. C.

Rooms, F.

F. Rooms, A. Pizurica, W. Philips, “Estimating image blur in the wavelet domain,” in Proceedings of the Fifth Asian Conference on Computer Vision (ACCV) [Asian Federation of Computer Vision Societies (AFCV), Victoria, Australia, 2002], pp. 210–215.

Schnars, U.

Sedgewick, R.

R. Sedgewick, Algorithms, 2nd ed. (Addison-Wesley, Reading, Mass., 1988).

Sica, L.

Smith, K. C. A.

S. J. Erasmus, K. C. A. Smith, “An automatic focusing and astigmatism correction system for the SEM and CTEM,” J. Microsc. 127, 185–199 (1982).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Unser, M.

M. Liebling, Th. Blu, M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004).
[CrossRef]

M. Liebling, Th. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

M. Unser, Th. Blu, “Wavelet theory demystified,” IEEE Trans. Signal Process. 51, 470–483 (2003).
[CrossRef]

Vetterli, M.

D. L. Donoho, M. Vetterli, R. A. DeVore, I. Daubechies, “Data compression and harmonic analysis,” IEEE Trans. Inf. Theory 44, 2435–2476 (1998).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vorontsov, M. A.

Watson, J.

P. R. Hobson, J. Watson, “The principles and practice of holographic recording of plankton,” J. Opt. A Pure Appl. Opt. 4, 34–49 (2002).
[CrossRef]

Weaver, J. B.

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magn. Reson. Med. 21, 288–295 (1991).
[CrossRef] [PubMed]

Welsh, B. M.

Widjaja, J.

J. Widjaja, S. Jutamulia, “Use of wavelet analysis forimproving autofocusing capability,” Opt. Commun. 151, 12–14 (1998).
[CrossRef]

Yansun, X.

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magn. Reson. Med. 21, 288–295 (1991).
[CrossRef] [PubMed]

Yaroslavskii, L. P.

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, New York, 1980).

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