Abstract

We introduce two filtering methods for near-field diffuse light diffraction tomography based on the angular spectrum representation. We then combine these filtering techniques with a new method to find the approximate depth of the image heterogeneities. Taken together these ideas improve the fidelity of our projection image reconstructions, provide an interesting three dimensional rendering of the reconstructed volume, and enable us to identify and classify image artifacts that need to be controlled better for tissue applications. The analysis is accomplished using data derived from numerical finite difference simulations with added noise.

© Optical Society of America

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References

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  1. A. G. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Physics Today 48, 34-40 (March 1995).
    [CrossRef]
  2. B. Chance, in Photon Migration in Tissues (Plenum Press, 1989)
  3. B. J Trombeg, L. O. Svaasand, T. Tsay and R. C. M. Haskell, "Properties of Photon Density Waves in Multiple-Scattering Media," Appl. Opt. 32, 607 (1993).
    [CrossRef]
  4. M. A. O' Leary, D. A . Boas, B. Chance and A. G. Yodh, "Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995).
    [CrossRef]
  5. 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]
  6. Y. Yao, Y. Wang, Y. Pei, W. Zhu and R. L. Barbour, "Frequency-domain optical imaging of absorption and scattering distributions using a born iterative mehod," J. Opt. Soc. Am. A 14, 325-342 (1997).
    [CrossRef]
  7. M. O'Leary, "Imaging with Diffuse Photon Density Waves," in PhD Thesis (Dept. Physics & Astronomy, U. of Pennsylvania, May 1996) .
  8. M. S Patterson, B. Chance and B. C. Wilson, "Time Resolved Re ectance And Transmittance for the Non-Invasive Measurement of Tissue Optical Properties," Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  9. S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller and B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg and P. van der Zee, Proc SPIE IS11, 35-64 (1993).
  10. D. A. Benaron, D. C. Ho, S. Spilman, J. P. Van Houten and D. K. Stevenson, "Tomographic time-of-flight optical imaging device," Adv. Exp. Med. Biol. 361, 609-617 (1994).
    [CrossRef]
  11. Gratton, E. and J. B. Fishkin, "Optical spectroscopy of tissue-like phantoms using photon density waves," Comments on Cell. and Mol. Biophys. 8(6), 309-359 (1995).
  12. J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner and B. J. Tromberg, "Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject," Appl. Opt. 36, 10 (1997).
    [CrossRef] [PubMed]
  13. W. Bank and B. Chance, "An Oxidative Effect in metabolic myopathies - diagnosis by noninvasive tissue oximetry," Ann. Neurol. 36, 830 (1994).
    [CrossRef] [PubMed]
  14. Y. Hoshi and M. Tamura, "Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks," Neuroimage. 5, 292 (1997).
    [CrossRef] [PubMed]
  15. A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain functions," Trends. Neurosci. 20, 435 (1997) .
    [CrossRef] [PubMed]
  16. B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. Roy. Soc. London B. 352, 707 (1997).
    [CrossRef] [PubMed]
  17. B. W. Pogue and K. D. Paulsen, "High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of apriori magnetic resonance imaging structural information," Opt. Lett. 23, 1716-1718 (1998).
    [CrossRef]
  18. R. M. Danen, Y. Wang, X. D. Li, W. S. Thayer and A. G.Yodh, "Regional imager for low- resolution functional imaging of the brain with diffusing near-infrared light," Photochem. Photobiol. 67, 33-40 (1998).
    [CrossRef] [PubMed]
  19. J. H.Hoogenraad, M. B.van der Mark, S. B.Colak, G. W.'t Hooft and E. S. van der Linden, "First Results from the Philips Optical Mammoscope," Proc.SPIE / BiOS-97 (SanRemo, 1997 ) .
  20. S. K. Gayen, M. E.Zevallos, B. B. Das, R. R. Alfano and "Time-sliced transillumination imaging of normal and cancerous breast tissues," in Trends in Opt. And Photonics, ed. J. G. Fujimoto and M. S. Patterson.
  21. X. D. Li, J. Culver, D. N. Pattanayak, A. G. Yodh and B. Chance, "Photon Density Wave Imaging With K-Space Spectrum Analysis: clinical studies - background substraction and boundary effects," Technical Digest Series - CLEO '98, 6, 88-89 (1998).
  22. S. Fantini, S. A. Walker, M. A. Franceschini, K. T. Moesta, P. M. Schlag, M. Kaschke, and E. Gratton. "Assessment of the size, position, and optical properties of breast tumors in vivo by non-invasive optical methods" Appl. Opt. 37, 1982-1989 (1998).
    [CrossRef]
  23. M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, and M. Kaschke. "Frequency-domain instrumentation techniques enhance optical mammography: Initial clinical results" Proc. Natl. Acad. Sci. USA, 94, 6468-6473 (1997).
    [CrossRef]
  24. S. Fantini, M. A. Franceschini, G. Gaida, E. Gratton, H. Jess and W. W. Mantulin, "Frequency- domain optical mammography: edge effect corrections," Med. Phys. 23, 149 (1996).
    [CrossRef] [PubMed]
  25. E. Wolf, "Principles and Development of Diffraction Tomography" in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996).
    [CrossRef]
  26. A. J.Devaney, "Diffraction Tomography," Inv. Meth. In Electromagnetic Imaging, 1107-1135 .
  27. E. Wolf, "Inverse Diffraction and a New Reciprocity Theorem," J. Opt. Soc. Am. 58, 1568 (1968).
  28. E. Wolf , "Three Dimensional Structure Determination of Semi-Transparent Objects From Holographic Data," Opt. Commun. 1 , 153-156 (1969).
    [CrossRef]
  29. B. Q. Chen, J. J. Stamnes, and K. Stamnes, "Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium," Pure Appl. Opt. 7, 1161-1180 (1998).
    [CrossRef]
  30. D. L. Lasocki, C. L. Matson and P. J. Collins, "Analysis of forward scattering of diffuse photon-density waves in turbid media: a diffraction tomography approach to an analytic solution," Opt. Lett. 23, 558-560 (1998).
    [CrossRef]
  31. D. N. Pattanayak, "Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media," GE Tech. Info. Series 91CRD241 (1991).
  32. 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]
  33. X. D. Li, in PhD Thesis (Dept. Physics & Astronomy, U. of Pennsylvania, May 1998).
  34. X. D. Li, D. N. Pattanayak, J. P. Culver, T. Durduran and A. G. Yodh, "Near-Field Diffraction Tomography with Diffuse Photon Density Waves," to be published (1999).
  35. X. Cheng and D. Boas, "Diffuse Optical Reflection Tomography Using Continous Wave Illumination," Opt. Express 3, 118-123 (1998), http://epubs.osa.org/oearchive/source/5663.htm.
    [CrossRef] [PubMed]
  36. J. C. Schotland, "Near-field Inverse Scattering: Microscopy to Tomography," SPIE 3597 (1999).
  37. 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, 214-219 (1997).
    [CrossRef] [PubMed]
  38. C. L. Matson, "A Diffraction Tomographic Model Of The Forward Problem Using Diffuse Photon Density Waves," Opt. Express 1, 6-11 (1997), http://epubs.osa.org/oearchive/source/1884.htm.
    [CrossRef] [PubMed]
  39. S. J. Norton and T. Vo-Dinh, "Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution," J. Opt. Soc. Am. A 15, 2670-2677 (1998).
    [CrossRef]
  40. J. C. Schotland, "Continous Wave Diffusion Imaging," J. Opt. Soc. Am. A 14, 275-279 (1997).
    [CrossRef]
  41. T. Durduran, J. Culver, L. Zubkov, M. Holboke, R. Choe, X. D. Li, B. Chance, D. N. Pattanayak and A. G. Yodh, "Diffraction Tomography In Diffuse Optical Imaging; Filters & Noise," SPIE 3597 (1999).
  42. J. Ripoll and M. Nieto-Vesperinas, "Re ection and Transmission Coefficients of Diffuse Photon Density Waves," in press.
  43. J. Ripoll and M. Nieto-Vesperinas, "Spatial Resolution of Diffuse Photon Density Waves," to be published in J. Opt. Soc. Am. A (1999).
  44. C. L. Matson, "Resolution, Linear Filtering, and Positivity," J. Opt. Soc. Am. A 15, 33-41 (1998).
    [CrossRef]
  45. F. J. Harris, "On The Use of Windows For Harmonic Analysis with the Discrete Fourier Transform," Proc. Of IEEE 66, 51-83 (1978).
    [CrossRef]
  46. A.Kak and M. Slaney, in Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  47. A. J. Devaney, "Linearised Inverse Scattering in Attenuating Media," Inv. Probs. 3, 389-397 (1987).
    [CrossRef]
  48. A. J. Devaney, "Reconstructive Tomography With Diffracting Wavefields," Inv. Probl. 2, 161-183 (1986).
    [CrossRef]
  49. Essentially we assume that the scattering contrast is slowly varying. For a detailed description we refer to [33] and [34].
  50. A. J. Banos, in Dipole Radiation In the Presence of a Conducting Half-Space (Pergamon Press, New York, 1966).
  51. J. W. Goodman, in Introduction To Fourier Optics, (McGraw-Hill, San Fransisco , 1968).
  52. We are aware of a similar normalization scheme by Hanli Liu and her collaborators ( private communications SPIE Jan 1999).

Other (52)

A. G. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Physics Today 48, 34-40 (March 1995).
[CrossRef]

B. Chance, in Photon Migration in Tissues (Plenum Press, 1989)

B. J Trombeg, L. O. Svaasand, T. Tsay and R. C. M. Haskell, "Properties of Photon Density Waves in Multiple-Scattering Media," Appl. Opt. 32, 607 (1993).
[CrossRef]

M. A. O' Leary, D. A . Boas, B. Chance and A. G. Yodh, "Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995).
[CrossRef]

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]

Y. Yao, Y. Wang, Y. Pei, W. Zhu and R. L. Barbour, "Frequency-domain optical imaging of absorption and scattering distributions using a born iterative mehod," J. Opt. Soc. Am. A 14, 325-342 (1997).
[CrossRef]

M. O'Leary, "Imaging with Diffuse Photon Density Waves," in PhD Thesis (Dept. Physics & Astronomy, U. of Pennsylvania, May 1996) .

M. S Patterson, B. Chance and B. C. Wilson, "Time Resolved Re ectance And Transmittance for the Non-Invasive Measurement of Tissue Optical Properties," Appl. Opt. 28, 2331 (1989).
[CrossRef] [PubMed]

S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller and B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg and P. van der Zee, Proc SPIE IS11, 35-64 (1993).

D. A. Benaron, D. C. Ho, S. Spilman, J. P. Van Houten and D. K. Stevenson, "Tomographic time-of-flight optical imaging device," Adv. Exp. Med. Biol. 361, 609-617 (1994).
[CrossRef]

Gratton, E. and J. B. Fishkin, "Optical spectroscopy of tissue-like phantoms using photon density waves," Comments on Cell. and Mol. Biophys. 8(6), 309-359 (1995).

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner and B. J. Tromberg, "Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject," Appl. Opt. 36, 10 (1997).
[CrossRef] [PubMed]

W. Bank and B. Chance, "An Oxidative Effect in metabolic myopathies - diagnosis by noninvasive tissue oximetry," Ann. Neurol. 36, 830 (1994).
[CrossRef] [PubMed]

Y. Hoshi and M. Tamura, "Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks," Neuroimage. 5, 292 (1997).
[CrossRef] [PubMed]

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain functions," Trends. Neurosci. 20, 435 (1997) .
[CrossRef] [PubMed]

B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. Roy. Soc. London B. 352, 707 (1997).
[CrossRef] [PubMed]

B. W. Pogue and K. D. Paulsen, "High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of apriori magnetic resonance imaging structural information," Opt. Lett. 23, 1716-1718 (1998).
[CrossRef]

R. M. Danen, Y. Wang, X. D. Li, W. S. Thayer and A. G.Yodh, "Regional imager for low- resolution functional imaging of the brain with diffusing near-infrared light," Photochem. Photobiol. 67, 33-40 (1998).
[CrossRef] [PubMed]

J. H.Hoogenraad, M. B.van der Mark, S. B.Colak, G. W.'t Hooft and E. S. van der Linden, "First Results from the Philips Optical Mammoscope," Proc.SPIE / BiOS-97 (SanRemo, 1997 ) .

S. K. Gayen, M. E.Zevallos, B. B. Das, R. R. Alfano and "Time-sliced transillumination imaging of normal and cancerous breast tissues," in Trends in Opt. And Photonics, ed. J. G. Fujimoto and M. S. Patterson.

X. D. Li, J. Culver, D. N. Pattanayak, A. G. Yodh and B. Chance, "Photon Density Wave Imaging With K-Space Spectrum Analysis: clinical studies - background substraction and boundary effects," Technical Digest Series - CLEO '98, 6, 88-89 (1998).

S. Fantini, S. A. Walker, M. A. Franceschini, K. T. Moesta, P. M. Schlag, M. Kaschke, and E. Gratton. "Assessment of the size, position, and optical properties of breast tumors in vivo by non-invasive optical methods" Appl. Opt. 37, 1982-1989 (1998).
[CrossRef]

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, and M. Kaschke. "Frequency-domain instrumentation techniques enhance optical mammography: Initial clinical results" Proc. Natl. Acad. Sci. USA, 94, 6468-6473 (1997).
[CrossRef]

S. Fantini, M. A. Franceschini, G. Gaida, E. Gratton, H. Jess and W. W. Mantulin, "Frequency- domain optical mammography: edge effect corrections," Med. Phys. 23, 149 (1996).
[CrossRef] [PubMed]

E. Wolf, "Principles and Development of Diffraction Tomography" in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996).
[CrossRef]

A. J.Devaney, "Diffraction Tomography," Inv. Meth. In Electromagnetic Imaging, 1107-1135 .

E. Wolf, "Inverse Diffraction and a New Reciprocity Theorem," J. Opt. Soc. Am. 58, 1568 (1968).

E. Wolf , "Three Dimensional Structure Determination of Semi-Transparent Objects From Holographic Data," Opt. Commun. 1 , 153-156 (1969).
[CrossRef]

B. Q. Chen, J. J. Stamnes, and K. Stamnes, "Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium," Pure Appl. Opt. 7, 1161-1180 (1998).
[CrossRef]

D. L. Lasocki, C. L. Matson and P. J. Collins, "Analysis of forward scattering of diffuse photon-density waves in turbid media: a diffraction tomography approach to an analytic solution," Opt. Lett. 23, 558-560 (1998).
[CrossRef]

D. N. Pattanayak, "Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media," GE Tech. Info. Series 91CRD241 (1991).

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]

X. D. Li, in PhD Thesis (Dept. Physics & Astronomy, U. of Pennsylvania, May 1998).

X. D. Li, D. N. Pattanayak, J. P. Culver, T. Durduran and A. G. Yodh, "Near-Field Diffraction Tomography with Diffuse Photon Density Waves," to be published (1999).

X. Cheng and D. Boas, "Diffuse Optical Reflection Tomography Using Continous Wave Illumination," Opt. Express 3, 118-123 (1998), http://epubs.osa.org/oearchive/source/5663.htm.
[CrossRef] [PubMed]

J. C. Schotland, "Near-field Inverse Scattering: Microscopy to Tomography," SPIE 3597 (1999).

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, 214-219 (1997).
[CrossRef] [PubMed]

C. L. Matson, "A Diffraction Tomographic Model Of The Forward Problem Using Diffuse Photon Density Waves," Opt. Express 1, 6-11 (1997), http://epubs.osa.org/oearchive/source/1884.htm.
[CrossRef] [PubMed]

S. J. Norton and T. Vo-Dinh, "Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution," J. Opt. Soc. Am. A 15, 2670-2677 (1998).
[CrossRef]

J. C. Schotland, "Continous Wave Diffusion Imaging," J. Opt. Soc. Am. A 14, 275-279 (1997).
[CrossRef]

T. Durduran, J. Culver, L. Zubkov, M. Holboke, R. Choe, X. D. Li, B. Chance, D. N. Pattanayak and A. G. Yodh, "Diffraction Tomography In Diffuse Optical Imaging; Filters & Noise," SPIE 3597 (1999).

J. Ripoll and M. Nieto-Vesperinas, "Re ection and Transmission Coefficients of Diffuse Photon Density Waves," in press.

J. Ripoll and M. Nieto-Vesperinas, "Spatial Resolution of Diffuse Photon Density Waves," to be published in J. Opt. Soc. Am. A (1999).

C. L. Matson, "Resolution, Linear Filtering, and Positivity," J. Opt. Soc. Am. A 15, 33-41 (1998).
[CrossRef]

F. J. Harris, "On The Use of Windows For Harmonic Analysis with the Discrete Fourier Transform," Proc. Of IEEE 66, 51-83 (1978).
[CrossRef]

A.Kak and M. Slaney, in Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

A. J. Devaney, "Linearised Inverse Scattering in Attenuating Media," Inv. Probs. 3, 389-397 (1987).
[CrossRef]

A. J. Devaney, "Reconstructive Tomography With Diffracting Wavefields," Inv. Probl. 2, 161-183 (1986).
[CrossRef]

Essentially we assume that the scattering contrast is slowly varying. For a detailed description we refer to [33] and [34].

A. J. Banos, in Dipole Radiation In the Presence of a Conducting Half-Space (Pergamon Press, New York, 1966).

J. W. Goodman, in Introduction To Fourier Optics, (McGraw-Hill, San Fransisco , 1968).

We are aware of a similar normalization scheme by Hanli Liu and her collaborators ( private communications SPIE Jan 1999).

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Figures (10)

Figure 1.
Figure 1.

The generic near-field diffraction tomography experiment. The detector is scanned in a 2D grid on the surface of the plane parallel to and displaced from the plane containing the source. The breast is embedded in the box along with Intralipid in order to match its average optical properties.

Figure 2.
Figure 2.

Simplified flow diagram of the image reconstruction algorithm. Dotted lines are used for optional steps. Brown and blue indicate real-m filtering, green indicates G-filtering steps.

Figure 3.
Figure 3.

(a)Amplitude of the scattered wave as a function of p for a fixed q and (b) amplitude of the tumor function plotted in the k-space for noiseless (only numerical noise) and (c) noise-added data from the single object (sec. 5.1) . The maximum frequency, |p| = π/∆x. The rising “wings” on the sides are due to noise. The noise effects are amplified by ≈ 103 relative to the noiseless case at large p.

Figure 4.
Figure 4.

Single Slice Phantom: Left figure shows a 3D rendering of the phantom. Gray region has background properties. The detector plane is assumed to be at z=5cm and the source is at the origin in z=0cm plane. Amplitude of the scattered field in the detector plane for the phantom shown in the middle (noiseless) and right (noise added) figures.

Figure 5.
Figure 5.

(a) and (b) projections at z=2.42, z=3.28 respectively, (c) Sj vs zj (cm) through the transverse center, peaks at z=2.71, (d) projection at z=2.71. All with real-m filter.

Figure 6.
Figure 6.

(a) projection at z=2.71 at real-m peak, (b) Sj vs zj (cm) through the transverse center peaks at z=4cm , (c) projection at z=4.0. All obtained with G-giltering.

Figure 7.
Figure 7.

Estimate of resolution(cm) vs distance from source plane (cm) . The changing depth dependent cut-off frequency results in the increase in resolution with distance from source plane (i.e decrease in resolution with depth from the detector plane).

Figure 8.
Figure 8.

Two slice Phantoms: Two leftmost figures show 3D renderings. Gray region has background properties. The detector plane is at z=5cm and the source is at the origin in z=0cm plane. Amplitude of the scattered field at the detector plane is shown in the two rightmost figures. The left is the noiseless data and the right shows the data after adding noise.

Figure 9.
Figure 9.

(a) Circles (crosses) show Sj vs zj (cm) through the center region of deeper (shallower) object obtained with m-filter. Peak for both objects are exact. Then with G-filter we get projections at (b) z=3.71 and (c) at z=4.41.

Figure 10.
Figure 10.

Three Dimensional Rendering of the G-filter reconstruction. An isosurface at Sj = 0.042 is shown in two different angles in (a) and (b). In (c) and (d) images of Sj in x-z plane through y=2cm and y=-2cm are shown respectively. Compare the results to that of fig.(8)

Equations (12)

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

Φ sc ( r ) = V T ( r ) G 0 ( r , r ) d 3 r .
T abs ( r ) = υ D 0 Φ 0 ( r ) δ μ a ( r ) ,
T sc ( r ) = 3 D 0 k 0 2 υ Φ 0 ( r ) .
G 0 ( r , r ) = exp ( i k 0 r r ) 4 π r r .
G 0 ( r , r ) = dpdq G ̂ 0 ( p , q , z d , z ) e i ( p ( x d x ) + q ( y d y ) ) ,
G ̂ 0 ( p , q , z d , z ) = i 2 m e i m z d z ,
Φ ̂ sc ( p , q , z d ) = d z G ̂ 0 ( p , q , z d , z ) T ̂ ( p , q , z ) .
Φ ̂ sc ( p , q , z d ) = j = 1 N G ̂ 0 ( p , q , z d , z ) T ̂ ( p , q , z j ) = j = 1 N i Δ z 2 m T ̂ ( p , q , z j ) e i m ( z d z j ) .
T ̂ ( p , q , z obj ) = Φ sc ( p , q , z d ) Δ z G ̂ 0 ( p , q , z d , z obj ) ,
T ( x , y , z obj ) = dpdq e i 2 π ( p x + q y ) Δ z G ̂ ( p , q , z obj , z ) dxdy e i 2 π ( p x + q y ) Φ sc ( x , y , z d ) ,
S j = δ μ a δ μ ¯ a xy ( δ μ axy δ μ ¯ a ) 2
p max 2 + q max 2 = ( 2 π L ) 2 .

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