## Blind deconvolution in optical diffusion tomography

Optics Express, Vol. 10, Issue 1, pp. 46-53 (2002)

http://dx.doi.org/10.1364/OE.10.000046

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### Abstract

We use blind deconvolution methods in optical diffusion tomography to reconstruct images of objects imbedded in or located behind turbid media from continuous-wave measurements of the scattered light transmitted through the media. In particular, we use a blind deconvolution imaging algorithm to determine both a deblurred image of the object and the depth of the object inside the turbid medium. Preliminary results indicate that blind deconvolution produces better reconstructions than can be obtained using backpropagation techniques. Moreover, it does so without requiring prior knowledge of the characteristics of the turbid medium or of what the blur-free target should look like: important advances over backpropagation.

© Optical Society of America

## 1. Introduction

_{o}plane. Other geometries could be considered, such as a semi-infinite medium or an infinite slab. The experimental results in this paper are generated using an essentially infinite slab, but the theory is presented for an infinite geometry to simplify the mathematics. The spatial distribution of the light in the turbid medium, ϕ(x, y, z) , is given by

_{hom}(x, y,z) is the spatial distribution of the light in the absence of the imbedded object (the homogeneous or background light) and ϕ

_{pert}(x, y, z) represents the perturbation of the light due to the presence of the object. In the case of an absorptive object, under the Born approximation [1

1. E. P. Zege, A. P. Ivanov, and I. L. Katsev, *Image transfer through a scattering medium*, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6. [CrossRef]

2. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express **1**, 6–11 (1997) http://www.opticsexpress.org/oearchive/source/1884.htm. [CrossRef] [PubMed]

_{o}plane, ϕ(x,y,z

_{o}), can be considered as the superposition of two components: a blurred component ϕ

_{pert}(x,y,z

_{o}) from the light that is scattered by the object, and a background component ϕ

_{hom}(x,y,z

_{o}) from the incident light scattered by the turbid medium. The inverse problem associated with imaging the imbedded object requires removing both the background component ϕ

_{hom}(x,y,z

_{o}) and the blurring h(x,y,z) from ϕ

_{pert}(x,y,z). Matson and Liu [3

3. C. L. Matson and H. Liu, “Backpropagation in turbid media,” J. Opt. Soc. Am. A **16**, 1254–1265 (1999). [CrossRef]

_{o}(see Fig. 1), Eq. (2) can be rewritten as [4

4. C. L. Matson, “Deconvolution-based spatial resolution in optical diffusion tomography,” Appl. Opt. **40**, 5791–5801 (2001). [CrossRef]

_{1}is the largest z value inside the object support, and the depth parameter Δz = z

_{o}-z

_{1}is the distance between the part of the object closest to the detection plane and the detection plane. The difference between Eq.(2) and Eqs.(3) and (4) is that the integration indicated in Eq.(2), which is over all space, is broken up into two pieces. The first piece is the integration over the object support (Eq.(4)), and the second piece is the propagation of the light perturbed by the object (o

_{2D}(x,y,z

_{1})) to the detection plane by using the angular spectrum solution to the homogeneous source-free scalar wave propagation [5]. The angular spectrum propagation operation is indicated by the convolution of o

_{2D}(x,y,z

_{1}) with h(x,y,Δz), where the latter term is the two-dimensional inverse Fourier transform of the angular spectrum transfer function for the distance Δz and is referred to as the system point spread function (PSF) throughout this paper. Equation (3) indicates that the image reconstruction problem can be viewed as 2D deconvolution, where the desired reconstructed image is o

_{2D}(x,y,z

_{1}); that is, the desired reconstruction is the projection of the 3D object structure into the z=z

_{1}plane. To obtain the absorptive properties of the object from the deconvolved data requires solving Eq.(4) and requires multiple views, in general [3

3. C. L. Matson and H. Liu, “Backpropagation in turbid media,” J. Opt. Soc. Am. A **16**, 1254–1265 (1999). [CrossRef]

3. C. L. Matson and H. Liu, “Backpropagation in turbid media,” J. Opt. Soc. Am. A **16**, 1254–1265 (1999). [CrossRef]

## 2. Blind deconvolution algorithm

6. R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP **104**, 1096–1103 (1992). [CrossRef]

_{2D}(x,y)] and PSF [h(x, y)] intensity distributions that minimize the cost function

_{d}(x, y) is a binary function that masks out bad pixels or regions of low signal-to-noise ratio in the recorded image, n(x, y) is the detector read noise, and ⊗ is the correlation operator. The weighting term (the denominator) allows for both Poisson and Gaussian noise [7

7. M. Lloyd-Hart, S. M. Jefferies, E. K. Hege, and J. R. P. Angel, “Wave Front Sensing with Time-of-flight Phase Diversity,” Opt. Lett. **26**, 402–404 (2001). [CrossRef]

_{1}in the argument list of o

_{2D}and the term Δz in the argument list of h for clarity. The iterative approach uses a conjugate gradient routine to find the intensity distributions that minimize ε.

_{o}(x,y)and s

_{h}(x,y)are binary functions that designate the spatial extent of o

_{2D}(x,y)and h(x,y),and θ(x,y) and ψ(x, y) are the parameters to be optimized. The PSF reparameterization includes a correlation (band-limit) function,

*τ*(x,y), [8

8. D. G. Sheppard, B. R. Hunt, and M.W. Marcellen, “Iterative multi-frame super-resolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A **15**, 978–992 (1998). [CrossRef]

1. E. P. Zege, A. P. Ivanov, and I. L. Katsev, *Image transfer through a scattering medium*, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6. [CrossRef]

2. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express **1**, 6–11 (1997) http://www.opticsexpress.org/oearchive/source/1884.htm. [CrossRef] [PubMed]

9. C. L. Matson, N. Clark, L. McMackin, and J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. **36**, 214–220 (1997). [CrossRef] [PubMed]

*α*depends on the scattering and absorption characteristics of the turbid medium. The incoherent, band-limited PSF is given by the inverse Fourier transform

*τ*(x, y) and N

^{2}is the number of pixels in the image. In practice, we model the PSF via the turbid medium OTF during the early part of the iteration, and then change to the pixel-by-pixel PSF model when the object reconstruction appears to be fairly well established. The reason for this is that Eq. (9) is quantitatively valid only for an infinite and homogeneous medium, and we are inherently dealing with a finite, inhomogeneous medium (due to the presence of an object). By switching to the pixel-by-pixel model during the later iterations, we help avoid artifacts in the reconstructed object due to shortcomings in the PSF model.

## 3. Experimental setup

11. C. L. Matson and H. Liu, “Resolved object imaging and localization with the use of a backpropagation algorithm,” Opt. Express **6**, 168–174 (2000), http://www.opticsexpress.org/oearchive/source/19570.htm. [CrossRef] [PubMed]

_{2}to give scattering properties, and a dye to give absorption properties [11

11. C. L. Matson and H. Liu, “Resolved object imaging and localization with the use of a backpropagation algorithm,” Opt. Express **6**, 168–174 (2000), http://www.opticsexpress.org/oearchive/source/19570.htm. [CrossRef] [PubMed]

_{a}) and reduced scattering (μ’

_{s}) coefficients for the phantoms were 0.01cm

^{-1}and 18 cm

^{-1}, respectively. The scattered light was recorded using a SpectraSource 16 bit 512×512 CCD camera and stored in a computer. The targets are shown in Fig. 3. The blurred images of the targets were produced by subtracting from the blurred target measurement a second measurement of the same turbid media without a target present. We realize that it is generally not practical to measure and remove the background component of the measured signal, either in the field (surveillance), or

*in vivo*(biomedical). As such, this approach has limited applicability. However, it does allow us to make a preliminary assessment of blind deconvolution as a potential tool for post processing imagery obtained through a turbid medium.

## 4. Reconstruction results

*α*, we have found that there are several {Δz, α} pairs that produce almost identical PSF profiles, and thus essentially the same values for the cost function. This degeneracy in the PSF is reminiscent of what Carasso [12

12. A. S. Carasso, “Direct blind deconvolution,” SIAM J. Appl. Math. **61**, 1980–2007 (2001). [CrossRef]

13. S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. **415**, 862–864 (1993). [CrossRef]

11. C. L. Matson and H. Liu, “Resolved object imaging and localization with the use of a backpropagation algorithm,” Opt. Express **6**, 168–174 (2000), http://www.opticsexpress.org/oearchive/source/19570.htm. [CrossRef] [PubMed]

**16**, 1254–1265 (1999). [CrossRef]

## 5. Discussion

*in vivo*(biomedical). In principle, simultaneous estimation of the homogeneous wave should be possible as it only requires the estimation of one extra parameter. Thus, we are currently investigating how to adapt our BD algorithm to estimate the background signal as well.

## Acknowledgements

## Footnotes

1 | Although a transmission geometry was used, a reflection geometry could just as easily have been employed [10 10. X. Cheng and D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express |

## References and Links

1. | E. P. Zege, A. P. Ivanov, and I. L. Katsev, |

2. | C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express |

3. | C. L. Matson and H. Liu, “Backpropagation in turbid media,” J. Opt. Soc. Am. A |

4. | C. L. Matson, “Deconvolution-based spatial resolution in optical diffusion tomography,” Appl. Opt. |

5. | J. W. Goodman, |

6. | R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP |

7. | M. Lloyd-Hart, S. M. Jefferies, E. K. Hege, and J. R. P. Angel, “Wave Front Sensing with Time-of-flight Phase Diversity,” Opt. Lett. |

8. | D. G. Sheppard, B. R. Hunt, and M.W. Marcellen, “Iterative multi-frame super-resolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A |

9. | C. L. Matson, N. Clark, L. McMackin, and J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. |

10. | X. Cheng and D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express |

11. | C. L. Matson and H. Liu, “Resolved object imaging and localization with the use of a backpropagation algorithm,” Opt. Express |

12. | A. S. Carasso, “Direct blind deconvolution,” SIAM J. Appl. Math. |

13. | S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(170.7050) Medical optics and biotechnology : Turbid media

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 30, 2001

Published: January 14, 2002

**Citation**

Stuart Jefferies, Kathy Schulze, Charles Matson, Kurt Stoltenberg, and E. Keith Hege, "Blind deconvolution in optical diffusion tomography," Opt. Express **10**, 46-53 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-46

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### References

- E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image transfer through a scattering medium, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6. [CrossRef]
- C. L. Matson, "A diffraction tomographic model of the forward problem using diffuse photon density waves," Opt. Express 1, 6-11 (1997) <a href="http://www.opticsexpress.org/oearchive/source/1884.htmF">http://www.opticsexpress.org/oearchive/source/1884.htm</a>. [CrossRef] [PubMed]
- C. L. Matson and H. Liu, "Backpropagation in turbid media," J. Opt. Soc. Am. A 16, 1254-1265 (1999). [CrossRef]
- C. L. Matson, "Deconvolution-based spatial resolution in optical diffusion tomography," Appl. Opt. 40, 5791-5801 (2001). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, 1996), Chap. 3.
- R. K. Pina and R. C. Puetter, "Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion," PASP 104, 1096-1103 (1992). [CrossRef]
- M. Lloyd-Hart, S. M. Jefferies, E. K. Hege, and J. R. P. Angel, "Wave Front Sensing with Time-of-flight Phase Diversity," Opt. Lett. 26, 402-404 (2001). [CrossRef]
- D. G. Sheppard, B. R. Hunt, and M.W. Marcellen, "Iterative multi-frame super-resolution algorithms for atmospheric-turbulence-degraded imagery," J. Opt. Soc. Am. A 15, 978-992 (1998). [CrossRef]
- C. L. Matson, N. Clark, L. McMackin, and J. S. Fender, "Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves," Appl. Opt. 36, 214-220 (1997). [CrossRef] [PubMed]
- X. Cheng and D. A. Boas, "Diffuse optical reflection tomography with continuous-wave illumination," Opt. Express 3, 118-123 (1998), <a href="http://www.opticsexpress.org/oearchive/source/5663.htm">http://www.opticsexpress.org/oearchive/source/5663.htm</a>. [CrossRef] [PubMed]
- C. L. Matson and H. Liu, "Resolved object imaging and localization with the use of a backpropagation algorithm," Opt. Express 6, 168-174 (2000), <a href="http://www.opticsexpress.org/oearchive/source/19570.htm">http://www.opticsexpress.org/oearchive/source/19570.htm</a>. [CrossRef] [PubMed]
- A. S. Carasso, "Direct blind deconvolution," SIAM J. Appl. Math. 61, 1980-2007 (2001). [CrossRef]
- S. M. Jefferies and J. C. Christou, "Restoration of astronomical images by iterative blind deconvolution," Astro. Phys. J. 415, 862-864 (1993). [CrossRef]

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