Abstract

We propose an algorithm to enhance diffraction-limited images based on pixel-to-pixel correlations introduced by the finite width of the Point Spread Function (PSF). We simulate diffraction-limited images of point sources by convolving the PSF of a diffraction-limited lens with simulated images, and enhance the blurred images with our algorithm. Our algorithm reduces the PSF width, increases the contrast, and reveals structure on a length scale half of that resolvable in the unenhanced image. Our enhanced images compare favorably with images enhanced by conventional Tikhonov regularization.

© 2006 Optical Society of America

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  1. M. Born, E. Wolf, and A. B. Bhatia, Principles of optics : electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, Cambridge [England]; New York, 1999).
    [PubMed]
  2. M. G. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J Microsc 198 (Pt 2), 82-7 (2000).
    [CrossRef] [PubMed]
  3. J. T. Frohn, H. F. Knapp, and A. Stemmer, "True optical resolution beyond the Rayleigh limit achieved by standing wave illumination," Proc Natl Acad Sci U S A 97, 7232-6 (2000).
    [CrossRef] [PubMed]
  4. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
    [CrossRef] [PubMed]
  5. M. Dyba and S.W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163,901 (2002).
    [CrossRef]
  6. V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143,903 (2005).
    [CrossRef]
  7. M. K. Sundareshan, S. Bhattacharjee, R. Inampudi, and H. Y. Pang, "Image preprocessing for improving computational efficiency in implementation of restoration and superresolution algorithms," Appl. Opt. 41, 7464-74 (2002).
    [CrossRef] [PubMed]
  8. C. H. Lee, H. Y. Chiang, and H. Y. Mong, "Sub-diffraction-limit imaging based on the topographic contrast of differential confocal microscopy," Opt. Lett. 28, 1772-4 (2003).
    [CrossRef] [PubMed]
  9. F. Q. Chen and D. Gerion, "Fluorescent CdSe/ZnS nanocrystal-peptide conjugates for long-term, nontoxic imaging and nuclear targeting in living cells," Nano Letters 4, 1827-1832 (2004).
    [CrossRef]
  10. S. W. Huang, H. Y. Mong, and C. H. Lee, "Super-resolution bright-field nanometer topographic optical microscopy based on contrast," Microsc. Res. Tech. 65, 180-185 (2004).
    [CrossRef]
  11. C. H. Lee and J. P. Wang, "Noninterferometric differential confocal microscopy with 2-nm depth resolution," Opt. Commun. 135, 233-237 (1997).
    [CrossRef]
  12. J . Högbom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astrophys. J. Suppl. Ser. 15,417-426 (1974).
  13. B. G. Clark, "An Efficient Implementation of the Algorithm CLEAN" Astron. Astrophys. 89, 377-378 (1980).
  14. P. C. Hansen, Rank-deficient and discrete ill-posed problems : numerical aspects of linear inversion, SIAM monographs on mathematical modeling and computation (SIAM, Philadelphia, 1997).
  15. P. C. Hansen, "Regularization tools Version 3.0 for Matlab 5.2," Numerical Algorithms 20, 195-196 (1999).
    [CrossRef]
  16. http://www2.imm.dtu.dk/˜pch/Regutools/regutools.html.
  17. D. Uttamchandani and S. McCulloch, "Optical nanosensors - Towards the development of intracellular monitoring," Advanced Drug Delivery Reviews 21, 239-247 (1996).
    [CrossRef]

2005 (1)

V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143,903 (2005).
[CrossRef]

2004 (2)

F. Q. Chen and D. Gerion, "Fluorescent CdSe/ZnS nanocrystal-peptide conjugates for long-term, nontoxic imaging and nuclear targeting in living cells," Nano Letters 4, 1827-1832 (2004).
[CrossRef]

S. W. Huang, H. Y. Mong, and C. H. Lee, "Super-resolution bright-field nanometer topographic optical microscopy based on contrast," Microsc. Res. Tech. 65, 180-185 (2004).
[CrossRef]

2003 (1)

2002 (2)

2000 (3)

M. G. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J Microsc 198 (Pt 2), 82-7 (2000).
[CrossRef] [PubMed]

J. T. Frohn, H. F. Knapp, and A. Stemmer, "True optical resolution beyond the Rayleigh limit achieved by standing wave illumination," Proc Natl Acad Sci U S A 97, 7232-6 (2000).
[CrossRef] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

1999 (1)

P. C. Hansen, "Regularization tools Version 3.0 for Matlab 5.2," Numerical Algorithms 20, 195-196 (1999).
[CrossRef]

1997 (1)

C. H. Lee and J. P. Wang, "Noninterferometric differential confocal microscopy with 2-nm depth resolution," Opt. Commun. 135, 233-237 (1997).
[CrossRef]

1996 (1)

D. Uttamchandani and S. McCulloch, "Optical nanosensors - Towards the development of intracellular monitoring," Advanced Drug Delivery Reviews 21, 239-247 (1996).
[CrossRef]

1980 (1)

B. G. Clark, "An Efficient Implementation of the Algorithm CLEAN" Astron. Astrophys. 89, 377-378 (1980).

1974 (1)

J . Högbom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astrophys. J. Suppl. Ser. 15,417-426 (1974).

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

Bhattacharjee, S.

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

Chen, F. Q.

F. Q. Chen and D. Gerion, "Fluorescent CdSe/ZnS nanocrystal-peptide conjugates for long-term, nontoxic imaging and nuclear targeting in living cells," Nano Letters 4, 1827-1832 (2004).
[CrossRef]

Chiang, H. Y.

Clark, B. G.

B. G. Clark, "An Efficient Implementation of the Algorithm CLEAN" Astron. Astrophys. 89, 377-378 (1980).

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

Dyba, M.

M. Dyba and S.W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163,901 (2002).
[CrossRef]

Frohn, J. T.

J. T. Frohn, H. F. Knapp, and A. Stemmer, "True optical resolution beyond the Rayleigh limit achieved by standing wave illumination," Proc Natl Acad Sci U S A 97, 7232-6 (2000).
[CrossRef] [PubMed]

Gerion, D.

F. Q. Chen and D. Gerion, "Fluorescent CdSe/ZnS nanocrystal-peptide conjugates for long-term, nontoxic imaging and nuclear targeting in living cells," Nano Letters 4, 1827-1832 (2004).
[CrossRef]

Gustafsson, M. G.

M. G. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J Microsc 198 (Pt 2), 82-7 (2000).
[CrossRef] [PubMed]

Hansen, P. C.

P. C. Hansen, "Regularization tools Version 3.0 for Matlab 5.2," Numerical Algorithms 20, 195-196 (1999).
[CrossRef]

Hell, S. W.

V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143,903 (2005).
[CrossRef]

Hell, S.W.

M. Dyba and S.W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163,901 (2002).
[CrossRef]

Högbom, J

J . Högbom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astrophys. J. Suppl. Ser. 15,417-426 (1974).

Huang, S. W.

S. W. Huang, H. Y. Mong, and C. H. Lee, "Super-resolution bright-field nanometer topographic optical microscopy based on contrast," Microsc. Res. Tech. 65, 180-185 (2004).
[CrossRef]

Inampudi, R.

Knapp, H. F.

J. T. Frohn, H. F. Knapp, and A. Stemmer, "True optical resolution beyond the Rayleigh limit achieved by standing wave illumination," Proc Natl Acad Sci U S A 97, 7232-6 (2000).
[CrossRef] [PubMed]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

Lee, C. H.

S. W. Huang, H. Y. Mong, and C. H. Lee, "Super-resolution bright-field nanometer topographic optical microscopy based on contrast," Microsc. Res. Tech. 65, 180-185 (2004).
[CrossRef]

C. H. Lee, H. Y. Chiang, and H. Y. Mong, "Sub-diffraction-limit imaging based on the topographic contrast of differential confocal microscopy," Opt. Lett. 28, 1772-4 (2003).
[CrossRef] [PubMed]

C. H. Lee and J. P. Wang, "Noninterferometric differential confocal microscopy with 2-nm depth resolution," Opt. Commun. 135, 233-237 (1997).
[CrossRef]

McCulloch, S.

D. Uttamchandani and S. McCulloch, "Optical nanosensors - Towards the development of intracellular monitoring," Advanced Drug Delivery Reviews 21, 239-247 (1996).
[CrossRef]

Mong, H. Y.

S. W. Huang, H. Y. Mong, and C. H. Lee, "Super-resolution bright-field nanometer topographic optical microscopy based on contrast," Microsc. Res. Tech. 65, 180-185 (2004).
[CrossRef]

C. H. Lee, H. Y. Chiang, and H. Y. Mong, "Sub-diffraction-limit imaging based on the topographic contrast of differential confocal microscopy," Opt. Lett. 28, 1772-4 (2003).
[CrossRef] [PubMed]

Pang, H. Y.

Stemmer, A.

J. T. Frohn, H. F. Knapp, and A. Stemmer, "True optical resolution beyond the Rayleigh limit achieved by standing wave illumination," Proc Natl Acad Sci U S A 97, 7232-6 (2000).
[CrossRef] [PubMed]

Sundareshan, M. K.

Uttamchandani, D.

D. Uttamchandani and S. McCulloch, "Optical nanosensors - Towards the development of intracellular monitoring," Advanced Drug Delivery Reviews 21, 239-247 (1996).
[CrossRef]

Wang, J. P.

C. H. Lee and J. P. Wang, "Noninterferometric differential confocal microscopy with 2-nm depth resolution," Opt. Commun. 135, 233-237 (1997).
[CrossRef]

Westphal, V.

V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143,903 (2005).
[CrossRef]

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

Advanced Drug Delivery Reviews (1)

D. Uttamchandani and S. McCulloch, "Optical nanosensors - Towards the development of intracellular monitoring," Advanced Drug Delivery Reviews 21, 239-247 (1996).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

B. G. Clark, "An Efficient Implementation of the Algorithm CLEAN" Astron. Astrophys. 89, 377-378 (1980).

Astrophys. J. Suppl. Ser. (1)

J . Högbom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astrophys. J. Suppl. Ser. 15,417-426 (1974).

J Microsc (1)

M. G. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J Microsc 198 (Pt 2), 82-7 (2000).
[CrossRef] [PubMed]

Microsc. Res. Tech. (1)

S. W. Huang, H. Y. Mong, and C. H. Lee, "Super-resolution bright-field nanometer topographic optical microscopy based on contrast," Microsc. Res. Tech. 65, 180-185 (2004).
[CrossRef]

Nano Letters (1)

F. Q. Chen and D. Gerion, "Fluorescent CdSe/ZnS nanocrystal-peptide conjugates for long-term, nontoxic imaging and nuclear targeting in living cells," Nano Letters 4, 1827-1832 (2004).
[CrossRef]

Numerical Algorithms (1)

P. C. Hansen, "Regularization tools Version 3.0 for Matlab 5.2," Numerical Algorithms 20, 195-196 (1999).
[CrossRef]

Opt. Commun. (1)

C. H. Lee and J. P. Wang, "Noninterferometric differential confocal microscopy with 2-nm depth resolution," Opt. Commun. 135, 233-237 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (3)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-6 (2000).
[CrossRef] [PubMed]

M. Dyba and S.W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163,901 (2002).
[CrossRef]

V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143,903 (2005).
[CrossRef]

Proc Natl Acad Sci U S A (1)

J. T. Frohn, H. F. Knapp, and A. Stemmer, "True optical resolution beyond the Rayleigh limit achieved by standing wave illumination," Proc Natl Acad Sci U S A 97, 7232-6 (2000).
[CrossRef] [PubMed]

Other (3)

M. Born, E. Wolf, and A. B. Bhatia, Principles of optics : electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, Cambridge [England]; New York, 1999).
[PubMed]

P. C. Hansen, Rank-deficient and discrete ill-posed problems : numerical aspects of linear inversion, SIAM monographs on mathematical modeling and computation (SIAM, Philadelphia, 1997).

http://www2.imm.dtu.dk/˜pch/Regutools/regutools.html.

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Figures (7)

Fig. 1.
Fig. 1.

A plot of the PSF for a diffraction-limited lens, identifying the node (located at r 1) and point of steepest slope (located at r 2). Both of these features are exploited by our algorithm.

Fig. 2.
Fig. 2.

An illustration of our imaging geometry. A region O in the object plane is imaged onto a pixel P in the image plane. When enhancing the image we examine pixels along the rings R1 and R2, which correspond to the node and inflection point of the PSF, respectively.

Fig. 3.
Fig. 3.

(a) Diffraction-limited image of a point source. (b) Enhanced image. (Scale bars indicate wavelength of light). (c) Intensity profile of diffraction-limited and enhanced images, taken along the horizontal axis through the center of each image. Features and asymmetry due to noise and pixellation.

Fig. 4.
Fig. 4.

Images of two point objects of equal intensity, with spacings noted on scale bars: (a), (b), (c) Diffraction-limited images; (d), (e), (f) Enhanced images.

Fig. 5.
Fig. 5.

Aspect ratios of diffraction-limited and enhanced images of two point objects of equal intensity, plotted as a function of the object spacing.

Fig. 6.
Fig. 6.

Images of two point objects of unequal intensity, with spacings indicated by scale bars and intensity ratios noted below. (a),(c),(e) Diffraction-limited images. (b),(d),(f) Enhanced images.

Fig. 7.
Fig. 7.

Images of two point sources separated by 320 nm (scale bar). (a) Original diffraction-limited image. (b) Image enhanced by our method. (c)–(e) Images enhanced by Tikhonov regularization with regularization parameter λ = (c)1.0, (d) 4.3, and (e) 10.0.

Equations (1)

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I ( r ) = ( J 1 ( NA k 0 r M ) NA k 0 r M ) 2

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