Linearization of Characteristic Curves in Photographic Photometry

Gerard de Vaucouleurs

After a survey of various definitions of the photometric characteristic curve of photographic emulsions,this paper reintroduces a long forgotten photometric parameter, the opacitance w = opacity - 1 = OD - 1and describes its application to a characteristic function I = AWn(n 4 2 ), which is linear in log-log co-ordinates over the whole range of densities of practical interest in photographic photometry. In a specialcase n = 1 and J = I/A = .

Introduction

It is frequently asserted that the response to lightexposure of ordinary photographic emulsions is non-linear and that linearity is one of the important ad-vantages of electronic and image conversion techniques,especially for applications to astronomical photometry.The second half of this statement is justified if it refersto the well documented linear relationship betweenexposure and developed density of the electronographicplate.",2 The first half of the statement, however, isnot warranted, tradition notwithstanding.

Apart from the nonadditivity property known asreciprocity failure between the action of differentexposure times, the photographic effect of differentflux densities in the same exposure time is nonlinearonly if one insists on measuring it by the conven-tional optical density of the developed image. There isno good reason, other than historical, for optical densityto be used in photographic photometry to the exclusionof other useful-and often preferable-functions of thetransmission of a negative.

The H and D Curve

The relation between the blackening of the developedemulsion and incident energy density or intensity isexpressed by the so-called characteristic curve of theemulsion for the experimental conditions at hand. Forhistorical and technical reasons this relation has becomealmost synonymous with the Hurter and Driffield, orH and D, curve relating the optical density of the photo-graphic silver deposit to the decimal logarithm of therumination or exposure It (or intensity I, if t = con-stant). The conceptual justification for the use ofoptical density was, according to Hurter and Driffield,3

the assumed proportionality between density and"amount of silver deposited per unit area." However,

The author is with the Astronomy Department, University ofTexas, Austin, Texas 78712.

Received 12 January 1968.

astronomers and physicists who use the photographicemulsion as a photometric device are not concernedwith weighing reduced silver, but rather with measuringincident flux density ratios in the most direct, precise,and convenient manner, especially at low flux levels anddensities where the H and D curve is particularly ill-suited. The well known shape of this curve (Fig. 1) ismainly responsible for the concept that the photographicresponse is inherently nonlinear. Apart from adjacencyeffects and other small scale distortions that can beminimized or avoided by suitable choice of exposure anddevelopment conditions (obviously a must for correctphotographic photometry), the nonlinearity associatedwith the shape of the characteristic curve can be over-come by redefining the photographic response. Thenonlinearity of the H and D curve arises because of thespecial choice of variables and, as we shall see, it isquite easy to produce other nonlinear or linear charac-teristic curves over various intervals of plate darkening;the one to choose depends on the problem at hand.

Advantages of a Linear Characteristic CurveIn particular, a long forgotten parameter first in-

troduced by Baker 4 in 1925 gives a linear relation overthe whole range of medium and low densities (down tothe detection threshold) of practical interest in astro-nomical photometry.

Because of its simplicity and convenience and thefact that it seems to have escaped the attention of mostpotential users (including this writer who rediscoveredit independently in connection with a program of sur-face photometry of galaxies), a brief review of theproblem of linearization of the photographic character-istic curve and examples of the favored solution arepresented in the following sections.

There are several advantages in the use of a linearor quasilinear relation between flux density and a mea-sure of photographic darkening: (1) simplicity of themathematical representation and convenience in thenumerical calculations, in particular conservation of theproperty of additivity, (2) avoidance of the systematic

August 1968 / Vol. 7, No. 8 / APPLIED OPTICS 1513

D

Fig. 1. Conventional H and D ciin a narrow range x at medium

0 0s *

Fig. 2. Dependence of photograpangle Q and (b) divergence angle

grained emulsion, after

errors that often arise near tl

the optical density is

D = logO = cologt = logo/0' (3)

---- '/B (the natural logarithms and corresponding concepts ofoptical thickness or depth are not used in photographicpractice). Because a developed photographic image is astrongly scattering medium, the transmitted flux qk'is made up of two parts: (1) direct transmission, (2)diffuse transmission. The former depends only on Q,the latter depends on both Q and '. Thus, t, 0, and D

, are functions of both Q and ' (and to a lesser degree ofB 2 log I wavelength because the majority of developed silver

-irve D(log I) is roughly linear grains are larger than the wavelength of light in mosta densities near D 1. photographic emulsions of astronomical interest).

Thus, different types of photographic density may be de-fined (in parallel light, in diffuse light, in convergentlight, etc.) as explained in most standard photographictexts.'' In what follows we assume that all transmissionmeasurements are made on a given instrument with fixed

1.5 - optics providing constant values of Q and '. Forsimplicity also, we may assume that the measuring de-

, g , , | vice gives a response or deflection that is proportional toI I 2 s a the flux received and, therefore, to the transmission

'hic density on (a) convergence factor t as in most direct-reading photoelectric plateQ' (in steradians) for a coarse- photometers. If a double beam, self-balancing systemG. A. Boutry (1935). is used, the reading is generally the abscissa of an

optical wedge and it is presumed proportional to thedensity; if so, it is a simple matter to return to the

ie threshold because of the corresponding linear deflection.asymptotic approach of the H and D curve of the logIaxis, (3) possibility of taking unbiased averages ofmeasured intensities down to the noise level, whereassome arbitrary smoothing and interpolation is neededwith the conventional characteristic because of the im-possibility of computing logI at points where fluctua-tions associated with granulation cause the density to beless than the average fog level.

DefinitionsTo present the argument clearly some fundamental

definitions must be recalled. Consider an area A of adeveloped photographic plate or film P receiving insome measuring photometer or densitometer a fluxq. through a solid angle Q (Fig. 2).

Let 4' be the flux transmitted by the plate and col-lected in the solid angle Q' by the receiving system.Frequently Q is defined by the objective 0 forming onP an image of the scanning spot S and ' by the objec-tive O' collecting the transmitted flux; but more gen-erally a material diaphragm in contact with the emul-sion may be used to define A and Q, Q' may have anytwo independent values from 0 to 2 7r.

By definition, the transmission factor of the area A ofthe plate is

t = ' (1)

the opacity is

0 = 1/ =/+, (2)

Various Types of Characteristic Curves

Consider now an astronomical photograph that hasreceived a calibration exposure, for example, in a tubesensitometer or similar device, on a part of the plate notpreviously exposed to the sky. The incorrect practiceof printing the calibration spots on top of the partalready exposed to the sky light, or to pre-expose thecalibrating area to artificial light "to simulate the skyfog" has been generally abandoned since 1945 when itbecame clear that the reasoning behind the procedureswas fallacious.7 Simultaneous exposure to sky andcalibration spots would be correct, but is not yet incommon practice. No significant error results fromcalibration exposure before or after the sky exposureas long as the age difference AT between the twoexposures is negligible with respect to the time delayT between the last exposure and development. Inpractice, AT/T 0.1 is sufficient for most emulsionswith normal latent image evolution properties.6

Let A0 be the deflection in an unexposed part of theplate corresponding to the chemical fog level, and a thedeflections in the calibration spots exposed to knownrelative flux densities I. A characteristic curve is arelation between some suitable function of t = fo(b) =

8/Ao and some function of I.

y(I) = f(S/AO). (4)

1514 APPLIED OPTICS / Vol. 7, No. 8 / August 1968

,.

The conventional H and D curve uses the relation

yI(I) = h1(D) (5)

astronomical surface photometry, 0 the photometricfunction

between the variables,

yI(I) = logI, (6)

.A(3) = logAio/ = D, (7)

where D is the density above fog. It is also possible tomeasure densities above (or below) the sky backgroundlevel A if part of the plate carries a direct photo-graph of the sy, then

fs(5) = logA,/3 = d, (8)where d = D - D.

The form of the characteristic curve is not changed,but the zero point is merely shifted along the densityaxis. This form has some advantages when the skydensity level D8 = logAo/A3 is in the range (1 < D < 2)of the so-called linear part (inflection tangent) of the Hand D curve (Fig. 1) and, in particular, for the surfacephotometry of galaxies.8

Characteristic Curves for Low DensitiesIt often happens, however, that the astronomical ex-

posure is in the low density range corresponding to thetoe of the H and D curve (D < 1) where the gradient- = D/b (logI) is low and tends to zero with D (andlogI -- - co). In such a case there is no particular ad-vantage in using the conventional form of the char-acteristic curve. Better forms have been proposed andeffectively used that have a linear part in this densityrange; for example6

Y2 = logI = h(t) = 2[f0(6)l. (9)

This curve (Fig. 3) has an inflexion tangent near t 0.5 (D = 0.3) and it is often very nearly symmetricabout its inflection point. The quasilinear range is 0.3< f < 0.7 (corresponding to 0.15 < D < 0.65). Atstill lower densities and down to the threshold there isempirical evidence with some theoretical justificationsfor a quadratic relation of the form

fi&() = D = a2 + .. ., (10)where the higher order terms are egligible for D <0.15. In this range a characteristic curve in the co-ordinates (D, 12) will be linear and pass through theorigin. The representation is useful for very low densi-ties (generally too low for precision photometry) (Fig. 4).

Definition of Opacitance and Linearization ofthe Photographic Response

In all the above cases a quasi-linear relationship isachieved over a small range between t or D and logIor 12. Other functions of I might be found with similarresults, but this is not really what we mean when wedescribe the response of a receiver as linear. Here werequire a linear relation (over a large range) betweenthe response and I itself, not some function of I.During the development of a computer program de-signed to apply numerical methods of mapping to

f3w = [(A/6) - I] = (11)

was introduced by the writer to help represent thecharacteristic curve in the region of low densities(D < 1) as a series expansion of the form

mI = s_ Ajil2.

j=i (12)

Experience showed the first term to be dominant.Later developments led to the replacement of the seriesexpansion by a single term of the form

I = Ac-, = Af4(B), (13)

where A is a proportionality factor and n is an exponentoften in the range 0.4 < n < 0.7 for fast astronomicaland spectroscopic emulsions measured in convergentlight.

The photometric quantity,

W = (AO/b) - 1 = 0 - 1, (14)

which the writer proposes to call the opacitance be-cause of its relation to opacity, is actually an old,though forgotten concept. It was first introducedover forty years ago in the form

DI = loglo(10D - 1) = D + logio(1 - 10 D) (15)

by Baker4 and applied by Sampson" to the photo-graphic spectrophotometry of bright stars. It is un-fortunate that the valuable properties of Baker'sdensity, as DI has sometimes been referred to, have notbeen generally recognized and adopted by astronomers.This writer, for one, after thirty years as a practitionerof photographic photometry, was not aware of Baker's

9IT- I I 'o I

Fig. 3. Characteristic curve t(log I) is roughly linear in a smallrange of low densities near D 0.3.

Fig. 4. Characteristic curve D(12 ) is roughly linear in a shortrange of very low densities D < 0.2.

August 1968 / Vol. 7, No. 8 / APPLIED OPTICS 1515

a.,

w

.1 I 3 10 I 30

Fig. 5. Characteristic curves w(I) are linear in log-log coordin-ates over a large range of densities D < 2. The relation I = A-is shown for various common values of n = 0.4 to 0.7. The rela-tion between opacitance co and density D is illustrated by the two

scales of ordinates.

work until an alert referee called his attention to thenearly forgotten papers and to a brief reference tothem in Mees' book.' By analogy to the relation be-tween density and opacity, DI = logc might be calledthe densitance.

The relation between opacitance , density D, andintensity I is shown in Fig. 5 for common values of n.Examples of applications to some emulsion types com-monly used at observatories are shown in Fig. 6 inlog-log coordinates in which a least-squares solutionfor the constants A and n of Eq. (13) is simplest.

Note also that since Eq. (13) has just two constants,a minimum of only two calibration spots (say, nearD 0, 3 and D 1) is, in principle, sufficient to de-termine the characteristic; this may be an advantagewhere space is limited and, if not, it offers the possi-bility of reducing calibration errors due to the photo-metric local errors'2 by placing several pairs of spots indifferent parts of the plate.

In linear coordinates the relation must pass throughthe origin since when I = 0 it is obvious that = A0and co = 0 (no exposure, no photochemical response).This is a very important advantage for it gives an exactpoint in the region of the lowest densities where mea-surements are increasingly inaccurate because of granu-lation noise and photometric local errors. 2 This ad-vantage is lost in the characteristic curves with log asvariable for which it is not uncommon to find that anempirical fit of the calibration points (either by freehand interpolation or by a least-squares fit of a poly-nomial expansion) converges to a small, but finitedensity when I 0.

A linear characteristic curve brings a second im-portant benefit in helping to reduce the effects of localerrors. Calibration spots are often arranged in two ormore parallel rows, the first with densities increasing,say, from left to right, the second from right to left,etc. in order to avoid cumulative effects of local errorson the characteristic curve. The differential localerrors along each row lead to systematic departuresfrom the linear characteristic that are immediately inevidence [Fig. 7(a)]; on the contrary when the H andD curve is used, such departures are often hidden bythe free hand interpolation or polynomial curve fitting,since they merely change the curvature of the char-acteristic [Fig. 7(b)].

ApplicationsThe linearity of the relation between and Co"

through the origin has another important advantagewith respect to the problem of smoothing the granula-tion noise near the detection threshold. With theusual logarithmic characteristic curve, it is necessaryto smooth the data points, often by free hand interpola-tion, before I can be computed when 8 -- A0, since Iis not defined when D = logAo/5 < 0. This happenswhenever > A because of statistical fluctuations(granulation). With the linear characteristic, andprovided I is assigned the sign of , i.e., by writingI = A (w/lw cJ.J , negative values of I cause no problemand unbiased numerical smoothing procedures can beapplied to the measurements. This is especially help-ful when the data merge very slowly into the back-ground field, e.g., in the faint outer fringes of a galaxy.For example, closed isophotes of a spiral galaxy could betraced by numerical mapping down to a level of 1% ofthe superimposed night sky light.' 0

This representation is also helpful for the photometryof in-focus star images measured with a fixed aperturein a Schilt type photometer. In this application, therelation between densitance = logo and stellar magni-tude m was found to be linear over a range of more thanfive magnitudes, and possibly down to the detectionthreshold, on fast, blue sensitive emulsions commonlyused in stellar photography. Sensitometer tests ofseveral other emulsion types, including ammonia hy-persensitized ir plates, indicate that the linearizationprocedure has fairly general validity.

Special Case: Direct Intensity ReadingsWhen Ao/8 >> 1, log co -) D, hence

o(logw)/(log I) = n-' = * -y = D/b(logI). (16)

By a suitable choice of developer and photometer(s2, '), it is usually possible to develop a fast emulsionto -y = 1. It appears, therefore, possible to selectstandard processing and measuring procedures suchthat y* = 1 and n = 1. Then the characteristic willbe strictly a linear function of w:

I = At. (13a)

Further, without loss of generality, we may take asunit of I, the value I1 corresponding to c = 1 (D =

1516 APPLIED OPTICS / Vol. 7, No. 8 / August 1968

0.1

RN 6725 Cordoba 1962)103a -0, = 10 min.DI 9, 6 min., 19C

/

0/

0

I 1.0

I II

/0

/

C.

I I . . I .

0.05 0.1

30

W

10

0.5 I

Fig. 6. Examples of linear characteristic curves in log -log I coordinates for various batches of Kodak 103a-0 and IIa-0 emulsionsused at several observatories between 1954 and 1967.

log2 = 0.301), i.e., I1 = A; then, and always for y* =1/n = 1, the characteristic reduces to

J = I/I, = ; (13b)

Eq. (13b) is equivalent to

(I + )/I = 0 = A,/a, (13c)

or again,

I,/(I + 1) = t = 1/(J + 1). (13d)

Either of these several forms permits a direct intensityreading of the photograph since most photometers canbe arranged or used so as to record directly the platetransmission t or the opacity 0.

August 1968 / Vol. 7, No. 8 / APPLIED OPTICS 1517

w

1000

100

10

.01

3ORN-262 Mt Stromlo (1954)103a -0, t = 60 min./D19, 6min., 18°C

/

/

/

/f0 a.

0 / .............

W

100

10

5

0.1

I I I I I I i

82C -629 McDonald (1964) /H - 0, t = 30 min.DI9, 6min., 19°C

0

0

b.I I I I I I

0.5 I 1.0

w30

10

0.5

0_

0.02 0.03 0.1 0.3 I

% I I I I I I I I I | I I I I {; I

. . . . . .. . . .

.... ....... . . . . . .l l l l l l lll

., , I I , , I I I I I I , I .

Fig. 7. Eect of local errors on characteristic curve. 'Itortion of the charateristic curve is immediately obviouslinear w(I) or logw-(log I) representation (a), but it is rdent in the S-shaped H and D curve whose correct curv,not known a priori (b). The true characteristic is showndashed curves, the apparent characteristic by the full line,corresponding observed calibration points are plotted c

graph for this fictitious example.

We arrive at the following simple result: Aphotographic emulsion is developed to y* =

relative intensity J = I/1, is equal to the opacitover the whole range of validity of Eq. (13)intensity unit is that corresponding to a plate trasion t =2

The author is most grateful to the anonymous referee/; :D who called his attention to A. E. Baker's papers and

for helpful comments on the original manuscript.

References

1. G. E. Kron and I. I. Papiashvili, Pub. Astron. Soc. Pacific' 79,9 (1967).

2. M. F. Walker and G. E. Kron, Pub. Astron. Soc. Pacific79,551 (1967).

3. F. Hurter and V. C. Driffield, J. Soc. Chem. Ind. London 9,455 (1890).

4. A. E. Baker, Proc. Roy. Soc. Edinburgh, 45, 166 (1925); 4734 (1926); 48, 106(1928).

'he dis- 5. C. E. K. Mees, Theory of the Photographic Process (The Mac-s in the millan Company, New York, 1954), 2nd ed., p. 168.not evi- 6. G. de Vaucouleurs, J. Dragesco, and P. Selme, Manuel de

iture is Photographie Scientifique (Revue d'Optique, Paris, 1956).by the 7. G. de Vaucouleurs, J. des Observ. 31, 113 (1948).

The 8. G. de Vaucouleurs, Ann. Astrophys. 11, 247 (1948).On each 9. W. J. Albersheim. J. Motion Pict. Eng. 29, 417 (1937).

10. W. B. Jones, D. L. Obitts, It. M. Gallet, and G. de Vaucoul-

Vhen a1, thetance X

; theusmis-

eurs, Astronomical Surface Photometry by Numerical Map-ping Techniques, Publ. Dept. Astron., Univ. Texas, (II) Nr.8 (1967).

11. R. A. Sampson, Mon. Not. Roy. Astron. Soc. 85, 212(1925).

12. G. de Vaucouleurs, Sci. Ind. Photo. 14, 149, 193 (1943); 17,257 (t946).

OSA Corporation Members continued fromn page 1498

WILLEY OPTICAL CORPORATION, P.O. Box 101, Melbourne, Florida32901, specializes in lens and optical instrument design and fabrication,including aspherics, unconventional optics, high precision fabrication ofdifficult components, optical retroreflectors, and telephoto catadioptrics.

XEROX CORPORATION, P.O. Box 1540, Rochester, New York 14603,manufactures and markets office copying equipment and supplies, photo-copy equipment and supplies, and photographic sensitized materials.It specializes in xerography, a unique, dry electrostatic process for copyreproduction. Principal plants are in Rochester and IVebster, New York.

CARL ZEISS, Oberkochen, Wuertt., Germany, manufactures microscopesfor all types of application in science and industry, electron micro-scopes, instruments for particle-size analyses, ophthalmological and othermedical-optical instruments, optical-analytical instruments, interferomet-ric instruments, precision measuring instruments for machine shop,metrology, and quality control, high voltage units, surveying instruments,astronomical instruments, planetarium projectors, binoculars, photographiclenses, magnifiers, spectacles lenses, sunglasses.

ZOOMAR, INC., 55 Sea Cliff Ave, Glen Cove, New York 11542, pioneeredin varifocal optics, mirror optics, and other high-precision optics of originaldesign. Its products include Zoomar lenses for still motion picture cameras,industrial TV, and broadcasting TV; special optics for missiles andsatellites; tracking units for motion picture and TV; optical instrumentfor lens testing, optics for gun cameras; training devices; microrecordingequipment and modulation transfer function meter.

TINSLEY LABORATORIES, INC., 2448 6th St, Berkeley California 94710,founded in 1926, is engaged in the design and mnanufacture of precisionoptical components and optical instruments. These include missile andsatellite components, special cameras, data acquisition equipment, andtracking and astronomical telescopes.

TROPEL, INC., 52 West Ave, Fairport, New York 14450, provides optical de-sign and consulting services, constructs prototype optical systems, andmanufactures specialized photographic and spectroscopic instruments.Optical engineering and development is concentrated in lens design, comput-ing methods, and image analysis. Commnercial instruments manufacturedinclude laser accessory instruments, vacuum usv monochrosnators, andaccessories, high-time resolution spectrographs, imwterferometers.

TRW SYSTEMS GROUP, TRW INC., 1 Space Park, Redondo Beach,California 90278, as an industrial contractor to government agencies andindustry, performs research, development, engineering, fabrication, andtesting in missile and space programs. It employs more than 4500 scien-tists and engineers in the performance of its current contracts and subcon-tracts. Its TRIV Instruments Division manufactures technical devicesdeveloped from company-sponsored advance-technology research.

THE UNITED LENS COMPANY, INC., 259 Worcester St, Southbridge,Massachusetts 01550, snanufactures lens blanks of all types. Blanks areground ito finished lenses by manufacturers of cameras, projectors,telescopes, periscopes, microscopes, range finders, gun sights, fire controlinstruments, tracking camneras, aerial cameras, bomb sights, and guidancesystems. Other lens blanks are widely used in making single vision,bifocal, trifocal, and sunglass spectacle lenses.

VALPEY CORPORATION, 1244 Highland St, Holliston, Massachusetts01746, established in 1931, has production facilities for precision fabricationsuch as cutting, grinding, shaping, and polishing of most known natural andsynthetic optical materials. Production disciplines include piezoelectrictransducer crystals, specialized optical elements, and optical elements forir and laser instrumentation. The company also formed two new divisionsin late 1967: the Optical Coating Division, which produces productionquantity standard coatings and state-of-the-art custom optical coatings, andthe Electronics Division which designs and manufactures packaged piezo-electric assemblies and crystal transducer devices.

VARO OPTICAL INC., Varo Industrial Park, Mount Prospect, Illinois60056, designs, develops, and manufactures precision optical components,assemablies and systems fros prototype through high volume, and supplieslens benches and ir scopes.

THE WARNER & SWASEY COMPANY, CONTROL INSTRUMENT DIVI-SION, 32-16 Downing St, Flushing, New York 11354, has been active inoptics since 1880, when it began to build astronomical telescopes. The Con-trol Instrument Division develops and manufactures rapid scan spec-trometers and other high speed optical instruments for measurement oftransient and high temperature phenomena. Research is carried on inspectroscopy of hot gases, plasmas, and shocks.

THE WHITE DEVELOPMENT CORPORATION, 80 Lincoln Ave, Stam-ford, Connecticut 06902, consulting physicists and instrument designers, isconcerned with the design and development of optical instruments for thelaboratory. Mrost of its products are manufactured by other instrumentcompanies under license. The firsa also designs and builds custom instru-ments and accessories.

1518 APPLIED OPTICS / Vol. 7, No. 8 / August 1968