Monte Carlo calculations of the modulation transfer function of an optical system operating in a turbid medium Piero Bruscaglioni, Paolo Donelli, Andrea Ismaelli, and Giovanni Zaccanti 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-pLm diameter) that are not small in comparison with the wavelength = 0.6328 plm. 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. Key words: Modulation transfer function, point-spread function, small-angle approximation, Monte Carlo code for evaluating the point-spread function and the modulation transfer function, Rayleigh scattering, scattering by spheres with diameters of 1 pLm. 1. Introduction Dense media can considerably affect images that are formed by a certain optical system. The scattering effect on radiation propagating in a turbid medium causes a loss of quality in the transmitted images that an optical system can produce. Imaging in sea- water, in turbulent clear air, and in adverse atmo- spherical conditions (rain, fog, clouds) are only a few examples of areas of research for which image propa- gation through random media is of interest. The quantitative knowledge of degrading effects on images caused by a turbid medium is of great impor- tance for developing imaging techniques to improve the quality of images produced. Many efforts have been made in recent years to improve resolution in images formed through biological tissues (for exam- ple, time gating in transillumination techniques for breast cancer detection'). Many effects are involved in the deterioration of the transmitted image: for a natural medium we must take into account the presence ofaerosols, turbulence,backgroundlights,and the perfor- mance of the optical detection system used, of course. The authors are with the Department of Physics, University of Florence, via S. Marta 3, 50139 Florence, Italy. Received 3 June 1992. 0003-6935/93/152813-12$05.00/0. c 1993 Optical Society of America. For the case of scattering and turbulence effects of a random turbid medium, we can factorize the modu- lation transfer function 2 4 (MTF) in different terms, each of which represents a separate aspect of the problem. In many situations the effect of light scat- tering (and, if present, absorption) by randomly sus- pended particles is predominant. In this paper we treat cases of this type and assume a turbid medium that consists of a random distribution of spherical particles interposed between the object (a point source) and the receiver (a simple thin lens), as shown in Fig. 1. This kind of problem is usually dealt with by means of the MTF, which is expressed as a function of the spatial frequency f (in cycles per millimeter or cycles per milliradian). For an isoplanatic system, the MTF is defined as the modulus of the optical transfer function, which is the double Fourier transform of the point-spread function (PSF), S(p), and which we consider to be normalized as follows: f S(p)exp(i2irf' p/q)dp MTF(f) = - (1) J S(p)dp 20 May 1993 / Vol. 32, No. 15 / APPLIED OPTICS 2813
Transcript
Monte Carlo calculations of the modulationtransfer function of an optical system operatingin a turbid medium
Piero Bruscaglioni, Paolo Donelli, Andrea Ismaelli, and Giovanni Zaccanti
Using a Monte Carlo method, we investigate the effect of a turbid medium on image transmission bymeans of the modulation transfer function approach. We present results that refer to a medium thatconsists of a random distribution of water spherical particles in air. We analyze the effect of geometricconditions (medium width and position) and source characteristics (Lambertian, beam emission). Wepresent results for small spheres (Rayleigh scattering) and spheres (1.0-pLm diameter) that are not small incomparison with the wavelength = 0.6328 plm. Numerical data show a large modulation transferfunction dependence on the source emission aperture and a substantial independence of the mediumwidth for a fixed value of the optical depth. In accordance with reciprocity principles, we test an inversescheme of Monte Carlo calculation, the advantage of this scheme being a substantial reduction incalculation time.
Key words: Modulation transfer function, point-spread function, small-angle approximation, MonteCarlo code for evaluating the point-spread function and the modulation transfer function, Rayleighscattering, scattering by spheres with diameters of 1 pLm.
1. Introduction
Dense media can considerably affect images that areformed by a certain optical system. The scatteringeffect on radiation propagating in a turbid mediumcauses a loss of quality in the transmitted images thatan optical system can produce. Imaging in sea-water, in turbulent clear air, and in adverse atmo-spherical conditions (rain, fog, clouds) are only a fewexamples of areas of research for which image propa-gation through random media is of interest.
The quantitative knowledge of degrading effects onimages caused by a turbid medium is of great impor-tance for developing imaging techniques to improvethe quality of images produced. Many efforts havebeen made in recent years to improve resolution inimages formed through biological tissues (for exam-ple, time gating in transillumination techniques forbreast cancer detection'). Many effects are involvedin the deterioration of the transmitted image: for anatural medium we must take into account the presenceof aerosols, turbulence, background lights, and the perfor-mance of the optical detection system used, of course.
The authors are with the Department of Physics, University ofFlorence, via S. Marta 3, 50139 Florence, Italy.
Received 3 June 1992.0003-6935/93/152813-12$05.00/0.c 1993 Optical Society of America.
For the case of scattering and turbulence effects ofa random turbid medium, we can factorize the modu-lation transfer function 2 4 (MTF) in different terms,each of which represents a separate aspect of theproblem. In many situations the effect of light scat-tering (and, if present, absorption) by randomly sus-pended particles is predominant. In this paper wetreat cases of this type and assume a turbid mediumthat consists of a random distribution of sphericalparticles interposed between the object (a point source)and the receiver (a simple thin lens), as shown in Fig.1. This kind of problem is usually dealt with bymeans of the MTF, which is expressed as a function ofthe spatial frequency f (in cycles per millimeter orcycles per milliradian).
For an isoplanatic system, the MTF is defined asthe modulus of the optical transfer function, which isthe double Fourier transform of the point-spreadfunction (PSF), S(p), and which we consider to benormalized as follows:
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:|00SSSS.S:$g$:S72:X)X;00SSSSSSS:SSS:fA:X ASUSASASSSSSS07:V40S::2j$j)2j$j$Xif004SSSaSSSSSSStAV$:S)J$)S)?:f.:SSSSS::;:SSASaS :D :ffAS ;: ;:;4111X1iC;;titES 418144i:Xt\yEt: 11j4Xit10Et t f40:0000000 )100dyt ld X: X
:00: )j1j:Xt00E 0 40Xtltty 0000tt;0L X D4:tE:L : Va :0:
OBJECTPLANE
L
THINLENS
IMAGE PLANE
Fig. 1. Schematic of the geometry considered in Sections 2 andand 3: S, Lambertian source; S', image of S; L, thin lens with focallength f = 5 cm and a radius of 1 cm.
where f (in cycles per radian) is the angular spatialfrequency, p is the coordinate vector on the imageplane, and q is the lens-image plane distance, and theintegral is extended over the image plane. In caseswith axial symmetry, such as those we examine here,expression (1) becomes a Fourier-Bessel transformand the MTF is a function of the modulus f = I f only.
Multiple scattering effects on the MTF of an opticalsystem can be well described analytically by thesmall-angle approximation (SAA) scheme. In thisscheme one can derive the following expression forthe MTF term that is due to particle scattering in theabsence of absorption5-7:
M(f)
= exp-2rs fL dx' f [I - Jo(2rs Lf)] p(s)sdsl,
(2)
wherep(s) is the phase function; ur, (in inverse meters)is the scattering coefficient; s = 2 sin(0/2), where 0 isthe scattering angle; L is the medium width; and J0 isthe Bessel function. However, the SAA scheme is oflimited application. In particular, it is not adequatefor large optical depths and particles that are smallcompared with the wavelength (see Ref. 6 and Sect. 4of Ref. 8). Numerical solutions assume great impor-tance for those situations that cannot be treatedanalytically. A numerical Monte Carlo-based proce-dure, which is called the SEMIM code, has beendeveloped to overcome the SAA scheme limitations. 9
We successfully tested SEMIM code results by compar-ing them with both theoretical and experimental datain different and well-controlled situations.10 Thebasis of the code is the consideration of the turbidmedium as an ensemble of secondary sources whosedefocused images add to the image of the primarypoint source to constitute the PSF on the imageplane. Details of the procedure can be found in Refs.9 and 10. However, Appendix A gives an explana-tion and a formal justification of the procedure imple-mented by the code.
We investigated a situation with a source and adetection optical system operating in a turbid me-
dium consisting of a random suspension of sphericalwater particles in air. This situation can help usmodel a real atmospheric case. Our aim was to usethe MTF approach to make a quantitative study ofthe effect of geometric parameters and source charac-terization on images produced by the optical systemin a turbid medium. Our results refer to the com-pound system of the source, the medium, and thelens. The medium deteriorates transmitted imagesin a different way for different values of particle sizeand concentration. Therefore we conducted thestudy by considering particles that were both smalland not small in comparison with the wavelength.Figure 1 illustrates the situation examined. Theturbid medium is infinitely extended laterally, andconsists of a homogeneous random distribution ofmonodisperse water spheres in the air. The SEMIMcode requires as inputs the scattering characteristicsof the medium [optical depth T and phase functionp(O)], which are obtainable by the Mie theory. Wepresent results obtained by the SEMIM code forspherical nonabsorbing particles with diameters of0.002 ,um (Rayleigh scattering) and 1.0 ,um for parti-cles that are, respectively, small and that are compa-rable with the wavelength of the emitted radiation(X = 0.6328 jim). In particular, we analyzed theeffect of the source-detector distance and the sourceemission aperture on the MTF. Results are pre-sented and discussed as follows: Section 2 is devotedto numerical results obtained for different values ofsource-detector distance d; in Section 3, we showfurther numerical results but for different values ofthe emitted beam semiaperture ,; in Section 4, wepresent the inverse Monte Carlo calculation method;and in Section 5 we give our conclusions.
2. Effect of the Distance between the Source and theLens on the MTF
The multiple scattering effects on the propagation ofradiation depend largely on the geometry of thesituation. In particular, if a spatially limited turbidlayer is interposed between the source and receiver,the relative position of the layer has a great influenceon the transmitted image quality; we can obtainbetter images for the case in which the medium isadjacent to the source than for the opposite case (inwhich the medium is adjacent to the receiver); this isthe well-known shower-curtain effect (this effect hasbeen quantified in terms of MTF in the two differentcases of wide7 and narrow" beams emitted by asource and for different geometries).
Here we present MTF numerical results for anonabsorbing medium that fills the whole spacebetween the source and the receiver. We performedcalculations by varying source-lens distance d for afixed value of optical depth r. It is easy to show thatthe MTF in the SAA scheme is independent ofdistance d. Indeed, if the medium homogeneouslyfills the whole space between the source and the lens,the simple changes c - r8', and d -> d' for a fixedvalue of the optical depth T = usd = c'd' leaveexpression (2) for the MTF unchanged.
In this section we show that this interesting prop-erty can also be extended to cases for which the SAAscheme is inadequate. We present results for twodifferent values of the particle size (RS and sphereswith a diameter of 1.0 aim) and for two values of theoptical depth (T = 2 and T = 5). Henceforth we indi-cate the Rayleigh scattering case as RS and the case ofthe spheres with a diameter of 1.0 jlm as S. In Fig.2, the MTF (in decibels) versus the spatial frequency f(in cycles per milliradian) is plotted. The figurerefers to the IS case and to an optical depth of T = 2.Curves A, B, C, and D are relative to four values ofdistance d (see caption) ranging from 12.5 cm to 1 m.We can see that the three curves B, C, and A arealmost exactly superimposed and that curve Apresents only a small discrepancy with respect to theothers. This result is not surprising if we recognizethat a moderate optical depth and particles that arenot small compared with the wavelength are condi-tions for the SAA scheme to be applicable. A nonex-haustive but indicative criterion9 is to examine theMTF asymptotic value M for high frequencies: M =limf a MTF(f). The SAA scheme, assuming an iso-tropic point source, predicts that M approaches thevalue exp(-7) independently of particle size andgeometry. For a Lambertian source, we can intro-duce a correction term a9:
M = exp(-a7). (3)
An a value approximately equal to 0.935 was indi-cated in Ref. 9 for spheres that are not small incomparison with the wavelength (for these particles,the SAA predicts that a = 1). For T = 2, M _ -8.1dB, and this value agrees well with the numericalresults of Fig. 2.
In order to test the MTF dependence on d in
conditions far from the SAA scheme, we also per-formed calculations for particles that are small incomparison with the wavelength (the RS case).Appendix B deals with the simple relationship be-tween M and the total amount of scattered powerreaching the receiver aperture. In Figs. 3 and 4 theplotted MTF 's refer to the RS case; again the numeri-cal results are relative to different values of distance d(see caption). Figure 3 (curves A, B, C, and D) refersto the optical depth = 2, while Fig. 4 refers to ahigher optical depth of = 5. A sufficiently large(even more than 10) number of scattering orders wereconsidered by the code to obtain these results. Forthese small spheres (Rayleigh scattering) the SAAscheme is largely inadequate, as we can see from theasymptotic value M, which is quite different from theSAA scheme prediction (M _ -8.1 dB for T = 2 andM _ -20.3 dB for T = 5). Again, we can still see asubstantial independence of the distance d, includingthose cases for which the SAA scheme is not adequate.A possible explanation of this independence is men-tioned at the end of Section 4.
Figure 5 compares the MTF 's obtained for the twokinds of particle considered (at the same optical depthT = 2): the IS case (curve A) and the RS case (curveB). The difference between the two asymptotic pla-teau values is due to the different total diffusedreceived powers in the two cases, and implies that theSAA scheme is not applicable for the smaller parti-cles.
The general result that the MTF is independent ofd (apart from short distances) has the followingimportant effect with regard to the Monte Carloprocedure. Not only can we make calculations forone distance and apply the result to other distances,but we can carry out calculations for the simplestcase, i.e., the far object, whose image is on the focal
MTF (dB)
2
0
-2
-4
-6
-8
-10
-6 -5 -4 -3 -2 -l 010 10 10 10 10 10 10
SPATIRL FREQUENCY (cycles/mrad)
Fig. 2. MTF 's versus spatial frequency are plotted for the IS case,T = 2. Curves A, B, C, and D refer to four different values ofd: 12.5, 25, 50, and 100 cm, respectively. The geometry is thesame as that of Fig. 1.
MTF (dB)
2
0
-2
-4
-6
-8
-10 -- I - I I I
-6 -5 -4 -3 -2 -l 010 10 10 10 10 10 10
SPRTIAL FREQUENCY (cycles/mrad)
Fig. 3. MTF's versus spatial frequency are plotted for the RScase, T = 2. Curves A, B, C, and D refer to four different values ofd: 12.5, 25, 50, and 100 cm, respectively.
3. Effect of Source Emission Characteristics on theMTF
0
Fig. 4. MTF's versus spatial frequency are plotted for the RScase, T = 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.
plane. In this case we can implement the simplestprocedure to calculate the incoming radiation on thelens aperture. For the case of a large distance, wecan consider radiance distribution to be the same overthe whole aperture. Thus it is necessary only todetermine radiation arriving at the center, and to sortscattered power in different intervals of arrival angle0 (with respect to the optical axis). Thus we cancalculate a function P(a), where P(a) indicates scat-tered power that is received in a range of 0 between 0and a. Taking into account the relationship for
(4)
In this section, we present numerical results toexamine the MTF dependence on the characteristicsof the emitted light beam. In a recent paper,9 wecalculated the influence of a turbid medium on theMTF in the two cases of an isotropic emission and aLambertian emission from a surface element in orderto make comparisons of our numerical results withtheoretical and experimental data. Here we investi-gate the aperture effect of the beam emitted by thesource. The effect of the beam aperture is alsoconnected to other parameters, such as particle sizeand optical depth. Thus, again, particles with smalldiameters and particles with diameters that are com-parable with the wavelength X are considered.
It is clear that, by decreasing the emitted beamaperture, we can obtain better resolution in imagesproduced by an optical system operating in a turbidmedium (i.e., a narrower spread function) because ofthe smaller illumination region of the medium. Inorder to analyze this dependence quantitatively, weperformed some numerical Monte Carlo calculationsfor some values of the semiaperture angle of aconically emitted beam ranging from 5 to 90° andwith Lambertian emission inside the cone. Figure 6shows some MTF (in decibels) curves versus spatialfrequency f (in cycles per milliradian) plotted for the
MTF (dB)
2MTF (dB)
-6 -5 -4 -3 -2 -110 10 10 10 10 10 10
SPATIAL FREQUENCY cycles/mrad)
0
-2
-4
-6
-10
0
Fig. 5. MTF 's versus spatial frequency. Curve A, 1S case, T = 2;curve B, RS case, X = 2. The geometry is the same as that of
Fig. 1.
I I I I I
-6 -5 -4 -3 -2 -l 010 10 10 10 10 10 10
SPATIAL FREQUENCY cycles/mrad)
Fig. 6. MTF's versus spatial frequency are plotted for differentvalues of beam semiaperture in the IS case. The seven curvesrefer, respectively from the upper to the lower, to , = 5, 100, 200,30°, 40°, 500, and 85°. The geometry is the same as that of Fig. 1,and X = 2 and d = 25 cm.
IS case; each curve corresponds to a value of semiap-erture (3, which increases from the upper to the lowercurve (see caption). The optical depth is T = 2 for allcurves. We can note a saturation effect in the MTFfor 2 3
sat with I3sat 50°. This means that, forapertures greater than a certain saturation value (3satithe turbid medium does not further affect the images.The value Psat depends on particle diameter but isindependent of 7, as we see below.
Figure 7 is similar to Fig. 6, but the MTF curveshere are for the RS case. We can observe a behaviorthat is analogous to that outlined in the IS case butnote some differences: the saturation effect is nowshifted to a larger value of semiaperture angle fiat 700, and the MTF asymptotic levels M are higher thanin the preceding case. The latter fact can be easilyexplained if we consider that the SAA scheme is notapplicable for these small particles and the amount ofreceived diffuse power is less than in the S case.M is related to the total diffuse received power. Thelarger value of the saturation angle 3sat can beexplained by the following argument: for the IScase, the Mie theory gives a more pronounced phasefunction p(O) in the forward direction, and thus thedetection probability that is due to a scattering eventfor photons emitted with a substantial angle (withrespect to the optical axis) can become negligible incomparison with photons emitted and scattered inthe near-axis forward direction. This saturationeffect is more visible in Fig. 8, where the MTFhigh-frequency level M versus the angle is plottedfor the two kinds of particles considered and for thesame optical depth 7 = 2.
We now present a series of Monte Carlo numericalresults in order to examine the MTF dependence on
MTF (dB)
M (dB)0 -
-2 -
-4 I-
-84..
I --* I -*- - I0 10 20 30 40 50 60 70
I I8o 9o
BETA (dog)
Fig. 8. High-frequency MTF limit value M (in decibels) versus thesemiaperture angle of emission . The triangles denote the RScase, and the squares denote the S case. The data are relative toresults presented in Figs. 6 and 7.
optical depth T in the IS case. We obtain results fordifferent values of T from the same calculation usingthe scaling relationships defined in Refs. 9 and 12,which allow us to scale (within a certain range) theresults to the desired a. Figures 9 and 10 showgroups of six MTF curves that refer to optical depths1, 2, 3, 4, 5, and 6 (from the upper to the lower curve,respectively) for the IS case and for a beam semiaper-ture (3 = 20° (Fig. 9) and ( = 50° (Fig. 10).
MTF (dB)
0 =
-1
-2 .
-3 _
-4 .
-5
-6
-6 -510 10
SPATIAL
-4 -3 -2 -110 10 10 10FREQUENCY (cycles/mrad)
Fig. 7. MTF 's versus spatial frequency are plotted for differentvalues of beam semiaperture P in the RS case. The nine curvesrefer, respectively from the upper to the lower, to P = 50, 100, 200,30,400,50°,60°,70°, and 855. The geometry is the same as that ofFig. 1, and T = 2 and d = 25 cm.
4.
0
-4
-12
-16
-20
-24 - I A I
-6 -5 -4 -3 -2 -1 010 10 10 10 10 10 10
SPATIAL FREQUENCY (cycles/mr ad)
Fig. 9. MTF 's versus spatial frequency are plotted for the IS casewith P = 200. The six curves are related to six different values ofthe optical depth T = 1, 2, 3, 4, 5, and 6, from the upper to the lowerrespectively. The geometry is the same as that of Fig. 1, and d25cm.
Fig. 10. MTF 's versus spatial frequency are plotted for the 1Scase with = 50°. The six curves are relative to six differentvalues of the optical depth T = 1, 2, 3, 4, 5, and 6, from the upper tothe lower respectively. The geometry is the same as that of Fig. 1and d = 25 cm.
We also performed calculations for other values of(3and obtained MTF curves similar to those of Figs. 9and 10. From these curves we can see, according tothe two-frequency model proposed by Kuga and Ishi-maru,13 that the slope depends on the optical depth T
but the roll-off frequency fo is independent of T andalso of (3 (fo is the frequency for which a MTFapproaches a constant limit value at high frequencies).In Fig. 11 asymptotic values M (in decibels) versus theoptical depth T for some values of the semiaperture (
M (dB)0
-4
-8
-12
-16
-20
-24
0 1 2 3 4 5 6 7
OPTICAL DEPTH
Fig.11. High-frequency MTF limit value M versus optical depth Tis plotted for the iS case. The geometry is the same as that of Fig.1 and d = 25 cm. Curves A, B, C, and D refer to P = 10°, 20°, 30°,and 40°, respectively. Curves for higher values of [3 are exactlysuperimposed to the D curve.
(see caption) are plotted. We can observe a lineardependence on X so that we can express M by means ofEq. (3) (with a depending on A). Again, the satura-tion effect is visible and the value (3sat seems to also bethe same (3sat 40°, 50°) for larger optical depths.We can conclude that, if the SAA holds, Eq. (3) is validwith a independent of T and the particle diameter butdependent on the type of source emission type (isotro-pic, Lambertian, or other) and beam semiaperture for(3 < (3sat; the saturation value Sat is independent ofoptical depth T within the range investigated.
4. Inverse Scheme of Numerical Monte CarloCalculation of the MTF
In Ref. 8, Sect. 5.2.2, it is stated, as a deduction frombasic principles, that for a generic medium the contri-bution of a source at a point r to the irradiance at asecond point r' is the same as the contribution of thesame source from point r' to the irradiance at r.Thus we investigated the possibility of employingreciprocity principles that allow us to invert thepositions of object and image plane for calculating thePSF in the case of an optical system in a turbidmedium. For our numerical procedure this inver-sion has the advantage of a substantial reduction incalculation time.
A comparison of the results we obtained by usingboth the direct and the inverse procedures showedthat the calculated MTF's coincide for particles thatare not small in comparison with the wavelength.However, for small (Rayleigh) particles and for cer-tain situations, differences of the order of 0.5-1.0 dBcan arise for optical depths that are larger than a fewunits. We considered different geometric situationsfor testing the validity of the inverse scheme, which iscombined with our procedure based, as described inRef. 9, on modeling the turbid medium as an ensem-ble of scattering centers, the latter producing defo-cused images on the image plane of the primary objectpoint source.
To show how the use of the inverse scheme reducescalculation time when it is inserted in our procedure,we refer to Figs. 12(a) and 12(b), which describe thedirect and the inverse calculation schemes, respec-tively. A point source at S emits radiation with acertain angular distribution. A Monte Carlo codeselects, on the basis of probability laws, the directionof emission of a photon and the positions of thesubsequent scattering point of its trajectory in theturbid medium. By applying the semianalyticscheme, from each scattering point (A and B in Fig.12), the code calculates the probability of the photonreaching a series of predetermined points on plane X',which is the image plane of the source S. Geometricoptics is followed for the latter calculations (see alsoAppendix A).
The inverse scheme considers a source with thesame characteristics of S (i.e., both are Lambertiansources) at point S' (the image of S). We determinethe trajectories within the medium of the photonscoming out from the lens by the Monte Carlo code.
Fig. 13. Geometry for results that are presented in Fig. 14. Thethin lens has a focal length f = 5 cm and a radius of 1 cm. We takethe source-medium and medium-lens distances to be equal to 10%of source-lens distance d.
we examined: the turbid medium fills a large part ofthe space between the source and the lens. Corre-spondingly, Fig. 14 shows the comparison betweenthe results of the direct and the inverse scheme forthe IS spheres, 7 = 5. We obtained the uppermost of
, 5 U the four curves (curve A) by the direct scheme. Wecalculated the other three curves with the inversescheme; each of these three curves refers to a differ-ent value of d, showing again, therefore, the indepen-dence of d. Apart from the small difference for themiddle frequency range, which could be ascribed todifferent numerical accuracies, the validity of theinverse scheme is demonstrated in this case. We can
?Lge note that, since the MTF is plotted versus the spatialne angular frequency, the PSF's are the same apart
from a magnification factor. We can also see that in
Fig. 12. Geometry for (a) the direct and (b) the inverse scheme ofcalculation. In the direct procedure the defocused images A', B' ofthe 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 isreached by radiation scattered at points A, B reached by radiationemitted at S' (image of S).
It can be seen that the probability of reaching anypoint in the object plane from any scattering point isalways nonzero, while for the direct scheme, depend-ing on the particular geometry considered, manyscattering points are not in view of the chosen pointson the image plane. As with the direct scheme, wecalculate the MTF by a double transform of theirradiance on the plane I (the object plane).
In comparison with the direct method, the inversescheme shows a better calculation efficiency andneeds a lower number of Monte Carlo photon draw-ings to reach an acceptable statistical convergence ofthe result. In the inverse scheme, a reduction factorof 10 for calculation time was generally found incomparison with the direct scheme.
Figure 13 shows the geometry of the first situation
MTF (dB)4-
0,-4 -
-8 -
-12
-16
-20
-24
-6 -5 -4 -3 -2 -110 10 10 10 10 10
SPATIAL FREQUENCY (cycles/mrad)
Fig. 14. MTF's versus spatial frequency are plotted for the IScase, =5. Curve A is relative to the direct scheme, with d = 50cm; the other three curves are derived in the inverse scheme withd=50cm, Im, and lOm.
Fig. 15. Geometry for results that are presented in Figs. 16, 17,and 18. Turbid layer of width: 18 mm, d = 190 mm, thin lenswith focal length f = 5 cm and a radius of 2.5 mm.
the asymptotic part of the curves at higher frequen-cies the MTF has the value predicted by the SAAtheory, i.e., -20.3 dB, taking into account the correc-tion factor introduced for a Lambertian source (seeSection 2 and Table 1 of Appendix B).
In the second geometric case, the turbid medium isconcentrated near the object point (Fig. 15). Figures16, 17, and 18 show the results obtained. We noticethat the inverse and the direct schemes give the sameresults for the iS spheres (Fig. 18), while a differenceof 0.7 dB is present for the RS spheres in theasymptotic value of the MTF for T = 5 (Fig. 17). TheRS case at = 2 (Fig. 16), however, does not show asubstantial difference between the results of the twoschemes.
Finally, the results for the third geometry of Fig.19, in which the turbid medium fills the space be-tween object and lens, are shown in Figs. 20 and 21.
MTF (dB)
2
0
-2
-4
-6
-8
-10I I I I
-5 -4 -3 -2 -1 010 10 10 10 10 10
SPATIAL FREQUENCY (cycles/mrad)
Fig. 16. MTF's versus spatial frequency are plotted for the RScase, T = 2. The two curves are relative to the direct (lower curve)and the inverse (upper curve) schemes.
-16
-20 1 *l ... * . l - J ..
-5 -4 -3 -2 -1 010 10 10 10 10 10
SPATIL FREQUENCY cycles/mrad)
Fig. 17. MTF's versus spatial frequency are plotted for the RScase, r = 5. The two curves are relative to the direct (lower curve)and the inverse (upper curve) scheme.
For X = 2 the two methods give the same results forboth the iS and the RS spheres.
The inverse scheme can also justify the resultsshown in Figs. 2, 3, 4, and 14, which showed that theMTF remained the same when the distance of theobject from the optical system changed, while theoptical depth was unchanged. In the inverse scheme,the medium is illuminated by a beam exiting from thelens and converging toward the object point. In theapproximation of geometric optics, which gives a firstmodel of the beam, the beam exiting from the lens hasa semiaperture equal to a/d, where a is the lensradius. For the Monte Carlo procedure, if a << d, aphoton exiting at a certain point of the lens could be
MTF (dB)4
0
-4
-12
-16
-20
-24
-5 -4 -3 -2 -1 010 10 10 10 10 10
SPATIAL FREQUENCY (cycles/mrad)
Fig. 18. MTF 's versus spatial frequency are plotted for the Scase, = 5. The two curves are relative to the direct and theinverse schemes.
Fig. 19. Geometry for results that are presented in Figs. 20 and21: d = 245 mm, thin lens has a focal lengthf= 5 cm and a radiusof 1 cm.
substituted by a photon exiting from the lens center,since the inclinations of the trajectories differ littleand the positions of the scattering points are almostthe same for the two photons. Thus, with a goodapproximation, we could consider a thin collimatedbeam from the lens center to the object plane. Thelatter case presents a PSF which, apart from thedifferent magnification factors, does not depend on d,given 7.
However, we can give an (approximate) justificationof the independence of d, given 7, of our results byconsidering that the radiative transfer equation canbe written in terms of the optical distances. Thus ifwe scale all the geometric distances by the samefactor, leaving the optical depth unchanged, we findthat the results of MTF (in terms of angular spacefrequencies) do not change. Actually, in the casesexamined, the lens size and the focal length did notchange together with d, and so scaling relationships
MTF
2
0
-2
-4
-6
-8
-10
(dB)
-6 -5 -4 -3 -2 -110 10 10 10 10 10
SPRTIAL FREQUENCY (cycles/mriad)
Fig. 20. MTF 's versus spatial frequency are plotted for the RScase, r = 2. The two curves are relative to the direct and theinverse schemes.
-8 -
-10
(dB)
-6 -5 -4 -3 -2 -110 10 10 10 10 10 10
SPATIAL FREQUENCY (cycles/mrad)
0
Fig. 21. MTF's versus spatial frequency are plotted for the 1Scase, T = 2. The two curves are relative to the direct and theinverse schemes.
for the transfer equation can be invoked as approxi-mate relationships (which, in our cases, work well,nevertheless).
5. Conclusions
With the many problems connected with the propaga-tion of radiation in turbid media we need to takemultiple scattering effects into account. The theoret-ical approaches for the calculation of the scatteringmedium contributions to the whole MTF of theimaging system require the applicability of the SAA ofthe scattered radiation (SAA scheme) in order toobtain useful analytical solutions. The proposednumerical Monte Carlo procedure (the SEMIM code)was¢.successfully applied to a more general range ofsituations. We used the SEMIM code to generatenumerical, data in order to analyze the MTF depen-dence on the source-lens distance d (Section 2) andon the source beam aperture (Section 3). Our re-sults extend the property (at least for the examinedsituations) that the MTF is independent of a variationof the distance d when the optical depth is keptunchanged. The property had already been shown[see Eq. (2)] in the framework of the SAA for amedium that fills the whole space between the sourceand the lens homogeneously. A simple explanationof the property is, however, given at the end ofSection 4.
We found and investigated a large MTF dependenceon the source characteristics for spherical particleswith a size that is small and comparable with thewavelength, and identified some features, such as asaturation effect for high values of the beam semiap-erture. The role of optical depth 7 was examined.Finally, we presented an inverse Monte Carlo calcula-tion scheme (Section 4). The method, which is basedon the application of reciprocity principles, allows usto obtain a substantial reduction in calculation time.
Here we give an explanation of the principle on whichthe SEMIM code is based. We refer to Fig. 22, whereS is a point object and S' is its image on image plane
.li Ps(xs ys, -D) is a generic scattering point whosedistance from lens plane El is D and whose image isthe point PA(X1, Y1, A), where A is the distance of PAfrom E. P(xi, yi, d) is a certain point on image planeYi on which the code calculates the irradiance. If Uindicates the field of the spherical wave arriving at Elfrom the scattering point Ps, and U(xi, yi) is theresulting field at Pi, we have, in the Fresnel-Kirchhoff approximation,
Ui(x, y k exp(ikd) i U(x, Y)
such that
(1 1 1 xs XiXD d fD d '
(1 1 1 +Y
Y~D + - DM.+ dr
By putting x,/D = -X/A and ys/D = - Y/A and byusing the lens law 1/f = (1/D) + (1/A), we then have
Axi - dXA -d
Ayi - dY° A-d (A3)
(Al)
where pi and p indicate the positions of points (xi, A),and (x, y) on Yi and on the lens aperture El, respec-tively, and f is the lens focal length. If the angle ofincidence of the field arriving at the lens is small wecan write
. (x - xJ)2 + (y - y )2U(p) = A(x, y)exp k D (A2)
Equation (A3) shows that the point (x0, yo, 0) is theprojection on the lens plane of the point (xi, yi, d) fromthe point (X, Y, A), which is the image of P. We alsohave a2 p/axay = 0 at (x0,y0, 0).
By applying the stationary phase method, we have,for the integral factor on the right-hand side of Eq.(Al),
27ri exp(iq)k A(xo, yo) pa
2p a2 p\ 1/2
Iax2 y2)
where q4 is a real function ofx0, yo, xi, yi and the secondderivatives are evaluated at (xo, yo). Then we have
By inserting Eq. (A2) into Eq. (Al), we can see thatthe resulting exponential factor in the integrand is arapidly fluctuating function. We can use the station-ary phase method of integration. Let us write theexponent of the integrand as
ik- P(Xi, yi, x, y).
We see that the point on the lens plane for whichap/ax = ap/ay = 0 is the point of coordinates (x0 , yo)
A
TURBID MEDIUM
Pi
S..............S
ID*
D
d
Fig. 22. Geometric scheme for the SEMIM code.
AUi(xi, yi) -~ A(x0 , yo)exp(i4i') A- d with i' = I + kd.
Thus, for the intensity I at (xi, yi), we have
A2
(A -~~~A4I(xi, A = X0, Y°) (A - d)2 ' (4
where I(xo, y) indicates the intensity of radiationscattered at S and arriving at (x0, yo).
Equation (A4) shows that the contribution of radia-tion scattered at P, to the intensity on the imageplane at a certain point is related to the intensity thatis incident upon the lens at a definite point by meansof a proportionality factor. The factor on the right-hand side of Eq. (A4) is the inverse of the ratio of theelement of area on 1i around (xi, yi) and the projectionof this element on the lens plane from the point (X, Y,A).
This led us to the scheme of calculating the spreadfunction on the plane Yi mentioned in Sections 2 and4 (see also Fig. 12). As we explained, the intensity onthe plane Yi is the sum of the intensities that wecalculated by using a Monte Carlo procedure todetermine the positions of the scattering points. Wecan see that the point (x0, yo) can be outside the part ofthe lens plane occupied by the lens. In this case thecode puts the corresponding contribution to the inten-sity at (xi, yi) equal to zero.
In summary, using the SEMIM code, we firstcalculate the positions of the scattering points accord-ing to probability laws, taking into account the scatter-ing and absorption coefficients of the medium and therelevant phase function. From each scattering pointand following the so-called semianalytic scheme, wethen calculate the scattered intensity at the point (x0,yo) on the lens plane corresponding to a chosen point(xi, yi) of the image plane by taking into account thescattering angle toward the point (xo, yo), the phasefunction, and the attenuation of the medium betweenthe scattering point and the point (x0, yo). Finally,we calculate the intensity at (xi, yi) according to Eq.(A4). The intensities deriving from all the scatteringpoints chosen by the Monte Carlo procedure aresimply added.
We can add, however, that some particular situa-tions of the geometry source-medium optical systemallow us to use some simplified versions of theSEMIM code, as was indicated by an example at theend of Section 2., in which we mentioned the case of asource placed at a large distance from the lens, withits image on the focal plane. Another case is that inwhich we can assume that the scattered radiance isuniform over the lens aperture. Then the geometricprocedure gives a distribution of irradiance on theimage plane, which is uniform in a circle that is theintersection of the cone subtended by the lens, as seenfrom the image of the scattering point. In this case asimple routine can distribute this received scatteredpower between different annuli centered at the pri-mary image point. For instance, we encounteredthis case while dealing with a medium concentratednear the source, relatively far from the lens.
Appendix B: Received Scattered Power andHigh-Frequency Behavior of the MTF
As was remarked [see Eq. (3) in Section 2, and Eq. (5)of Ref. 9] the turbid medium MTF reaches, for highspatial frequencies, a constant asymptotic value M.Within the SAA scheme, Eq. (3) follows for theisotropic point source case. For a different kind ofsource emission this asymptotic value changes the aparameter of Eq. (3), depending on the aperture andcharacteristics of the source emission, as was shownin Section 3. For those situations (in which particlesare small in comparison with the wavelength) thatcannot be dealt with in the SAA scheme, measure-ments13 and calculations9"10 have shown that themedium MTF again approaches a high-frequencyasymptotic value, which is higher than that predictedby SAA scheme.
As was indicated by Ishimaru [see Eq. (9) of Ref. 5],Eq. (3) with a = 1 means that the total received poweris equal to that received in the absence of scattering,with an absorbing-only medium. Within the SAAscheme, the amount of power emitted toward thereceiver, which is lost by scattering, is regained byforward scattering and scattering of radiation emit-ted in other directions. Actually, in the SAA scheme,the total received power is the same for a fixed value
of the optical depth T and for every particle size, thedifferences arising only in the distribution of theirradiance on the image plane (i.e., PSF) but not inthe total amount of received power. We can also seethe connection of M to received scattered radiationwith the following simple consideration. We canwrite the PSF S(p) as the sum of two components:the fractions SO(p) (unscattered) and S8 (p) (scattered)of the received power. Assuming an ideal thin-lensimaging system (a hypothesis that is justified whenthe spatial frequency cut-off introduced by the me-dium is predominant in comparison with that relativeto the imaging system performance), we have S0 =8(p)exp(-'r), T(So) = exp(-r), where T indicates adouble Fourier transform, T is the optical depth thatis due to scattering (as in this paper), and a unitaryreceived power is assumed in the absence of thescatterers. Indicating with P, and P0 the total re-ceived scattered and unscattered powers, respec-tively, and starting from MTF definition
MTF(f) = IT(So + S)/(Po + PJ),
we find, since limfi. T(PJ) = 0 and P0 = exp(-T), that
1M =imrnMTF(f) 1 (Bi)
Equation (B1) shows a direct link between MTFhigh-frequency asymptotic value M and the totalamount of scattered received power Ps. For a fixedvalue of T, P, determines M univocally. P does notdepend on scatterer size only when the SAA scheme isapplicable.
If we combine Eq. (B1) and Eq. (3) with a = 1, as forthe SAA scheme, we have
1 + P5 /p0 = exp(-7) = P0
Then
Ps= 1 - exp(-r) = 1 - Po. (B2)
Equation (B2) indicates that the total received scat-tered power Ps is equal to the fraction of poweremitted toward the receiver, which is lost for scatter-ing, as we remarked above. In a more general case,the ratio of scattered-to-unscattered received powerP,/Po is less than what we would obtain if the SAAscheme were applicable because of substantial scatter-ing losses of the directly received beam that are notcompletely compensated by rescattering of radiationtoward the receiver. As a consequence, we can find ahigher MTF plateau at high frequencies for thosecases for which the SAA scheme is not applicable, theasymptotic value M depending on the particle size(besides the optical depth, of course). In Fig. 5 ofSection 2 (as well as in Fig. 8, Section 3), this behaviorof the MTF at high frequencies appears clearly; Fig. 5shows a difference of almost 3 dB between the
aM represents the value found by means of the SEMIM code (Lambertian source). ML, Ml, and M are expressed in decibels. The lasttwo columns show the value of M given by Eq. (3) (a = 0.935, Lambertian source).
plateaus of curves A and B, the latter referring to theRS case for which the SAA scheme is inadequate.
In order to give an example of this high-frequencyMTF behavior, we performed some calculations toevaluate the total scattered received power P, for RSand S scatterers and for = 2 and 5. For thesecalculations we used a simple MC code, consideringthe lens aperture as the receiver. For both the RSand the S cases we considered a Lambertian as wellas an isotropic source in order to show some differ-ences in the results. We inserted the scattered re-ceived power P, obtained from these calculations intoEq. (Bi). In Table 1 the calculated values of PS/Poand M are reported for the cases considered. Thegeometry is the same as that illustrated in Fig. 1, withd = m and a lens-receiver with a radius of 1 cm.As we can see, the value P,/PO is smaller in the case ofsmall particles, and the corresponding M is higher.These values agree well with those found by theSEMIM code and also with the SAA predictions forthe case of IS scatterers, while in the RS case the SAAscheme is not applicable and the M value calculatedwith Eq. (Bi) is higher than exp(-T).
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