Low-angle elastic light scattering ~LAELS! is one ofthe most suitable and widely used techniques for per-forming particle sizing1 on samples with particlesbigger than the wavelength of light. LAELS findsapplications in many industrial problems and is alsoprofitably used in many fields of applied sciences,such as atmospheric and aerosol science, or for emul-sions and powder characterization.2 As with otheroptical techniques such as dynamic light scatteringand spectral extinction techniques, LAELS allowsthe characterization of the sample being carried outin situ and almost in real time. Probably the mostimportant feature of LAELS is the very wide range ofthe accessible scattering angles, typically two de-cades with the currently available photodetectors.This is essential for recovering particle-size distribu-tions over a correspondingly wide range of radii.
The main difficulty in retrieving the particle-size
F. Ferri and G. Righini are with the Istituto di Scienze Mate-matiche, Fisiche e Chimiche and with the Istituto Nazionale diFisica della Materia, University of Milan at Como, via Lucini 3,22100 Como, Italy. E. Paganini is with the Centro InformazioniStudi Esperienze, P.O. Box 12081, 20134 Milan, Italy.
Received 3 March 1997; revised manuscript received 6 June1997.
distribution from LAELS measurements is the solu-tion of a first-kind Fredholm integral equation thatdescribes the experimental data. This classic ill-posed problem, whose characteristics are common tomany other indirect-sensing experiments, invariablyleads to highly unstable solutions, because even arbi-trarily small noise components in the measured quan-tities can give rise to extremely large spuriousoscillations in the solution.3,4 Quite a number ofmethods have been developed over the past few yearsto tackle this problem, and a thorough review on thistopic can be found in Ref. 5 and the references therein.
One of the most popular inversion algorithms usedin LAELS6–8 is a nonlinear iterative algorithm orig-inally proposed in 1968 by Chahine.9,10 The mainadvantages of the Chahine method are that no apriori assumptions are needed for the distribution tobe recovered, no constraints are imposed on the so-lutions, which are always positive, and large amountsof data can be processed efficiently. On the otherhand, because the technique is sensitive to experi-mental noise, it is difficult to find a reliable criterionfor stopping the inversion procedure, and noisy un-stable solutions may occur. When the distributionsrecovered with the Chahine method are expressed asnumber distributions, they usually have a typical in-dented appearance characterized by the presence ofmany spurious peaks localized toward the small par-ticle side of the radius range. If the same recovereddistributions are expressed as weight distributions,
20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7539
their appearance is somewhat smoother and defi-nitely much more reliable. The consequence of thissusceptibility to noise therefore limits the use of theChahine method for retrieving only weight distribu-tions. However, even with weight distributions, thetendency of the Chahine method to become unreliablewith noise persists, and its capability of fitting thedata and accurately recovering the sample distribu-tion deteriorates with increasing noise levels.
In this paper we propose the use of a new algorithmfor the inversion of LAELS data. Such an algorithmwas devised by modification of the classical Chahinemethod, and it has already been applied successfully tothe inversion of spectral extinction data, on both sim-ulated11 and experimental data.12 In Ref. 11, we haveshown that, for the inversion of spectral extinctiondata, such an algorithm is stable and reliable withrespect to experimental noise and converges to a stablesolution independently of the starting distribution.The same algorithm, whose effectiveness depends onthe particular kernel of the integral equation, is inves-tigated and tested here by means of computer simula-tions. Its accuracy, stability, and reliability arestudied as functions of noise level, and a thoroughcomparison with the Chahine algorithm is carried out.The criterion adopted to stop the inversion procedureis also tested and critically discussed. The resultsshow that the method proposed here allows a largevariety of particle distributions and their parametersto be accurately reconstructed, both in weight and innumber. Moreover, it greatly improves the stabilityof the solutions against noise and is able to suppressthe typical indented and spiky-shaped appearance ofthe distributions obtained with the classical Chahinemethod. Finally, for this algorithm, the stopping cri-terion is rather objectively determined and is not leftas an arbitrary parameter of the method.
2. Theory
The intensity distribution of light scattered by anisotropic sample is a function of two angles, the scat-tering angle u, which is defined as the angle betweenthe scattered wave vector k and the incident wavevector k0, and the polar angle w, which is the anglebetween the polarization of the incident electric fieldand the projection of k onto a plane orthogonal to k0.Although the dependence on u carries information onthe sample structure and size, the w dependence isdue only to polarization effects and can be easilytaken into account and worked out of the equations.Consequently the scattered intensity can be ex-pressed as a function of u only, or, as is customarilydone in the theory of light scattering, as a function ofthe wave vector q that is equal to the difference be-tween k and k0, i.e., q 5 k 2 k0. The magnitude ofq is related to the scattering angle u by the relation
q 54pn
lsinSu
2D , (1)
where l is the vacuum wavelength of the light and nis the refractive index of the medium.
7540 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997
When elastic light-scattering data are used for per-forming particle sizing, they are usually taken at verylow scattering angles, and the scattered intensity ismeasured with bidimensional sensors, such as annu-lar arrays of photodiodes8 or CCD’s.13 These sensorsdetect the scattered intensity over their entire sensi-tive area to which an overall scattering angular range~umin, umax! and a polar angular range ~w1, w2! corre-spond. As shown in Fig. 1, the sensitive area is usu-ally subdivided into rings, each of them delimited bypolar angles w1 and w2 and characterized by an aver-age scattering angle u and an angular spread du.The angles w1 and w2 are the same for all the rings,and for most of the commonly used detectors they arew1 5 0, w2 5 90°, which corresponds to quarters ofrings, or w1 5 0, w2 5 180°, which corresponds tosemirings. The range of the scattering angles is de-termined by the physical distance of the innermostand the outermost rings from the optical axis and bythe optics used for collecting the scattered light. Atypical sensor is made up of 50 rings or less andcollects scattering angles from fractions of degrees totens of degrees, with a dynamic range of approxi-mately two decades. This implies that the angularspread corresponding to each ring is rather high ~duyu; 10% or larger! and only the scattered intensityaveraged over the solid acceptance angle associatedwith each ring can be detected.
If the sample is a diluted homogeneous suspensionof noninteracting polydisperse spheres, the averagescattered intensity is given by
I~q! 5 * Ip~q, r!N~r!dr, (2)
where N~r!dr is the number concentration of particles~in inverse cubic centimeters! with radii between r
Fig. 1. Schematic diagram of a typical detector for LAELS. Thesensor plane is subdivided into portions of concentric rings, each ofthem delimited by polar angles w1 and w2 and characterized by anaverage scattering angle u.
and r 1 dr and Ip~q, r! is the average intensity scat-tered by a particle with radius r. This is given by
Ip~q, r! 51V *
V
dPdV
~u, w, r!dV, (3)
where V is the acceptance solid angle that corre-sponds to q, dPydV is the scattered intensity per unitsolid angle provided by the Mie theory,14 and thecorrespondence between u and q is given by Eq. ~1!.In Eq. ~3! dPydV also depends on l and on the refrac-tion indexes of the particle and of the medium, but,for the sake of simplicity, it has not been reportedexplicitly.
Equation ~2! is a first-kind Fredholm integral equa-tion in which I~q! is provided by the experiment, N~r!is the distribution to be recovered, and Ip~q, r! is theknown kernel. This is a typical example of an ill-posed problem, meaning that rather different distri-butions can fit the experimental data to the samelevel of accuracy, provided that there is some noise inthe data. Consequently the solutions might behighly unstable and unreliable, and it is crucial toadopt a suitable inversion algorithm.
In this paper we carry out the inversion of Eq. ~2!by using a nonlinear inversion algorithm that wehave recently proposed for the inversion of spectralextinction data.11 Although a detailed description ofsuch an algorithm can be found in Ref. 11, here werecall only how this algorithm works and how it wasdevised from the original algorithm proposed by Cha-hine.10,11 According to our algorithm, the particle-size distribution is approximated by a histogram witha number of classes equal to the number of wavevectors qi at which the scattered intensity I~qi! ismeasured. Then the integral of Eq. ~2! is trans-formed into a set of linear algebraic equations:
I~qi! 5 (j
NjAij, j 5 1, 2, . . . , M, (4)
where Nj is the number concentration of the particlesthat belong to the jth class and Aij is a M 3 M matrixgiven by
Aij 5 *rj21
rj
Ip~qi, r!dr, (5)
where ~rj21, rj! is the interval of the jth class andIp~qi, r! is given by Eq. ~2!. The algorithm works inthe following way: If a particle distribution hasbeen recovered after p iterations, at the next iterationone has
Njp11 5 Nj
p (i51
M
Hij
Imeas~qi!
Icalcp ~qi!
, (6)
where Imeas~qi! and Icalcp ~qi! are, respectively, the
measured intensity and the intensity calculated on
the basis of the distribution Njp. Hij is a normalized
weight function given by
Hij 5Aij
(i
Aij
. (7)
Note that the retouching of the jth class is done by themultiplication of the population of the same class atthe previous iteration by a factor that is given by theaverage value of the ratios Imeas~qi!yIcalc
p ~qi! averagedover the entire q range through the weight functionHij. Because Hij, in the limit of infinitesimal classes,has the same shape of intensity scattered by the jthclass of particles, it weights the most the qi at whichthe scattered intensity given by the jth class presentsits maximum.
Equation ~6! was devised by modification of thealgorithm originally proposed by Chahine,10,11 whichworks in the following way:
Njp11 5 Nj
p Imeas~qj!
Icalcp ~qj!
. (8)
Note that, differently from Eq. ~6!, in which thewhole signal sequence I~qi! is used to correct thepopulation Nj
p, in this case the correction of Njp is
carried out by comparison of the measured and thecalculated intensities only at a given qj. Thereforeit is necessary to establish a one-to-one mappingbetween the class being corrected and the correspon-dent qj. In Chahine’s original algorithm this corre-spondence was based on the existence of a maximumin the kernel of Eq. ~2! when reported as a function ofq. When this maximum exists, each class of parti-cles with radii in the range ~rj21, rj! can be associatedwith a qj, chosen so that the behavior of I~q! due tothis class presents a maximum in correspondence ofq 5 qj. However, in the case of elastic light scatter-ing, such correspondence cannot be established be-cause, regardless of the particle radius, themaximum is always located at q 5 0. Nevertheless,the Chahine algorithm has been widely used in thepast years for inverting scattering data.6–8 A way toovercome this difficulty is to rewrite Eq. ~4! in termsof the integrated intensities, i.e., the product I~qi!Vi,which is, incidentally, exactly the quantity measuredin the experiment. In this case the kernel of theintegral equation does exhibit a maximum at a valueq*, which depends on the particle radius and also onthe particular arrangement of the rings of the detec-tor. As reported in Ref. 11, such a correspondencepermits us to estimate the range of radii over whichthe particle-size distribution can be recovered. If~qmin, qmax! denotes the range of the wave vectors ofthe measurement, then the smallest radius rmin isthat one for which q* 5 qmax, whereas the largestradius rmax is that one for q* 5 qmin. All the simu-lations reported in this work have been carried outwith a radius range determined with this criterion.
One of the most critical problems that arises wheniterative inversion algorithms are used is how to con-
20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7541
trol the convergence of the method and how to find areliable criterion for stopping the iterative process.In our case the inversion procedure was stopped onthe basis of the behavior, as a function of the numberof iterations, of the root mean error ~rme!:
rme 5 H 1M (
i51
M @Imeas~qi! 2 Icalcp ~qi!#
2
@Icalcp ~qi!#
2 J1y2
. (9)
The rme describes the average rms relative deviationof the retrieved intensities Icalc~qi! from the measuredintensities Imeas~qi!.
In order to determine the stop criterion, we alsomonitored, as a function of the iteration number, therelative variation of the rme parameter, the averageradius, the standard deviation, and the overall con-centration of the reconstructed distribution. Inmost of the tests carried out by using our algorithmwe observed that the inversion procedure is ratherstable, meaning that the rme parameter monotoni-cally decreases with the number of iterations andthat the retrieved distribution matches better andbetter the expected one. However, in some cases, ifthe inversion procedure is pushed too far away, itmay happen that an instability appears in the re-trieved distribution. This occurs mainly when thereis some noise present on the data, but it was alsoobserved for noiseless data that corresponded to verynarrow ~syr & 3%! or monodisperse distributionswhose signal sequence exhibited secondary maximaand deep oscillations. This instability develops in apeculiar way and, in all the investigated cases, italways had the same appearance, regardless of theparticular case that was being studied. After theinversion procedure has reached a point at which therme parameter does not vary appreciably any more,the algorithm starts adding particles with very smallradii. This produces a distribution with a high spu-rious peak at small particles, and in the next sectionwe give an example of this instability. Here wepoint out that the onset of such instability was foundto be strongly correlated to a slowdown in the rate ofconvergence, as we estimated by studying the behav-ior of the rme as a function of the iteration number.Indeed, soon after the instability has started to de-velop, either the derivative of the rme with respect tothe iteration number or the rme itself attains a min-imum. In the latter case the minimum is alwaysshallow, and, if the inversion procedure is kept going,the rme eventually decreases again.
The above picture was observed in the large ma-jority of the performed tests, and only a few excep-tions were encountered. In these cases, in which theinstability grows without the rme and its derivativereaching any minimum, the inversion procedure isstopped in connection with a maximum number ofiterations, which is typically 10,000 iterations.
In conclusion, although we have no theoretical ex-planation for supporting this choice, the above crite-rion was adopted and used in all the simulationscarried out in this work. As is shown in Section 3,the value of the rme at the stopping point is always
7542 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997
comparable with the rms amount of noise present inthe data, which is a fair indication that the data havebeen reconstructed as well as possible.
As far as the stopping criterion used in the classicalChahine algorithm is concerned, the same sort of con-sideration applies, but the situation is much more com-plex and the recovered distributions are characterizedby much more irregular instabilities. The reasonprobably is because the Chahine algorithm is, at thesame time, much faster and much more sensitive tonoise than ours, with the instabilities developing at arate comparable with the rate of convergence. Forthis algorithm there is only a loose correlation betweenthe instability and a minimum in the rme and itsderivative. With noisy data, the rme often reaches aminimum very soon ~;10–100 iterations! in the courseof the inversion procedure and then starts to increaseindefinitely. Thus is it difficult to find a valid andreliable stopping criterion. In the simulations re-ported in this paper, we imposed a minimum numberof iterations on the inversion method, and afterwardswe adopted the same stopping criterion and applied itto our algorithm.
As a final comment for this section, we point outthat Eq. ~2! and successive formulas referring to theinversion algorithm have been worked out forparticle-size number distributions. For characteriz-ing the sample in terms of weight distributions, be-cause the particles have a spherical shape, integralEq. ~2! becomes
I~q! 5 * Ip~q, r!S43
pr3rD21
W~r!dr, (10)
where W~r!dr is the mass of particles with radii be-tween r and r 1 dr and r is the particle density.Note that the kernel Eq. ~10! has the same q depen-dence as the kernel of Eq. ~2!. This makes Eqs. ~2!and ~10! completely equivalent as far as the inversionalgorithm is concerned. Indeed, it is easy to showthat, in the limit of infinitesimal classes, the algo-rithm of Eqs. ~4!–~8! is dependent on only the q de-pendence of the kernel and works in the samemanner with either number or weight distributions.Therefore, once one of the two distributions is recov-ered, we can obtain the other one by simply multi-plying or dividing the first one by the factor ~4y3pr3r!. However, it should be recalled that, becausethe scattered intensity is strongly dependent on theradius of the particles, the largest contribution to thescattering comes from the larger particles. Conse-quently it may happen that the recovered numberdistribution exhibits spurious peaks at particle radiiso small that they do not substantially affect thequality of the reconstructed data. On the otherhand, the presence of these peaks usually altersrather strongly the quality of the recovered distribu-tion, giving rise to a distribution with a shape, aver-age radius, and width that might be far from theexpected ones. However, if the same distribution isexpressed in terms of a weight distribution, becauseof the factor ~4y3 pr3r!, the presence of these spurious
peaks is strongly dampened and the weight distribu-tion may accurately match the expected one. Thusweight distributions are somewhat intrinsically morestable than number distributions and definitely moresuitable for reconstructing the data. This sort ofdichotomy between number and weight distributionsis a consequence of an ill-posed problem and it is whatmakes the classical Chahine algorithm work wellonly with weight distributions. Conversely, as isshown in Section 3, our method overcomes this limi-tation and allows us to recover accurately both num-ber and weight distributions. Depending on theparticular system being investigated, this can be arelevant advantage.
3. Computer Simulations
The computer simulations used for testing our inver-sion algorithm consist of two steps. First, simulatedaverage scattering intensity ~input data! are gener-ated by computer according to Eq. ~2! for a givennumber distribution ~input distribution!. Second,the input data are processed, and the retrieved dis-tribution ~output distribution! is compared with theinput one. The comparison is carried out either onthe basis of the number distribution or of the corre-sponding weight distribution. In particular, thefirst three moments of each distribution are evalu-ated: the average radius, the standard deviation,and the overall sample concentration. Below we usethe symbols rinp
nb , sinpnb , and cinp
nb to indicate the number-average parameters of the input distribution, and thesymbols rout
nb , soutnb , and cout
nb refer to the correspondingparameters of the output distribution.
The retrieved intensities ~output data! recon-structed on the basis of the output distribution @ac-cording to Eq. ~4!# are compared with the input onesby means of the rme parameter defined in Eq. ~9!.In most cases random noise is added to the inputdata, with the rms amount of noise being propor-tional to the input data. The noise was chosen tohave a flat probability distribution with no correla-tion between the data points. For each level ofnoise, the procedure was repeated with 100 differentsamples of noise. When the analysis against noise iscarried out, the output distributions and the outputdata are characterized by means of the average andthe standard deviations of the above parameters.The symbols ^rout
nb &, ^soutnb &, and ^cout
nb & indicate the av-erage values of the parameters characterizing theoutput distributions, and the symbol ^rme& representsthe average relative root mean deviation of the outputto the input data. Here ^. .& stands for an average over100 samples of noise. When weight-average param-eters are considered, all the above symbols are usedwith the superscript wt instead of nb.
The simulations were carried out with a detectormade of M 5 50 ring quarters ~w1 5 0, w2 5 90°! andcapable of collecting the light scattered by the sampleover a range of two decades in q vectors, from qmin 52.5 3 102 cm21 to qmax 5 2.5 3 104 cm21. The ringswere supposed to be spaced so that the corresponding
average wave vectors qi scale as a geometric progres-sion:
qi 5 q1ai21, i 5 1, 2, . . . , 50, (11)
where q1 5 qmin and a is the ratio of the geometricprogression; a 5 ~qmaxyqmin!1yM21 ; 1.10. Thespread dqi of each ring was chosen to be proportionalto qi, i.e., dqiyqi 5 10% for all the rings. This valuecorresponds to having the outer radius of each ringalmost coincident with the inner radius of the nextring, so as to fill the entire area of the detector ~fillingratio ;100%!. The input data were generated byMie theory14 with a vacuum wavelength of 0.6328 mmand refraction indexes of the particles and of themedium equal to 1.588 and 1.33, respectively. Allthe above figures were chosen in view of a possiblecomparison with the experiment in which the light ofa linearly polarized He–Ne laser is scattered by adiluted aqueous suspension of latex spheres and de-tected with a standard photodetector for LAELS.The range of radii used for the inversion was chosenwith the criterion of the maximum ~see Section 2!according to which rmin 5 0.70 mm and rmax 5 77 mm.Within this range, the 50 classes were scaled accord-ing to a geometric progression, giving rise to classeswith the same relative accuracy djyrj ; 10%, wheredj 5 rj 2 rj21 and rj 5 ~rj21 1 rj!y2. The zero-iteration distribution was chosen to be constant ~Nj 5constant!, and the final results were checked to beindependent of this starting distribution.
In the first test we show how the stopping criterionworks. The input data corresponding to an inputGaussian distribution with rinp
nb 5 10 mm, sinpnb 5 0.3
mm, and cinpnb 5 106 cm23 were processed, and the
output distribution, as well as its number-averageparameters rout
nb , soutnb , and cout
nb , was studied as a func-tion of the number of iterations. In this test no noisewas added to the input data. Figure 2 shows the
Fig. 2. Results of our inversion method in the case of a narrowinput Gaussian distribution with rinp
nb 5 10 mm, sinpnb yrinp
nb 5 3%, andcinp
nb 5 106 cm23 ~solid curve!. The three output distributions wererecovered when the inversion procedure was stopped after 2000,2638, and 5000 iterations. No noise was added to the input data.
20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7543
input distribution and the three output distributionsrecovered after 2000, 2636, and 5000 iterations. Itcan be noted that, although the input distribution~solid line! is well recovered for all three cases, onlythe output distribution corresponding to 2000 itera-tions accurately matches the input one over the en-tire radius range. If the inversion process is keptgoing, an instability arises with a spurious peak ap-pearing at very small radii and growing with thenumber of iterations, as shown by the output distri-butions recovered after 2638 and 5000 iterations.This instability grows regardless of the fact that thematching between the input and the output data doesnot change anymore or improves slightly as the num-ber of iterations increases. This is shown by Fig. 3,in which the input and the output data correspondingto the recovered distributions of Fig. 2 are plotted asfunctions of q. In Fig. 3~a! the input data correspondto circles; all three output data are shown as a singlesolid curve, being indistinguishable on the scale ofthe figure. To distinguish the small differences be-tween them and also to emphasize their deviationsfrom the input data, the percentage deviations be-tween the input and the output data are plotted inFig. 3~b!. The reconstruction is accurate for low qvalues, whereas it is worse for high q values. This isbecause the distribution of the scattered intensity fora narrow distribution is structured and, for large q,exhibits deep oscillations with a peak-to-peak ampli-tude as high as almost an order of magnitude.
To investigate the onset of the instability, we showin Fig. 4, as functions of the iteration number, thebehaviors of rout
nb , soutnb , and cout
nb , together with the rme
Fig. 3. Input and output data with our inversion method for thethree distributions of Fig. 2: ~a! the input data are shown as opencircles, while all three output data, being indistinguishable on thescale of the figure, are shown as a single solid curve; ~b! percentagedeviations between the output and the input data corresponding tothe three distributions recovered after 2000 iterations ~squares!,2638 iterations ~diamonds!, and 5000 iterations ~triangles!.
7544 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997
and the derivative of the rme with respect to theiteration number, ]~rme!y]~#it.!. At the beginning,the parameters rout
nb , soutnb , and cout
nb vary smoothly withthe iteration number and correctly converge with theexpected values ~indicated by the arrows!. How-ever, after ;2000 iterations, they suddenly start tochange and, within a few hundreds iterations, theydevelop large deviations that are consistent with theappearance and growth of a large number of particleswith very small radii. No sign of this instability isfound in the behavior of the rme, which decreasesmonotonically and smoothly with the iteration num-ber. Conversely, the rate at which the rme de-creases changes and ]~rme!y]~#it.! exhibits aminimum at 2638 iterations, which is reasonablysoon after the onset of the instability. Although thisminimum occurs when the instability has alreadystarted to grow, it represents a strong marker for theinstability and is a fair indication that the inversionprocedure has to be stopped. As described in Section2, the behavior shown in Figs. 2–4 is a typical exam-ple of the kind of instabilities observed when noise-less data that correspond to very narrow ~syr & 3%!or monodisperse distributions are inverted. Fornoisy data the situation is somewhat different. Theinstabilities are much more frequent, do not dependon the distribution’s being recovered, and appear ear-
Fig. 4. Behavior, as functions of the iteration number, of thenumber-average output parameters characterizing the sameGaussian distribution of Fig. 2: ~a! average radius, ~b! standarddeviation, ~c! concentration, ~d! rme parameter, ~e! derivative of therme with respect to the iteration number, ]~rme!y]~#it.!. The ar-rows represent the values of the parameters that correspond to theinput distribution; the vertical dashed–dotted line at 2638 itera-tions corresponds to the minimum in ]~rme!y]~#it.!.
Fig. 5. Comparison between the output distribu-tions recovered with our algorithm ~solid curvehistograms! and with the Chahine algorithm ~dot-ted curve histograms! for six input Gaussian dis-tributions. All the input distributions ~solidlines! were characterized by the same relativestandard deviation sinp
nb yrinpnb 5 10% and the same
number concentration cinpnb 5 106 cm23. The av-
erage radii were, from left to right, top to bottom,1, 2, 4, 10, 20, and 50 mm. No noise was added tothe input data.
lier in the course of the inversion procedure. Theironset is either correlated to a minimum in the rme orin its derivative. The higher the noise level on thedata, the higher the chances of finding an instabilityearlier during the inversion procedure.
The second test compares our inversion algorithmwith the classical Chahine algorithm. Six Gaussianinput distributions with average radii of 1, 2, 4, 10, 20,and 50 mm were used to test the two algorithms overthe entire range of recoverable radii. All the distri-butions were characterized by the same relative stan-dard deviation sinp
nb yrinpnb 5 10% and the same
concentration cinpnb 5 106 cm23. No noise was added to
the input data. The results of this test are given inFigs. 5 and 6 and in Table 1. The input distributions~solid lines!, the output distributions recovered with ouralgorithm ~solid curve histograms!, and the output dis-tributions recovered with Chahine algorithm ~dotted
curve histograms! are shown in Fig. 5. Although thereconstruction of all the input distributions is fairlyaccurate with our method over the entire radius range,the Chahine method correctly recovers only the distri-butions characterized by small average radii ~1 and 2mm!. For bigger particles, the output distributionsare reconstructed accurately only with regard to thepeak of the input distribution, but high spurious peaksappear at small particle radii. This indented shape istypical of the classical Chahine method and it is fre-quently observed when the data to be inverted exhibitsecondary maxima and deep oscillations. This is es-sentially due to the limited capability of the Chahinealgorithm to reconstruct the input data that corre-spond to these deep oscillations, as shown in Fig. 6.The figure shows, for each of the distributions of Fig. 5,the input and the output data obtained with the twomethods as a function of q. In part ~a! of each frame
Fig. 6. Comparison between the reconstructedand the input data for the same distributions ofFig. 5. ~a! of each frame shows the input data~circles! and the reconstructed data obtained withour method and the Chahine method as a singlesolid curve. The differences between the twomethods can be appreciated in ~b! of each frame inwhich the deviations between output and inputdata are reported. Our method is indicated byopen squares, and the Chahine method is indi-cated by filled squares.
20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7545
the input data are given as circles; the two output data,being indistinguishable on the scale of the figure, areshown as a single solid curve. To appreciate thesmall differences between the two methods, in part ~b!of each frame we show the corresponding deviationsobtained with our method ~open squares! and with theChahine method ~filled squares!. The data relative tothe smaller particles ~1 and 2 mm! are smooth and areaccurately reconstructed by both methods with devia-tions of approximately a few percent. Differences be-tween the two methods begin to appear when the inputdata become more structured with one or moreoscillations that correspond to secondary maxima.Whereas our method is substantially immune to thisfeature and reconstructs the data with deviations al-ways #5% over the entire q range, the Chahinemethod becomes inaccurate precisely in correspon-dence to these oscillations, at which the deviationsreach values of the order of ;10%–20% or even higher.This behavior is confirmed by the value of the rme thatcorresponds to the different distributions ~as given inTable 1!. For the smaller particles the two methodsare equivalent, with rme of the order of a few tenths ofa percent. For bigger particles, the rme remains ofthe order of a few percent for our method, whereas itincreases up to values of almost 10% for the Chahinemethod. These behaviors are correlated with the dif-ferent accuracies by which the parameters of the out-put distributions are retrieved. With the exception ofthe 2-mm particle, our method is able to recover, for allthe distributions, the rout
nb and coutnb with accuracies of
better than 1% and 2%, respectively. The standarddeviation is also recovered fairly well, with only anapproximately 10% increase of the distribution width.On the other hand, the Chahine method works excel-lently for small particles ~1 and 2 mm!, with accuraciesof the order of tenths of a percent. However, as soonas larger particles are considered, the accuracy of allthe recovered parameters deteriorates consistently,
Table 1. Comparison between our Method and the Classical ChahineMethod for Six Input Gaussian Distributions with Average Radii rinp
aAll the distributions are characterized by the same relativestandard deviation sinp
nb yrinpnb 5 10% and same number concentra-
tion cinpnb 5 106 cm23. The test was performed with no noise added
to the input data.
7546 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997
reaching values of approximately 40% as in the case ofthe parameter cout
nb for the 50-mm particle.All the results shown in Figs. 5 and 6 refer to
number-recovered distributions, and the correspond-ing output parameters reported in Table 1 arenumber-averaged parameters. However, as hasbeen already recalled at the end of Section 2, theChahine method is more commonly used for retriev-ing weight distributions. In this case the methodbecomes more accurate because the spurious peakspresent in the number output distributions are lesspronounced or disappear. Correspondingly, theweight-average output parameters are recoveredmuch better, with accuracies of the same order ofmagnitude of those attained for the number-averageparameters, when our method is used.
So far we have considered ideal input data, i.e.,noiseless data. In order to investigate how themethod works with noisy data, a Gaussian input dis-tribution characterized by rinp
nb 5 10 mm, sinpnb 5 1 mm,
and cinpnb 5 106 cm23 was exploited and a 3% rms noise
was added to the input data. The results obtainedwith our method are shown in Fig. 7, in which theinput distribution ~solid curve! and the output distri-butions that correspond to the first 30 samples of noiseare shown ~dotted lines!. The output distributionsare always retrieved excellently, as is also confirmedby the number-average output parameters: ^rout
nb & 5~9.99 6 0.03! mm, ^sout
nb & 5 ~1.02 6 0.05! mm, and ^coutnb &
5 ~1.004 6 0.01! 3 106 cm23. When the same test~with the same 100 samples of 3% rms noise! is carriedout with the Chahine method, the recovered distribu-tions are much noisier, with spurious peaks appearingat small radii. This is shown in Fig. 8, in which theinput distribution ~solid curve! and the output distri-butions that correspond, for sake of clarity, to onlythe first three samples of noise ~dashed, dotted, anddashed–dotted lines! are reported. Figure 8 showsthe typical indented appearance of the number dis-
Fig. 7. Results of our inversion method when 3% rms noise wasadded to the input data, corresponding to a Gaussian input distri-bution ~solid curve! characterized by rinp 5 10 mm, sinpyrinp 5 10%,and cinp 5 106 cm23. The output distributions that correspond tothe first 30 different samples are shown as dotted lines.
tributions retrieved with the Chahine method andalready encountered in the noiseless case. Thecorresponding output parameters are in this casemuch more poorly recovered: ^rout
nb & 5 ~7.6 6 1! mm,^sout
nb & 5 ~3.2 6 0.6! mm, and ^coutnb & 5 ~1.7 6 0.4! 3 106
cm23. If the comparison between the two methodsis performed on the basis of the weight distribu-tions, the differences are not so striking, but arestill appreciable. This can be inferred from Fig. 8,in which the number distributions recovered withthe Chahine method, besides the spurious peaks,show a somewhat indented appearance, which alsocorrespond to the main peak of the input distribu-tion. When our method is used, the weight-average parameters are recovered excellently withaccuracies given by ;0.1% for ^rout
wt &, ;0.5% for^cout
wt &, and ;5% for ^soutwt &. When the Chahine
method is used, the parameters are recovered muchmore poorly, with accuracies given by ;2% for ^rout
wt &,;7% for ^cout
wt &, and ;130% for ^soutwt &. Therefore,
although our method is able to recover all threeparameters quite accurately, the Chahine method isfairly accurate for only the average radii, slightly over-estimates the sample concentration, and completelyfails in recovering the width of the distribution.
To investigate the overall effect of noise on the re-trieved parameters obtained with the two methods, weadded different levels of noise to the input data thatcorresponded to the same Gaussian distribution of Fig.8. The results for the number-average parametersobtained with our method are shown in Fig. 9, in whichthe ratios ^rout
nb &yrinpnb , ^sout
nb &ysinpnb , ^cout
nb &ycinpnb , and their
respective error bars are plotted as functions of thenoise level. Figure 9 shows that there are two effectsof increasing the noise level. First, the mismatch be-tween input and output parameters grows larger, im-plying that the accuracy of the method is deterioratingat high noise levels; the systematic trend to reduce theaverage radius and to increase both the standard de-
Fig. 8. Results of the Chahine inversion method in the case of thesame Gaussian input distribution ~solid curve! of Fig. 7. Theinversions were carried out on the same input data as those of Fig.7. For the sake of clarity, we show the output distributions foronly the first three samples.
viation and the concentration is due to the presence ofspurious peaks in the recovered distribution, whosedensity and height increase with the noise level. Sec-ond, the error bars associated with the output param-eters become larger, meaning that the method isbecoming more and more sensitive to noise, with sout
nb
being the most critical parameter. This implies that,depending on the particular sample of noise being con-sidered, the output distributions will be somewhat dif-ferent from the input ones and their number willincrease statistically with an increase in the noise level~for noise levels of 10% rms, this number is of the orderof 20%!. Figure 9 also shows that the above effects,i.e., the systematic and the statistical deviations be-tween input and output parameters, are of the sameorder of magnitude. Therefore they are both to betaken into account when the performances of themethod are to be ascertained on noisy data. In par-ticular, the parameters ^rout
nb & and ^coutnb & can be recov-
ered quite well and, even with 10% rms noise, theiraccuracy is of the order of 1% and 2%, respectively.Differently, at the same noise level of 10% rms, theparameter ^sout
nb & is recovered with an accuracy of lessthan 20% and with poor reliability, as indicated by thelarge error bars associated with this parameter. Thisis partially because the adopted stopping criterion isstill to be optimized. As described above, when theinversion is stopped, although it is at its beginning, theinstability has already started to develop, and this issufficient to alter significantly the value of ^sout
nb &, asthis parameter is the most sensitive to the presence ofspurious peaks at small particles. When the sameinput data are inverted with the Chahine method, therecovered distributions are so spiky and indented ~see,for example, Fig. 8! that all the parameters are recov-ered with huge errors, even at very low noise levels.As an example, a noise level of 5% rms is enough to
Fig. 9. Comparison between the output and the input number-average parameters, recovered with our method, as functions ofthe noise level. The test was performed with the same Gaussiandistribution as that of Fig. 7, and the results are expressed in termsof the ratios ^rout
nb &yrinpnb , ^sout
nb &ysinpnb , and ^cout
nb &ycinpnb . The error bars
show the spread of the results obtained over 100 different noisesamples. The lines through the symbols are guides to the eye.
20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7547
produce, on the recovered average radius and concen-tration, errors of the order of 30% and 100%, respec-tively.
A much more meaningful comparison between thetwo methods is performed when weight distributionsare considered. Figure 10 shows the ratio ^rout
wt &yrinpwt
and respective error bars as a function of the noiselevel for both our method ~open circles! and theChahine method ~filled circles!. Both methods re-cover ^rout
wt & with rather high accuracy, even at veryhigh levels of noise, but our method is definitely muchmore accurate with no systematic deviations andsomewhat smaller error bars. The different perfor-mances of the two methods with respect to noise be-come more pronounced in Fig. 11, in which the ratiosof ^sout
wt &ysinpwt are shown. In this case the Chahine
method is highly inaccurate even at very low noiselevels ~1%, 3%! or for noiseless data. Conversely,our method is rather stable and accurate, with devi-ations that become of the order of 25% for noise levelonly near 10% or higher. Finally, when the recov-ering of the sample concentration is considered ~Fig.12!, the two methods exhibit astonishingly differentbehaviors. Our method is able to recover the sampleweight concentration even at noise levels of 20% rms,with no systematic deviations and error bars of theorder of a few percent. The Chahine method, on theother hand, is extremely sensitive to the noise andsystematically introduces errors that scale almostlinearly with the noise, i.e., they are approximatelytwice the noise level. This result is paralleled by thesimilar behavior shown by the stopping value of the^rme& parameter as function of the rms noise ~see Fig.13!. Indeed, although for our method the ^rme& isremarkably equal, within the error bars, to the rmsnoise for the entire range of noises, for the Chahinemethod, the ^rme& is by far larger than the rms noise
Fig. 10. Comparison between the output and the input weight-average radius ^rout
wt & as a function of the noise level for our method~open circles! and the Chahine method ~filled circles!. The testwas performed on the same Gaussian distribution as that of Fig. 7.The error bars show the spread of the results obtained over 100different samples of noise. The lines through the symbols areguides to the eye.
7548 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997
level. Asymptotically, for large noise levels, it scaleslinearly with the noise, reaching values that are ap-proximately twice the noise level. One can explainthis apparent coincidence between the deviations of^cout
wt & and ^rme& by looking at Fig. 14 and recallinghow the Chahine methods works. Figure 14 shows,as functions of q, the input data that correspond to asample of 20% rms noise ~circles!, and the recon-structed data obtained with our method ~solid curve!and the Chahine method ~dotted curve!. The curvecorresponding to our method passes reasonably wellthrough the data, whereas for the Chahine method it
Fig. 11. Comparison between the output and the input weight-average standard deviation ^sout
wt & as a function of the noise level forour method ~open squares! and the Chahine method ~filledsquares!. The test was performed on the same Gaussian distri-bution as that of Fig. 7. The error bars show the spread of theresults obtained over 100 different samples of noise. The linesthrough the symbols are guides to the eye.
Fig. 12. Comparison between the output and the input weight-average concentration ^cout
wt & as a function of the noise level for ourmethod ~open diamonds! and the Chahine method ~filled dia-monds!. The test was performed on the same Gaussian distribu-tion as that of Fig. 7. The error bars show the spread of theresults obtained over 100 different samples of noise. The linesthrough the symbols are guides to the eye.
passes well above the input data. In the latter case,the input data are fitted correctly to only a few valuesof q, at which, actually, they are just barely largerthan the reconstructed ones. This is a typicalsteady-state situation for the data reconstructed withthe Chahine method and it is due to the fact that thealgorithm corrects the population of each class byusing only the information coming from one wavevector. Consequently the only stable condition ofconvergence is the one for which the ratio Imeas~q!yIcalc~q! is either ,1 or slightly .1 and approaches 1with an increase in the number of iterations. In thisway the population of the classes of particles forwhich Imeas~q!yIcalc~q! , 1 will be constantly lowered,whereas those for which Imeas~q!yIcalc~q! ; 1 will re-
Fig. 13. Behavior of the stopping value of ^rme& as a function of thenoise level for our method ~open circles! and the Chahine method~filled circles!. The test was performed on the same Gaussian dis-tribution as that of Fig. 7. The error bars show the spread of theresults obtained over 100 different samples of noise. The straightline through the open circles corresponds to ^rme& 5 rms.
Fig. 14. Comparison between the input data ~open circles! andthe output data reconstructed with our method ~solid curve! andthe Chahine method ~dotted curve! for one sample of 20% rmsnoise. The input distribution was the same input Gaussian dis-tribution as that of Fig. 7.
main constant. This is the mechanism that accountsfor the spikelike appearance of the distributions recov-ered with the Chahine method and also accounts forthe huge deviations of Figs. 12 and 13. Indeed, whenthere is some noise in the input data, the reconstructedcurve will pass through only the few points whoseintensity has been strongly affected and enhanced bythe effect of the noise. In the case of Fig. 14, the rmslevel of noise was 20%, corresponding to peak fluctua-tions of ;35%, which is almost the same value of thedeviations observed in Figs. 13 and 14.
As a final test, we investigated the ability of ourinversion algorithm to retrieve bi-Gaussian distribu-tions. The two distributions were characterized byrinp
nb-a 5 5 mm, sinpnb-a 5 0.5 mm, cinp
nb-a 5 0.5 3 106 cm23
and by rinpnb-b 5 10 mm, sinp
nb-b 5 1 mm, cinpnb-b 5 0.5 3 106
cm23. The test was done by the addition of 3% rmsnoise to the input data. The results are shown inFig. 15 in which the input distribution ~solid curve!and the output distributions that correspond to thefirst ten samples of noise ~dotted lines! are shown.The peaks corresponding to the two Gaussians areclearly resolved and are characterized by ^rout
nb-b& 5 ~0.53 60.09! 3 106 cm23. It can be noted that all the pa-rameters of the two distributions are recovered withsimilar accuracies, namely a few percent for the av-erage radius and concentration and 10%–20% for thestandard deviations. These accuracies are some-what worse than those obtained with a single Gauss-ian distribution at the same noise level, but are stillremarkably good.
4. Conclusions
In this paper we have applied an innovative algorithmto the inversion of low-angle elastic light scattering
Fig. 15. Results of our inversion method in the case of a bi-Gaussian input distribution ~solid curve!. The two Gaussians arecharacterized by rinp
nb-a 5 5 mm, sinpnb-a 5 0.5 mm, cinp
nb-a 5 0.5 3 106
cm23 and by rinpnb-b 5 10 mm, sinp
nb-b 5 1 mm, cinpnb-b 5 0.5 3 106 cm23.
A 3% rms noise was added to the input data. The output distri-butions for ten different samples of noise are shown as dotted lines.
20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7549
~LAELS! data. We devised the algorithm by modify-ing the nonlinear iterative method originally proposedby Chahine in 1968 and we tested it by computer sim-ulations. The simulations were carried out within awave-vector range of 2.5 3 102–2.5 3 104 cm21; theparticle-size distributions were recovered in the 0.70–77-mm range of radii. All the tests were performed bysimulation of the conditions of an experiment in whichthe light of a linearly polarized He–Ne laser is scat-tered by a diluted aqueous suspension of latex spheresand detected with a standard photodetector for LAELS.
A thorough comparison between our method and theoriginal Chahine method was carried out by investi-gation of the accuracy, reliability, and stability of thetwo methods as functions of the noise level present inthe data. Compared with the Chahine algorithm, ourmethod is much more stable and reliable, reconstructsthe input data much better, with rms deviations al-ways of the order of the noise level, and allows bothnumber and weight particle-size distribution to be re-trieved accurately. Its major drawback is the some-what reduced rate of convergence, which, on the otherhand, is partially compensated for by the ever-increasing velocity of modern computers ~1000 itera-tions take less than 20 s on a 150-MHz Pentium PC!.
Our method has one more relevant advantage overthe Chahine method. Indeed, thanks to its high sta-bility, its convergence is always smooth and whensome instabilities occur, they develop in the large ma-jority of the cases after the recovered distribution hasalready attained its final shape. Moreover, the onsetof such instabilities is strongly correlated to a changein the rate of convergence, i.e., when the rme param-eter or its derivative with respect to the iteration num-ber reaches a minimum. Consequently, the stoppingpoint is fairly clearly determined and is not a user-dependent parameter of the inversion algorithm.Conversely, for the Chahine method the convergenceis, at the same time, much faster and highly irregular.Indeed the convergence rate and the rate at whichinstabilities develop are fairly comparable, making thedetermination of the stopping point much more trou-blesome and somewhat arbitrary.
The results of the computer simulations show thatour method can accurately recover narrow and broadbell-shaped distributions over the entire range of re-coverable radii. Polydisperse distributions, such asbi-Gaussian distributions can be recovered as well.When noisy data are processed, the first three mo-ments of the particle distributions, i.e., concentra-tion, average radius, and standard deviation, can berecovered fairly well even at noise levels of severalpercent. For example, for a Gaussian distributionwith an average radius of 10 mm and a standarddeviation of 1 mm, the accuracies on the recoveredaverage radius and concentration are always within afew percent, even at noise levels of 10%–20% rms.The overall shape of the input distribution is alsoreconstructed fairly well, without any indication ofthe noisy and indented appearance typical of the dis-tributions recovered with the Chahine method. The
7550 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997
distributions retrieved with our method are alwayssmooth and their tendency to become unstable, withthe appearance of a spurious peak at the small par-ticle side of the range, is efficiently discriminatedagainst by the adopted stopping criterion. More-over, when weight distributions are considered, thisresidual deviation disappears completely.
As a final comment, we point out that our method,differently from the Chahine method, does not requireany strict relation between the wave-vector range ofthe measure and the range of recoverable radii. Inprinciple, for a given q range, one should be able torecover the particle-size distribution over a range ofradii much larger than the one used in this work.Our preliminary results on this topic show that therange of recoverable radii can be substantially ex-tended without affecting the performances of the algo-rithm too much. These findings will be the subject ofa different paper.
We thank U. Perini, S. Musazzi, and P. Martinelli,Centro Informazioni Studi Esperienze, Milan, forhelpful discussions.
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