Abstract

Using a Monte Carlo method, we investigate the effect of a turbid medium on image transmission by means of the modulation transfer function approach. We present results that refer to a medium that consists of a random distribution of water spherical particles in air. We analyze the effect of geometric conditions (medium width and position) and source characteristics (Lambertian, beam emission). We present results for small spheres (Rayleigh scattering) and spheres (1.0-μm diameter) that are not small in comparison with the wavelength λ = 0.6328 μm. Numerical data show a large modulation transfer function dependence on the source emission aperture and a substantial independence of the medium width for a fixed value of the optical depth. In accordance with reciprocity principles, we test an inverse scheme of Monte Carlo calculation, the advantage of this scheme being a substantial reduction in calculation time.

© 1993 Optical Society of America

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References

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  1. H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
    [Crossref] [PubMed]
  2. R. L. Lutomirski, “Atmospheric degradation of electro-optical system performance,” Appl. Opt. 17, 3915–3921 (1978).
    [Crossref] [PubMed]
  3. N. S. Kopeika, S. Solomon, Y. Gencay, “Wavelength variation of visible and near-infrared resolution through the atmosphere. Dependence on aerosols and meteorological conditions,” J. Opt. Soc. Am. 71, 892–901 (1981).
    [Crossref]
  4. N. S. Kopeika, “Spatial frequency and wavelength dependence effects of aerosols on the atmospheric modulation transfer function,” J. Opt. Soc. Am. 72, 1092–1094 (1982).
    [Crossref]
  5. A. Ishimaru, “Limitation on image resolution by a random medium,” Appl. Opt. 17, 348–352 (1978).
    [Crossref] [PubMed]
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Press, New York, 1978), Vol. II, Sec. 20.20.
  7. Y. Kuga, A. Ishimaru, “Modulation transfer function of layered inhomogeneous random media using the small-angle approximation,” Appl. Opt. 25, 4382–4385 (1986).
    [Crossref] [PubMed]
  8. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
    [Crossref]
  9. P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
    [Crossref]
  10. P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
    [Crossref]
  11. L. R. Bissonette, “Calculation method of the MTF in aerosol media,” in Propagation Engineering: Third in a Series, L. R. Bissonette, W. B. Miller, eds., Proc. soc. Photo-Opt. Instrum. Eng.1312, 148–156 (1990).
  12. E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of two scaling relations in the study of multiple-scattering effects on the transmittance of light beams through a turbid atmosphere,” J. Opt. Soc. Am. A 2, 903–911 (1985).
    [Crossref]
  13. Y. Kuga, A. Ishimaru, “Modulation transfer function and image transmission through distributed spherical particles,” J. Opt. Soc. Am. A 2, 2330–2335 (1985).
    [Crossref]

1991 (3)

H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
[Crossref] [PubMed]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

1986 (1)

1985 (2)

1982 (1)

1981 (1)

1978 (2)

Battistelli, E.

Bissonette, L. R.

L. R. Bissonette, “Calculation method of the MTF in aerosol media,” in Propagation Engineering: Third in a Series, L. R. Bissonette, W. B. Miller, eds., Proc. soc. Photo-Opt. Instrum. Eng.1312, 148–156 (1990).

Bruscaglioni, P.

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of two scaling relations in the study of multiple-scattering effects on the transmittance of light beams through a turbid atmosphere,” J. Opt. Soc. Am. A 2, 903–911 (1985).
[Crossref]

Davies, E. R.

H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
[Crossref] [PubMed]

Donelli, P.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

Gencay, Y.

Ishimaru, A.

Ismaelli, A.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of two scaling relations in the study of multiple-scattering effects on the transmittance of light beams through a turbid atmosphere,” J. Opt. Soc. Am. A 2, 903–911 (1985).
[Crossref]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[Crossref]

Jackson, P. C.

H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
[Crossref] [PubMed]

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[Crossref]

Key, H.

H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
[Crossref] [PubMed]

Kopeika, N. S.

Kuga, Y.

Lutomirski, R. L.

Solomon, S.

Wells, P. N. T.

H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
[Crossref] [PubMed]

Zaccanti, G.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of two scaling relations in the study of multiple-scattering effects on the transmittance of light beams through a turbid atmosphere,” J. Opt. Soc. Am. A 2, 903–911 (1985).
[Crossref]

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[Crossref]

Appl. Opt. (3)

J. Mod. Opt. (2)

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Phys. Med. Biol. (1)

H. Key, E. R. Davies, P. C. Jackson, P. N. T. Wells. “Monte Carlo modeling of light propagation in breast tissue,” Phys. Med. Biol. 36, 591–602 (1991).
[Crossref] [PubMed]

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Press, New York, 1978), Vol. II, Sec. 20.20.

L. R. Bissonette, “Calculation method of the MTF in aerosol media,” in Propagation Engineering: Third in a Series, L. R. Bissonette, W. B. Miller, eds., Proc. soc. Photo-Opt. Instrum. Eng.1312, 148–156 (1990).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
[Crossref]

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Figures (22)

Fig. 1
Fig. 1

Schematic of the geometry considered in Sections 2 and and 3: S, Lambertian source; S′, image of S; L, thin lens with focal length f = 5 cm and a radius of 1 cm.

Fig. 2
Fig. 2

MTF's versus spatial frequency are plotted for the 1S case, τ = 2. Curves A, B, C, and D refer to four different values of d: 12.5, 25, 50, and 100 cm, respectively. The geometry is the same as that of Fig. 1.

Fig. 3
Fig. 3

MTF's versus spatial frequency are plotted for the RS case, τ = 2. Curves A, B, C, and D refer to four different values of d: 12.5, 25, 50, and 100 cm, respectively.

Fig. 4
Fig. 4

MTF's versus spatial frequency are plotted for the RS case, τ = 5. The curves refer to four different values of d: 12.5, 25, 50, and 100 cm. The geometry is the same as that of Fig. 1.

Fig. 5
Fig. 5

MTF's versus spatial frequency. Curve A, 1S case, τ = 2; curve B, RS case, τ = 2. The geometry is the same as that of Fig. 1.

Fig. 6
Fig. 6

MTF's versus spatial frequency are plotted for different values of beam semiaperture β in the 1S case. The seven curves refer, respectively from the upper to the lower, to β = 5°, 10°, 20°, 30°, 40°, 50°, and 85°. The geometry is the same as that of Fig. 1, and τ = 2 and d = 25 cm.

Fig. 7
Fig. 7

MTF's versus spatial frequency are plotted for different values of beam semiaperture β in the RS case. The nine curves refer, respectively from the upper to the lower, to β = 5°, 10°, 20°, 30°, 40°, 50°, 60°, 70°, and 85°. The geometry is the same as that of Fig. 1, and τ = 2 and d = 25 cm.

Fig. 8
Fig. 8

High-frequency MTF limit value M (in decibels) versus the semiaperture angle of emission β. The triangles denote the RS case, and the squares denote the 1S case. The data are relative to results presented in Figs. 6 and 7.

Fig. 9
Fig. 9

MTF's versus spatial frequency are plotted for the 1S case with β = 20°. The six curves are related to six different values of the optical depth τ = 1, 2, 3, 4, 5, and 6, from the upper to the lower respectively. The geometry is the same as that of Fig. 1, and d = 25 cm.

Fig. 10
Fig. 10

MTF's versus spatial frequency are plotted for the 1S case with β = 50°. The six curves are relative to six different values of the optical depth τ = 1, 2, 3, 4, 5, and 6, from the upper to the lower respectively. The geometry is the same as that of Fig. 1 and d = 25 cm.

Fig. 11
Fig. 11

High-frequency MTF limit value M versus optical depth τ is plotted for the 1S case. The geometry is the same as that of Fig. 1 and d = 25 cm. Curves A, B, C, and D refer to β = 10°, 20°, 30°, and 40°, respectively. Curves for higher values of β are exactly superimposed to the D curve.

Fig. 12
Fig. 12

Geometry for (a) the direct and (b) the inverse scheme of calculation. In the direct procedure the defocused images A′, B′ of the scattering points A, B may not intercept a generic point P' where the procedure considers the irradiance. On the contrary, with the inverse procedure, every point P of the object plane is reached by radiation scattered at points A, B reached by radiation emitted at S′ (image of S).

Fig. 13
Fig. 13

Geometry for results that are presented in Fig. 14. The thin lens has a focal length f = 5 cm and a radius of 1 cm. We take the source—medium and medium–lens distances to be equal to 10% of source–lens distance d.

Fig. 14
Fig. 14

MTF's versus spatial frequency are plotted for the 1S case, τ = 5. Curve A is relative to the direct scheme, with d = 50 cm; the other three curves are derived in the inverse scheme with d = 50 cm, 1 m, and 10 m.

Fig. 15
Fig. 15

Geometry for results that are presented in Figs. 16, 17, and 18. Turbid layer of width: 18 mm, d = 190 mm, thin lens with focal length f = 5 cm and a radius of 2.5 mm.

Fig. 16
Fig. 16

MTF's versus spatial frequency are plotted for the RS case, τ = 2. The two curves are relative to the direct (lower curve) and the inverse (upper curve) schemes.

Fig. 17
Fig. 17

MTF's versus spatial frequency are plotted for the RS case, τ = 5. The two curves are relative to the direct (lower curve) and the inverse (upper curve) scheme.

Fig. 18
Fig. 18

MTF's versus spatial frequency are plotted for the 1S case, τ = 5. The two curves are relative to the direct and the inverse schemes.

Fig. 19
Fig. 19

Geometry for results that are presented in Figs. 20 and 21: d = 245 mm, thin lens has a focal length f = 5 cm and a radius of 1 cm.

Fig. 20
Fig. 20

MTF's versus spatial frequency are plotted for the RS case, τ = 2. The two curves are relative to the direct and the inverse schemes.

Fig. 21
Fig. 21

MTF's versus spatial frequency are plotted for the 1S case, τ = 2. The two curves are relative to the direct and the inverse schemes.

Fig. 22
Fig. 22

Geometric scheme for the SEMIM code.

Tables (1)

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Table 1 Found Values for Ratio Ps/P0 in the Two Cases of Lambertian (L) and Isotropic (I) Sources and the Corresponding Values of M[Eq. (B1)] a

Equations (16)

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MTF ( f ) = | S ( ρ ) exp ( i 2 π f ρ / q ) d ρ S ( ρ ) d ρ | ,
M ( f ) = exp { 2 π σ s 0 L d x 0 1 [ 1 J 0 ( 2 π s x L f ) ] p ( s ) s d s } ,
M = exp ( α τ ) .
S ( ρ ) = 1 2 π f 2 α d P ( α ) d α ,
U i ( x i , y i ) = k exp ( ikd ) 2 π i d Σ l U ( x , y ) × exp [ i k 2 ( | ρ i ρ | 2 d ρ 2 f ) ] d ρ ,
U ( ρ ) = A ( x , y ) exp [ i k ( x x s ) 2 + ( y y s ) 2 2 D ] .
i k 2 φ ( x i , y i , x , y ) .
x 0 ( 1 D + 1 d 1 f ) = x s D + x i d , y 0 ( 1 D + 1 d 1 f ) = y s D + y i d .
x 0 = A x i d X A d , y 0 = A y i d Y A d .
2 π i k A ( x 0 , y 0 ) exp ( i ψ ) ( 2 φ x 2 2 φ y 2 ) 1 / 2 ,
U i ( x i , y i ) A ( x 0 , y 0 ) exp ( i ψ ) A A d with ψ = ψ + k d .
I ( x i , y i ) = I ( x 0 , y 0 ) A 2 ( A d ) 2 ,
MTF ( f ) = | T ( S 0 + S s ) / ( P 0 + P s ) | ,
M = lim f MTF ( f ) = 1 1 + P s / P 0 .
1 1 + P s / P 0 = exp ( τ ) = P 0
P s = 1 exp ( τ ) = 1 P 0 .

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