Abstract

Microscopy has become a de facto tool for biology. However, it suffers from a fundamental problem of poor contrast with increasing depth, as the illuminating light gets attenuated and scattered and hence can not penetrate through thick samples. The resulting decay of light intensity due to attenuation and scattering varies exponentially across the image. The classical space invariant deconvolution approaches alone are not suitable for the restoration of uneven illumination in microscopy images. In this paper, we present a novel physics-based field theoretical approach to solve the contrast degradation problem of light microscopy images. We have confirmed the effectiveness of our technique through simulations as well as through real field experimentations.

© 2009 Optical Society of America

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References

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  1. JamesB.  Pawley ed. Handbook of Biological Confocal Microscopy Third Edition (Springer, New York, 2005).
  2. D. Kundur, D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. pp. 43-64, May 1996).
  3. P. Shaw, "Deconvolution in 3-D optical microscopy," Histochem. J. 261573-6865 (1994).
    [CrossRef]
  4. P. Sarder, and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Process. Mag. pp. 32-45, May 2006.
  5. J. P. Oakley and B. L. Satherley, "Improving image quality in poor visibility conditions using a physical model for degradation," IEEE Trans. Image Process 7(2), 167-179 (1998).
    [CrossRef]
  6. K. Tan and J. P. Oakley, "Enhancement of color images in poor visibility conditions," Proc. Int’l Conf. Image Process. 2, 788-791 (2000).
  7. K. Tan and J. P. Oakley, "Physics Based Approach to color image enhancement in poor visibility conditions," J. Optical Soc. Am. 18(10), 2460-2467 (2001).
  8. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, "Polarization based vision through haze," Appl. Opt. 42(3), 511-525 (2003).
    [CrossRef]
  9. Y. Y. Schechner, and N. Karpel, "Clear underwater vision," Proc. IEEE Conf. Computer Vision and Pattern Recognition 1, 536-543 (2004).
  10. S. G. Narasimhan and S. K. Nayar, "Contrast restoration of weather degraded images," IEEE Trans. Pattern Anal. Mach. Intell 25(6), 713-724 (2003).
    [CrossRef]
  11. S. G. Narasimhan and S. K. Nayar, "Vision and the atmosphere," Int’l J. Computer Vision 48(3), 233-254 (2002).
    [CrossRef]
  12. S. G. Narasimhan and S. K. Nayar, "Removing weather effects from monochrome images," Proc. IEEE Conf. Computer Vision and Pattern Recognition 2, 186-193 (2001).
  13. S. G. Narasimhan and S. K. Nayar, "Chromatic framework for vision in bad weather," Proc. IEEE Conf. Computer Vision and Pattern Recognition 1 598-605 (2000).
    [CrossRef]
  14. R. Kaftory, Y. Y. Schechner, and Y. Y. Zeevi, "Variational distance-dependent image restoration," Proc. IEEE Conf. Computer Vision and Pattern Recognition (2007).
  15. S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of mages," IEEE Trans. Pattern Anal. Mach. Intell 6, 721-741 (1984).
    [CrossRef]
  16. L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992).
    [CrossRef]
  17. P. L. Combettes, and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
    [CrossRef] [PubMed]
  18. A. R. Patternson, A first course in fluid dynamics (Cambridge university press 1989).
  19. J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer 1995).
  20. M. Capek, J. Janacek, and L. Kubinova, "Methods for compensation of the light attenuation with depth of images captured by a confocal microscope," Microscopy Res. Tech. 69, 624-635 (2006).
    [CrossRef]
  21. P. S. Umesh Adiga and B. B. Chaudhury, "Some efficient methods to correct confocal images for easy interpretation," Micron. 32, 363-370 (2001).
    [CrossRef]
  22. K. Greger, J. Swoger, and E. H. K. Stelzer, "Basic building units and properties of a fluorescence single plane illumination microscope," Rev. Sci. Instrum 78, 023705 (2007).
    [CrossRef] [PubMed]
  23. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
    [CrossRef] [PubMed]
  24. P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).
  25. P. J. Keller, F. Pampaloni, E. H. K. Stelzer, "Life sciences require the third dimension," Curr. Opin. Cell Biol. 18, 117-124 (2006).
    [CrossRef] [PubMed]
  26. J. G. Ritter, R. Veith, J. Siebrasse, U. Kubitscheck. "High-contrast single-particle tracking by selective focal plane illumination microscopy," Opt. Express 16(10), 7142-7152 (2008).
    [CrossRef]

2008 (1)

2007 (2)

K. Greger, J. Swoger, and E. H. K. Stelzer, "Basic building units and properties of a fluorescence single plane illumination microscope," Rev. Sci. Instrum 78, 023705 (2007).
[CrossRef] [PubMed]

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

2006 (3)

P. J. Keller, F. Pampaloni, E. H. K. Stelzer, "Life sciences require the third dimension," Curr. Opin. Cell Biol. 18, 117-124 (2006).
[CrossRef] [PubMed]

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Process. Mag. pp. 32-45, May 2006.

M. Capek, J. Janacek, and L. Kubinova, "Methods for compensation of the light attenuation with depth of images captured by a confocal microscope," Microscopy Res. Tech. 69, 624-635 (2006).
[CrossRef]

2004 (2)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

P. L. Combettes, and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

2003 (2)

S. G. Narasimhan and S. K. Nayar, "Contrast restoration of weather degraded images," IEEE Trans. Pattern Anal. Mach. Intell 25(6), 713-724 (2003).
[CrossRef]

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, "Polarization based vision through haze," Appl. Opt. 42(3), 511-525 (2003).
[CrossRef]

2002 (1)

S. G. Narasimhan and S. K. Nayar, "Vision and the atmosphere," Int’l J. Computer Vision 48(3), 233-254 (2002).
[CrossRef]

2001 (2)

P. S. Umesh Adiga and B. B. Chaudhury, "Some efficient methods to correct confocal images for easy interpretation," Micron. 32, 363-370 (2001).
[CrossRef]

K. Tan and J. P. Oakley, "Physics Based Approach to color image enhancement in poor visibility conditions," J. Optical Soc. Am. 18(10), 2460-2467 (2001).

2000 (1)

K. Tan and J. P. Oakley, "Enhancement of color images in poor visibility conditions," Proc. Int’l Conf. Image Process. 2, 788-791 (2000).

1998 (1)

J. P. Oakley and B. L. Satherley, "Improving image quality in poor visibility conditions using a physical model for degradation," IEEE Trans. Image Process 7(2), 167-179 (1998).
[CrossRef]

1996 (1)

D. Kundur, D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. pp. 43-64, May 1996).

1994 (1)

P. Shaw, "Deconvolution in 3-D optical microscopy," Histochem. J. 261573-6865 (1994).
[CrossRef]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992).
[CrossRef]

1984 (1)

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of mages," IEEE Trans. Pattern Anal. Mach. Intell 6, 721-741 (1984).
[CrossRef]

Capek, M.

M. Capek, J. Janacek, and L. Kubinova, "Methods for compensation of the light attenuation with depth of images captured by a confocal microscope," Microscopy Res. Tech. 69, 624-635 (2006).
[CrossRef]

Chaudhury, B. B.

P. S. Umesh Adiga and B. B. Chaudhury, "Some efficient methods to correct confocal images for easy interpretation," Micron. 32, 363-370 (2001).
[CrossRef]

Combettes, P. L.

P. L. Combettes, and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

Del Bene, F.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992).
[CrossRef]

Geman, D.

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of mages," IEEE Trans. Pattern Anal. Mach. Intell 6, 721-741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of mages," IEEE Trans. Pattern Anal. Mach. Intell 6, 721-741 (1984).
[CrossRef]

Greger, K.

K. Greger, J. Swoger, and E. H. K. Stelzer, "Basic building units and properties of a fluorescence single plane illumination microscope," Rev. Sci. Instrum 78, 023705 (2007).
[CrossRef] [PubMed]

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

Hatzinakos, D.

D. Kundur, D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. pp. 43-64, May 1996).

Huisken, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Janacek, J.

M. Capek, J. Janacek, and L. Kubinova, "Methods for compensation of the light attenuation with depth of images captured by a confocal microscope," Microscopy Res. Tech. 69, 624-635 (2006).
[CrossRef]

Keller, P. J.

P. J. Keller, F. Pampaloni, E. H. K. Stelzer, "Life sciences require the third dimension," Curr. Opin. Cell Biol. 18, 117-124 (2006).
[CrossRef] [PubMed]

Kubinova, L.

M. Capek, J. Janacek, and L. Kubinova, "Methods for compensation of the light attenuation with depth of images captured by a confocal microscope," Microscopy Res. Tech. 69, 624-635 (2006).
[CrossRef]

Kubitscheck, U.

Kundur, D.

D. Kundur, D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. pp. 43-64, May 1996).

Marcello, M.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

Narasimhan, S. G.

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, "Polarization based vision through haze," Appl. Opt. 42(3), 511-525 (2003).
[CrossRef]

S. G. Narasimhan and S. K. Nayar, "Contrast restoration of weather degraded images," IEEE Trans. Pattern Anal. Mach. Intell 25(6), 713-724 (2003).
[CrossRef]

S. G. Narasimhan and S. K. Nayar, "Vision and the atmosphere," Int’l J. Computer Vision 48(3), 233-254 (2002).
[CrossRef]

Nayar, S. K.

S. G. Narasimhan and S. K. Nayar, "Contrast restoration of weather degraded images," IEEE Trans. Pattern Anal. Mach. Intell 25(6), 713-724 (2003).
[CrossRef]

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, "Polarization based vision through haze," Appl. Opt. 42(3), 511-525 (2003).
[CrossRef]

S. G. Narasimhan and S. K. Nayar, "Vision and the atmosphere," Int’l J. Computer Vision 48(3), 233-254 (2002).
[CrossRef]

Nehorai, A.

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Process. Mag. pp. 32-45, May 2006.

Oakley, J. P.

K. Tan and J. P. Oakley, "Physics Based Approach to color image enhancement in poor visibility conditions," J. Optical Soc. Am. 18(10), 2460-2467 (2001).

K. Tan and J. P. Oakley, "Enhancement of color images in poor visibility conditions," Proc. Int’l Conf. Image Process. 2, 788-791 (2000).

J. P. Oakley and B. L. Satherley, "Improving image quality in poor visibility conditions using a physical model for degradation," IEEE Trans. Image Process 7(2), 167-179 (1998).
[CrossRef]

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992).
[CrossRef]

Pampaloni, F.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

P. J. Keller, F. Pampaloni, E. H. K. Stelzer, "Life sciences require the third dimension," Curr. Opin. Cell Biol. 18, 117-124 (2006).
[CrossRef] [PubMed]

Pesquet, J. C.

P. L. Combettes, and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

Ritter, J. G.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992).
[CrossRef]

Sarder, P.

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Process. Mag. pp. 32-45, May 2006.

Satherley, B. L.

J. P. Oakley and B. L. Satherley, "Improving image quality in poor visibility conditions using a physical model for degradation," IEEE Trans. Image Process 7(2), 167-179 (1998).
[CrossRef]

Schechner, Y. Y.

Shaw, P.

P. Shaw, "Deconvolution in 3-D optical microscopy," Histochem. J. 261573-6865 (1994).
[CrossRef]

Siebrasse, J.

Stelzer, E. H. K.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

K. Greger, J. Swoger, and E. H. K. Stelzer, "Basic building units and properties of a fluorescence single plane illumination microscope," Rev. Sci. Instrum 78, 023705 (2007).
[CrossRef] [PubMed]

P. J. Keller, F. Pampaloni, E. H. K. Stelzer, "Life sciences require the third dimension," Curr. Opin. Cell Biol. 18, 117-124 (2006).
[CrossRef] [PubMed]

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Swoger, J.

K. Greger, J. Swoger, and E. H. K. Stelzer, "Basic building units and properties of a fluorescence single plane illumination microscope," Rev. Sci. Instrum 78, 023705 (2007).
[CrossRef] [PubMed]

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Tan, K.

K. Tan and J. P. Oakley, "Physics Based Approach to color image enhancement in poor visibility conditions," J. Optical Soc. Am. 18(10), 2460-2467 (2001).

K. Tan and J. P. Oakley, "Enhancement of color images in poor visibility conditions," Proc. Int’l Conf. Image Process. 2, 788-791 (2000).

Umesh Adiga, P. S.

P. S. Umesh Adiga and B. B. Chaudhury, "Some efficient methods to correct confocal images for easy interpretation," Micron. 32, 363-370 (2001).
[CrossRef]

Veith, R.

Verveer, P. J.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

Wittbrodt, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Appl. Opt. (1)

Curr. Opin. Cell Biol. (1)

P. J. Keller, F. Pampaloni, E. H. K. Stelzer, "Life sciences require the third dimension," Curr. Opin. Cell Biol. 18, 117-124 (2006).
[CrossRef] [PubMed]

Histochem. J. (1)

P. Shaw, "Deconvolution in 3-D optical microscopy," Histochem. J. 261573-6865 (1994).
[CrossRef]

IEEE Signal Process. Mag. (2)

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Process. Mag. pp. 32-45, May 2006.

D. Kundur, D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. pp. 43-64, May 1996).

IEEE Trans. Image Process (1)

J. P. Oakley and B. L. Satherley, "Improving image quality in poor visibility conditions using a physical model for degradation," IEEE Trans. Image Process 7(2), 167-179 (1998).
[CrossRef]

IEEE Trans. Image Process. (1)

P. L. Combettes, and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

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

S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of mages," IEEE Trans. Pattern Anal. Mach. Intell 6, 721-741 (1984).
[CrossRef]

S. G. Narasimhan and S. K. Nayar, "Contrast restoration of weather degraded images," IEEE Trans. Pattern Anal. Mach. Intell 25(6), 713-724 (2003).
[CrossRef]

Int’l J. Computer Vision (1)

S. G. Narasimhan and S. K. Nayar, "Vision and the atmosphere," Int’l J. Computer Vision 48(3), 233-254 (2002).
[CrossRef]

J. Optical Soc. Am. (1)

K. Tan and J. P. Oakley, "Physics Based Approach to color image enhancement in poor visibility conditions," J. Optical Soc. Am. 18(10), 2460-2467 (2001).

Micron. (1)

P. S. Umesh Adiga and B. B. Chaudhury, "Some efficient methods to correct confocal images for easy interpretation," Micron. 32, 363-370 (2001).
[CrossRef]

Microscopy Res. Tech. (1)

M. Capek, J. Janacek, and L. Kubinova, "Methods for compensation of the light attenuation with depth of images captured by a confocal microscope," Microscopy Res. Tech. 69, 624-635 (2006).
[CrossRef]

Nature Methods (1)

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E. H. K. Stelzer. "High-resolution threedimensional imaging of large specimens with light sheet-based microscopy," Nature Methods 4(4), 311-313 (2007).

Opt. Express (1)

Physica D (1)

L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992).
[CrossRef]

Proc. Int’l Conf. Image Process. (1)

K. Tan and J. P. Oakley, "Enhancement of color images in poor visibility conditions," Proc. Int’l Conf. Image Process. 2, 788-791 (2000).

Rev. Sci. Instrum (1)

K. Greger, J. Swoger, and E. H. K. Stelzer, "Basic building units and properties of a fluorescence single plane illumination microscope," Rev. Sci. Instrum 78, 023705 (2007).
[CrossRef] [PubMed]

Science (1)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Other (7)

JamesB.  Pawley ed. Handbook of Biological Confocal Microscopy Third Edition (Springer, New York, 2005).

Y. Y. Schechner, and N. Karpel, "Clear underwater vision," Proc. IEEE Conf. Computer Vision and Pattern Recognition 1, 536-543 (2004).

A. R. Patternson, A first course in fluid dynamics (Cambridge university press 1989).

J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer 1995).

S. G. Narasimhan and S. K. Nayar, "Removing weather effects from monochrome images," Proc. IEEE Conf. Computer Vision and Pattern Recognition 2, 186-193 (2001).

S. G. Narasimhan and S. K. Nayar, "Chromatic framework for vision in bad weather," Proc. IEEE Conf. Computer Vision and Pattern Recognition 1 598-605 (2000).
[CrossRef]

R. Kaftory, Y. Y. Schechner, and Y. Y. Zeevi, "Variational distance-dependent image restoration," Proc. IEEE Conf. Computer Vision and Pattern Recognition (2007).

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Figures (9)

Fig. 1.
Fig. 1.

Attenuation phenomenon due to incident of light on a specimen. dl and dA indicate infinitesimal length and area of the specimen, respectively.

Fig. 2.
Fig. 2.

Total number of photons emitted by an infinitesimal volume d r′ is given by n(r′)ρem(r′)d r′. A fraction of this light reaches point r and the number of photons per unit time is given by Eq. (8).

Fig. 3.
Fig. 3.

Geometry for confocal microscopy. Light is focused into a point and scattered light is collected along the same path as the incident light. Hence in a confocal geometry, the attenuation term sums over all light rays to and from the focusing lens to the focal point r f .

Fig. 4.
Fig. 4.

In the image acquisition process, the sample is scanned in discrete locations to generate z-stacks. The mean value of ρ can be calculated over the light cone area for each z-stack as shown by the shaded area in this figure.

Fig. 5.
Fig. 5.

Geometrical arrangement of side scattering, the light source originates from the side and illuminates one plane of the sample. Scattered light is collected in an orthogonal direction by a CCD camera.

Fig. 6.
Fig. 6.

Max. intensity projection (intensity profile) of 3D images. The top left figure shows that uniform illumination can be restored. Parameter calibration for the microscope setup is α′βn 0=181.27 (insert). Lower values, 121.51 and 90.02, resulted in over compensation. The bottom left figure shows that we can use the calibrated values of the top figure to restore images taken with different laser intensities, i.e. 1.5 and 2.0 times the laser intensity used for the top figure. Right figures show the 2D projections of original and restored images.

Fig. 7.
Fig. 7.

Restoration results of a set of confocal microscopy images, which has 155 z-stacks. The parameter 1/α′βn 0=0.014995 gives the best restoration. In the original views the illumination is not uniform, where as in our restored view the illumination becomes uniform. Maximum intensity projection graphs (averaging over the brightest 0.1% voxels of each z-stack) of all z-stacks are also shown. Solid- and dashed-lines indicate original and restored data, respectively.

Fig. 8.
Fig. 8.

Restoration results of another set of confocal microscopy images, which consists of 163 z-stacks. The parameter 1/α′βn 0=0.014995 gives the best restoration. In the original views the illumination is not uniform, where as in our restored view the illumination becomes uniform. Maximum intensity projection graphs (averaging over the brightest 0.1% voxels of each z-stack) of all z-stacks are also shown. Solid- and dashed-lines indicate original and restored data, respectively.

Fig. 9.
Fig. 9.

Restoration of a synthetically degraded image to an image of uniform illumination. The degraded image shows non-uniform illumination with maximum intensity projection falling off exponentially (assuming light source comes from the left). The parameter 1/α′n 0=0.0095 gives the best restoration in which the maximum intensity projection is flat.

Equations (31)

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

n(r)dA=f(r)dldtdA=f (r)cdAn(r)=f(r)(c)
dn(r)dl=n(r)ρab(r)
n(r)=n (rs)exp(γ(rs:r)ρab(r')dl)
nA(r)=ΣrsΩs,γ(rs:r)n(rs)exp(γ(rs:r)ρab(r')dl)
f(r)t+.f(r)v̂c=0
f(r)t=dn(r)dl.
f(r)t=n (r)ρab(r)
dns(r)=(n(r)ρem(r)dr4πrr2)(eγ(r:r)ρab(r)dl)
ns(r)=Ω,rr'n(r)ρem(r)eγ(r:r)ρab(r)dl4πrr2dr
n(r)=nA(r)+ns (r)
= ΣrsΩs,γ(rs:r) n (rs)eγ(rs:r)ρab(r)dl
+ rr n(r)ρem(r)eγ(r:r)ρab(r)dl4πrr2 d r
u0(rp)=rΩ,γ(r:rp)αγ ρem (r)n(r)eγρab(r)dldr
G(ri,rj)={exp(Σrkγ(rj:ri)ρab(rk)Δrk)4πrirj2ΔVij1ρem(ri)i=j
uiρem(ri)=bi + Σji (+exp(Σrkγ(rj:ri)ρab(rk)Δrk)4πrirj2ΔV) uj
G · u = b
ρ (z) = diskdxdyρxyzdiskdxdy
nA (r)=n0ΣrsΩs;γ(rs:r)exp (γ(rs:r)ρ(r)dl)
n0 exp (z=0rfρ(z)dz) ΣrsΩs;γ(rs:r)
= β n0 exp (z=0rfρ(z)dz)
u0(rp)=rΩ,γ(r:rp)αγq−1ρ(r)n(r)eγρ(r)dldr
n (rf)ρ(rf)ez=0rfρdzΣγ(rf,rp)q1αγΔvf
=α u (rf)ez=0rfρ(z)dz
ρA(ri)=u0iαβn0exp (2z=0ziρ(z)dz)
ρA(ri,z=1)=u0iαβn0exp [2ρA(z=0)Δz]
ρA(ri,z=k)=u0iα'βn0exp (2Σz=0k1ρA(z)Δz)
J(ρ)=bG·u
ρ* = argminρJ(ρ)
nA(r)=n0exp(γ(rs:r)ρ(r)dl)
u0(r)=αρ(r)n(r)=αu(r)
ρA(ri)=1α'n0 u0 (ri) exp (Σrkγ(rs,ri)ρ(rk)Δrk)

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