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Particle positioning from charge-coupled deviceimages by the generalized Lorenz–Mie theory andcomparison with experiment

David Moreno, Fernando Mendoza Santoyo, J. Ascencion Guerrero, andMarcelo Funes-Gallanzi

Three-dimensional position and velocity information can be extracted by direct analysis of the diffractionpatterns of seeding particles in imaging velocimetry with real-time CCD cameras. The generalizedLorenz–Mie theory is shown to yield quantitatively accurate models of particle position, such that it canbe deduced from typical experimental particle images with an accuracy of the order of 20 mm and an errorof 11 gray levels rms, data obtained by comparison of theoretical and experimental images. Both thetheory and an experimental verification of the problem presented here are discussed. © 2000 OpticalSociety of America

OCIS codes: 120.7250, 290.4020, 350.4990, 070.2580, 100.2000, 040.1520.

1. Introduction

Particle positioning is a crucial aspect of modern ve-locimetry techniques. Conventional particle imagevelocimetry ~PIV!, for instance, suffers from an in-trinsic limitation because it yields two-dimensionaldata, hence neglecting the third velocity component.For this reason, techniques for three-dimensional~3D! PIV have recently evolved as an area of intensiveesearch,1–15 yielding some success at the cost of in-reasing complexity. The increased complexity andhe limited optical access found in industrial appli-ations mean that many of these techniques, thoughhey are of academic interest, cannot be used in prac-ical industrial applications. In short, these tech-iques involve scanning light sheets, stereoscopiciews, holographic recording, parallel light sheets,raded intensity light sheets, or mixtures thereof.For practical applications, restricted viewing elim-

inates stereoscopic approaches. Robustness andease of experimental performance make holography a

D. Moreno ~dmh@andromeda.cio.mx!, F. Mendoza Santoyo~fmendoza@foton.cio.mx!, and J. Ascencion Guerrero are with theCentro de Investigaciones en Optica, Loma del Bosque 115, Colo-nia Lomas del Campestre, Leon, Guanjuato, Mexico. M. Funes-Gallanzi is with the Instituto Nacional de Astrofısica, Optica yElectronica, Luis Enrique Erro 1, Tonantzintla, Puebla, Mexico.

Received 24 January 2000; revised manuscript received 16 June2000.

0003-6935y00y285117-08$15.00y0© 2000 Optical Society of America

relatively unattractive option because it involves alengthy developing process and therefore is slow toyield results, although its large depth of field andstorage capacity make it somewhat attractive.Scanning light sheets are difficult to operate withrestricted optical access and high speeds, so theyhave not been employed in industrial environments.Moreover, practical application errors reported for allthese techniques are in general unacceptably large.

CCD cameras have shown their worth in industrialapplications and are in increasingly widespread useas image-recording media. Restricted optical accessto flows of interest, such as those found in turboma-chinery, mean that often only one camera can beaccommodated. An extension of the high-accuracytechnique16 to three dimensions, exploiting the digi-tal representation of particle images in intensity aswell as spatially, has been developed.17 In thisstudy, due consideration is given to the need for 3Dparticle positioning at low magnification, such thatan adequate portion of the particle-distribution re-gion can be interrogated. For a pattern-matchinglow-magnification approach to be successful, detailedmodels of the behavior of particles illuminated bymonochromatic light must be derived. A theoreticalmodel is also useful in other applications such asparticle sizing as well as for examination, for in-stance, of the effect of changing aperture size, thenature of image variation with the degree of particledefocus, and the required adjustment of parameters

1 October 2000 y Vol. 39, No. 28 y APPLIED OPTICS 5117

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5

for CCD cameras. An experimentally validatedmathematical model is the subject of this paper.

We consider a spherical absorbing particle illumi-nated with monochromatic coherent light. By de-signing a mathematical model that calculates theimage produced by the particle when it is illuminatedwith a Gaussian light beam, a light sheet, or a planelight wave, by including aberration effects and con-sidering digitization parameters one can position thespherical particle in three dimensions. Calculatedresults based on the theoretical model are comparedwith experimentally observed images for an 18-mm-diameter glass sphere, in forward scattering for aGaussian beam with a He–Ne laser for illumination.The model is also capable of calculating the particleimage for an arbitrary particle position within theilluminated region.

2. Brief Remarks on the Generalized Lorenz–MieTheory

The theory of a plane wave scattered by a spherical,isotropic, homogeneous, nonmagnetic particle is aclassic solved problem. More-advanced theoreticaland experimental techniques depend on the ability tohandle a more-general problem in which the scattercenter is illuminated by a laser beam and have led tothe so-called generalized Lorenz–Mie theory ~GLMT!.

he GLMT is a generalization of Lorenz–Mie theoryor arbitrarily incident shaped beams18 that can also

be applied to particle sizing, for instance. Gaussianbeam techniques have been extensively explored andapplied in the analysis and improvement of practicaldevices such as phase Doppler systems. However,these techniques were not previously applied tovelocimetry, for which particle images need to becalculated and then propagated through an aperture-and-lens system to an image plane. PIV, for in-stance, requires the application of GLMT techniquesfor a Gaussian beam with an elliptical cross section,19

which is the configuration of laser light sheets.The geometry used is illustrated in Fig. 1. Con-

sider two Cartesian coordinate systems ~Oguvw andOpxyz! attached to the incident field and to the par-icle center, respectively, with Ogu parallel to Opx

and Ogv parallel to Opy. Og and Op are the beamwaist center and the particle center, respectively.The incident field is linearly polarized, propagatingtoward the positive w axis with the electric field Eparallel to the u axis. The scattered light is ob-served at point M9~r, u, w!, where r, u, and w arespherical coordinates attached to the particle system~Opxyz!.

The scattering field components in the near fieldare described20 as

Er~s! 5 2 (

n51

`

(m52n

1n

an gn,TMm @jn~ar!

1 j0n~ar!#Pnumu~cos u!exp~imw!, (1)

118 APPLIED OPTICS y Vol. 39, No. 28 y 1 October 2000

Eu~s! 5 2

1ar (

n51

`

(m52n

1n Fan gn,TMm j9n~ar!tn

umu~cos u!

1 mbn gn,TEm jn~ar!)n

umu~cos u!Gexp~imw!, (2)

where

Ef~s! 5 2

iar (

n51

`

(m52n

1n Fman gn,TMm j9n~ar! )n

umu~cos u!

1 bn gn,TEm jn~ar!tn

umu~cos u!Gexp~imw!; (3)

a 5 ka (4)

s the size parameter, k is the wave number of thencident beam, a is the particle radius, an and bn arehe partial-wave scattering amplitudes of plane-waveie theory, given by

an 5i~n21!~1!n~2n 1 1!

n~n 1 1!

c9n~n# a!cn~a! 2 n# cn~n# a!c9n~a!

c9n~n# a!jn~a! 2 n# cn~n# a!j9n~a!,

(5)

bn 5i~n21!~1!n~2n 1 1!

n~n 1 1!

n# c9n~n# a!cn~a! 2 cn~n# a!c9n~a!

n# c9n~n# a!jn~a! 2 cn~n# a!j9n~a!,

(6)

where cn, xn, and jn 5 cn 2 ixn are the Riccati–Besselfunctions, which are related to the Bessel functions ofhalf-integer order; Pn

umu~cos u! are the associated Leg-endre polynomials, which depend uniquely on theobservation direction. n# is the complex refractiveindex of the sphere, and r 5 rya is the normalized

Fig. 1. System geometry for GLMT calculations.

pt

wi

sieaGaaHttamtlwti

distance from the particle to the scattered-light ob-servation point. All variables that have the sym-bol ; are normalized to the particle radius. A

rimed quantity denotes differentiation with respecto the argument of the function. The gn

m coefficientsare beam shape coefficients that are specific to GLMTand involve the characterization of the incidentbeam.

The coefficients gnm are dependent on the beam

waist’s diameter, wavelength, and Og coordinates u,v, and w. These beam coefficients can be numeri-cally evaluated by quadrature techniques,21 whichare, however, time consuming, preventing their ex-tensive application. After significant work with theuse of finite series,22 the best beam evaluationmethod was found to arise from so-called localizedapproximation23 for on-axis Gaussian beams that

as later successfully generalized to off-axis Gauss-an beams24 without any rigorous justification, which

was eventually produced for both on-axis and off-axisbeams.25,26 The localized approximation allows theevaluation of beam shape coefficients to be speededup by orders of magnitude compared with the earlierquadrature techniques.

Although a rigorous justification exists only forGaussian beams, the localized approximation hasbeen found empirically to be remarkably efficient forother kinds of beams, namely, laser sheets,27 top-hatbeams,28 and doughnut laser beams.29 This leads tothe conjecture that the localized approximation isactually valid for arbitrarily shaped beams.30

Therefore we used this method for the research de-scribed in this paper by utilizing previously foundexpressions.27

The technique used to compute the Ricatti–Besselfunctions and coefficients an and bn is described else-where.31 Computation of the associated Legendrefunction is accomplished with standard recursion for-mulas such as those presented by Abramowitz andStegun.32 All electric field quantities are normal-ized based on an assumed uniform incident electricfield of unit amplitude.

In the forward direction, the observed field is notonly the scattered field but a total field obtained asthe sum of the incident field and the scattered field.The incident field for a light sheet is given by, viz.,19

Er~i! 5 w0

shF~cos f!sin~u!S1 22Qx

lxr cos uD

12Qx

lxx0 cos uGexp~K!, (7)

Eu~i! 5 w0

shFcos fScos u 12Qx

lxr sin2 uD

22Qx

lxx0 sin uGexp~K!, (8)

Ew~i! 5 2w0

sh sin w exp~K!, (9)

where

K 5 2ia~r cos u 2 z0!, (10)

w0sh 5 ~2Qx Qy!

1y2 expF2iQx

v0x2 ~r cos f sin u 2 x0!

2

2iQy

v0y2 ~r sin f sin u 2 y0!

2G , (11)

Qx 51

i 1 ~2ylx!~r cos u 2 z0!, (12)

Qy 51

i 1 ~2yly!~r cos u 2 z0!, (13)

lx 5 av0x2 ,

ly 5 av0y2 , (14)

and v0x and v0y are the beam waist radii in the x andy directions, respectively. The incident field for aGaussian beam may be recovered as a special casefrom a laser sheet when v0x 5 v0y and for a waveplane by allowing v0x and v0y to become large.

3. Image Calculation

A. Approximate Image Calculation

Adrian and Yao33 proposed using a quadrature toderive a nominal particle image diameter de accord-ing to

de 5 @~M2dp2 1 ds

2!#1y2, (15)

where

ds 5 2.44~M 1 1!lfy# (16)

is the diameter of the point response function of adiffraction-limited lens measured at the first darkring of the Airy disk intensity distribution; the par-ticle diameter is dp, the wavelength is l, and thefocal length of the lens divided by the aperturediameter is fy#. As defined here, ds is also equal totwice the mean diameter of a laser speckle. Equa-tion ~15! is an approximate quadrature that repre-ents the combined effects of magnification andmage blurring on determination of the final diam-ter of the image. If the point response functionnd the geometric image distribution were bothaussian, this equation would be exact, and in factGaussian distribution was used in the past as an

dequate approximation to the Airy distribution.ence Eq. ~15! was used as an approximation, with

he assumption that the geometric image distribu-ion is not much different from a Gaussian. Thispproach represents a compromise between geo-etrical optics and a simplified approximation to

he Lorenz–Mie theory. However, the former is aimit theory that has some validity only when theavelength of the light used is much smaller than

he scatterer size and for light beams close to themaging system’s optical axis, and the latter condi-

1 October 2000 y Vol. 39, No. 28 y APPLIED OPTICS 5119

tstpv

b

5

tion is valid only when the particle is illuminated bya plane wave. The plane-wave assumption holdswhen the size of the scatterer is small enough rel-ative to the characteristic dimensions of the inci-dent beam, but in many cases this condition is notsatisfied. Even if the plane-wave assumptionholds in a velocimetry setup, the orientation of thiswave front relative to the viewing camera generallychanges substantially across the field of view, suchthat major corrections are required for each parti-cle’s scattering image. An accurate analysis of thefield must rely on the GLMT for any position in theilluminated field.

Simplifications in formulating the physical prob-lem to be solved, such as considering a particle to bea point function, were evaluated. However, such atreatment does not consider a particle’s complexrefractive index or representative laser power orthe different intensity distributions exhibited byparticles that are an equal distance out of focus buton opposite sides of the focal plane. The last-named particle property is of crucial importance inmaking a modeling approach worthwhile, because ifparticle positions behind and in front of the focalplane could not be distinguished, such an approachwould be severely deficient.

Therefore a theoretical model was initially de-vised that permits calculation of the image pro-duced by a spherical absorbing particle illuminatedby monochromatic light, including the amount oflight energy falling upon the CCD sensor.

B. Exact Image Calculation

The specific imaging system under consideration hasbeen described in detail by other authors34,35 and canbe seen simplified in Fig. 2. This system was used asa starting point to validate the software, which weobtained from Dennis R. Alexander of the Universityof Nebraska, and experimental results. In this sys-tem an object located at plane z0 is illuminated by alaser, which in previous research was a N2 laser andhere is a Nd:YAG or a He–Ne laser. One forms theimage by passing the scattered light through a plano–convex lens ~ f 5 50 mm, D 5 25 mm! onto a sensor~plane z3!, from where the image is sent to a framegrabber.

Assuming that the laser–droplet interaction canbe modeled as a linearly polarized plane wave inci-

Fig. 2. Initial layout: z0–z3, planes.

120 APPLIED OPTICS y Vol. 39, No. 28 y 1 October 2000

dent upon a spherical absorbing particle, one canobtain the electromagnetic field distribution every-where from classic Lorenz–Mie theory.36,37 Theseequations yield exact knowledge of the electromag-netic field at every point within the aperture locatedat plane z1. To remain consistent with the scalardiffraction theory outlined in Ref. 38, we assumedthat the x component of the electric field will be theonly significant electric field component through-out the imaging system and use the Fresnelapproximation to propagate the field to the imageplane.

In addition, the actual physical problem depictedin Fig. 2 can be simplified based on several exper-imental observations. For the aperture size of 8mm used in the imaging system, diffraction effectsthat are due to the aperture may be neglected, sothe aperture serves only to decrease the effectivelens diameter and hence to increase the fy#. Ashe aperture size becomes small, however, this as-umption is no longer valid. The mathematical de-ails that incorporate the above discussion areresented below, with an extension to GLMT forelocimetry applications.Let the x component of the external electric field

e Em, where the subscript refers to the field eval-uated at plane zm. The electric field at plane z1is known exactly. We can relate E2 to E1 by as-suming that the paraxial thin lens expression isvalid:

E2~x2, y2! 5 exp~iknD0!expF2ik2f

~x12 1 y1

2!GE1~x1, y1!

5 exp~iknD0!expF2ik2f

~x22 1 y2

2!GE1~x2, y2!,

(17)

where f is the focal length of the lens, D0 is the thick-ness of the lens, and n is the lens’s refractive index.By a thin lens, we mean that the thickness of the lensis such that a light ray enters and leaves the lens atessentially the same x–y coordinates, thus making~x1, y1! and ~x2, y2! interchangeable. Note that thelens in the imaging system is assumed to be ideal andfree from any aberrations. Using the Fresnel ap-proximation in proceeding from plane z2 to plane z3yields

E3~x3, y3! 5exp@ik~z3 2 z2!#

il~z3 2 z2! **A

E2~x2, y2!

3 expH ik2~z3 2 z2!

@~x3 2 x2!2

1 ~y3 2 y2!2#Jdx2dy2, (18)

Itau

w

where A is the integrating area. Substituting Eq.~17! into Eq. ~18! yields

E3~x3, y3! 5exp@ik~z3 2 z2!#

il~z3 2 z2! **A

exp~iknD0!

3 expF2ik2f

~x22 1 y2

2!GE1~x2, y2!

3 expH ik2~z3 2 z2!

@~x3 2 x2!2

1 ~y3 2 y2!2#Jdx2dy2. (19)

For convenience Eq. ~19! can be written in spherical~r, f! coordinates ~i.e., xm 5 rm cos fm and ym 5 rm sinfm for plane zm!:

E3~r3, f3! 5exp@ik~z3 2 z2!#

il~z3 2 z2! *0

2p

*0

ra

exp~iknD0!

3 expH2ik2f

@~r2 cos f2!2

1 ~r2 sin f2!2#JE1~r2, f2!

3 expH ik2~z3 2 z2!

@~r3 cos f3 2 r2 cos f2!2

1 ~r3 sin f3 2 r2 sin f3!2#Jr2dr2df2.

(20)

Based on the experimental observations discussedabove, the assumption for a field symmetry about theaxis, i.e., E1~r2, f2! 5 E1~r2!, is used. Also, becauseonly the intensity will be considered at plane z3, con-stant phase factors can be omitted, yielding

E3~r3, f3! 51

il~z3 2 z2! *0

ra

E1~r2!r2 expS2ik2f

r22D *

0

2p

3 expH ik2~z3 2 z2!

@r32 1 r2

2 2 2r2

3 cos~f2 2 f3!#Jdf2dr. (21)

n addition to simplifying the electric field descrip-ion, the symmetry axis assumption also permits ex-ct evaluation of the angular integral in Eq. ~21!. Byse of orthogonality along with the identity

exp~ia sin x! 5 (k52`

`

Jk~a!exp~ikx!, (22)

Eq. ~21! can be simplified, yielding

E~r3! 52p

il~z3 2 z2!expF ikr3

2

2~z3 2 z2!G *

0

ra

E1~r2!J0~b!

3 expH2ikr2

2

2 F1f

21

~z3 2 z2!GJr2dr2, (23)

where J0 is the Bessel function of the first kind oforder zero and

b 5kr2 r3

~z3 2 z2!. (24)

The intensity at the image plane can be writtenfrom Eq. ~23!:

I3~r3! 5 E3~r3!E3*~r3! 5 F 2p

l~z3 2 z2!G2 U*

0

ra

E1~r2!J0~b!

3 expH2ikr2

2

2 F1f

21

~z3 2 z2!GJr2dr2U2

, (25)

where the asterisk denotes the complex conjugate.To remain consistent with the Lorenz-Mie theory de-velopment, we write Eq. ~25! in terms of dimension-less variables:

I3~r3! 5 F a

~ z3 2 z2!G2U*

0

ra

E1~r2!J0~b!

3 expH2iar2

2

2 F1f

21

~ z3 2 z2!GJr2dr2U2

,

(26)

here

b 5r2r3a

~ z3 2 z2!,

r 5ra

, z 5za

, f 5fa

. (27)

Finally, experiments indicated the need to includeSeidel aberrations of a typical lens used for velocim-etry, such as a 90-mm Sigma Macro, to increase theaccuracy of the model and thereby the accuracy of theparticle positioning algorithm. The spherical andcoma forms of aberrations play the major roles inpattern matching; and therefore it was these thatwere included in the model. Here for completenessthe five forms of aberration are expressed in the fol-lowing equation:

W~S, u! 5 F 1 Er2 cos u 1 Dr22 1 Cr2

2~1 1 cos2 u!

1 Br23cos u 1 Ar2

4, (28)

where the coefficients given indicate aberrations forthe following properties: A, sphericity; B, coma; C,astigmatism; D, defocusing; E, tilt about the x axis;and F a constant or piston term.

1 October 2000 y Vol. 39, No. 28 y APPLIED OPTICS 5121

5

This Seidel aberration correction is included in thegeneral treatment as a supplementary phase term inEq. ~17! above.

4. Experimental Arrangement

The experimental setup and experimental resultscompared with the calculations are fully described inRef. 39, so here they are briefly mentioned just to givean indication of the experimental conditions and re-sults. For model verification purposes, calculationswere made for a 50-mm particle at a laser wavelengthl 5 337 nm ~Ref. 35! and were found to be in closeagreement with other authors’ experimental–computational results as well as with results for aGaussian wave front.

The experimental setup included provision for re-cording forward scattering, backscatter, and themore conventional orthogonal viewing of a thin lightsheet of PIV. Particles were deposited upon a mi-croscope slide held by an x–y–z micrometer holder.The slide was antireflection coated to optimize thesignal-to-noise ratio. Three modes of illuminationwere investigated: a plane wave front as normallyconsidered by classic Lorenz–Mie theory, a Gaussianwave front as normally employed in particle sizingapplications, and a light sheet as in conventional PIV.Two lasers were used for these tests: a stableHe–Ne laser and a Nd:YAG laser for viewing smallparticles. A range of particle sizes was investigated;these ranged from water to glass spheres and pollen.Actual particle sizes were verified by microscopictechniques. Available sizes ranged from 4 to 150mm.

Image data for a 18-mm glass sphere, with all thecomputational details, taken for the three illumina-tion schemes and the three viewing positions men-tioned, are available from either D. Moreno or F.Mendoza Santoyo, together with calculations withthe GLMT. Therefore this data set can be used tovalidate other scattering models for applications suchas velocimetry, phase-Doppler anemometry, and par-ticle sizing.

5. Results and Discussion

The difference in intensity distribution exhibited byparticles that are an equal distance out of focus buton opposite sides of the object plane of the lens wasobserved and reported previously.34,35 It becomesmore significant as particle size decreases, contribut-ing to the accuracy of the 3D positioning model pre-sented here, a feature that can be clearly seen in thecalculations. Figure 3 shows a plot of the image ofan 18-mm particle at a magnification of 7.853 for arange of defocus from 25.0 to 15.0 mm. Note thatno size-related simplification is included in the mod-el; therefore the calculations are valid formicrometer-sized particles, as employed in unsteady-flow research that uses velocimetry. Larger parti-cles were used for convenience only.

Using the GLMT allows particle position to be mod-eled accurately anywhere in the beam by use of thethree most commonly used wave fronts: plane

122 APPLIED OPTICS y Vol. 39, No. 28 y 1 October 2000

waves, Gaussian beams, and light sheets. Thismodel was verified against experimental data by theauthors of this paper and against numerical calcula-tions obtained by other authors. For an illustrationof this point see Fig. 4, where an experimental imageof a particle with an 18-mm diameter is comparedwith the image of the particle calculated with theGLMT toolbox ~shown in Fig. 5! for forward scatter-ing and Gaussian illumination, as in other publisheddata and calculations. The difference between thecalculated and the experimental images was 11 graylevels rms, with a positional accuracy of 5 mm at amagnification of 7.853. The particle position accu-racy has been estimated as 20 mm at low magnifica-tions. For comparison, calculations by classicLorenz–Mie theory yielded positional accuracies ofthe order of 50 mm. One can improve the position

Fig. 3. Mesh plot of particle image versus defocus ~experimental-ly verified Gaussian-beam generalized Lorenz–Mie calculation:18-mm glass particle!.

Fig. 4. Comparison of radial intensity for experimental ~dashedcurve! and theoretical ~solid curve! particle images.

w

Fplops

accuracy by increasing the magnification, at the costof reducing the size of the region that is investigated.For high-speed unsteady-flow research by velocim-etry techniques, areas of interest are small but re-quire magnifications in the range of 13–33 at most,

ith current CCD technology.Figure 5 shows the front end of the GLMT toolbox.

acilities are included for changing particle physicalarameters as well as the illumination source, theens parameters, the wave-front parameters, and therientation. Calculations can be compared with ex-erimental results and visualized by use of user-pecified digitization parameters.

6. Conclusions

The theoretical model described in this paper caneasily be extended to a more-complex imaging systemgeometry involving multilens or fiber-optic systems,for instance, as it is well known that a multilenssystem can be approximated to a single lens withcharacteristics equal to the lens system. A code hasbeen developed from the mathematical model thatincludes, for what we believe is the first time, not onlythe theoretical considerations of GLMT but also itspractical implementation to yield realistic particleimages. Micrometer-sized particle diffraction im-ages have been quantitatively compared with exper-

Fig. 5. Typical GLMT

imental data, for the first time to the best of ourknowledge.

Much work remains to be done to in refine the code,speed it up, and extend it so all forms of aberrationsare considered, with various aperture shapes alsotaken into account. However, having an accuratequantitative particle image model is an importantstep forward for velocimetry applications. Beingable to find 3D particle positioning from CCD cameraimages opens the way for routine accurate 3D veloc-ity estimation.

D. Moreno thanks the Consejo Nacional de Cienciay Tecnologıa ~Mexico! for his doctoral study grant.We thank Dennis R. Alexander of the University ofNebraska, Lincoln, Nebraska, for his help and adviceas to computational techniques used in this paper.M. Funes-Gallanzi thanks the Royal Society of Lon-don for supporting the initial phase of this project,which was later completed with the kind support ofPhoton Imaging Inc. ~USA!, and Arquitectura en Sis-temas Comptacionoles Integrales, Sociedad Anonimade Capital Variable ~Mexico!.

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box front-end display.

tool

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2

2

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