## Optical image reconstruction based on the third-order diffusion equations

Optics Express, Vol. 4, Issue 8, pp. 241-246 (1999)

http://dx.doi.org/10.1364/OE.4.000241

Acrobat PDF (118 KB)

### Abstract

This paper presents a third-order diffusion equations-based optical image reconstruction algorithm. The algorithm has been implemented using finite element discretizations coupled with a hybrid regularization that combines both Marquardt and Tikhonov schemes. Numerical examples are used to compare between the third- and first-order reconstructions. The results show that the third-order reconstruction codes are more stable than the first-order codes, and are capable of reconstructing void-like regions. From the examples given, it has also been shown that the first-order codes fail to both qualitatively and quantitatively reconstruct the void-like regions.

© Optical Society of America

## 1. Introduction

1. R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

12. D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B **352**, 649–659 (1997). [CrossRef]

## 2. Reconstruction algorithm

16. H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE **2389**, 608–614 (1995). [CrossRef]

^{(1)}, Φ

^{(2)}, Φ

^{(3)}, and Φ

^{(4)}are the first four components in the spherical harmonic expansion of the photon radiance where the first component, Φ

^{(1)}, is the average diffused photon density. μ

_{a}is the absorption coefficient. μ′

_{t}= μ

_{a}+ (1 - g)μ

_{s}, where μ

_{s}is the scattering coefficient and g is the average cosine of the scattering angle. D = 1 / 3μ′

_{t}is the diffusion coefficient. x̂ and ŷ are the unit vectors along x and y axes, respectively. S is the light source term.

^{(1)–(4)}, D, and μ

_{a}have been expanded as the sum of coefficients multiplied by a set of locally spatially-varying Lagrangian basis functions ϕ ϕ

_{p}, and ϕ

_{q}. ∮ expresses integration over the boundary surface. N is the node number of a finite element mesh. The expansions used to represent D and μ

_{a}are P and Q terms long where P ≠ Q ≠ N in general; however, in the study reported here P=Q=N.

_{a}images from presumably uniform initial estimates of the spatial D and μ

_{a}distributions. Thus, we need a way of updating D and μ

_{a}from their starting values. Following the inverse procedures outlined in [8

8. K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. **22**, 691–702 (1995). [CrossRef] [PubMed]

9. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A **13**, 253–266 (1996). [CrossRef]

_{a}is obtained:

_{1}, ∆D

_{2},…∆D

_{N},∆μ

_{a,1}, ∆μ

_{a,2},…∆μ

_{a, N})

^{T}is the update vector for the optical property profiles. Φ

^{o}= (

^{T}and Φ

^{c}= (

^{T}, where

^{(1)}, is used in Eq. (9) since it is the dominant component [16

16. H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE **2389**, 608–614 (1995). [CrossRef]

^{(2)–(4)}, are set to zeros at the boundary. In Eq. (9), the decomposition of the ill-conditioned matrix

*ℑ*

^{T}

*ℑ*is stabilized by a synthesized Marquardt and Tikhonov regularization scheme [8

8. K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. **22**, 691–702 (1995). [CrossRef] [PubMed]

9. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A **13**, 253–266 (1996). [CrossRef]

## 3. Numerical examples of reconstruction

_{s}=2.0 mm

^{-1}, μ

_{a}=0.012 mm

^{-1}; the optical properties for the background are μ′

_{s}=1.0 mm

^{-1}, μ

_{a}=0.006 mm

^{-1}. For the second case, the optical properties for the target are μ′

_{s}=0.01 mm

^{-1}, μ

_{a}=0.005 mm

^{-1}; the optical properties for the background are μ′

_{s}=1.0 mm

^{-1}, μ

_{a}=0.01 mm

^{-1}. The first case is used to just demonstrate the implementation of our third-order reconstruction codes. The purpose of the second case is to test if the third-order codes can reconstruct a void-like region and if it can provide more stable reconstructions than the first-order codes when noisy data is used. The optical properties assigned to the void-like region is similar to that in the CSF layer in brain tissue [14

14. M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. **41**767–783 (1996). [CrossRef]

_{a}in place. Fig. 1 (b, c) shows the D and μ

_{a}images for the first case reconstructed under conditions of no noise. As can be seen, the images are clearly recovered.

_{a}images for the second case. In order to provide a comparison, D and μ

_{a}images reconstructed using our first-order codes are displayed in Fig. 2 (c, d). A number of observations can be made from Fig. 2. The almost non-scattering, void-like target can be qualitatively recovered for both D and μ

_{a}images using the third-order codes [Fig. 2 (a, b)], whereas it cannot be correctly recovered using the first-order codes [Fig. 2 (c, d)]. Interestingly, the target location for both D and μ

_{a}images recovered is incorrectly “swapped” to the left when the firs-order codes were used. From Fig. 2 (a, b), one can see that the third-order codes can produce correct reconstructions of the target location and shape. While the reconstructed target size for D image is correct [Fig. 2 (a)], the recovered target size for μ

_{a}image is larger than the exact target size. When the first-order codes were used, the recovered target size, location and shape for both D and μ

_{a}images are totally incorrect. From Fig. 2, it can be seen that both D and μ

_{a}images can be quantitatively reconstructed using the third-order codes [Fig. 2 (a, b)], whereas the recovered values of both D and μ

_{a}in the target region are all “swapped” with respect to the exact values when the first-order codes were used [Fig. 2 (c, d)]. However, it is interesting to note that the first-order codes produce better background region reconstruction than the third-order codes. Given the facts that the third-order codes can reconstruct the void-like regions from noisy data and the first-order codes cannot do so, one can also see that the third-order codes are more stable than the first-order codes.

## 4. Conclusions

10. K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. **35**, 3447–3458 (1996). [CrossRef] [PubMed]

11. H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. **25**, 183–193 (1998). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120. |

2. | S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. |

3. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. |

4. | X. D. Li, T Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction Tomography for biomedical imaging with diffuse-photon density waves,” Opt. Lett. |

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

6. | S. A. Walker, S. Fantini, and E. Gratton, “Image reconstruction by backprojection from frequency domain optical measurements in highly scattering media,” Appl. Opt. |

7. | S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, and N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. |

8. | K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. |

9. | H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A |

10. | K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. |

11. | H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. |

12. | D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B |

13. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. |

14. | M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. |

15. | In Mathematics and Physics of Emerging Biomedical Imaging (National Academy Press, Washington, D.C., 1996). |

16. | H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE |

17. | D. A. Boas et al, ”Photon Migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, Proc. SPIE |

18. | D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Optics Express |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.6960) Imaging systems : Tomography

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3660) Medical optics and biotechnology : Light propagation in tissues

**ToC Category:**

Focus Issue: Biomedical diffuse optical tomography

**History**

Original Manuscript: March 1, 1999

Published: April 12, 1999

**Citation**

Huabei Jiang, "Optical image reconstruction based on the third-order
diffusion equations," Opt. Express **4**, 241-246 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-8-241

Sort: Journal | Reset

### References

- R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, "A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data," in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87-120.
- S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modeling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997). [CrossRef] [PubMed]
- M. A. OLeary, D. A. Boas, B. Chance, and A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography," Opt. Lett. 20, 426-428 (1995). [CrossRef]
- X. D. Li, T Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, "Diffraction Tomography for biomedical imaging with diffuse-photon density waves," Opt. Lett. 22, 573-575 (1997). [CrossRef] [PubMed]
- C. L. Matson, "A diffraction tomographic model of the forward problem using diffuse photon density waves," Optics Express 1, 6-12 (1997); http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]
- S. A. Walker, S. Fantini, and E. Gratton, "Image reconstruction by backprojection from frequency domain optical measurements in highly scattering media," Appl. Opt. 36, 170-179 (1997). [CrossRef] [PubMed]
- S. B. Colak, D. G. Papaioannou, G. W. t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, and N. A. A. J. van Asten, "Tomographic image reconstruction from optical projections in light- diffusing media," Appl. Opt. 36, 180-213 (1997). [CrossRef] [PubMed]
- K. D. Paulsen, H. Jiang, "Spatially-varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-702 (1995). [CrossRef] [PubMed]
- H. Jiang, K. D. Paulsen and U. L. Osterberg, B. W. Pogue and M. S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996). [CrossRef]
- K. D. Paulsen and H. Jiang, "Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization," Appl. Opt. 35, 3447-3458 (1996). [CrossRef] [PubMed]
- H. Jiang, K. D. Paulsen, U. L. Osterberg and M. S. Patterson, "Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms," Med. Phys. 25, 183-193 (1998). [CrossRef] [PubMed]
- D. T. Delpy, M. Cope, "Quantification in tissue near-infrared spectroscopy," Phil. Trans. R. Soc. Lond. B 352, 649-659 (1997). [CrossRef]
- A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998). [CrossRef] [PubMed]
- M. Firbank, S. Arridge, M. Schweiger and D. Delpy, "An investigation of light transport through scattering bodies with non- scattering regions," Phys. Med. and Biol. 41 767-783 (1996). [CrossRef]
- In Mathematics and Physics of Emerging Biomedical Imaging (National Academy Press, Washington, D.C., 1996).
- H. Jiang, K. D. Paulsen, "A finite element based higher-order diffusion approximation of light propagation in tissues," Proc. SPIE 2389, 608-614 (1995). [CrossRef]
- D. A. Boas et al, "Photon Migration within the P3 approximation," in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, Proc. SPIE 2389, pp. 240-247 (1995). [CrossRef]
- D. A. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Optics Express 1, 404-413 (1997), http://epubs.osa.org/oearchive/source/2831.htm [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1. |
Fig. 2. (a) |

« Previous Article | Next Article »

OSA is a member of CrossRef.