## A field theoretical restoration method for images degraded by non-uniform light attenuation : an application for light microscopy

Optics Express, Vol. 17, Issue 14, pp. 11294-11308 (2009)

http://dx.doi.org/10.1364/OE.17.011294

Acrobat PDF (356 KB)

### 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

## 1. Introduction

3. P. Shaw, “Deconvolution in 3-D optical microscopy,” Histochem. J. **26**1573–6865 (1994).
[CrossRef]

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 **60**259–268 (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]

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]

8. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization based vision through haze,” Appl. Opt. **42(3)**, 511–525 (2003).
[CrossRef]

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]

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]

## 2. Field theoretical formulation

^{3}that contains the whole imaging system, including the image object, the attenuation medium, light sources and detectors (e.g. camera). The light sources originate from infinity, we consider them to originate from the boundary of the region of interest

*∂*Ω. However, this is not a requirement in our formalism. Let

**r**

_{s}∊Ω

*be the set of points in the light sources and*

_{s}**r**

*be the locations of the voxels in the detector.*

_{p}### 2.1. Photon density and light intensity

*f*(

**r**) be the number of photons per unit volume and

*n*(

**r**) be the light intensity at a point

**r**∊Ω. Total number of photons in an infinitesimal volume

*dV=dldA*(Fig. 1) is

*f*(

**r**)

*dldA*. In the time interval

*dt*, the number of photons passing through the area

*dA*is,

*c*is the speed of light in the medium,

*n*(

**r**) is the number of photons passing through a unit area per unit time.

### 2.2. Attenuation and the absorption coefficient

*x*-axis at a point

**r**. The differential change of intensity through the medium with an infinitesimal thickness

*dl*is given by,

*ρ*

_{ab}(

**r**) is the absorption coefficient of light at a point r. In several papers [5

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]

8. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization based vision through haze,” Appl. Opt. **42(3)**, 511–525 (2003).
[CrossRef]

*ρ*

_{ab}is in general a function of the wavelength of light

*ρ*

_{ab}=

*ρ*

_{abλ}. For clarity in our derivation, we shall leave out the subscript

*λ*. Generalization of our equations to handle multiple

*λ*is tedious but trivial. For now, we shall assume a monochromatic formalism.

**r**

*to a point*

_{s}**r**, integrate Eq. (2) from

**r**

*to*

_{s}**r**,

*γ*(

**r**

*:*

_{s}**r**) is a light ray joining

**r**

*and*

_{s}**r**. Summing again over all rays from light sources to

**r**,

*A*stands for the intensity due to attenuation component.

*n*(

_{A}**r**) is the total light intensity at

**r**due to all light sources (

**r**

*∊Ω*

_{s}*). Eq. (4) states that light intensity decays exponentially in general, but the exponent may vary at different points in the image.*

_{s}### 2.4. Scattering and the emission coefficient

**r**due to scattering of the light from an infinitesimal volume

*d*

**r**′ is given by,

*S*stands for scattering component, and

*γ*(

**r**′ :

**r**) is a light ray from

**r**′ to

**r**. The denominator in the first term is a geometric factor that reflects the geometry of 3D space. The numerator is the amount of photons emitted per unit time by the volume element

*d*

**r**′. The second term represents the attenuation of light from

**r**′ to

**r**. Integrating over all

**r**′∊Ω,

**r**′=

**r**, the total scattered light received at

**r**is

### 2.5. Image formation

**r**∊Ω as a sum of attenuation and scattering components.

*d*

**r**is

*ρ*

_{em}(

**r**)

*n*(

**r**)

*d*

**r**. Suppose the detector detects a part of this light to form pixel

**r**

*in the 3-dimensional observed image*

_{p}*u*

_{0}(for example, in confocal microscopy), the pixel intensity at

**r**

*is given by,*

_{p}**r**∊Ω to

**r**

*. The attenuation term appears again in this equation as light is attenuated when traveling from the medium location*

_{p}**r**to the pixel location

**r**

*.*

_{p}*αγ*=

*α*(

_{γ}**r,r**

*) is a function that depends on the lensing system of the detector. The subscript*

_{p}*γ*is used to indicate that

*α*depends on the path of the light.

_{γ}*ρ*

_{ab}(

**r**) and

*ρ*

_{em}(

**r**). Thus we want to estimate

*ρ*

_{ab}(

**r**) and

*ρ*

_{em}(

**r**) given the observed image

*u*

_{0}(

**r**

*). To this end, we want to solve Eq (11) for*

_{p}*ρ*

_{ab}(

**r**) and

*ρ*

_{em}(

**r**). At first sight, this equation seems non-invertible and its solution may not be unique. In the latter sections, we will illustrate examples in which we solve Eq. (11) for confocal microscopy and for a side scattering configuration. At this point, we would like to highlight several observations:

*Geometry*: All geometrical information is embedded in the paths

*γ*(

**r**:

**r**′), which represents light rays from point

**r**to

**r**′.

*Light Source*: Light source information is given by the summation over Ω

*and*

_{s}*γ*(

**r**

*:*

_{s}**r**) in Eq. (4).

*Airlight*: The airlight effect [10

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]

*Non-unique solution*: The solution of Eq. (11) is non-unique in general. Consider a case when Ω contains an opaque box and an image is taken of this box. Since the box is opaque, the values of

*ρ*

_{ab}and

*ρ*

_{em}within the box are undefined.

### 2.6. Discretization

**r**. We perform our discretization such that each finite-element corresponds to one voxel in the image data. Let

*b*=

_{i}*n*(

_{A}**r**

*),*

_{i}**b**=(

*b*

_{1}, ⋯,

*b*),

_{N}*N*is the total number of voxels. Next define

*u*/

_{i}*ρ*

_{em}(

**r**

*)=*

_{i}*n*(

**r**

*i*),

*u*=(

*u*

_{1}, ⋯,

*u*). Finally, define a matrix

_{N}*G*with components

*G*=(

_{ij}**r**

*,*

_{i}**r**

*),*

_{j}## 3. Confocal microscopy

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]

**r**

*. The summation over all light rays in Eq. (4) sums over all rays from the focusing lens*

_{f}*γ*(

**r**

*:*

_{s}**r**

*). We can take the area of the lens to be a set of points in the incident light sources Ω*

_{f}*. Detected light travels via the same paths through the focusing lens. Hence the summation over all light rays in Eq. (11) sums over the same paths*

_{s}*γ*(

**r**

*:*

_{f}**r**

*) but in the opposite direction. Lastly, for clarity, define the symbol*

_{s}*ρ*(

**r**)=

*ρ*

_{ab}(

**r**)=

*q*

^{-1}

*ρ*

_{em}(

**r**).

*ρ*(

**r**) over the disk area of light cones for each z-stacks (shaded gray in Fig. 4).

*n*

_{0}).

**r**

*is collected by the photomultiplier (see Fig. 3). Hence replacing the*

_{f}*ρ*in Eq. (11) by 〈

*ρ*〉 defined in Eq. (14), we have,

*V*is the confocal volume and

_{f}*α*′=∑

*(*

_{γ}**r**

*,*

_{f}**r**

*)*

_{p}*q*

^{-1}

*αγ*Δ

*V*is complicated but otherwise constant number. We have also use

_{f}*u*(

**r**)=

*ρ*(

**r**)

*n*(

**r**).

*n*≪

_{S}*n*,

_{A}*n*(

**r**)≅

*n*(

_{A}**r**)). Substituting Eqs. (15) and (16) into Eq. (13),

*ρ*is the true light emission coefficient we are solving for if we neglect the scattering terms. We now give a description of how

_{A}*ρ*can be calculated from the observed image slice-by-slice through the z-stack.

_{A}*ρ*is calculated starting from the first slice.

_{A}*z*=0, the integral in Eq. (17) evaluates to zero so that

*ρ*is proportional to the intensity in the observed image.a i.e.

_{A}*ρ*(

_{A}**r**

*,*

_{i}*z*=0)=

*u*

_{0i}/

*α*′

*β n*

_{0}.

*ρ*for the second slice depends on

_{A}*ρ*for the first slice,

_{A}*z*is the thickness of the discretized z-stack. 〈

*ρ*〉 (

_{A}*z*=0) is an average value calculated using values of

*ρ*for the first slice.

_{A}*ρ*for the

_{A}*k*-th slice is given by,

*ρ*slice-by-slice starting from the first slice, the values of

_{A}*ρ*had been calculated for all (

_{A}*k*-1)th slice and hence the exponential term, ∑

^{k-1}

_{z=0}2〈

*ρ*〉(

_{A}*z*)Δ

*z*can be calculated easily.

*ρ*from first to last slice in sequence to obtain the whole restored image.

_{A}*ρ*>0, we use the absolute sign to avoid the Karush-Kuhn-Tucker condition. In the above equation

*b*=

_{i}*b*(

**r**

*)=*

_{i}*n*(

_{A}**r**

*) (Eq. (15)) and*

_{i}*u*=

_{i}*u*(

**r**

*)=*

_{i}*u*

_{0}(

**r**

*)exp(∫*

_{i}

^{r}

^{f}_{z=0}〈

*ρ*〉

*dz*)/

*α*′(Eq. (16)). To reduce computation time, we do sampling in calculating the mean

*ρ*(

**r**) over the disk area of light cones for each z-stacks.

*k*can be evaluated numerically. In practice,

*ρ*can be used for the initial guess of

_{A}*ρ*in the gradient descend method. Through our numerical simulations, we found that

*ρ*is a good approximation to

_{A}*ρ*. Using

*ρ*as initial condition reduces the local minimum problem in the gradient descend method.

_{A}## 4. Side scattering geometry

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]

25. P. J. Keller, F. Pampaloni, and 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, and U. Kubitscheck. “High-contrast single-particle tracking by selective focal plane illumination microscopy,” Opt. Express **16(10)**, 7142–7152 (2008).
[CrossRef]

*n*

_{0}at a point

**r**=(

*x,y*),

*x*-direction as shown in Fig. 5. As in the case of confocal microscopy, we used

*ρ*=

*q*

^{-1}

*ρ*

_{em}=

*ρ*

_{ab}.

**r**

*and the sample location*

_{p}**r**. Then Eq. (11) may be written as,

*α*′ represents the integrated effects of quantum yield and the camera, including summations over all rays etc. An analytic solution can be obtained (Eq. (23)) if we further ignore the scattering term,

*A*is used to indicate that an approximated solution is obtained using attenuation term only. With this approximation, Eq. (23) can be easily solved numerically and their results are given in the next section.

## 5. Results

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]

8. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization based vision through haze,” Appl. Opt. **42(3)**, 511–525 (2003).
[CrossRef]

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]

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]

### 5.1. Validation and calibration

*α*′

*βn*

_{0}=181.27 gives the best result. The calibrated parameter value can be used for images taken with different laser intensities, as shown in the bottom row of Fig. 6 for 1.5

*n*

_{0}and 2.0

*n*

_{0}. Let the value of the parameter be

*C*=

*α′β*, the relationship between two parameters (

_{n}*C*

_{1},

*C*

_{2}) with different laser intensities (

*n*

_{1},

*n*

_{2}) is simply

*C*

_{1}/

*C*

_{2}=

*n*

_{1}/

*n*

_{2}. Figures on the right of Fig. 6 shows the 2D projections of original and restored images for 1.5

*n*

_{0}and 2.0

*n*

_{0}. We can see that the restored image is more uniformly illuminated.

### 5.2. Confocal microscopy

*µ*m/pixel. The original microscope images are of size 512×512×

*n*voxels with a resolution of 0.137

_{z}*µ*m in the x- and y-direction and 0.2

*µ*m in the z-direction.

*n*is the number of z-stacks in the images.

^{z}>*n*voxels by averaging the voxels in the x- and y- direction while maintaining the resolution in the z-direction. Fig. 7 and 8 show restoration results for a 256×256×

^{z}*n*voxels images. Here we used Eq. (17) to restore the images. The adjusted tuning parameter 1/

_{z}*α′βn*

_{0}=0.014995 gives optimal restoration results.

*n*=155 and

_{z}*n*=163 z-stacks, respectively. Fig. 7(a) and 8(a) show the maximum intensity projection onto the yz-plane of the respective original images. Similarly, Fig. 7(b) and 8(b) show the maximum intensity projection onto the yz-plane of the respective restored images. And Fig. 7(c) and 8(c) show the maximum intensity projection (averaged over top 0.1% of the brightest voxels in the xy-plane) onto the z-axis. The illuminating laser originates from the bottom and one could easily observed that for the original image, the voxels are much brighter at the bottom of the image and intensity drops towards the top of the image. After restoration, the illumination becomes uniform. The restored image is also darker on the average, however, many image processing techniques are robust against the average voxel intensity. The achievement in our work is to restore the image to have uniform illumination. Afterwhich, other image enhancement methods such as histogram equalization can be used. Fig. 7(c) and 8(c) clearly show the difference between the intensity profile of the original (solid lines) and restored (dashed lines) images. Over-exposed areas in the bottom z-stacks are correctly compensated by our restoration method.

_{z}### 5.3. Side scattering microscopy

*α*′

*n*

_{0}can be adjusted to obtain optimal results.

*n*

_{0}is the incident light intensity and

*α*′ is a geometric factor that is usually unknown. For small 1/

*α*′

*n*

_{0}, the image is hardly restored and for large 1/

*α*′

*n*

_{0}, there is over-compensation of the attenuation effect. The optimal value of 1/

*α*′

*n*

_{0}is 0.0095 for this image in which the restored image is almost perfectly (uniformly) illuminated.

## 6. Conclusions

**7(2)**, 167–179 (1998).
[CrossRef]

**42(3)**, 511–525 (2003).
[CrossRef]

**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]

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]

## References and links

1. | James B. Pawley ed. |

2. | D. Kundur and 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. |

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 |

6. | K. Tan and J. P. Oakley, “Enhancement of color images in poor visibility conditions,” Proc. Int’l Conf. Image Process. |

7. | K. Tan and J. P. Oakley, “Physics Based Approach to color image enhancement in poor visibility conditions,” J. Optical Soc. Am. |

8. | Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization based vision through haze,” Appl. Opt. |

9. | Y. Y. Schechner and N. Karpel, “Clear underwater vision,” Proc. IEEE Conf. Computer Vision and Pattern Recognition |

10. | S. G. Narasimhan and S. K. Nayar, “Contrast restoration of weather degraded images,” IEEE Trans. Pattern Anal. Mach. Intell |

11. | S. G. Narasimhan and S. K. Nayar, “Vision and the atmosphere,” Int’l J. Computer Vision |

12. | S. G. Narasimhan and S. K. Nayar, “Removing weather effects from monochrome images,” Proc. IEEE Conf. Computer Vision and Pattern Recognition |

13. | S. G. Narasimhan and S. K. Nayar, “Chromatic framework for vision in bad weather,” Proc. IEEE Conf. Computer Vision and Pattern Recognition |

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 |

16. | L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D |

17. | P. L. Combettes and J. C. Pesquet, “Image restoration subject to a total variation constraint,” IEEE Trans. Image Process. |

18. | A. R. Patternson, |

19. | J. B. Pawley, |

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. |

21. | P. S. Umesh Adiga and B. B. Chaudhury, “Some efficient methods to correct confocal images for easy interpretation,” Micron. |

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 |

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 |

24. | P. J. Verveer, J. Swoger1, F. Pampaloni, K. Greger, M. Marcello, and E. H. K. Stelzer. “High-resolution threedimensional imaging of large specimens with light sheet-based microscopy,” Nature Methods |

25. | P. J. Keller, F. Pampaloni, and E. H. K. Stelzer, “Life sciences require the third dimension,” Curr. Opin. Cell Biol. |

26. | J. G. Ritter, R. Veith, J. Siebrasse, and U. Kubitscheck. “High-contrast single-particle tracking by selective focal plane illumination microscopy,” Opt. Express |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(100.2000) Image processing : Digital image processing

(100.3020) Image processing : Image reconstruction-restoration

(110.0180) Imaging systems : Microscopy

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 29, 2009

Revised Manuscript: June 16, 2009

Manuscript Accepted: June 18, 2009

Published: June 22, 2009

**Virtual Issues**

Vol. 4, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Hwee K. Lee, Mohammad S. Uddin, Shvetha Sankaran, Srivats Hariharan, and Sohail Ahmed, "A field theoretical restoration method for images degraded by non-uniform light attenuation : an application for light microscopy," Opt. Express **17**, 11294-11308 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11294

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### References

- JamesB. Pawley ed. Handbook of Biological Confocal Microscopy Third Edition (Springer, New York, 2005).
- D. Kundur, D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. pp. 43-64, May 1996).
- P. Shaw, "Deconvolution in 3-D optical microscopy," Histochem. J. 261573-6865 (1994). [CrossRef]
- P. Sarder, and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Process. Mag. pp. 32-45, May 2006.
- 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]
- K. Tan and J. P. Oakley, "Enhancement of color images in poor visibility conditions," Proc. Int’l Conf. Image Process. 2, 788-791 (2000).
- 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).
- Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, "Polarization based vision through haze," Appl. Opt. 42(3), 511-525 (2003). [CrossRef]
- Y. Y. Schechner, and N. Karpel, "Clear underwater vision," Proc. IEEE Conf. Computer Vision and Pattern Recognition 1, 536-543 (2004).
- 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]
- 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).
- 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]
- L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60259-268 (1992). [CrossRef]
- 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]
- A. R. Patternson, A first course in fluid dynamics (Cambridge university press 1989).
- J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer 1995).
- 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]
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