2052 J. Opt. Soc. Am. A/Vol. 15, No. 8 /August 1998 Gan et al.
Image formation in turbid media under amicroscope
X. S. Gan, S. P. Schilders, and Min Gu
Optoelectronic Imaging Group, School of Communications and Informatics, Victoria University of Technology,P.O. Box 14428, MCMC, Victoria 8001, Australia
Received January 9, 1998; revised manuscript received March 24, 1998; accepted April 23, 1998
1. INTRODUCTIONImaging an object embedded in a turbid medium has at-tracted substantial interest in the past decade. Researchwork in this field can be classified into two categories:transillumination imaging, in which case a parallel beamprobe is used1–10 and microscopic imaging, in which casea microscope objective is used for illumination.11–16 Mostpublications have focused on technological developmentsto reduce scattered light, while few papers have discussedthe mechanisms for actually forming an image, particu-larly in the field of microscopic imaging through turbidmedia. A variety of optical gating methods, based onphysical differences between the scattered light and theunscattered (ballistic) light have been developed. Suchphysical differences include the degree of coherence,4,5 thedegree of polarization,6,7 the difference in propagatingtime1–3,8,9 through a turbid medium, and the deviationfrom the path of unscattered photons.15,16 There hasbeen discussion about which gating method is better,15
but the criterion that can be used to determine the effi-ciency of a gating method has not been discussed thor-oughly.
Using an objective lens in a microscopic imaging sys-tem raises the question of which numerical aperture of anobjective is suitable for imaging.14 According to the im-aging theory based on Born’s approximation,17 which ig-nores the multiple scattering in a turbid medium, a high-numerical-aperture objective lens can provide highdiffraction-limited resolution. Born’s approximation isapplicable to the case in which the optical thickness n, de-fined as the thickness of a turbid medium divided by thescattering mean-free-path length, is less than 1. On theother hand, a low-numerical-aperture objective can sup-press scattered photons, which statistically travel at highangles.14,16,18,19 Both arguments are based on the as-sumption that ballistic light is dominant in forming animage. When a turbid medium is thick, e.g., when n
. 10, unscattered light may be too weak to be detected,particularly in the presence of detector noise. In thissituation scattered light has to be included in construct-ing an image. An important question raised here is whatrole the scattered light plays in constructing an image.To answer this question, the relationship of scatteredphotons to image resolution should be investigated. Inthis paper we investigate the role of scattered light in con-structing an image under a microscope and propose a cri-terion for determining the efficiency of a gating method.
This paper is organized as follows. In Section 2 a de-tailed analysis of the resolution of an image constructedby scattered photons experiencing different numbers ofscattering events is presented. The effect of the numeri-cal aperture of an objective and a finite-sized detector onresolution is discussed in Section 3. The relationship ofresolution to signal strength is discussed in Section 4,which leads to a criterion for determining the efficiency ofa gating method. Experimental results are presented inSection 5 to demonstrate the new theoretical prediction.Finally, a discussion and conclusion are presented in Sec-tion 6.
2. INFLUENCE OF SCATTERED PHOTONSON RESOLUTIONThe Monte Carlo simulation method is one of the mostuseful methods for studying light propagation through aturbid medium.20–22 The advantage of the Monte Carlosimulation method is that it can track the change inphysical properties of a photon during each scatteringevent. Therefore it can predict the behavior of the scat-tered light and its role in constructing an image. Toidentify how many scattering events a photon has experi-enced before it leaves a turbid medium, the number ofscattering events N is recorded during the simulation
1998 Optical Society of America
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process. This modified Monte Carlo simulation methodwas described in detail in our previous paper.23
A schematic diagram of a scanning optical microscopeconsidered in this paper is shown in Fig. 1. Here L1 andL2 are the illumination and detection objectives, respec-tively. P is a finite-sized pinhole of diameter vd , placedin front of the detector D. Let us consider a thin sharpedge embedded in a scattering slab of thickness (d) 180
Fig. 1. Schematic diagram of a scanning optical microscope.
Fig. 2. (a) Image resolution as a function of the number of scat-tering events N or the optical thickness n (NA 5 0.25, vd → `).In the case of GN versus N, n 5 12. (b) Normalized photon-number distribution rN as a function of the number of scatteringevents N for different values of the optical thickness n (NA5 0.25, vd → `).
mm, which consists of 0.48-mm polystyrene beads sus-pended in water. According to Mie scattering theory,24,25
the anisotropy value g of the scattering medium is 0.81for a wavelength of 0.633 mm. From the image intensityof the sharp absorption edge scanned in the x direction,the transverse resolution G is defined as the distance be-tween the 90% and 10% intensity points. The definitionof the symbols is also shown in Appendix A.
All the photons collected by the detector are dividedinto N groups, according to the number of scatteringevents that a photon experiences. In this way the trans-verse resolution, GN of an image constructed by each in-dividual group of photons can be determined.
The transverse resolution GN shown in Fig. 2(a) in-creases with the number of scattering events N for agiven value of the optical thickness. It is seen that oncea photon is scattered, it contributes to an image with reso-lution much lower than the diffraction-limited resolution,which is less than 1 mm for a high-numerical-aperture ob-jective. For example, photons that are scattered onlyonce construct an image with resolution larger than 14mm. This feature shows that an image with resolutionnear the diffraction limit can be obtained only if the un-scattered light is dominant in the signal collected by thedetector. In other words, to obtain an image withdiffraction-limited resolution, most of the scattered pho-tons have to be suppressed by means of a gating process.
However, the strength of the unscattered light de-creases exponentially as the optical thickness n increases.In this situation the unscattered signal is too weak to bedetected in comparison with the noise generated in an im-aging system. So when an object is embedded in a thickturbid medium, the scattered light has to be regarded aspart of the signal in constructing an image. This featurecan be seen from the photon-number distribution rN , nor-malized by the total number of input photons, as a func-tion of the number of scattering events N for different val-ues of the optical thickness n [Fig. 2(b)]. As can beobserved, when a turbid medium becomes thick the num-ber of unscattered photons drops quickly, while the num-ber of scattered photons increases, particularly those thatare scattered multiple times. Owing to the increase ofthe multiply scattered photons in the total number of pho-tons, image resolution degrades quickly as a turbid me-dium becomes thick. The image resolution G, as a func-tion of the optical thickness n, of a turbid medium isshown in Fig. 2(a). As expected, the image resolution G ismuch lower than the diffraction-limited resolution evenfor n 5 1 and becomes worse as a turbid medium be-comes thicker.
3. EFFECTS OF THE NUMERICALAPERTURE AND THE PINHOLE SIZEIn an optical microscope—for example, in a confocalmicroscope—the aperture size of objectives along with thesize of the detector pinhole act as angular gates23 for sup-pressing scattered photons. Before a pinhole mask startsto play its role, photons emerging from a turbid mediumfirst go through a preselection process determined by theaperture of objectives. In Fig. 3, image resolution G, as afunction of the optical thickness n, for different values of
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the numerical aperture of objectives is illustrated. It canbe noticed that an improvement in image resolution oc-curs when a low-numerical-aperture objective is used. Alow-numerical-aperture objective rejects more photonstraveling at high angles than a high-numerical-apertureobjective. These photons statistically experience morescattering events and therefore contribute to an imagewith lower resolution. As a result, image resolution ishigher for a lower-numerical-aperture objective.
It should be mentioned that a high-numerical-apertureobjective gives a narrow diffraction spot at its focus andtherefore gives a high-resolution image if scattered pho-tons are negligible. But when scattering is strong, ahigh-numerical-aperture objective collects more scatteredphotons emerging from a turbid medium. Therefore, fora high-numerical-aperture objective, the degradation ofimage resolution caused by scattered photons is severe ifno other gating mechanism is used.
The improvement in resolution caused by the apertureof an objective comes from two aspects: one is from thereduction of scattered photons experiencing a given num-ber of scattering events at high angles, and the other isfrom the change in the photon-number distribution, rN .To explain these two aspects further, in Fig. 4 we showimage resolution GN for different values of the numericalaperture of objectives. These curves clearly show that for
Fig. 3. Image resolution G as a function of the optical thicknessn for different values of the numerical aperture of objectives(vd → `).
Fig. 4. Image resolution GN for different values of the numericalaperture of the objective (n 5 12, vd → `).
a given number of scattering events, photons traveling athigh angles carry less information about the object andcan be suppressed by the aperture of an objective.
The normalized photon-number distribution rN for dif-ferent values of the numerical aperture of an objective isshown in Fig. 5. It is seen that a high-numerical-aperture objective collects more scattered photons than alow-numerical-aperture objective. But the profiles of thenormalized photon-number distribution rN are similar inall cases. For example, the ratio r12 /r50 is 4.84, 4.7, and4.55 for numerical aperture (NA) 5 0.25, 0.5, and 0.75,respectively. It is clear from Figs. 4 and 5 that use of alow numerical aperture does show a sound improvementin the first aspect but that the change in the second as-pect is not so significant.
A pinhole mask in a confocal microscope has a strongeffect on suppressing scattered photons. How a pinholeselects photons depends on the deviation of a scatteredphoton from the path of unscattered photons. The imageresolution G, as a function of the optical thickness n, fordifferent pinhole sizes is shown in Fig. 6. It is noted thatwhen a finite-sized pinhole is placed in front of the detec-tor, image resolution is improved significantly comparedwith the case without a pinhole. A pinhole can suppressscattered photons because the photons are statisticallydistributed away from the center of the detector planeand the distance from the center reflects how much ascattered photon has deviated from the path of ballisticphotons. Therefore a finite-sized pinhole can reduce thedegradation of image resolution caused by scattered pho-tons.
Fig. 5. Normalized photon-number distribution rN as a functionof the number of scattering events N for different values of thenumerical aperture of objectives (n 5 12, vd → `).
Fig. 6. Image resolution G as a function of the optical thicknessn for different pinhole sizes (NA 5 0.25).
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To further analyze the effect of the pinhole size, weshow the transverse resolution GN in Fig. 7. It is seenthat when a finite-sized pinhole is used, the resolution ofan image formed by photons scattered N times is im-proved noticeably. Even when vd 5 100 mm, the im-provement in resolution is significant compared with thecase without a pinhole. This finding shows that a pin-hole selectively reduces the number of scattered photonsfor a given number of scattering events N. In otherwords, a pinhole actually suppresses scattered photonsthat are distributed farther away from the center of thedetector plane and carry less information about the ob-ject. The photon-number distribution rN emerging fromthe turbid medium and collected by a pinhole of differentsizes is shown in Fig. 8. As expected, scattered photonsare reduced significantly if a small pinhole is used. Itcan also be noticed that the reduction of scattered photonsis more significant when the number of scattering eventsN is larger. This feature is reflected by the fact that theratio r12 /r50 is 4.84, 20.5, and 28 for pinholes of vd→ `, vd 5 100 mm, and vd 5 40 mm, respectively.Thus the use of a finite-sized pinhole reduces the weight-ing of more scattered photons. Therefore the overall im-age resolution G can be improved, since photons experi-encing more scattering events contribute to an imagewith lower resolution. It is clear from Figs. 7 and 8 thata finite-sized pinhole not only reduces the number of scat-tered photons according to their path deviation in the de-tector plane but also changes the weighting of the scat-tered photons in the total number of photons collected bya detector.
Fig. 7. Image resolution GN for different pinhole sizes (n5 12, NA 5 0.25).
Fig. 8. Normalized photon-number distribution rN as a functionof the number of scattering events N for different pinhole sizes(n 5 12, NA 5 0.25).
4. RELATIONSHIP OF RESOLUTION TOSIGNAL LEVELAccording to the discussion in Section 3, an image of highresolution can be acquired by use of a low-numerical-aperture objective or a pinhole of small size, which leadsto low signal strength. This trade-off between imageresolution and signal strength actually occurs in mostgating methods. In this section we propose a criterion fordetermining the efficiency of a gating method.
In Fig. 9 the image resolution G as a function of signalstrength, defined as the ratio of the number of photonscollected by a detector to the number of input photons, isillustrated for different values of the numerical apertureof an objective. It is seen that for a given resolution, thesignal strength for a high-numerical-aperture objective ishigher than that for a low-numerical-aperture objective.More interestingly, for a given signal strength, imageresolution G is much higher for a high-numerical-apertureobjective. To understand this feature, we consider thepoints A and B in Fig. 9, which correspond to two combi-nations of the numerical aperture and the pinhole size.The signal strength in the two cases is the same. Forpoint A an objective of NA 5 0.25 and a pinhole of vd5 4000 mm are used, while for point B an objective of
NA 5 0.75 and a pinhole of vd 5 110 mm are used. Theimage resolutions G in these two cases are 59 and 43 mm,respectively, which shows a better resolution in the lattercase.
The reason for this phenomenon can be drawn from thephoton-number distribution rN and the transverse reso-lution GN in the two cases, which are illustrated in Figs.10(a) and 10(b), respectively. It is noted from Fig. 10(a)that although the total number of photons collected by thedetector is almost the same in the two cases, the weight-ing of the less scattered photons in case B is larger thanthat in case A. This feature shows that a small pinholein case B is more efficient in suppressing highly scatteredphotons than a low-numerical-aperture objective in caseA. Another contribution from a pinhole can be seen fromFig. 10(b). The transverse resolution GN in case B ishigher than that in case A, which means that for eachgroup of photons that undergo a given number of scatter-ing events, a pinhole selects photons that are distributedclose to the center of the detector plane and are thereforemore related to the object. The results shown in Figs. 9and 10 suggest that the gating mechanism provided by an
Fig. 9. Image resolution G as a function of signal strength fordifferent values of the numerical aperture of objectives (n5 12).
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objective, which removes scattered photons on the basis oftheir propagation angle, is not as efficient as that by apinhole, which selects photons on the basis of the devia-tion from the path of ballistic photons.
Fig. 10. A comparison of the normalized photon-number distri-bution rN and image resolution GN between cases A and B: (a)normalized photon-number distribution, (b) image resolution.
5. EXPERIMENTAL RESULTSThe conclusions shown in Sections 2–4 still hold for a re-flection scanning system with highly reflective objects.For convenience, experimental work based on the prin-ciple described in Fig. 1 was performed in a reflection op-tical scanning microscope. The turbid sample consistedof polystyrene microspheres (diameter 5 0.48 mm, refrac-tive index 1.59) suspended in water with a volume con-centration of 2.5%. Its scattering mean-free-path lengthwas approximately 19.2 mm (anisotropy value g 5 0.81),according to Mie scattering theory.24,25 The thickness ofthe sample cell was approximately 100 mm, which corre-sponds to the optical thickness n of 10.4. The illumina-tion beam was from a 7-mW He–Ne laser.
A thin highly reflective metal bar of width 46 mm wasplaced at the bottom of the sample cell. Images of a barrecorded under different criteria are shown in Fig. 11.These images are formed mainly by scattered photons.This feature can be seen from Fig. 11(f), which demon-strates that the image recorded by a small pinhole (vd5 50 mm) is almost invisible. In this case the signal col-lected by the detector is contributed mainly by the unscat-tered photons and some scattered photons that are scat-tered back to the path of unscattered photons. Since thesignal strength is so weak, owing to the use of a small pin-hole, the noise generated in the imaging system becomesa major factor in degrading the image. When the pinholesize is increased, e.g., when vd 5 150 mm [Fig. 11(e)], animage of transverse resolution G of 14 mm is obtained. Inthis situation the signal strength is greatly increasedbecause of the contribution from scattered photons.Although this image resolution is much lower than thediffraction-limited resolution, which is approximately
Fig. 11. Measured images of a bar embedded in a turbid medium of an optical thickness of n 5 10.4: (a) NA 5 0.25, vd5 4000 mm, G 5 24.1 mm; (b) NA 5 0.25, vd 5 150 mm, G 5 13.4 mm; (c) NA 5 0.75, vd 5 4000 mm, G 5 25.5 mm; (d) NA 5 0.75,vd 5 500 mm, G 5 15.6 mm; (e) NA 5 0.75, vd 5 150 mm, G 5 14 mm; (f) NA 5 0.75, vd 5 50 mm.
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1 mm for an objective of NA 5 0.75, it is still approxi-mately 50 times higher than the result obtained in tran-sillumination imaging through a turbid medium with asimilar optical thickness.7
When NA 5 0.75, G 5 25.5, 15.6, and 14 mm for vd5 4000, 500, and 150 mm, respectively [see Figs. 11(c),11(d), and 11(e)]. In the case of NA 5 0.25, G 5 24.1 and13.4 mm for vd 5 4000 and 150 mm, respectively [see Figs.11(a) and 11(b)]. These behaviors qualitatively demon-strate the theoretical conclusion that image resolutioncontributed by scattered photons becomes poor as the pin-hole size increases. A comparison of Fig. 11(a) with Fig.11(c) and a comparison of Fig. 11(b) with Fig. 11(e) alsoconfirm that for a given pinhole size, resolution is im-proved as the numerical aperture of an objective is re-duced. Spherical aberration that is due to refractive-index mismatch is another factor that may degrade imageresolution, but its influence is quite small (a few microme-ters) compared with the degradation caused by scattering(tens of micrometers) in our experimental condition.
In Figs. 11(a) and 11(d) the signal strength is almostthe same. The former corresponds to NA 5 0.25 and apinhole of vd 5 4000 mm, and the latter corresponds toNA 5 0.75 and a pinhole of vd 5 500 mm. The trans-verse resolution G (15.6 mm) is better in the latter case,compared with that (24.1 mm) in the former. This resultconfirms our theoretical prediction that the combinationof a high-numerical-aperture objective and a small-sizedpinhole is a better choice than the combination of a low-numerical-aperture objective and a large-sized pinhole.In other words, gating by a small pinhole is more efficientthan gating by a low-numerical-aperture objective. Theabove experimental phenomena were also observed whenthe optical thickness n of the sample was 20, although theimage contrast degraded appreciably.
6. DISCUSSION AND CONCLUSIONWe have explored both theoretically and experimentallythe role of scattered photons in forming an image.Diffraction-limited resolution in imaging through a turbidmedium can be achieved only under the condition thatballistic light is dominant in the detected signal or thatthe majority of scattered light can be removed. In prac-tice the condition for imaging with ballistic photons orscattered photons is dependent on whether ballistic lightis strong enough for detection in the presence of noisegenerated from an imaging system and electronic devices.Figure 11(f) shows that in our experimental condition, theballistic light is too weak compared with the noise gener-ated from the imaging system. The strength of noise isthe key factor in determining the condition for imagingwith ballistic photons.
However, even in the case of a highly sensitive detec-tor, the ballistic light decays exponentially as the opticalthickness of a turbid medium becomes large. Therefore,for a thick turbid medium, scattered photons have to beincluded as part of the signal in order for the noise level tobe overcome. The results in this paper show that al-though image resolution contributed by scattered photonsis lower than diffraction-limited resolution, it is 50 timeshigher than resolution in transillumination imaging,
which is determined by the size of a parallel beam probe.7
It is also demonstrated that resolution contributed byscattered photons can be improved by use of a low nu-merical aperture or a small-sized pinhole in front of a de-tector. These results would be useful in skin biopsy withan endoscope.
In the case of imaging with scattered photons in a tur-bid medium we have proposed a criterion for determiningthe efficiency of a gating method. The criterion is basedon the comparison of image resolution for a given signalstrength. It would be interesting to use the criterion pro-posed in this paper to compare other gating methods withthe gating method by a finite-sized pinhole. A compari-son between finite-sized pinhole and polarization gatingmethods will be described in a forthcoming paper.
APPENDIX A: DEFINITION OF VARIABLES
N Number of scattering events that a photon experi-enced
n Optical thickness, which is the sample thickness interms of the scattering mean free path
G Transverse resolution, defined as the distance be-tween the 90% and 10% intensity points
GN Transverse resolution of an image constructed byphoton scattered N times
rN Photon-number distribution as a function of thenumber of scattering events N
NA Numerical aperture of an objectivevd Diameter of a pinholeg Anisotropy value
ACKNOWLEDGMENTSThe authors thank the Australian Research Council forits support. S. P. Schilders is supported by an Australianpostgraduate award.
*Address correspondence to Min Gu.
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