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Bassi et al. Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2833

Spatial shift of spatially modulated light projectedon turbid media

Andrea Bassi,1,* David J. Cuccia,2 Anthony J. Durkin,3 and Bruce J. Tromberg3

1IIT, ULTRAS-INFM-CNR and IFN-CNR, Dipartimento di Fisica, Politecnico di Milano,Piazza Leonardo da Vinci 32, 20133 Milano, Italy

2Modulated Imaging Inc, Technology Incubator Office, 1002 Health Sciences Rd., Irvine, California 92612, USA3Beckman Laser Institute and Medical Clinic, University of California Irvine, 1002 Health Sciences Road, Irvine,

California 92612, USA*Corresponding author: andreabassi@polimi.it

Received July 22, 2008; revised September 9, 2008; accepted September 9, 2008;posted September 15, 2008 (Doc. ID 99282); published October 23, 2008

This work extends modulated imaging, a recently developed technique based on the projection of structuredlight on a turbid medium that is able to measure optical properties of the high-scattering medium and performtomography. We observe that structured light obliquely projected on a turbid medium undergoes a spatial shiftduring propagation. We propose a method to measure the spatial phase shift of a sinusoidal fringe patternprojected in a turbid medium, and we present a model derived from the diffusion approximation to describe thelight propagation. Experimental validation by measurements performed on liquid phantoms is presented.© 2008 Optical Society of America

OCIS codes: 290.7050, 290.1990.

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. INTRODUCTIONodeling the propagation of light in turbid media is im-

ortant in various scientific fields. In biomedical opticsodels are useful to estimate the penetration of light in

issues. Furthermore, models are used to estimate the op-ical properties of tissues. In particular the measurementf the absorption ��a� and the reduced scattering ��s�� co-fficients of tissue is the basis of several diagnostic appli-ations [1].

Methods used to model light propagating in turbid me-ia have been applied in the frequency domain [2] to char-cterize the time dependence of a photon density wave inerms of its amplitude attenuation and phase retardation.n this paper we apply this concept to the spatial fre-uency domain through the use of obliquely incidenttructured illumination.

Structured light methods have been used extensivelyor profilometric measurements [3] and for machine vision4]. Recently, the utility of structured illumination for im-ging and quantification of optical properties in turbidystems such as biological tissues has been demonstrated5]. In this paper, we extend this method, called modu-ated imaging (MI), to model the propagation of spatiallyeriodic waves with oblique illumination. In this geom-try the propagating photon density wave undergoes bothhase shift and amplitude attenuation.The spatial shift of light obliquely incident in turbidedia is a well-known process: if a collimated light beam

s injected into a scattering medium, the light re-emittedt the surface will be spatially shifted with respect to itstate at the injection point. Obliquely incident lightources have been used successfully in diffuse optics forhe measurement of optical properties [6], for fluorescence

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ensing [7], and for measurement of the anisotropy ofighly scattering media [8].Here, we investigate this phenomenon in the spatial

requency domain through spatially periodic wide-field il-umination. During propagation in the turbid medium,tructured light undergoes scattering and absorption. Asresult, when a sinusoidal fringe pattern is obliquely pro-

ected on a sample, the reflectance at the surface will bettenuated and shifted in phase. In order to characterizehis reflectance we investigated both analytically and ex-erimentally the behavior of the phase shift that a sinu-oidal fringe pattern undergoes during propagation in aurbid medium.

In Section 2, a model—derived from diffusionheory—is presented for the analytic prediction of the re-ectance of a turbid medium illuminated with a sinu-oidal fringe pattern. In Section 3 an experimentalethod for the measurement of the phase shift is illus-

rated, and measurements on liquid phantoms are shown.n Sections 4 and 5 the theoretical and experimental re-ults are compared and potential applications are dis-ussed.

. THEORYs described in Fig. 1, when a sinusoidal fringe pattern isbliquely projected on a turbid medium, the light willropagate along the direction given by Snell’s law andventually it will be absorbed and scattered. The back-cattered light at the surface will be a combination of theight diffusively reflected at all depths in the sample and,ssuming a linear homogenous medium, it will be sinuso-ally modulated at the same spatial frequency of the in-

008 Optical Society of America

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2834 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Bassi et al.

ident light. Compared to the incident light, it will be at-enuated and spatially displaced. In the followingections, we will denote as PS the phase difference (orhase shift) between the incident light and the scatteredight, detected at the surface.

According to the diffusion approximation, in a turbidedium the fluence rate � and a light source q are related

y the equation [9]

�2� − �eff2 � = − 3�trq, �1�

here �tr=�a+�s� and �eff=�3�a��a+�s��. In our geometryhe light source is a fringe pattern spatially modulatedlong the x direction and can be modeled by the complexxponential q= q̃0 exp�i�2�fxx��. Here q̃0 is a complexumber that describes the light source.Assuming a linear medium, the fluence rate will beodulated at the same spatial frequency as the source

nd will be in the form �= �̃0 exp�i�2�fxx��. Inserting �nd q in Eq. (1) we obtain

�2�̃0

�2z− �eff�2�̃0 = − 3�trq̃0, �2�

here

�eff� = ��eff2 + �2�fx�2. �3�

quation (2) therefore extends the diffusion approxima-ion in the case of a sinusoidally modulated source [5].

A planar illumination in a turbid medium can be de-cribed as an extended source [9,10], decaying exponen-ially. Therefore, q̃0 can be described as a decreasing ex-onential along the direction of propagation:

q̃0 = P0

�tr

cos �exp�−

�trz

cos ��exp�i�q�. �4�

ere P0 is the incident intensity, z is the distance fromhe surface, � is the angle of propagation, and �q is thehase of the source. According to Fig. 1 �q can be ex-ressed as a function of the propagation distance by �q2�f �x=2�f z tan �. The diffusion equation becomes

ig. 1. (Color online) Scheme of structured light propagation inturbid medium. The light source (a sinusoidal fringe pattern)

ropagates obliquely. During propagation it is attenuated andndergoes a phase shift.

x x

�2�̃0

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�tr

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�trz

cos ��exp�i2�fxz tan ��.

�5�

ts solution is

�̃0 = � exp�− � �tr

cos �− i2�fx tan ��z + C exp�− �eff� z�,

�6�

here

� =3P0�tr�tr/cos �

���eff� �2 − � �tr

cos �− i2�fx tan ��2 , �7�

nd C depends on the choice of the boundary conditions.t should be noted that to find this solution a secondoundary condition has been implicitly chosen: the flu-nce has been set to zero at positive infinity. Using theartial current boundary condition [11] at the surface z0 the fluence is proportional to the module of the flux:

0�0�=−�J� �0�. Here J� =−�� /3�tr, �= �1−Reff� /2�1+Reff�,nd Reff�0.0636n+0.668+0.710/n−1.440/n2.We obtain

C = −

��� �tr

cos �− i2�fx tan �� + 3��tr3��tr + �eff�

. �8�

hus the fluence rate at the surface is

�̃0�0� = � + C

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�tr

cos �

�eff� +�tr

cos��+ i2�fx tan �

���eff� +�tr

cos ��2

+ �2�fx tan ��2 .

�9�

he amplitude of the reflectance at the surface is propor-ional to the fluence rate and its value is

R = ��̃0�0� =3P0��tr

�3��tr + �eff� �

�tr

cos ����eff� +

�tr

cos ��2

+ �2�fx tan ��2−1/2

. �10�

he phase of the fluence at the surface is described as

� = arctan�2�fx tan �

�eff� +�tr

cos � . �11�

. MATERIALS AND METHODS. Experimental Setuphe MI system shown in Fig. 2 consists of a digital pro-

ector (BenQ PB8260) that creates a sinusoidal pattern on

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Bassi et al. Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2835

he surface of the sample and a 16-bit CCD cameraRoper Cascade 512F, 512512 pixels) to image the dif-usely reflected light. An interference filter in front of theamera enables the selection of a narrow wavelengthand (�=660 nm, ��=20 nm). The spatial frequency ofhe pattern is varied from 0 to 0.2 mm−1, and 38 spatialrequencies are sequentially acquired within this range. Ailtable mirror between the digital projector and thehantom enables us to perform the measurement at dif-erent angles of incidence.

. Amplitude and Phase Measurementhe sample (with surface positioned in a plane x ,y) is il-

uminated with a sinusoidal fringe pattern modulatedlong the x direction at the spatial frequency fx. The re-ectance is recorded for three projected fringe patterns.or each of the three projections the sinusoidal pattern ishase shifted along the x direction by 2� /3 from the oth-rs. The intensities Ii of the backscattered light at any lo-ation �x ,y� will be the sum of the planar �I�� and sinusoi-ally modulated components along the x direction �I��:

I1�x,y� = I��x,y� + I��x,y�cos�2�fxx + ��x,y� − 2�/3�,

I2�x,y� = I��x,y� + I��x,y�cos�2�fxx + ��x,y��,

I3�x,y� = I��x,y� + I��x,y�cos�2�fxx + ��x,y� + 2�/3�.

�12�

The measurement of the amplitude of the reflectanceas been discussed in recent publications [5]. Briefly, itan be determined using a demodulation algorithm by theormula

A�x,y� =21/2

3��I1�x,y� − I2�x,y��2 + �I2�x,y� − I3�x,y��2

+ �I3�x,y� − I1�x,y��2�1/2. �13�

n principle the measurement of the PS could be obtainedy projecting a sinusoidal pattern on the object and calcu-ating the phase of the 2-dimensional Fourier transforma-ion of the diffusively reflected light. However thisethod, like those based on the projection of a single pat-

ern, is very challenging due to the uncertainty in the de-ermination of the modulation frequency.

ig. 2. Experimental setup: a sinusoidal fringe pattern is pro-ected on the liquid phantom. The scattered light is filtered at60 nm and acquired by a CCD.

Here we propose to use a method previously developedor phase-shifting profilometry [12]. In this technique ainusoidal fringe pattern is projected on the sample. Inhe presence of a curved surface the sinusoid undergoes aS that is proportional to the height of the surface. There-

ore the measurement of the phase gives a direct estima-ion of the height of the object. Instead, in our measure-ents the sample is flat and the PS is due only to the

ropagation through the turbid medium. The phase is de-ived from the intensity measured at the three phases as

��x,y� = arctan��3I1�x,y� − I3�x,y�

2I2�x,y� − I1�x,y� − I3�x,y�� . �14�

his method takes advantage of the fact that the acquisi-ion of three shifted images is already required for theeasurement of the amplitude of the modulated light;

herefore no additional acquisition time is needed to per-orm this measurement. The measurement is performedndependently at each location (x ,y position) on theample. Thus the amplitude and phase information cane extracted for every point, creating an image of theample. In the present work only homogeneous samplesre considered and amplitude and phase are calculated byveraging over a 1010 pixel area (corresponding to 11 cm) at the center of the image. It is worth noting that

imilar results could have been achieved using a struc-ured light source and a single photodiode.

. Calibration of Amplitude and Phasecalibration is required in order to determine the true re-

ectance of the sample, and to separate it from the re-ponse of the experimental apparatus. This can bechieved by use of a phantom with known optical proper-ies. The amplitude of the reflected light from the samplender study �ASAMPLE� can be calculated after measuringhe reflectance of the sample itself �ASAMPLE

MEASURED� and thehantom �APHANTOM

MEASURED�, and calculating—by use of the de-cribed model or Monte Carlo simulations—the phantomxpected reflectance �APHANTOM

PREDICTED�:

ASAMPLE =ASAMPLE

MEASURED

APHANTOMMEASUREDAPHANTOM

PREDICTED. �15�

Similarly, the measured value of the PS is relative tohe phase of the fringe pattern incident on the sample;herefore in order to measure the PS of the sample�SAMPLE� a calibration is required. Using Eq. (11) we canredict the PS of the calibration phantom �PHANTOM

PREDICTED.fter measuring the PS of the sample ��SAMPLE

MEASURED� andhe phantom ��PHANTOM

MEASURED�, we can determine �SAMPLE as

�SAMPLE = �SAMPLEMEASURED − �PHANTOM

MEASURED + �PHANTOMPREDICTED.

�16�

herefore particular attention must be paid to the preciselignment of the optical setup: an error can occur if theample and the reference phantom are not placed in theame position because an additional phase shift ��2�fx�z tan �inc can be introduced by the different dis-

ances �z from the light projector. For the measurements

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2836 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Bassi et al.

escribed later a displacement of 0.1 mm would give aaximum error of about �0.1 rad.

. Measurement Protocoleasurements have been performed on liquid phantomsade of scattering Intralipid diluted in water and absorb-

ng Nigrosin (Sigma, St.Louis, Missouri) in a total volumef 300 ml contained in a plastic holder. The scatteringroperties of the Intralipid were characterized using lit-rature reports [13]. The Nigrosin absorption was charac-erized using a spectrophotometer.

In a first experiment the measurement of the reflec-ance at varying scattering coefficients was performed atn angle of incidence of 50° from the normal. A phantomith absorption coefficient �a=0.0068 mm−1 and reduced

cattering coefficient �s�=0.29 mm−1 was initially pre-ared and measured. The amount of Intralipid was se-uentially increased in order to vary the reduced scatter-ng coefficient from �s�=0.29 mm−1 to 8.03 mm−1 in sixteps. An amount of liquid equal to the added Intralipidas removed from the container to maintain the sameolume of liquid in all measurements, and the absorptionoefficient is therefore decreased from �a0.0068 mm−1 to 0.0045 mm−1. The phantom with highercattering was used to calibrate the measurement.

In a second experiment the absorption coefficient of theample was varied from �a=0.0004 mm−1 to 0.102 mm−1

n six steps by adding Nigrosin to a sample with reducedcattering coefficient �s�=0.6 mm−1. A last experiment waserformed to test the dependence of the reflectance on thengle of incidence. A sample with �a=0.013 mm−1 and

s�=0.67 mm−1 was measured at the three angles of pro-ection of 25°, 45°, and 65° from the normal. These valuesf optical properties and projection angles were chosen toest the model both within the limits of diffusion theorynd in other cases where diffusion theory fails with veryow values of scattering.

. RESULTS AND DISCUSSION. Discussion of the Modelefore discussing the measurements we focus on the PS

eatures. The behavior of the PS as a function of spatialrequency and optical properties is shown in Fig. 3. Theurves are calculated from Eq. (11), and they show thathe PS increases nonlinearly with the spatial frequency.n Fig. 3(a) the transport coefficient �tr is varied and theeduced albedo �s� /�a is kept constant. A strong depen-ence of the phase shift on �tr is shown: for each spatialrequency the PS decreases as �tr is increased. This is be-ause the light source decays more rapidly: if the attenu-tion in the tissue is increased (i.e., the interactionengths decreased) the backscattered light is dominatedy the part of the sample that is close to the surface,here the PS is smaller. In the extreme case of infinite �ar �s� the PS will reduce to zero because of total absorp-ion or total reflection at the surface.

Conversely in Fig. 3(b) the ratio �s� /�a is varied and theS is shown as a function of the normalized frequency

f /� �. For constant � a weak dependence of the phase

x tr tr

n the albedo is shown. The reason is that the PS is aunction of the sum �eff� +�tr / cos �, and since �eff� is gener-lly smaller than �tr, the value of the phase is mainly de-ermined by the transport coefficient, independently ofhe albedo.

. Experimental Resultscomparison between the model and the experimental

ata for varying optical properties is shown in Fig. 4 andig. 5. The dotted curves in Figs. 4(a) and 4(b) represent,espectively, the amplitude and the PS of the reflectancebtained for the phantoms with increasing scattering.he solid curves represent the values predicted by the

heoretical model using the expected values for the re-uced scattering shown in the legend. For the highestalue of scattering the agreement is perfect, as this phan-om is used for calibration. The agreement between themplitude predicted by the diffusion approximation andhe measurement is better that 5% for all the reducedcattering values with �s��0.2 mm−1. In Fig. 4(b) the er-or between the measured and predicted PS is smaller

ig. 3. (Color online) Effect of the transport coefficient (a) and ofhe reduced albedo (b) on the phase. The values of �tr and �s� /�are indicated in the legend.

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Bassi et al. Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2837

han 10% for �s��0.6 mm−1, but at lower scattering val-es the model fails to accurately predict the phase. Thisemonstrates that the phase is more difficult to describeith the diffusion theory than the amplitude. This result

s reasonable since the PS is strongly dependent on theorward propagation of the light. More advancedodels—such as DeltaP1 [14] and Monte Carlo

imulations—will be more appropriate to describe propa-ation at these values of �eff� , as well as at high spatial fre-uencies. Moreover at increasing frequencies the ampli-ude decreases, giving smaller appreciable errors.

In Fig. 5 the experimental results (dotted curves) andhe theoretically predicted values (solid curves) for themplitude and the PS are shown for phantoms with vary-ng absorption (the expected values of absorption arehown in the legend). The dependence of the phase on ab-orption is weak because, as mentioned previously, the PSs strongly dependent on �tr=�a+�s�, and here �a �s�.he amplitude can be used to estimate the optical prop-rties of tissues by using an inversion algorithm based on

ig. 4. (Color online) Calculated amplitude (a) and phase (b) ofhe reflectance at different scattering. The values of the reducedcattering coefficient �s� are indicated in the legend. The expectedalues for the amplitude and the phase are indicated by the solidurves.

iffusion theory [5]. Instead, the curves shown in Fig. 5(b)uggest that the estimation of absorption will be challeng-ng if only phase information is used. However the mea-urement of the PS could potentially increase the robust-ess of the estimation of �a and �s�.In Fig. 6 the dependence of the phase on the angle of

ncidence is shown. Here, in Fig. 6(a) a very small differ-nce in the amplitude is found between the three angles.n particular the backscattered light from the sample in-reases with angle, and the difference reaches approxi-ately 4%. This is in accordance with the fact that if the

ight is injected at larger angles, it will more easily arrivet the surface. Figure 6(b) shows that the PS increasesith the angle, as expected. It is worth remembering that

he propagation angles are smaller than the projectionnes (they are �18°, 32°, and 43°). A good agreement (bet-er than 5%) is found in all three cases.

ig. 5. (Color online) Calculated amplitude (a) and phase (b) ofhe reflectance at different absorption. The values of the absorp-ion coefficient �a are indicated in the legend. The expected val-es for the amplitude and the phase are indicated by the solidurves.

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2838 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Bassi et al.

. CONCLUSIONSn this paper we have proposed a new analysis for theroblem of propagation of light in turbid media, showinghat obliquely projected spatially modulated light is at-enuated and phase shifted during propagation. We pro-osed a method to measure the PS that sinusoidallyodulated light undergoes in a turbid medium, and we

resented a theoretical model based on the diffusion ap-roximation that allowed us to predict the amplitude andhe PS of the reflectance. Experiments were performed toest the theoretical model: the agreement between theorynd experiments is better than 10% for typical opticalroperties of biological tissues ��s��0.6 mm−1� and spatialrequencies �0.2 mm−1. For lower scattering values atrong inaccuracy ��20% � was found, showing a failure ofiffusion theory for these optical properties.

ig. 6. (Color online) Calculated amplitude (a) and phase (b) ofhe reflectance at different projection angles (indicated in the leg-nd). The expected values for the amplitude and the phase arendicated by the solid curves.

Potential application of this method could be the im-rovement of imaging and tomography of turbid media.he combined use of amplitude and phase could improve

he robustness of 3-dimensional reconstruction of opticalroperties of the medium.The in vivo application of the technique to biological

amples will be subject to future investigation. The cur-ature of biological samples might be a serious limitationo this technique if profilometry is not considered. Con-ersely the model could be used to improve the accuracyf profilometric techniques based on structured light [3]pplied to biological tissues or other highly diffusive me-ia when optical properties are known.Another potential application of the technique is theeasurement of anisotropy of tissues. Anisotropy

trongly influences nondiffusive propagation of collimatedight at positions that are close to the injection point [8];herefore we expect a dependence of the PS on anisotropyf the sample, particularly at high spatial frequencies.onte Carlo simulations are under development to sup-

ort and investigate this argument.

CKNOWLEDGMENTShis work was partially supported by the Ministeroell’Istruzione, dell’Università e della Ricerca (MIUR) In-erlink Project (prot. II04CGM5MG); the Laser Micro-eam and Medical Program (LAMMP), a National Insti-utes of Health (NIH) Biomedical Technology Resource,rant P41-RR01192; and the U.S. Air Force Office of Sci-ntific Research (USAFOSR) Medical Free-Electron Laserrogram (F49620-00-2-0371 and FA9550-04-1-0101). D. J.uccia would also like to acknowledge support from theational Science Foundation (NSF) Graduate Researchellowship Program.

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1

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