Abstract

Imaging structures within a turbid medium using Angular Domain Imaging (ADI) employs an angular filter array to separate weakly scattered photons from those that are highly scattered. At high scattering coefficients, ADI contrast declines due to the large fraction of non-uniform background scattered light still within the acceptance angle. This paper demonstrates various methods to enhance the image contrast in ADI. Experiments where a wedge prism was used to deviate the laser source so that scattered photons could be imaged and subtracted from the image obtained by standard ADI provided the greatest improvement in image contrast.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2008 (3)

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

F. Vasefi, B. Kaminska, P. K. Y. Chan, and G. H. Chapman, "Multi-spectral angular domain optical imaging in biological tissues using diode laser sources," Opt. Express 16, 14456-14468 (2008).
[CrossRef] [PubMed]

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

2007 (2)

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
[CrossRef]

2005 (1)

H. Hielscher, "Optical tomographic imaging of small animals," Curr. Opin. Biotechnol. 16, 79-88 (2005).
[CrossRef] [PubMed]

2003 (2)

A. F. Fercher, "Optical coherence tomography-principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

G. H. Chapman, M. Trinh, N. Pfeiffer, G. Chu, and D. Lee, "Angular Domain Imaging of Objects Within Highly Scattering Media Using Silicon Micromachined Collimating Arrays," IEEE J. Sel. Top. Quantum Electron. 9, 257-266 (2003).
[CrossRef]

2001 (1)

J. G. Fujimoto, "Optical coherence tomography," C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1099-111 (2001).

1999 (1)

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, and G. Muller, " Optical Properties of Circulating Human Blood in the Wavelength Range 400--2500 nm," J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

1995 (2)

V. V. Tuchin, "Fundamentals of low-intensity laser radiation interaction with biotissues: dosimetry and diagnostical aspects," Bull. Russ. Acad. Sci. Phys. Ser. 59,120-143 (1995).

L. Wang, S. L. Jacques and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146, 7 (1995).
[CrossRef] [PubMed]

1993 (1)

D. A. Benaron and D. K. Stevenson, "Optical time-of-flight and absorbance imaging of biologic media," Science 259,1463-1466 (1993).
[CrossRef] [PubMed]

1991 (2)

H. G. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nanometers," Appl. Opt. 30, 4507-4514 (1991).
[CrossRef] [PubMed]

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253, 769-771 (1991).
[CrossRef] [PubMed]

1990 (1)

W. F. Cheong, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

1988 (1)

M. R. Arnfield, J. Tulip, and M. S. McPhee, "Optical propagation in tissue with anisotropic scattering," IEEE Trans. Biomed. Eng. 35, 372-381 (1988).
[CrossRef] [PubMed]

1987 (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, "Angular Dependence of HeNe laser Light Scattering by Human Dermis," Laser Life Sci. 1, 309-333 (1987).

1940 (1)

L. Henyey, and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophysics. J. 93, 70 (1940).
[CrossRef]

Alfano, R. R.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, "Angular Dependence of HeNe laser Light Scattering by Human Dermis," Laser Life Sci. 1, 309-333 (1987).

Arnfield, M. R.

M. R. Arnfield, J. Tulip, and M. S. McPhee, "Optical propagation in tissue with anisotropic scattering," IEEE Trans. Biomed. Eng. 35, 372-381 (1988).
[CrossRef] [PubMed]

Benaron, D. A.

D. A. Benaron and D. K. Stevenson, "Optical time-of-flight and absorbance imaging of biologic media," Science 259,1463-1466 (1993).
[CrossRef] [PubMed]

Chan, P.

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

Chan, P. K.

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
[CrossRef]

Chan, P. K. Y.

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

F. Vasefi, B. Kaminska, P. K. Y. Chan, and G. H. Chapman, "Multi-spectral angular domain optical imaging in biological tissues using diode laser sources," Opt. Express 16, 14456-14468 (2008).
[CrossRef] [PubMed]

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

Chapman, G. H.

F. Vasefi, B. Kaminska, P. K. Y. Chan, and G. H. Chapman, "Multi-spectral angular domain optical imaging in biological tissues using diode laser sources," Opt. Express 16, 14456-14468 (2008).
[CrossRef] [PubMed]

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
[CrossRef]

G. H. Chapman, M. Trinh, N. Pfeiffer, G. Chu, and D. Lee, "Angular Domain Imaging of Objects Within Highly Scattering Media Using Silicon Micromachined Collimating Arrays," IEEE J. Sel. Top. Quantum Electron. 9, 257-266 (2003).
[CrossRef]

Cheong, W. F.

W. F. Cheong, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Chu, G.

G. H. Chapman, M. Trinh, N. Pfeiffer, G. Chu, and D. Lee, "Angular Domain Imaging of Objects Within Highly Scattering Media Using Silicon Micromachined Collimating Arrays," IEEE J. Sel. Top. Quantum Electron. 9, 257-266 (2003).
[CrossRef]

Dorschel, K.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, and G. Muller, " Optical Properties of Circulating Human Blood in the Wavelength Range 400--2500 nm," J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Fercher, A. F.

A. F. Fercher, "Optical coherence tomography-principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Friebel, M.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, and G. Muller, " Optical Properties of Circulating Human Blood in the Wavelength Range 400--2500 nm," J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Fujimoto, J. G.

J. G. Fujimoto, "Optical coherence tomography," C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1099-111 (2001).

Greenstein, J. L.

L. Henyey, and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophysics. J. 93, 70 (1940).
[CrossRef]

Hahn, A.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, and G. Muller, " Optical Properties of Circulating Human Blood in the Wavelength Range 400--2500 nm," J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Henyey, L.

L. Henyey, and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophysics. J. 93, 70 (1940).
[CrossRef]

Hielscher, H.

H. Hielscher, "Optical tomographic imaging of small animals," Curr. Opin. Biotechnol. 16, 79-88 (2005).
[CrossRef] [PubMed]

Ho, P. P.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Jacques, S. L.

L. Wang, S. L. Jacques and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146, 7 (1995).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, "Angular Dependence of HeNe laser Light Scattering by Human Dermis," Laser Life Sci. 1, 309-333 (1987).

Kaminska, B.

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

F. Vasefi, B. Kaminska, P. K. Y. Chan, and G. H. Chapman, "Multi-spectral angular domain optical imaging in biological tissues using diode laser sources," Opt. Express 16, 14456-14468 (2008).
[CrossRef] [PubMed]

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
[CrossRef]

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

Lee, D.

G. H. Chapman, M. Trinh, N. Pfeiffer, G. Chu, and D. Lee, "Angular Domain Imaging of Objects Within Highly Scattering Media Using Silicon Micromachined Collimating Arrays," IEEE J. Sel. Top. Quantum Electron. 9, 257-266 (2003).
[CrossRef]

Liu, C.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253, 769-771 (1991).
[CrossRef] [PubMed]

McPhee, M. S.

M. R. Arnfield, J. Tulip, and M. S. McPhee, "Optical propagation in tissue with anisotropic scattering," IEEE Trans. Biomed. Eng. 35, 372-381 (1988).
[CrossRef] [PubMed]

Moes, C. J. M.

Muller, G.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, and G. Muller, " Optical Properties of Circulating Human Blood in the Wavelength Range 400--2500 nm," J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Pfeiffer, N.

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
[CrossRef]

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

G. H. Chapman, M. Trinh, N. Pfeiffer, G. Chu, and D. Lee, "Angular Domain Imaging of Objects Within Highly Scattering Media Using Silicon Micromachined Collimating Arrays," IEEE J. Sel. Top. Quantum Electron. 9, 257-266 (2003).
[CrossRef]

Prahl, S. A.

H. G. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nanometers," Appl. Opt. 30, 4507-4514 (1991).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, "Angular Dependence of HeNe laser Light Scattering by Human Dermis," Laser Life Sci. 1, 309-333 (1987).

Roggan, A.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, and G. Muller, " Optical Properties of Circulating Human Blood in the Wavelength Range 400--2500 nm," J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Stevenson, D. K.

D. A. Benaron and D. K. Stevenson, "Optical time-of-flight and absorbance imaging of biologic media," Science 259,1463-1466 (1993).
[CrossRef] [PubMed]

Trinh, M.

G. H. Chapman, M. Trinh, N. Pfeiffer, G. Chu, and D. Lee, "Angular Domain Imaging of Objects Within Highly Scattering Media Using Silicon Micromachined Collimating Arrays," IEEE J. Sel. Top. Quantum Electron. 9, 257-266 (2003).
[CrossRef]

Tuchin, V. V.

V. V. Tuchin, "Fundamentals of low-intensity laser radiation interaction with biotissues: dosimetry and diagnostical aspects," Bull. Russ. Acad. Sci. Phys. Ser. 59,120-143 (1995).

Tulip, J.

M. R. Arnfield, J. Tulip, and M. S. McPhee, "Optical propagation in tissue with anisotropic scattering," IEEE Trans. Biomed. Eng. 35, 372-381 (1988).
[CrossRef] [PubMed]

van Gemert, M. J. C.

van Marle, J.

van Staveren, H. G.

Vasefi, F.

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

F. Vasefi, B. Kaminska, P. K. Y. Chan, and G. H. Chapman, "Multi-spectral angular domain optical imaging in biological tissues using diode laser sources," Opt. Express 16, 14456-14468 (2008).
[CrossRef] [PubMed]

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
[CrossRef]

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

Wang, L.

L. Wang, S. L. Jacques and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146, 7 (1995).
[CrossRef] [PubMed]

Wang, L. M.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Zhang, G.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Zheng, L.

L. Wang, S. L. Jacques and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146, 7 (1995).
[CrossRef] [PubMed]

Appl. Opt. (1)

Astrophysics. J. (1)

L. Henyey, and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophysics. J. 93, 70 (1940).
[CrossRef]

Bull. Russ. Acad. Sci. Phys. Ser. (1)

V. V. Tuchin, "Fundamentals of low-intensity laser radiation interaction with biotissues: dosimetry and diagnostical aspects," Bull. Russ. Acad. Sci. Phys. Ser. 59,120-143 (1995).

C. R. Acad. Sci. Ser. IV Phys. Astrophys. (1)

J. G. Fujimoto, "Optical coherence tomography," C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1099-111 (2001).

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146, 7 (1995).
[CrossRef] [PubMed]

Curr. Opin. Biotechnol. (1)

H. Hielscher, "Optical tomographic imaging of small animals," Curr. Opin. Biotechnol. 16, 79-88 (2005).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

W. F. Cheong, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

F. Vasefi, P. K. Y. Chan, B. Kaminska, G. H. Chapman, N. Pfeiffer, "An Optical Imaging Technique Using Deep Illumination in the Angular Domain," IEEE J. Sel. Top. Quantum Electron. 13, 1610-1620 (2007).
[CrossRef]

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Proc. SPIE (3)

N. Pfeiffer, P. Chan, G. H. Chapman, F. Vasefi, and B. Kaminska, "Optical imaging of structures within highly scattering material using a lens and aperture to form a spatiofrequency filter," Proc. SPIE 6854,68541D (2008).
[CrossRef]

F. Vasefi, G. H. Chapman, P. K. Y. Chan, B. Kaminska, and N. Pfeiffer, "Enhanced angular domain optical imaging by background scattered light subtraction from a deviated laser source," Proc. SPIE 6854,68541E (2008).
[CrossRef]

P. K. Chan, F. Vasefi, G. H. Chapman, B. Kaminska, and N. Pfeiffer, "Angular Domain Optical Tomography in Scattering Media with Multi-spectral Diode," Proc. SPIE 6435,64350M (2007).
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Figures (7)

Fig. 1.
Fig. 1.

Angular domain imaging in transillumination mode.

Fig. 2.
Fig. 2.

Silicon micromachined angular filter array (etched bottom section).

Fig. 3.
Fig. 3.

Illustration of limited field of view in one dimensional AFA.

Fig. 4.
Fig. 4.

ADI scan of sample including resolution target (L-shaped targets with lines and spaces of 150 µm, 200 µm, 300 µm, and 400 µm) placed in the middle position of 1 cm thick optical cell filled (a) Water, (b) 0.6% Intralipid s=4.8 cm-1, µa=0.01cm-1), and (c) 0.75% Intralipid s=6 cm-1, µa=0.01cm-1) using an 808 nm laser diode.

Fig. 5.
Fig. 5.

ADI setup with 808 nm laser diode and aspheric and cylindrical lens collimation system.

Fig. 6.
Fig. 6.

Contrast enhancement of ADI scan of L-shape target (line and space width of 400 µm) at 808nm for a scattering medium composed of 0.3% Intralipid™ in a 2 cm thick optical cell. (a) Original ADI scan, (b) Wedge inserted ADI scan (c) Wedge subtraction image (d) Parallel polarized ADI scan (PADI) (e) cross polarized ADI scan (f) subtraction result of parallel and cross orientation (PADI-CS).

Fig. 7.
Fig. 7.

Comparison of image contrast enhancement using polarization contrast enhancement and wedge subtraction methods in different scattering ratio levels.

Tables (1)

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Table 1. Summary of the contrast enhancement methods using ADI

Equations (7)

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Ω I ( x , Ω ) = ( μ a + μ s ) I ( x , Ω ) + μ s 4 π I ( x , Ω ) S ( Ω Ω ) d Ω + Q ( x , Ω )
g = cos θ = 2 π 0 π S ( Ω Ω ) cos θ sin θ d θ
S ( Ω Ω ) = 1 4 π 1 g 2 ( 1 + g 2 2 g cos θ ) 3 2
μ s = μ s ( 1 g )
Ω = 4 arcsin ab ( 4 d 2 + a 2 ) ( 4 d 2 + b 2 ) [ sr ]
SR max = 4 π Ω = 3.5 × 10 5
Contrast Ratio = mean ( I max ) mean ( I min ) mean ( I max ) + mean ( I min )

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