Characteristic curves for photographic emulsions fromnonlinear fitting: a study of statistical andmodel error
Joel Tellinghuisen
The intensity calibration of photographic emulsions is examined from the standpoint of nonlinear least-squares fits to analytic expressions. A simple but widely neglected expression for computing the propagatedstatistical error is used to examine the error in the relative exposure as determined from the fitted calibrationfunction. The statistical error is also compared with the dependence on the choice of fit function, or model er-ror. In the present applications, which replicate those obtainable routinely in emission spectroscopy, therelative intensity is obtainable to better than 10% over a 2-order-of-magnitude change in intensity. Keywords: Photometry, photographic emulsion, characteristic curve, nonlinear least-squares, error propagation.
1. Introduction
The photographic emulsion is the venerable medi-um for recording image information from radiationsources. Although this medium is now being sup-planted by electronic array technology, there are stillapplications where photographic recording remainsthe method of choice.' These include cases which takeadvantage of the large dynamic range and noise-inte-grating properties of film, such as emission spectrosco-py of electronically noisy discharges. In addition tothese continuing uses of photographic recording,thereexists a considerable body of archival astronomical andspectroscopic data on photographic plates and films.
The biggest drawback to quantitative intensity workwith photographic emulsions is the inherently verynonlinear interdependence between the exposure andthe resultant darkening of the developed film or plate.A graphic display of photographic optical density (de-fined as the base-10 logarithm of the reciprocal trans-mittance of the developed emulsion) vs the logarithmof exposure (irradiance X time) constitutes the charac-teristic curve, 2 also called the Hurter-Driffield curve
The author is with Vanderbilt University, Chemistry Depart-ment, Nashville, Tennessee 37235.
after its originators.3 The characteristic curve typi-cally displays a sigmoid shape, the details of whichdepend on the specifics of the emulsion and its devel-opment as well as on the wavelength and intensity(reciprocity effects) involved in its exposure.
The problem of obtaining such characteristic curvesand of extracting reliable intensity information fromthem has preoccupied workers for a century. Theadvent of the computer has enabled this problem to beattacked with moderate sophistication, inspiring anumber of contributions which have described variouscomputational procedures for accomplishing the cali-bration.1' 4-' 2 Some of these approaches have empha-sized functional transformations designed to renderthe dependence of blackening on exposure linear overat least some range of exposure, thus facilitating least-squares fits to a straight line.569 Another popularmethod is the use of piecewise-fitted polynomials orcubic splines, sometimes in combination with suchvariable transformations 8 '1, 2 These methods haveall been demonstrated to work; however, informationabout the precision with which intensities can be ob-tained has generally been only qualitative.
In this work I have investigated the use of nonlinearfitting to analytic representations of characteristiccurves, an approach pursued earlier by Green andMcPeters.10 Modern least-squares methods 3"14 makethe fitting of data to complex functions little morecomplicated than the linear fitting preferred in most ofthe work in Refs. 4-12. In what I belive is the first suchapplication to this problem of photographic intensitycalibration, I have also specifically investigated thepropagation of error into the determined intensities.
The latter computation is almost trivial but is none-theless not widely recognized by practitioners of theart of least-squares fitting. Results of the presentstudy show that even routine treatment of photo-graphic spectra can yield relative intensities that arereliable to better than 10% over a range of 100 inexposure.
II. Experiment
The calibration spectra were photographed on Ko-dak 103a-O plates in conjunction with work on theemission spectrum of BrCl.15 The source was aquartz-halogen lamp. The filament image was inten-tionally defocused on the slit of the spectrometer (stig-matic, JY model HR1500, 1.5-m focal length with a3600-groove/mm holographic grating). Spectra werephotographed near 328 nm using 5-,um slits and expo-sures of 0.25, 0.5, 1, 2, and 4 min at constant intensity.The five images were each -2 mm wide and werepositioned close together on the plate, spanning a totalvertical distance of 17 mm. Development time was 8min in Kodak D-19 at a temperature of 210C.
The use of varying exposure times to determinevarying intensities at constant exposure time neglectsreciprocity failure. The extent of reciprocity failure isa function of both time and total exposure; for thisemulsion and the factor-of-16 range of exposure times,the maximal effective errors in the apparent exposureamount to _10%.2 However, because of the manner inwhich optical densities from different exposures over-lap, this systematic error becomes largely randomizedin the present data treatment methods. Still, in preci-sion work constant exposure times are preferred.
Another potential problem in the present experi-ments is the unknown effect of nonuniformity in theemulsion and its development,' since the several im-ages are physically separated on the plate. However,such nonuniformity problems are inherent in the pho-tographed band spectra. Their magnitude is crudelytested through multiple sampling and alternative fit-tings of the calibration spectra, as described below.
Optical densities were recorded and logged on flop-py diskettes using a Photometric Data Systems model1050 microdensitometer, which was interfaced to aTRS80 model 4 microcomputer by Salter.16 The den-sities were read every 2 Am, using the same narrow slit(6 X 600,Lm) used to photometer the line spectra buttraversing across the bands in a direction perpendicu-lar to that used to measure the line spectra. Theresults (Fig. 1) show that the grain structure in thisrelatively grainy emulsion contributes considerablenoise to the recorded images at this high resolution.Since all the fine structure in Fig. 1 is just noise, thelogged data were binned in groups of twenty pointseach and averaged for the analysis described below.Note that the defocused filament image on the slit haseffectively produced five continuous tone images,which offers advantages over the more conventionalprocedure for obtaining characteristic curves fromdata recorded with a discretely stepped neutral densi-ty filter (see below).'
E=4.0
2 2.0 w\2
0.25
0
0 1 2
x (mm)
Fig. 1. Microdensitometer tracings of tungsten filament imagesrecorded spectrographically near 328 nm using the indicated fiveexposures (in minutes) at constant intensity. The five images havebeen displaced along the ordinate to avoid confusing overlap. Verti-cal dotted lines indicate regions where data were taken for the
Method 1 analysis described in the text.
Ill. Least-Squares Fits
A. Functional Forms
We seek to fit the density vs exposure data to suit-able closed-form relations. Since there is no simpletheoretical expression for this dependence, the fittingbecomes an exercise in model testing. The empiricallyderived transformation relations that have been usedin the past to achieve a linear dependence betweenplate darkening and exposure are effective only inlimited regions of the characteristic curve, for exam-ple, in the toe-to-linear region69 or the shoulder-to-saturation regime.7 To span the region from toe toshoulder requires a sigmoid type function, one of thesimplest of which is the hyperbolic tangent function.In the present work I have focused my attention on thisfunction and simple variations thereon and on theeffective opacitance defined by Green and McPeters.10
The function y = ex/(ex + 1) has the desired sigmoidshape and ranges from 0 to 1 with inflection at x = 0.For application to typical characteristic curves, thequantity x must be defined as
x = a(lnE - nE0) ln(Era), (1)
where Er is the relative exposure, referenced to theinflection point, and a appropriately scales the x-axis.Thus ex in the defined quantity y is just Era. Adding abackground b and an overall density scaling factor c,the optical density D can be expressed as
D = b + cy. (2)
Equation (2) forces the toe and shoulder regions ofthe characteristic curve to behave symmetrically.This symmetry can be broken through incorporationof other parameters, for example,
Another possibility might be a polynomial in y. I haveexamined all of these forms in the present work.
Equation (2) may be solved for the relative exposurethrough
Era = dI(c-d),
2
(4)
where d D - b. Equation (4) emphasizes the connec-tion with the effective opacitance defined by Greenand McPeters,'0 which may be written as
El/n = ed 11 - ea(D-c)
M
i
(5)
In Eq. (5) the additional parameter a serves the sametoe/shoulder symmetry breaking function as does theexponent p in Eq. (3). [Equations (4) and (5) becomeequivalent if one represents the exponentials in Eq. (5)by their linear expansions, sets a = 1, and identifies ain Eq. (4) as the reciprocal of n in Eq. (5).]
To solve for Er from Eq. (3) with p #d 1, I have usedNewton's method.
0-i 0 1
Log (E)
Fig. 2. Optical density as a function of the logarithmic exposure, asrecorded at four of the six positions indicated in Fig. 1 (omitting the
fourth and sixth from the left to avoid confusion).
B. Computational Procedures
The fitting of data to such expressions as Eqs. (2)-(5), which are nonlinear in the adjustable parameters,is by now only slightly less routine than the linearfitting methods which have been emphasized in pastwork on photographic characteristic curves. Themethodology is well documented in the litera-ture,13 14"17 and programs are readily available.18 Theone additional complication is that convergence on anabsolute minimum in x2 is never assured in nonlinearfitting, whereas such convergence failure occurs in lin-ear fitting only from redundancy-caused roundoffproblems. Thus, to facilitate the search for minimalx2, it is extremely useful to be able to freeze selectedparameters and vary them manually in trial and errorfashion.18
In the perfectly general case of nonlinear leastsquares, there is no need to distinguish between inde-pendent (hence error free) and dependent (uncertain)variables. In the parlance of Deming,17 the fittingprocess is an adjustment of the data to the (hopefully)true curve, the minimized quantity being the sum ofweighted squared residuals (i.e., the adjustments) ineach of the uncertain variables. To ensure that theadjustable parameters are independent of the chosenrepresentation of the fitted function, the weights areevaluated on the adjusted curve. This is the formal-ism developed by Lybanon13 and employed in thepresent calculations. For example, it means that fitsto
F(a,b,c,EO;E,D) D - b - cy (6)
from Eq. (2) and to
G - a lnEr - ln[d/(c - d)] (7)
from Eq. (4) will yield identical values of the parame-ters a, b, c, and E0 and their variance-convariance
matrix. (However, the convergence properties of thetwo representations will not be identical.)
Considerably less well known to data analysts is avery simple relation for computing the propagatederror in a function of the least-squares parameters. Iff(fl) represents some linear function of the parametersA3, the estimated variance in f is' 7 "19
gg, (8)
in which the ith element of the vector g is the quantityaf/Oa3i and V is the variance-covariance matrix fromthe fit. (Note that neglect of the off-diagonal ele-ments in V yields the usual textbook expression for theerror propagation of uncorrelated variables.14) Thissame expression holds also for nonlinear functions of fi,provided the errors in the parameters are suitablysmall. Experience with other cases in which the ad-justable parameters are determined with precisioncomparable to that obtained here suggests that thiscondition is satisfied in the present work; I will assumeso and use Eq. (8) to calculate the error in Er as afunction of D. The required partial derivatives in gare obtained numerically by a finite difference calcula-tion (as are also the partial derivatives required in thefitting itself). This approach has proved fully reliablein past applications.20'2'
C. Data Treatment
If we choose several x values in the images of Fig. 1and plot D vs log(E) for each, we obtain curves likethose shown in Fig. 2. Each of these contains informa-tion about the adjustable parameters in the expressionfor the characteristic curve but requiring differentshifts along the abscissa (i.e., different Eo). Thus, forexample, a fit of six such sets to Eq. (2) requires a totalof nine adjustable parameters, six being the differentE0 values.
What variables are to be considered uncertain here?
Fig. 3. Optical density in the top image in Fig. 1 plotted vs thedensity at the same x in the bottom image using data binned in
groups of twenty points.
In truth both D and E are uncertain, but with care theuncertainty in E can be reduced to a negligible level.Also the uncertainty in E is such that any residual in aparticular E should be the same in all the curves in Fig.2. With this in mind one might incorporate additionalparameters in the form of adjustments to the five Evalues. Since the uncertainty in E is approximatelyconstant in the present work, its effect is more impor-tant for small E. (Note that adjustments to E affectsingle points on each curve in Fig. 2, so they do notlump with the adjustable E0 values.)
This approach, namely, of translating the severalcurve segments along the abscissa until they coincide,has been referred to as the "process on the Fourierdomain" by Collados and Bonet.12 It has the advan-tage of yielding directly the relative exposures of se-lected regions of the image (through the several E0values) and, via Eq. (8), their errors. This might be themethod of choice in, for example, the calibration of linespectra from multiple exposures. For future referenceI call this Method 1.
An alternative approach, called the half-filter ormultiple filter method' and referred to as "process onthe measuring domain" in Ref. 12, involves sets of Dvalues from two (or more') images related by knownexposure differences. As usually implement-ed,14 8"1 '1 2 pairs of such D values from correspondingimage points are used to obtain an empirical function,D2 = f(D,), from which the characteristic curve is sub-sequently constructed. For illustration, Fig. 3 dis-plays the optical density of the darkest image in Fig. 1as a function of that at corresponding points in theweakest image. All such points are relatable throughtheir known exposure difference, in this case a factor of16. If the characteristic curve is assumed to be one ofthe functions given above, one can obtain it directlyfrom such pairs of points. For example, if the data in
2
ii -1 0 1
Log (E)
Fig. 4. Characteristic curve obtained from the fit of the data dis-played in Fig. 2 and Table I to the hyperbolic tangent type represen-
tation of Eq. (2) (Method 1 analysis).
Fig. 3 are assumed to follow Eq. (2), they can be ex-pressed as
16a d2 (C-dl)d(c-(d2)
with di = Di- b as defined after Eq. (4). A similarrelationship can be obtained assuming Eq. (5) for thecharacteristic curve. Both of these have been tested incomputations described below. Equation (9) can alsobe expanded to include all five images, but this is notdone here.
The fitting to Eq. (9) and its counterpart from Eq.(5), which I call Method 2, assumes perfect knowledgeabout the relative exposure, which can be experimen-tally reasonable, as noted above. Also, it is clear thatboth variables (D1 and D2) are now inherently uncer-tain, which means that a proper treatment should al-low for adjustment of both. This point has been ne-glected in previous work with this method, presumablybecause of the perceived complexity of nonlinear fit-ting. (Least-squares fits in which more than one vari-able are treated as uncertain are inherently nonlinear,even for linear fitted functions.)
In principle the noise or granularity in densitome-tered photographic images may depend on the density,which means that the data included in the fits shouldbe weighted accordingly. The expected behavior22 isan increase roughly as D'13 to Dl/2. Such a dependenceis not obvious in the images in Fig. 1. Moreover,statistical analyses of binned data in regions where D ischanging only slowly with image position yield no sta-tistically significant change in noise from the lowest tothe highest density regions. Consequently equalweighting of all measured D values is justified in thepresent least-squares fits. It is worth noting further inthis regard that the reduced x2 values from the various
Fig. 5. Error bands (1-ar) on derived exposure, plotted as percentdeviation, as a function of optical density. The long dashed bandsrepresent the error in Er, the narrow solid bands that in Er/E,(D =1.35). The short dashed curve shows the deviation obtained from asecond data set treated identically. Vertical dotted lines demarcate
the range sampled by the data from Fig. 1.
Table I. Binned (20) and Averaged Optical Densities (XIOOO) from Fig. 1and Residualsa from Least-Squares Fit to Eq. (2) by Method lb
Background 0.25 min 0.5 min 1.0 min 2.0 min 4.0 min
a Second entry in each double column, observed-calculated.b Least-squares parameters and errors (1-a in parentheses): a =
1.082(35), b = 0.050(11), c = 2.442(68), E values-1.401(84),3.94(24), 3.12(18), 1.247(75), 1.030(63), 1.303(78). Estimated stan-dard deviation in D = 0.028.
fits discussed below are consistently too large by anorder of magnitude. However, this is not a deficiencyof the fit models, because a similar figure is obtainedfrom statistics of the binned data in the backgroundregions between exposed images. The implication isthat the statistical noise observed on the fine scale ofthe binned data (twenty points spanning 40 Atm ofimage) is an overly optimistic indicator of the noise onthe coarse scale. In the error propagation calculationsdescribed below, the variance-covariance matrix V isobtained using the more pessimistic external esti-mates17 of 0.2, which should properly reflect the coarse-scale noise.
In all fits I have included a few of the above-men-tioned background points. These are fitted just to therelation b = D and thereby determine b much betterthan it would be determined from the image dataalone. (Note that all measured data contribute to thedetermination of b, but these background points deter-mine only b.)
10
C:04 J
D
0
1 2
Fig. 6. Percent deviation in derived values of Er/E(D = 1.35)relative to results of Fig. 4 and Table I. All results involve the dataof Table I. Solid curve-Method 1, Eq. (5); long dashed curve-Method 2, Eq. (2); short dashed curve-Method 2, Eq. (5). Vertical
dotted lines delimit the experimental sampling region.
IV. Results and Discussion
Figure 4 shows the characteristic curve determinedby fitting the appropriately binned and averaged datafrom the six indicated positions in Fig. 1 (plus sixbackground points not shown) to Eq. (2) with only Dtaken as uncertain (Method 1). The relevant data andfit results are also summarized in Table I.
Figure 5 displays the 1-. error bands on the expo-sure, as determined from Eqs. (4) and (8). Here therelative error is computed two ways which effectivelybracket the quantities generally desired: The broaderband is the relative error in Er, while the narrower isthat in ErIEr(D = 1.35), as might be appropriate fordetermination of intensities relative to a feature ofmedium density on the image. The computed error inthe latter goes to zero atD = 1.35, as it should; however,this result is optimistic, since it ignores the error inher-ent in measuring D. On the other hand, the error in Eralone tends to be pessimistic, since it effectively in-cludes uncertainty in the inflection point, which can-cels in a determination of relative intensities. In thisregard, the six Eo values, which are highly correlated,nonetheless yield mutual relative intensities with er-rors of -3%. Over the two units of D covered experi-mentally, which coincidentally translates into a rangeof 100 in exposure, Er is almost everywhere determinedto better than 10% by just the 36 D values used in thisfit.
Figure 5 also includes results from a second data setobtained in a separate densitometer scan of the fiveimages, illustrating consistency within the computederrors. On the other hand, Fig. 6 shows that the modelerror can exceed the statistical error. The three curvesdepicted here represent deviations from the results ofTable I and Fig. 4 when the same data set is fitted toEq. (5) by Method 1 and to both Eqs. (2) and (5) by
Method 2. In fact, the error bands obtained via Meth-od 2 (using about sixty points) are a factor of -2 moreoptimistic than those obtained from Method 1, whichis mainly a consequence of the much greater number ofdegrees of freedom in the Method 2 fits. Of course,increasing the number of points included in the Meth-od 1 fits would further narrow the computed errorbands. However, Fig. 6 shows that any further suchstatistical gain is illusory for this particular data set,since the reliability of the determination is alreadylimited by the model error. Thus in this case there islittle point in including more than thirty to fortypoints, except to extend the range of sampled D ifpossible.
The Method 1 fit to the effective opacitance func-tion of Eq. (5), with its one additional parameter, yield-ed a reduction by 18% in the summed squared residualscompared with the fit of the same data to Eq. (2).However, the present data do not sample the shoulderregion very well, which introduces convergence prob-lems. Convergence was achieved by first locating theshallow minimum in x2 by externally varying a andthen fixing any one of the six very highly correlated E0values. The extra parameter p in Eq. (3) led to an evengreater reduction in x2 (20%) and no convergence prob-lems; however, most of the resultant parameters, in-cluding the six E0 values and p itself, were statisticallyindeterminate at the 1-r level, raising questions aboutthe propriety of including the extra parameter. (Nev-ertheless, the relative Eo values remained well definedto -3-4%.) Inclusion of "corrections" to E for the 15-and 30-s exposures in the fits to Eq. (2) produced noconvergence problems but also no real statistical im-provement. A quadratic polynomial in y proved simi-larly ineffective. Comparable results were obtainedfrom fits of the second data set in all cases where it wastried.
V. Conclusion
The intensity calibration of photographic emulsionshas been examined through nonlinear least-squaresfits of optical density vs exposure data to analyticexpressions for the characteristic curve. An examina-tion of the propagated statistical and model error inthe derived relative exposure as a function of densityshows that exposures and hence intensities can bedetermined to better than 10% over a 2-order-of-mag-nitude variation. The calibration involves only thirtyto forty density values, at which point the largest un-certainty is the model error. Since the experimentaleffort used to obtain the sample data was far fromheroic, it is likely that the precision of the determina-tion can be improved by a factor of 2-3 and the range ofexposures by perhaps another factor of 5. Better andmore extensive data might also justify the incorpora-tion of additional adjustable parameters in more com-plex fit functions. However, the present reliability isalready adequate for many applications and betterthan generally recognized by spectroscopists.
The kinds of computational routines used here toperform the nonlinear fits are now widely available and
computationally efficient. Convergence (or the lackthereof) was always clearly evident within 10 cycles,which typically required only a few seconds of CPUtime on a VAX 8800 mainframe computer. On theother hand, Eq. (8), which is the key to the estimationof propagated error for correlated quantities (in thepresent case the least-squares parameters), is grosslyunderutilized by data analysts. In view of its almosttrivial implementation, it deserves more attention.
I thank a referee for calling to my attention Ref. 22and the matter of the dependence of granularity ondensity. This work was supported by the Air ForceOffice of Scientific Research, AFSC, under contractAFOSR-90-0030.
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