Particle image fields and partial coherence

Robert Rubinstein and Paul S. Greenberg

The accuracy of Young's fringe method for reducing velocity field data is compromised by a spatiallyincoherent background field which originates in the random locations of the seeding particles. The probabili-ty density function for the spatial frequency cutoff of this background is derived as a function of the particlecount, the distribution governing the power in each frequency interval is derived, and conditions are foundunder which the Van Cittert-Zernike theorem applies. The background field resembles the far field of apartially coherent source in the high particle count limit, but departs significantly at low and moderatecounts.

1. Introduction

Despite the increasing importance of particle imagevelocimetry (PIV)' as a planar full field flow measure-ment procedure, the statistical properties of PIV mea-surements have not yet been analyzed completely.Therefore, fundamental questions like the optimalseeding density and interexposure interval for a giveninterrogation region size and the velocity resolutionlimits in the presence of decorrelating mechanismssuch as local velocity gradients have not been an-swered, leaving the experimentalist to rely on trial anderror procedures to obtain the most satisfactory mea-surements. Answers to such questions require a sta-tistical characterization of the noise introduced intoPIV measurements by the random locations of theseeding particles. Since this particle image noise isthe sum of incoherent contributions, it resembles phe-nomena like laser speckle, but the possibility of smallor moderate numbers of particles makes a more carefulinvestigation mandatory.

This paper presents an analysis of particle imagenoise as it arises in the Young's fringe method ofvelocity data reduction (see Fig. 1) in PIV. In PIV, amultiply exposed photograph is taken of a seeded flowfield illuminated by a thin light sheet. In what follows,the plane of the photograph will be called the particleimage plane. Coherent illumination of a small portionof the photograph, the interrogation region, producesin the far field the Fourier transform of the multiplyexposed particle images in this region. The plane of

Robert Rubinstein is with Sverdrup Technology, Inc., NASA Lew-is Research Center Group, Middleburg Heights, Ohio 44130, andPaul S. Greenberg is with NASA Lewis Research Center, Cleveland,Ohio 44135.

Received 1 March 1989.

the far field will be called the transform plane. Itcontains a grainy, spatially incoherent backgroundmodulated by Young's fringes. In principle, the fringespacing determines the coherent particle displace-ments in the interrogation region. The spatially inco-herent background field, the particle image noise, con-tains information about the random (relative) particlepositions.

It is evident that any measurement of the fringespacing will be degraded by the spatial noise spectrumof the incoherent background field. This situation haslong been recognized in the application of convention-al2 and holographic3 interferometry, and more recentlyin speckle interferometry4 and specifically in particleimage velocimetry.5 ,6 Therefore, a more complete un-derstanding of this measurement process must beginwith a characterization of this background field.Three properties of this field will be derived and ap-plied to the problem of Young's fringe resolution. InSec. II, the probability density function for the spatialfrequency cutoff is derived as a function of particlepopulation. As the particle population increases, thisdistribution becomes concentrated at a deterministicvalue inversely proportional to the sample volume size,the value characteristic of laser speckle. By analogywith Goodman's definition7 of speckle size, the inversespatial frequency cutoff can be taken as a measure ofthe grain size of particle image noise. Since resolutionof a fringe will entail averaging the spatially randomnoise field over some portion of a fringe, the accuracyof the fringe measurement must depend on the numberof grains per fringe. Whereas, the effectively infinitenumber of radiators in laser speckle leads to a deter-ministic grain size, the finite number of sources ofparticle image noise cause grain size to be a randomvariable. Fluctuations in the grain size entail a finiteprobability that any given fringe spacing will be poorlyresolved due to the absence of a sufficient number of

5282 APPLIED OPTICS / Vol. 29, No. 35 / 10 December 1990

Probe'

laser

Fig. 1. Young's fringe geometry. Scattering from randomly dis-

tributed particles in the image plane gives rise to a spatially incoher-

ent background field in the transform plane.

grains. The probability density for the spatial fre-quency cutoff, therefore, determines the probabilitythat, because of finite particle population effects, thefringe resolution will be reduced below the value whichwould occur in a laser speckle field.

In Sec. III, it is shown that a Fourier decompositionof the noise, the power in each frequency band is Pois-son distributed and that, independent of particle pop-ulation, the expected power at each frequency followsthe triangular shape of the power spectral densityfunction of laser speckle. The question arises whetherthis spectrum can be dominated by a noise frequencyinstead of by the frequency corresponding to the inter-exposure displacement. Using the noise spectrum de-rived in this section, a simple model is derived for theSNR as a function of particle population. It should benoted, that this analysis is closely related to the tech-nique applied by Keane and Adrian5 and by Coplandand Pickering,6 in which the interexposure displace-ments are derived from the image plane autocorrela-tion function.

In Sec. IV, the question is addressed of when theparticle image noise is statistically identical to the farfield of a partially coherent source. It is argued thatwhen subregions of characteristic length A all containapproximately the same number of particles, then theparticle image field acts like a quasihomogeneoussource in the sense of Wolf8 9 with correlation length A.In this case, the generalized Van Cittert Zernike theo-rem of Goodman'0 and Wolf8 9 can be applied. Esti-mates are derived for the number of particles neces-sary to bring about this behavior. The generalizedVan Cittert Zernike theorem implies that the trans-form plane intensity is slowly modulated; the lengthscale for the modulation is proportional to 1/A. As Aapproaches zero, the modulations disappear and theimage plane acts like a partially coherent source. Thislong wavelength behavior is applied to the possibilitydiscussed for example by Hinsch, Schipper, andMach," that local reduction in fringe visibility can beused to diagnose the presence of gradients or random

motion in the interrogation region. This requires thedetection of (deterministic) variations of fringe visibil-ity, but the Van Cittert Zernike theorem suggests thepresence of random visibility fluctuations as well. Ac-cordingly, a problem arises at the long wavelength levelof extracting a signal from noise. A simple model isdeveloped to determine the seeding density requiredto permit this type of measurement.

The goal of the present analysis is to develop com-plete probability density functions whenever possibleand to exhibit their dependence on particle populationexplicitly. This approach, therefore, differs some-what from that of Meynart' 2 and of Keane and Adrian5

in which certain fundamental mean properties of thenoise field are derived. Despite the importance ofspatially averaged properties, they may not provide anaccurate description of a single realization, especiallywhen the particle population is low.

II. Derivation of the Grain Size Distribution Model

The particle image field will be modelled by randompoints with a 2-D Poisson distribution. Justificationfor this simplification will be given later. Thus, theimage plane for a single exposure is described as

I,= I - (1)

where (to) are coordinates in the image plane, (ini)are the particle locations, and I is the intensity, as-sumed constant, scattered by a single particle. Thenumber of particles g has the Poisson distribution,

exp(-s)sm (2)

where s is the Poisson parameter: the expected num-ber of particles, or the mean particle source densitytimes the area of the interrogation region. Given thevalue of m, the points (i,mq) can be considered to beuniformly distributed over the interrogation region.

As usual, the Poisson model requires spatial homo-geneity and lack of interaction between particles. Themost important restriction is that the particle imagesdo not overlap. Significant overlap would cause inter-ference between the particle images resulting in imageplane speckle; this is the laser speckle velocimetrylimit of Adrian."'1 3 An analysis of this case woulddiffer from the present analysis only in the choice of animage plane model.

The far field of the coherently illuminated imageplane is given by the modulus of the Fourier transformof the image plane' 4

I(x,y) = C m + E exp27ri[x( - tj) + y(ni - ?j)]/Xzie (3)

where xy are coordinates in the far field, X is theillumination wavelength, and z is the distance betweenthe image and transform planes. (If a lens is used toproduce the spatial transform, then z becomes thedistance to the front focal plane.) The constant C isdetermined by the properties of the medium used torecord the particle images, the intensity of the illumi-

10 December1990 / Vol. 29, No. 35 / APPLIED OPTICS 5283

nating beam (assumed constant), and the exposuretime. An analysis by Pickering and Halliwell'5 hasestablished that additional noise terms due to the filmitself are negligible when high resolution emulsions orliquid gates are used, hence this effect will be ignored.This formula contains the justification for modelingthe particle images as points: because of the Fouriertransform relation between Eqs. (1) and (3), if theparticle images are not points but any arbitrary shapewith identical orientation, expression (3) for the trans-form plane intensity would just be multiplied by themodulus of the Fourier transform of one image. Spa-tial intensity variations in the illuminating beam and aconstant offset term in the recording medium trans-mission function would cause the same modification ofthe image plane model. Such modifications would beinessential in what follows.

Important differences between particle images andpartially coherent sources arise even in the definitionof grain size. In the theory of laser speckle, for exam-ple, grain size is defined by the first zero in the intensi-ty autocorrelation function7

r(Ax,Ay) = (I(xy) I(x + Axy + 4y)). (4)

Since the far field of a partially coherent source can beconsidered to arise from contributions from an infinitenumber of differential regions, it is reasonble to invokeergodicity. Thus, the average in Eq. (4) can be inter-preted either as a spatial average or as an ensembleaverage. The far field intensity correlation is deter-mined by the Van Cittert Zernike theorem2 7 whichimplies, for example, that if the interrogation volumeis a circle of diameter L, the transform plane grain sizeis a constant times Xz/L.

That particle images behave quite differently is atonce evident from consideration of a three particlesystem. In this case, if the lattice generated by theparticles is formed in the image plane, the transformplane is a doubly periodic function with period latticegiven by the reciprocal lattice. The grain size is deter-mined by the particle positions and can take any valuebetween infinity and the inverse sample volume size.Moreover, even the term grain size is misleading be-cause the projections of the grains onto the coordinateaxes need not have the same size even if the interroga-tion region is circular. Of course, these distinctionsarise simply because the particles need not completelyfill the interrogation region. Grain size is, therefore,not a deterministic quantity defined by the interroga-tion region geometry alone, but a random variabledependent on the arrangement of particles in the inter-rogation region.

A simple measure of grain size for particle imagesbased on Eq. (3) is

kx = maxXz/2rl - jI,

ky = maxXz/27r1 - jl.

Grain size is thus not only a random variable, but also avector quantity. Note that the grain size componentsare determined by the particle pairs with the largest xand y separations. This definition of grain size is

unrealistic for some special distributions of points: asimple example would arise if there were a large num-ber of closely spaced particles and a small number ofdistant outliers. However, such configurations onlyarise when there are a large number of particlespresent, and it will be seen that the probability thatsuch a configuration will occur is extremely low. Thispossibility will therefore be ignored.

The problem then arises of computing the grain sizedistribution. The solution requires computing thedistribution of the maximum projected point separa-tions in a collection of Poisson distributed points. Be-cause the Poisson points have independent x and y-coordinates, this problem reduces immediately to the1-D case. Suppose, first, that the number of particlesis known. The problem is: given m points chosenindependently with uniform distribution in a givenfixed interval, what is the distribution of the maximumpoint spacing? As in the derivation of the Poissondistribution, this distribution will be evaluated firstfor a discrete version of the problem in which theparticles occupy a finite number of cells of equallength, then this cell size will be taken to zero.

Let the number of cells be N and let exactly m cellsbe occupied. Choose N so large that there is negligibleprobability of finding more than one particle in a cell.The probability that the occupied cells are included inexactly k + m consecutive cells is the number of inter-vals consisting of k + m consecutive cells times thenumber of ways k of these cells can be vacant dividedby the total number of choices of m occupied cells, thatis

(N- k -m+1) (k + m-2) /()

Define the interval length 1 = k + m. Then the num-ber of arrangements in which the particles are locatedin an interval of length between M and N is given bythe sum:

E (N-1+1)( 2)MSISN m-2

= -m N(M -2) ... (M -m)/(m -1

+(M-1)...(M-m)/m(m-2)!

Note in particular that

E(N 1+ 1) (L-2)=QNm1N

because the left side is the total number of arrange-ments of m particles in N cells when N >> m. Theprobability that the m particles are in a number of cellsbetween M and N equals

1-m(M-2)...(M-m)/(N-1)...(N-m+l)+(m-l)(M-1) ... (M - m)/N ... 1(N-m+i).

Let M = N and let N -X in this expression. Theresult is that the probability that the m points are in aninterval of length at least a times the interval length

5284 APPLIED OPTICS / Vol. 29, No. 35 / 10 December 1990

equals 1 - maml + (m - 1)am = 1 - am. Equivalent-ly, let r denote the fraction of the interval that is filled.Then

0.8

p(r < a) = am. (5) Z 0.6

The expected fraction of the interval that is filled is = 0.4

easily evaluated as

(CZ) = 1 - 1/(m + 1),

which approaches 1 as m becomes infinite.It is a simple extension of this calculation to account

for randomness in the particle count governed by thePoisson distribution. Thus,

p(r < a) = , p(r < aI1A = m)p(o = m) = am exp(-s)sm /m!m22 m22

= exp(a - 1)s - (1 + as) exp(-s), (6)

where s is the Poisson parameter, the mean sourcedensity times the area of the interrogation region.Note that the grain size is not defined unless the sam-ple volume contains at least two particles. Thus, p(r <1) = 1 - (1 + s) exp(-s) < 1, because (1 + s) exp(-s) isthe probability that the sample volume contains zeroor one particle.

Extension of these results to 2-D distributions istrivial because the two 1-D Poisson distributions ob-tained by projecting the points along the two coordi-nate axes are independent. Thus, let r. and ry denotethe fractions of the projected interrogation region thatis filled. Then

p(r. < a and ry < fl) = p(ry < a and ry < A = m)p(y= m)

= E p(r2 < aA = m)p(ry < MA = m)p(L = m)

= exp[a3 - 1)s] - (1 + afs) exp(-s).

This distribution becomes concentrated at a = = 1 asthe mean particle population s increases. This corre-sponds to the speckle limit in which the spatial fre-quency cutoff is a function only of the interrogationregion dimensions. However, for low particle popula-tions, the frequency cutoff becomes more uniformlydistributed over all possible values of spatial frequen-cies.

The quantity a can be related to the signal to noiseratio in a Young's fringe measurement, or more pre-cisely, to the decrease in signal-to-noise ratio (SNR)due to the effect of finite particle population. Good-man's analysis7 of intensity resolution in a laser speck-le field leads to the conclusion ail (I) p112, where theleft side is the usual measure of resolvability and p isthe number of grains per fringe. If intensity is re-solved by averaging over the grains, this equation plau-sibly asserts that the measurement statistics are thoseof the average of a random sample of size p from a fixeddistribution. If a-' is provisionally accepted as a mea-sure of the ratio of grain size in particle image noise tolaser speckle size, then a gives the ratio of the numberof grains per fringe in particle image noise to the num-ber of speckles per fringe in laser speckle. Thus, a 1 /2

measures the reduction of fringe resolution due tofinite particle population effects. The distribution

s=10

S=15

S=20

S=25

0 0.2 0.4 0.6 0.8 1

Ratio of Laser Speckle Size to Grain Size

Fig. 2. Probability that the grain size exceeds the laser speckle sizefor various mean particle populations.

function Eq. (6) provides an exact analysis of the rela-tion between this degradation measure and the parti-cle population. Suppose that a probabilityp is chosenand the particle population parameter s is requiredwhich insures that the degradation is no greater than awith probability p. Ignoring the second term in Eq.(6), which is negligible for more than eight particles,

-lnp1-a

Note than s becomes infinite as a approaches 1. Forexample, with p = 1/10 and a = 90%,

s = 10 lnlO 23.

Thus, a mean particle population of 23 insures thatthere is only 10% probability that the fringe resolutionis less than V0TB - 95% of its speckle limit value.

The behavior of the cumulative density function Eq.(6) as a function of particle population is further illus-trated in Fig. 2. It is evident from this figure that asthe particle population increases, the values of theratio a become sharply concentrated near the laserspeckle limit value, a = 1. On the other hand, moder-ate population parameters produce considerable un-certainty about the grain size: for an expected popula-tion of 10, for example, there is a 10% chance that thegrain size will exceed 1/0.77 130% of its limitingvalue; this would reduce the fringe resolution below4-.77 - 88% of its speckle limit value.

The question remains whether the inverse spatialfrequency cutoff is a reasonable grain size measure forparticle image noise. There is some flexibility in in-troducing a grain size measure; the one proposed hasthe attribute of reducing to Goodman's familiar defini-tion7 of speckle size in the limit of infinite particlepopulation. On the other hand, using the highestnoise frequency as a characteristic frequency may un-derestimate the grain size when the particle count ismoderate. Furthermore, as is shown in Sec. IV, parti-cle image noise really resembles laser speckle onlywhen small subregions of the interrogation region allcontain approximately the same number of particles.It is shown that much higher particle populations arerequired to bring about this behavior. Thus, a grainsize measure which includes information about the

10 December 1990 / Vol. 29, No. 35 / APPLIED OPTICS 5285

so

A

entire spectrum is more realistic. Investigation of thisquestion is in progress.

111. Power Spectral Density of a PIV Noise Sample

The formula for the transform plane intensity, Eq.(3), exhibits the Fourier components of the noise fieldexplicitly. The question naturally arises of how thepower in each compdnent is distributed. As in thegrain size calculation, it is sufficient to consider the 1-D case first. Divide the image plane into N cells oflength so that NAl = L is fixed, where L is arepresentative linear dimension of the interrogationregion. The electric field in the transform plane inthis approximation is

E = E xk exp(2rixkA//Xz)

where XA denotes the number of particles in imageplane cell k. In this notation, the particle populationis the random variable p = x + ... + XN, with thePoisson distribution (2), and

I = XjXkexp(2,rix(k -j)A/Xz)j,k

= E XjXk exp(27rixlT/\z)I -k=l

so that the power at discrete frequency 1A/Xz is givenby the sum

SI = X1X 1l + X2XI+2 + + XN-IXN. (7)

The power between the discrete frequencies (1 -41)A/Xz and (1 + Al)Av/Xz is

SI-Al + S- +... + SI+AI. (8)

The power in the continuous frequency band (K -AK)L/Xz to (K + AK)L/Xz, 0 K 1 is found by takingthe limit A/ -> 0 holding

NA = L,

IA^ = KL, (9)

2A = AKL.

The Poisson distribution for the total particle popula-tion x + ... + XN arises by assuming that each xirepresents an independent Bernoulli trial with successprobability p = a + (zg2 ) where a = sL is theexpected particle density. Thus,

p(xi = 0) = 1 - aAt + O(A2)(10)

p(xi = 1) = aA + O(A 2

).

For the derivation of the distribution of SI in Eq. (7),the probabilities (10) to order O(A43 ) will be required.Thus, consider the x to arise from two independentBernoulli trials with success probability p = 12aAf +O(A 2). Then

p(X = ) = (1 _ /2aAt)2 + O(A 3 ),

P(x = 1) = t(l - /2oA~) + O(A 3), (11)

These equations agree with Eqs. (10) to order Ag2 .Since the xi are independent, to order A/ 3,

p(xixj = 0) = 2p(xi = & j 0d °) + p(xi = xj = 0)

= 1-a2At2

p(xxj = 1) = a2A\2

It follows that if the xixj were independent, then SIwould have a binomial distribution for which the Pois-son limit would follow as A/ -> 0 in standard fashion.16

This argument fails because the xixj are not indepen-dent. But it is easy to check that they are independentin the limit At - 0. Omitting terms of order A\3, Eqs.(11) imply

p(xiX = & XjXk = 0) = p(xj = 0) + p(xi Fd 0 & Xi = & Xk = 0)

= 1 -222\

p(XXj = O)p(xjxk = 0) = (1 -a 2 A 2)2 = 1 -22At2,

p(XiXJ = & jXk = 1) = p(Xji 0 & Xi = & xk = 1) = 2A\ 2 ,

p(XiXJ O)p(xjxk = 1) = (1 - a 2A42 )a2 A\2 = a2At2,

p(XiX, = 1 & XjXk = 1) = P(Xi = Xj = -1) = (At3)

p(xiXi = 1)p(xjXk = 1) = 0(Ae),

(12)

so that

p(xixj = n & XjXk = m) = p(xixj = n)p(xjxk = m) + O(A 3)

Product pairs like x1x2 and X3X4 with no equal indicesare independent.

The number of terms in the sum of Eq. (8) is

M = (N -I - A) +... + (N - 1 + Al)LQ2 [( KIK 2vK2].

Therefore, using Eqs. (12),

p(SI-A1 + ... + SI+, = n)

(n) (ar2)(n - a 2A 2)M-n + O(MAt3).

In the limit At - 0 subject to conditions (9):

(M) ( 2At 2 )n(1 - 2A2)Mn - [oi2L2(1 -K)AK]n

X exp[- 2L 2(1 -K)AK]O(MA~ 3) - 0,

where the first relation is the standard Poisson limit.16

It follows that the power at frequency LK/XZ is Poissondistributed with parameter (expected power per unitfrequency) 2 L2(1 - K) =

2(1 - K). The extension of

this result to two dimensions is, of course, immediate.The expected power as a function of frequency has

the triangular dependence characteristic of the powerspectral density of laser speckle. In this analysis, thetotal power increases linearly with particle populationsince the power scattered per particle is fixed. A dif-ferent limit could be formed in which the total power isheld constant; this limit would apply to laser speckle,in which the power scattered by a differential area isproportional to the area. It is easily verified that in

5286 APPLIED OPTICS / Vol. 29, No. 35 / 10 December 1990

lAK=25

laK=50...........

1/AK-100

l/AK=200

Mean Particle Population

Fig. 3. Tenth percentile SNR (coherent power/estimated noisepower) as a function of mean particle population.

this limit, the asymptotic power distribution is Gauss-ian.

This description of the noise spectrum applies tofringe resolution based on Fourier analysis of thetransform plane of a double pulsed photograph. Asemphasized by Keane and Adrian,5 this type of fringeanalysis is closely related to velocity detection by di-rect autocorrelation of the imaging plane. Discussionsof error in this autocorrelation procedure are present-ed both by Keane and Adrian5 and by Coupland andPickering.6 To make a connection with the presentanalysis, assume that the transform will be discretizedinto wavenumber bands of width AK. Fixing such afrequency resolution scale in the transform of thetransform plane implies that particles a distance oforder AK apart cannot be distinguished; this is compa-rable to the introduction in Refs. 5 and 6 of a finiteparticle size. Assume that the signal power is m,where m is the number of particles. This neglects thepossibility of particles leaving the interrogation regionbetween exposures. As a measure of SNR, take theratio of m to the peak noise power. It is not known inadvance at which discrete frequency this peak willoccur, so the assumption will be made that it occurs inthe lowest range of frequencies, which has the largestPoisson parameter. As shown above, this representa-tive noise power has Poisson distribution with parame-ter m2 AK. As a representative noise power take thevalue which is exceeded with probability 10%. TheSNR computed with this noise power represents the10th percentile SNR, the value which must be exceed-ed 90% of the time. It can be computed as a function ofparticle population for fixed frequency resolution AK.Typical results appear in Fig. 3.

Two important trends can be seen. First, for fixedparticle population, the SNR can be made arbitrarilylarge by increasing the frequency resolution. If thereare m particles and of the order of 1/AK possible parti-cle positions, then putting any fixed power in somenoise frequency band requires that some fraction a ofthe particle positions be determined in terms of theothers. Thus, on the order of (1 - a)m particle posi-tions can be assigned freely; provided that 1/AK >> m,the ratio of the number of such arrangements to thetotal number of arrangements is of order

Table 1. Fifth Percentile SNR for Eighteen Particle Image Pairs as aFunction of Resolution Parameter

1/AK SNR (Ref. 5)a SNR (Present approx.)

25 0.050 5.0 5.1

100 9.4 10.4200 12.8 16.3

a SNR = 20 logio (coherent power/peak noise power).

( 1/A )/41/AK) am,

which approaches zero rapidly if AK - 0.Second, for fixed resolution, as the particle popula-

tion increases, the SNR approaches zero. If m2 AK islarge, the normal approximation to the Poisson distri-bution shows that the noise power is (const.m2 AK) 1,where the constant is the relevant percentile of thenormal distribution. Therefore, the SNR scales as

SNR - m/m 2AK = 1/mAK. (13)

The average noise power in the lowest frequencyrange is just m2 AK. Therefore, this scaling would ap-ply regardless of the value of m2AK if the SNR had beendefined using the average noise power instead of apercentile. The scaling of Eq. (13) is derived by Keaneand Adrian5 ; it is noteworthy that this relation alsoarises here despite differences in the formulation of theproblem.

More quantitative comparisons can be made withrecent results of Coupland and Pickering6 based onMonte Carlo simulations of double pulsed photo-graphs with up to eighteen particle pairs. As notedearlier, Coupland and Pickering simulate velocity de-tection by measurement of image plane autocorrela-tion peaks; the frequency resolution parameter AK inthe transform of the transform plane will be set equalto the effective particle size (the size of the region overwhich a single particle image is correlated with itself)in this detection scheme. The results for the fifthpercentile SNR, the value which is exceeded 95% of thetime for m = 18 particles, appears in Table I. TheSNR measure specified in Ref. 6, 20 loglo (m/peaknoise power) is also shown. Whereas, Coupland andPickering's simulation gives access to the completenoise spectrum, the model applied here uses only thenoise power in one frequency band and, therefore,provides a somewhat cruder approximation to thepeak noise power. Nevertheless, the agreement is sur-prisingly good, especially for AK = 50 and AK = 100.Table I contains no value at m = 18, AK = 25 forCoupland and Pickering's simulation because, in theirformulation, the number of particles is limited by therequirement that the particle images not overlap.

There are significant differences, however. Thepresent results indicate a steady decrease of the SNRwith increasing particle population, while Couplandand Pickering describe their results as indicating anapproximately constant SNR plateau for sufficientlylarge m; although, it must be noted that their simula-

10 December 1990 / Vol. 29, No. 35 / APPLIED OPTICS 5287

12

10

8

6

0 20 40 60 80 100

......................... ..........................

tions do not extend beyond m = 18. In this connec-tion, we can only note that decrease in SNR is consis-tent with Keane and Adrian's5 scaling, see Eq. (13).Second, the present theory shows sharp increase of theSNR as m decreases <20. This is partly due to treat-ing the particle images as points, but must also reflect abreakdown in the approximation of the peak noisepower by the power in one frequency range. Indeed, asCoupland and Pickering point out, a measurementwith exactly two particles is completely ambiguous,and must give a SNR of zero. In summary, this pre-liminary comparison suggests that the present approx-imation is reliable at particle populations sufficientlylarge that the SNR curve reaches its asymptotic form,but that at lower particle populations, it will overpre-dict the SNR.

It must be emphasized that this SNR analysis differsfundamentally from the analysis of Sec. II because itintroduces a parameter characteristic of the measure-ment system, namely the frequency resolution AK.Nevertheless, the difference in conclusion about theeffect of increasing the particle population is striking.Considering the results of the previous section, it isnow clear that the experimentalist is faced with twocompeting mechanisms. A suitably large particle pop-ulation is required to adequately resolve the periodici-ty of the resulting fringe pattern with reasonably highprobability. This is accomplished, however, at theexpense of decreasing signal to noise power within thenecessarily fixed frequency bands in the Fourier plane.

IV. Generalized Van Cittert Zemike Theorem

The behavior of the grain size in the large particlecount limit suggests the question of when a coherentlyilluminated particle image field can be considered apartially coherent source. If the particle population islarge, phase incoherence due to the random particlelocations must cause the same type of interference thatproduces the far field of a partially coherent source.The question arises of how many particles are neces-sary to bring about this behavior.

The far field of a partially coherent source satisfiesthe Van Cittert Zernike theorem.27 The basic hy-pothesis in the derivation of this theorem is that theimage plane fields are uncorrelated at different points:

(EQt,1)EQn1)) = I( - ', -').Obviously, this equation cannot hold for particle

image fields. If a point P is not occupied, the imageplane intensity is trivially correlated with itselfthroughout the largest circle which can be drawn aboutPwhich contains no particle. This fact suggests that itis reasonable to subdivide the image plane by consider-ing the scattering from finite subregions, say A by Asquares. If these regions are large enough that theyare all occupied with high probability, then phase inco-herence makes the fields from disjoint regions uncorre-lated. But the fields must be considered autocorrelat-ed over the length A. This means that the particleimages act like a quasihomogeneous source in the senseof Wolf8'9 to which the generalized Van Cittert Zernike

theorem of Goodman' 0 and Wolf89 applies. Thistheorem states that the far field autocorrelation isobtained by multiplying the autocorrelation predictedby the Van Cittert Zernike theorem by a modulationterm which varies with the length scale z/A. Theproblem is to choose the length A correctly. An analy-sis of the far field based on this coarse graining proce-dure will lead to a general model for the transformplane. A form of the generalized Van Cittert Zerniketheorem will be derived from this model which appliesto particle image fields in the high particle populationlimit. Precise information about the number of parti-cles necessary to bring about this limit will also bederived for a typical case. It will be shown that whenthe particle population approaches infinity, the parti-cle images act like a partially coherent source. Ofcourse, this limit will not necessarily apply to particlesof finite size, since the analysis assumes that the parti-cle images never overlap.

As in the preceeding calculations, it suffices to con-sider the 1-D case obtained by projection on the coor-dinate axes. Let the length of the projected intervalbe L and suppose the interval divided into N equalcells of length LIN. Let (j denote the midpoint ofinterval number J and if the projected coordinate jlies in interval J, write

tj = ( + Atj.

Then the transform plane intensity can be written

I(x) = I E exp[2rix(# - AZj,k

= IX Z exp[27rix({ - K)/XzIJ.K jeJkeK

X exp[2ix(A - A)/z],

where JK range over the cells and j E J denotes thatparticle j lies in cell J. Define

IJK(X) = I exp[2irix(#j - K)/z]

ZJK(X) = E exp[2rix(Aj - Ak)/XZ],jeJ,keK

so that

I(X) = XIJK(X)ZJK(X).J,K

The deterministic function IJK can be rewritten:IJK(X) = I exp[2rixL(J - K)/NXzj.

(14)

(15)

The most rapidly varying of these functions has awavenumber proportional to 1L. The random quan-tities ZJK are functions of position, but since IA\ - AkI< LIN, they can be considered slowly varying relativeto function (15), provided N is sufficiently large.

Equation (14) achieves a scale separation by ex-pressing the transform plane as a Fourier series withspatially slowly varying coefficients. However, al-though the analysis is exact to this point, it is merely

5288 APPLIED OPTICS / Vol. 29, No. 35 / 10 December 1990

formal because there is no apparent reason to selectany particular number of cells N. In fact, if N equalsone, the scale separation disappears, and Eq. (14) be-comes nugatory. At the other extreme, if N is verylarge, many of the coefficients ZJK will be zero.

Equation (14) provides a useful description of thetransform plane when N is chosen so that, with highprobability, each cell contains approximately the samenumber of particles. In this case, the ZJK have ap-proximately the same variance for all x. Then Eqs.(14) and (15) predict that the power in wavenumber kj= j/NL,-N •j N, is proportional to 1-ljl/N. Thisis the power spectrum characteristic of laser speckle,7

for example, but here, as in the generalized Van CittertZernike theorem, there is also slow modulation of thetransform plane over the length scale XzN/L because ofthe spatial variation of the factors ZJK. Note, that if Ncan be taken infinitely large, the modulation scalediverges, and the particle image field simply acts like apartially coherent source.

Thus, Eq. (14) describes the far field of a quasihomo-geneous source provided each cell contains approxi-mately the same number of particles. Then the ZJKare approximately identically distributed. The distri-bution of ZJK for fixed x is easily expressed in terms ofthe first-order intensity distributions. Define

Ej = a, exp(27rixAj/X/z).jeJ

Then Ej + EK represents a random walk with as manysteps as the sum of the particle counts in cells J and K.The well-known distribution of the modulus of Ej isderived in the Appendix. Thus, the distributions ofEJEJ,EKEK, and (EJ + EK)(EJ + EK) are known. Fur-thermore,

(Ej + EK)(EJ + EK) = EjEJ + EKEK + 2ZJK.

This analysis reduces the question, when does theparticle image field act like a quasihomogeneoussource, to the question, what is the largest number ofcells N so that cells of length LIN all contain nearly thesame number of particles? The problem is to satisfythe uniformity condition with a large enough value ofN such that the modulations produced by the func-tions ZJK are on a length scale larger than the smallestgrain size produced by the function IJK.

As a measure of uniformity, the value

S2 = (N/M2) E (Xi - M/N) 2 (16)

is proposed, where Xi is the number of particles in celli, m is the number of particles, and N is the number ofcells. Note that

x2 = Ms, (17)

has the chi-square distribution with N - 1 degrees offreedom if the cell occupation numbers are large. Thenormalization adopted in Eq. (16) assigns the sameuniformity score to distributions with the same per-centage fluctuations.

The distribution of S is easily established. Let

M=Ml+M2 +...+MN,Ml> M2 ... rMN

be a partition of m and let

Mk+l =... = mN = rnk 0.

Then the number of arrangements of m particles intoN cells with occupation numbers given by this parti-tion equals

I(Ml, . . ,rm) ( l) m!/Ml! ... mk!,

where I(ml,. . ., ma) is the number of distinct permuta-tions of the numbers ml,. . ., ma. Each such arrange-ment obviously has the same S2 score. This distribu-tion is easily tabulated for small to moderate values ofm. For large particle populations, the distribution iswell approximated by the chi-square distributionthrough Eq. (17).

The signal to noise analysis of Sec. II was based onthe fine grained properties of particle image noise; thegeneralized Van Cittert Zernike theorem emphasizesinstead the long wavelength properties. These enterin the analysis of suggestions that spatial changes infringe visibility might be used to assess nonuniformi-ties in velocity throughout the interrogation regioncaused by spatial velocity gradients or turbulence.Hinsch et al.11 propose that the probability densityfunction of velocities in the interrogation region can berecovered in principle from spatial variations in fringevisibility. Thus, a problem of extracting signal fromnoise arises at the long wavelength level: it is neces-sary to separate these (deterministic) modulationsfrom the modulations described by the generalizedVan Cittert Zernike theorem.

Suppose that it is proposed to resolve a long wave-length phenomenon which takes place over some largenumber of grains n. Measurements of this type can-not be contemplated unless the grain size is close to itsminimum value; accordingly, the particle populationwill be assumed large enough that the grain size is closeto its speckle limit value with high probability. Eachparticle population m is associated with a characteris-tic scale of random long wavelength modulations,found for example by requiring S2 < 1/10 with proba-bility 90%. Denote this scale, measured in number ofgrains by 1. A graph of this scale as a function ofparticle population appears in Fig. 4. For large m, thenormal approximation to the x2 distribution can beapplied to show that I m/10 - 0.563 ml/ 2 . Let usadopt the hypothesis that resolution of a phenomenonat the scale n requires that 1 >> n. Thus, Fig. 4 demon-strates that even moderate particle populations areinadequate because there is random modulation over asmall number of grains. With -400 particles, thewavelength of random modulations of 30 grains shouldpermit resolution of long wavelength phenomena overup to -5-6 grains.

With such a large particle population, the SNR anal-ysis of Sec. III can be expected to impose significantfrequency resolution constraints. Let it be required,

10 December 1990 / Vol. 29, No. 35 / APPLIED OPTICS 5289

C9.e

(o0

E

z:0)C.5a)

0

0'it

35

30

25

20

15

10

Fig. 4. Tent}.

100 200 300

Particle Population

h percentile of long wavelength modulfunction of particle population.

for example, to determine AK so that the SNR with 400particles is at least 2 with probability 90%. Use thestandard approximations:

7, exp(-,U)Ah/k! _ Z (y C )1:5hSC-1 (

where Z denotes the cumulative distribution functionof the unit normal density. Set

= M2AK,

c - 1 = m/2,

where ,u is the Poisson parameter of the noise spec-trum, and the tenth percentile is set to m/2 so that theSNR equals 2. It follows that

m2A- (m + 1)/2 1.28/(m- + 1)/2,

since Z(1.28) = 0.9. With m = 400, AK - 730. In termsof the autocorrelation analysis of Refs. 5 and 6, thisimplies that achieving a SNR of 2 with probability 90%requires that the typical linear dimension of a particleimage be about 1/730 times the typical linear dimen-sion of the interrogation region. Although this analy-sis is preliminary in character, it clearly suggests thatmeasurements of this type will require extremelyheavy seeding and, therefore, very small particles orvery high frequency resolution.

V. Conclusions

The accuracy of velocity measurements using theYoung's fringe method in particle image velocimetry iscompromised by transform plane noise which origi-nates in the random particle locations. Three proper-ties characteristic of this noise were derived, and ap-plied to PIV measurement statistics. The exactprobability density function for spatial frequency cut-off was computed as a function of particle population.It was found that this probability density reduces tothe deterministic value predicted by coherence theoryat high particle population, but that at low particlepopulation, the distribution becomes more uniformlydistributed over all possible spatial frequencies. Thisdistribution was used to assess the reduction of fringeresolvability compared to the large population, or laser

speckle limit. Fourier analysis of the noise field showsthat the power at each frequency is Poisson distribut-ed. The Poisson parameter varies as a function offrequency like the power spectral density function oflaser speckle. This form of the spectrum was appliedto the problem of computing the SNR in a Fourieranalysis of the image plane of a double-pulsed photo-graph. The results depend on a frequency resolutionparameter AK: the SNR can be increased by decreas-ing AK, but at fixed resolution, indefinite increase ofthe particle population results in asymptotically de-creasing SNR. A generalized Van Cittert Zernike

ation scale as a theorem is derived for particle image noise. This theo-rem characterizes the long wavelength behavior of thenoise field. A preliminary application to the analysisof spatial variation of fringe visibility suggests thatextremely high seeding density is required to ade-quately resolve this phenomenon.

Appendix: First-Order Properties of the Transform Plane

The distribution of intensity at a point in the trans-form plane entered the analysis of transform planecorrelation properties through the distribution of thefunctions ZJK. Accordingly, a brief discussion of thesedistributionsl7"1 8 will be given.

Rewrite the expression for total power, Eq. (2), indimensionless variables as

2

I(x,y) = E exp[i(Xt + ynk)]1:k5m

where (Qknk) are the particle locations and (x,y) aretransform plane coordinates. Define k = X + Yk.Then away from the origin, k is uniformly distributedmod 27r for all (xy). The distribution of I is thedistribution of the square of the distance travelled in aplanar random walk of m steps. The cumulative dis-tribution function for the first power of the distance is9

p(x < rim steps) = J [Jo(p)] 0J 1(pr)rdp.

Poisson weighting gives

p(x < r) = r J 1(u) expfs[J0(u/r) - 1]1du

for the cumulative distribution when the number ofpoints obeys the Poisson distribution with parameters.17'18 Note, that this distribution governs the squareroot of intensity.

To investigate limiting forms of these distributions,it is useful to normalize to constant expected intensityby replacing r by sIt. Then,

P( = J 1(u) exps[J(u/VX - jdu.

Asymptotic forms, valid for large s are easily derivedby expanding the Bessel function Jo in a power seriesand integrating term by term. The result is

P(n = [1 - exp(-I/s)] + I exp(-I/s)(1 - I/2s)/4s2

- I exp(-I/s)(1 - I/s + I2/6s2)/6s + (1/s3).

5290 APPLIED OPTICS / Vol. 29, No. 35 / 10 December 1990

Io

Note, that the distribution reduces to the negativeexponential distribution to leading order in 1/s. Thisis the well known distribution for intensity in laserspeckle. 7

References

1. R. J. Adrian and C. S. Yao, "Development of Pulsed Laser

Velocimetry for Measurement of Turbulent Flow," in Proceed-

ings of the Eighth Biennial Symposium on Turbulence (Uni-versity of Missouri, Rolla, 1983).

2. M. Born and E. Wolf, Principles of Optics (Pergamon, London,

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4. K. A. Stetson, "Effect of Scintillation Noise in Heterodyne

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6. J. M. Coupland and C. J. D. Pickering, "Particle Image Veloci-

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7. J. W. Goodman, "Statistical Properties of Laser Speckle Pat-

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9. W. H. Carter and E. Wolf, "Coherence and Radiometry withQuasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785-796 (1977).

10. J. W. Goodman, "Some Effects of Target-Induced Scintillationon Optical Radar Performance," Proc. IEEE 53,1688 (1965).

11. K. Hinsch, W. Schipper, and D. Mach, "Fringe Visibility in

Speckle Velocimetry and the Analysis of Random Flow Com-ponents," Appl. Opt. 23, 4460-4462 (1984).

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Young's Fringes in Speckle Velocimetry," Appl. Opt. 24, 1448-

1453 (1985).13. R. J. Adrian, "Statistical Properties of Particle Image Veloci-

metry Measurements in Turbulent Flow," in ProceedingsFourth International Symposium on Applications of LaserAnemometry in Fluid Mechanics (Instituto Superior Tecnico,Lisbon, 1988).

14. J. W. Goodman, Fourier Optics (McGraw-Hill, New York,

1968).15. C. J. D. Pickering and N. A. Halliwell, "Laser Speckle Photogra-

phy and Particle Image Velocimetry: Photographic Film Noi-se," Appl. Optics 23, 2961-2969 (1984).

16. W. Feller, Introduction to Probability Theory and Its Applica-tions (Wiley, New York, 1950).

17. S. H. Chen, P. Tartaglia, and P. N. Pusey, "Light Scatteringfrom Independent Particles-Nongaussian Correction to theClipped Intensity Correlation Function," J. Phys. A. 6, 490(1973).

18. D. W. Schaefer and P. N. Pusey, "Statistics of Non-GaussianScattered Light," Phys. Rev. Lett. 29, 8 (1972).

19. G. N. Watson, Treatise on the Theory of Bessel Functions(Cambridge U.P., Cambridge, 1915).

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