critical frequency relations in scotopic vision

CRITICAL FREQUENCY RELATIONS IN SCOTOPICVISION

BY

HERBERT E. IVES

INTRODUCTION

In his "Studies of Flicker,"' T. C. Porter discovered that thestraight line representing the relation of critical frequency tologarithm of illumination underwent an abrupt change of slopeat about .25 meter candles. At this same point, according toother evidence, vision changes from color vision to gray vision,or from "cone" to "rod" vision, if we accept the correlationindicated by the "duplicity theory." In investigating criticalfrequency phenomena by monochromatic light, the present writerdiscovered 2 that this change of slope is of such magnitude withblue light as to constitute a complete change in the character ofthe illunination-critical frequency relation. The straight lineafter its change of direction becomes parallel to the log I axis,that is, critical frequency becomes a constant independent ofillumination. At the same time the blue hue of the light vanishesand is replaced by a colorless or gray appearance. The isolationof rod or scotopic vision appears to be complete, where in thecase of white light, or monochromatic light of long wavelengths,it is only partial. This phenomenon of constant critical fre-quency for blue radiation of low intensity has since been con-firmed by Frank Allen.3 In a later investigation by the writerand Mr. Kingsbury,4 testing a "diffusion" theory of intermittentvision, observations were made in this same region with discs ofvarying ratio of open to closed sectors. The constancy of criticalspeed was again found, and the ratios of speeds for differentopenings was entirely different from that holding at higherilluminations. Some support for the "diffusion" theory under

' T. C. Porter, Proc. R. S., 70, 313-329, 1902.2 Ives, Phil. Mag., Sept., 1912, p. 352.3 Allen, Phil. Mag., July, 1919, p. 82.4 Ives and Kingsbury, Phil. Mag., April, 1916, p. 290.

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SCOTOPIC CRITICAL FREQUENCIES

study was claimed because these new relations were clearlypredicted by the theory when the term used in its expression tocover the effects of change of diffusivity with illumination wasmade constant.

In subsequent consideration of this isolation of rod vision, it hasappeared to the writer that here was an exceptionally promisingopening for studying the nature of vision. It is highly probablethat colorless (rod) vision is the more primitive kind of vision,a survival of an earlier, less developed type. It should be easierto elucidate. When it is understood, and not until then, are wereally justified in attempting to formulate theories of the farmore complex phenomena of high intensity, color, vision.

4

FIG. 1. Apparatus used for study of low intensity flicker phenomena

1. Extended light source.2. Projection lenses.3. Sector disc.4. Slit.5. Motor.6. Diffusing screen.7. Neutral tint wedge.8. Artificial pupil.9. Variable resistance.10. Revolution counter.

The present study was, therefore, undertaken as forming apart of the study of "rod" vision, which from general con-

May, 1922] 255

HERBERT E. IVES [J.O.S.A. & R.S.I., VI

siderations appears a logical starting point for the investigationof vision. It was aimed more particularly at the time relationsof such vision, as deducible from the effects of intermittentillumination. More specifically still, it is an investigation chieflyof the effects on critical frequency of the form factor of the inter-mittent illumination, thus extending the work above mentioned,on the effects of variation in the ratio of exposure to obscuration,to variation in the manner of rise and fall of the stimulus.

APPARATUS AND METHOD OF OBSERVATION

The essential features of the apparatus used are easily graspedfrom the diagram, Fig. 1. An image of an extended light source

A Serles b eries C eries D Sertes E 5eries

Al c ,= - /,E (

A, °(= B zC =D,=2 E=2 1=2Az Pt o w E7 " = 6

A, '?/ ' / L E

A4 6 D4 -6 E. " -As d= , E*.

A, D=j, B,~=j- C, c~4 12 c(=- E 6 /

FIG. 2. Sector discs used to control the wave-form of the stimulus

1, is thrown by means of two lenses, 2, on a diffusing screen 6(several sheets of finely ground glass) which is viewed by the eyethrough an artificial pupil, 8, of 2 sq. mm. area. The onlyportion of the lenses, 2, utilized is that limited by a narrow slit,4, whose sides are radial from the axis about which discs 3 rotate.By varying the contour of the discs it is obvious that any desiredvariation of the brightness of the field 6 throughout a cycle may beobtained. The speed of rotation of the discs is altered roughlyby a set of pulleys, finely by a variable resistance 9, in series with

256


the motor 5, and is read by the electric tachometer 10. Thebrightness of the image at 6 is altered by means of a neutraltint wedge 7, having a range of transmission of approximately10 000 to 1.,

The discs upon whose shape depends the variation with time ofbrightness of the diffusing screen were cut from thin sheet alumi-num, and after being drilled to fit the motor axle, were sand-blasted and painted with dull black lacquer. They were madesymmetrical to give two cycles per revolution. The shapes chosenare shown in Fig. 2. They are arranged in five series, lettered A,B, C, D, and E. The first number of the A series gives equalintervals of light and darkness, changing abruptly from onecondition to the other; it is similar to the discs which have beenmost frequently used in experiments of this sort. The ampli-tude, a, is 2 the total opening. The rest of the series consistof the variants on the first disc obtained by decreasing the ampli-tude to 3/8, 4, 3/16, 1/8, 1/16, so that in place of an alternationof light and dark, the alternation is between 1/8 light and 7/8dark, etc. In the second (B) and the third (C) series the contourof the time-brightness distribution is changed from "squaretopped" to "saw toothed"; in the B series one edge of each toothis vertical, in the C both are equally inclined. Only the firstmembers (a= 2) of these two series were cut. Series D is similarto series A, except that the wave-form is sinusoidal, the ampli-tude range from 2 to 1/16, as before. Series E consists of thevariation of series A formed by altering the ratio of light to.darkness. The openings (4) made up were 1/12, 1/6, 1/3, 2, 2/3,5/6, 11/12. The only amplitude cut was 2. It will be notedthat E4 is identical with Al, and that A2, A3 , etc., are the variationsof E4 with respect to amplitude. Hence while every combina-tion of shape, amplitude and opening was not provided, thewhole set of discs covered fairly well the significant variations ofwave-form.

In order to secure low intensity blue light a mercury vapor lampwas used as light source, the blue and violet lines being isolatedby means of a blue filter6 placed at 6. An opaque screen in front

I Made and calibrated by the Eastman Kodak Co.6 Wratten monochromatic filter for isolating blue mercury lines.

May, 1922] 257


of the mercury arc was pierced with a circular aperture of approx-imately 112 cm diameter. A circular portion 11.1 mm. indiameter of the image of this at 6, as limited by a diaphragm, wasthe observed bright field. This had a diameter of 4.5 degrees,-considerably larger than the fovea, and the attention was directedto the center of the field. The variable neutral tint wedge wasmoved to a point where the field appeared gray, and where thecritical frequency is independent of brightness,-as determinedby measurements described below.

The method of making measurements was as follows: Afterthe five minutes or thereabouts in the dark laboratory, necessaryfor the eye to become dark adapted, the observer (H.E.I.) placedhis eye at the observing aperture 8, started the motor, and slowlyincreased the speed by moving the sliding contact of the variableresistance 9, until the flicker at first observed vanished. Thiswas announced by calling to an assistant who was simultaneouslywatching the speed, and who recorded the speed at the instant.The procedure of setting was then reversed, starting above thecritical frequency the speed was reduced until flicker appeared.This alternation of direction of setting was continued until acomplete group was obtained. Except where otherwise stated, agroup consisted of ten settings. Each point of a series (set ofdiscs) was measured twice in any run, the discs being put throughfirst in one order and then in the reverse order. Consequentlythe determination of a point involved twenty settings. Extremesettings frequently varied as much as ten per cent in speed toeither side of the mean, and the return series mean would oftendrift by as much as five per cent from the first series. Settingswere fairly reproducible from one day to another; less so if aninterval of several days intervened, although measurementsbelonging to the new period were in good agreement amongthemselves. It is therefore evident that all the measurementswhich are to be compared one with another for study, should bemade as nearly as possible at the same time, and that nothing isgained by multiplication of measurements over a lengthenedperiod during which a drift of values may occur, unless, of course,the period is so long and the measurements so numerous that all

258


the points to be determined have been equally affected. Themain series of measurements here recorded (Fig. 4) and used forintercomparison were those made on six consecutive mornings,at the same hours, each of the series, A, D, E being carried throughon two separate days. In spite of the factors above mentioned,which contribute to low precision, the critical speed values forthe various wave shapes are believed to be established by theforty settings involved to within 3 per cent, as was evidenced bythe fact that the values derived from the same points as theyoccur in the first and last series run (A8 and E4), are mutallyconsistent, indicating a reasonably constant state of the obser-ver's vision during the six day period. Certain series made dur-ing the assembling and test of the apparatus, while not sufficientlyconsistent to be intercompared, were in general in agreement intheir essential characteristics with the main series here chosen forpresentation.

EXPERIMENTAL RESULTS

As a necessary preliminary to the study of the constant criticalspeed region it was necessary to establish what setting of theneutral tint wedge was required to insure that this region wasactually being used, and, of course, to verify the non-dependenceof critical speed on intensity for all the disc shapes used. To coverthese points a series of observations was made early in the studyon typical discs at different values of the neutral wedge. Onlyfive settings were made on each point, as this number is sufficientto locate the reading well enough for the immediate purpose.The points, which are, for the reason just stated, somewhatscattering, are shown in Fig. 3 for the discs Al, A 5, A6 , D, D6,D6, E, E7 , B and C. Abscissae are wedge scale units (eachunit=a difference in log-brightness of .214), ordinates, criticalspeeds in cycles per second.

It will be seen that in every case the critical speed does becomea constant beyond some wedge scale value. The wedge valuenecessary to use for each series was thus easily picked from thisplot. Before leaving this figure, attention may be called to afeature which illustrates the discussion on precision above. By

May, 1922] 259

HERBERT E. IVES [J.O.S.A. & R.S.I.. VI

comparing the critical frequency values of the various discs withthe series shown in Fig. 4, which were made some time later, itwill be noted that these preliminary values are all somewhatlower. They are, however, as a family mutually interrelatedvery closely as are the more precise final results.

16 - - - - -- _ _ - - -

r _ _ _ . 859~ t - ~~~ E,J4…

-4C

2--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3

1 0 _ _CD_,_.o--As

1X ,, 10 9 8 7 6 5 4 3 2 , 0Absorbing Wedge 5cale

FIG. 3. Critical frequency-log brightness determinations for various wave-forms,showing wedge scale value necessary to insure observations falling in rod visionregion

Critical Speeds for Discs Al, B1, C,, D,.-The several differentshapes of amplitude =12, and average transmission 12 were pickedout as the first series to be studied. In all, three complete setsof measurements (each of 40 settings per disc as above explained)were made at intervals during two months. The experimentallydetermined critical speeds are shown in Table 1, in which arealso calculated their ratios of speeds, compared to that of thesimplest shape (sine curve "D"). In the last column are valuesof the ratios as computed by an empirical formula (to be dis-cussed later).

It is evident tha-t variation of wave-form unaccompaniedby change of amplitude or of mean transmission has a well markedeffect on critical speed. It is perhaps most striking that thelowest speed is not given by the disc whose shape changes most

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TABLE 1. Critical Speed in Cycles per Second

Disc 1st Set 2nd Set 3d Set

A, 12.6 12.4 13.30B1 10.6 10.3 11.5C1 11.5 11.2 11.8DI 11.9 11.7 12.6

Ratios to D1 Mean Calculated by EmpiricalFormula

A1 1.06 1.06 1.06 1.06 1.06B1 .89 .87 .91 .89 .88C1 .96 .95 .93 .95 .94Di 1.00 1.00 1.00 1.00 1.00

gradually, namely the sine curve (D1) but by that one whichcombines both the slowest variation with the fastest, namely,the "saw-tooth" with one vertical edge of the tooth. Even morestriking is the fact that the same speed is obtained whicheverway the disc is run, whether the abrupt transition leads or follows.This is shown by the following table of measured speeds, from thefirst set used in Table 1:

TABLE 2

Direction of Disc.

Abrupt Transition Leading, Following

Mean of first ten settings 10.35 10.40Mean of second ten settings 10.75 10.70Mean, cycles per second 10.55 10.55

From the latter fact it is to be inferred that the significant factorin the speed is some feature of the shape which is unaltered bydirection.

Speeds for the "A" Series.-These measurements constitutewith those that follow on the D and E discs the "main" series,made on consecutive days as described in a previous section.The results obtained (with wedge set at 7) for the square toppedwave forms of various amplitudes are shown in Fig. 4 (upper curve

May, 1922] 261


to left). As the amplitude is decreased the critical speed falls off,the two approaching zero together. The rate of decrease of speedwith amplitude is, as will be described below, logarithmic.

Critical Speeds for the "D" series.-These, determined with thewedge set at 8, are also shown in Fig. 4. For the higher amplitudesthe relation of critical speed to amplitude is similar to that of theA discs, except that the values are lower. When, however, theamplitude drops to 1/8, the critical speed has fallen to a value toolow to fit on a smooth curve continued to the origin. At ampli-tude 1/16, no flicker can be produced at any speed and hence nocritical speed exists.

14 -

H = .12 S 25 ' Z5 0=0ao Oe 60' /20' l80' 240' 300' 360'

Araplitude Opoetnitn

FIG. 4. Critical frequency against amplitude (left) and against opening (right). Fllines plotted from empirical formaula.

This failure of flicker would be most simply explained bysupposing that the amplitude of variation of visual sensationhad dropped below the threshold. This explanation is inadequate,however, since the square topped disc of the same amplitudebehaves normally. A more satisfactory idea of what happens isobtained by considering the phenomena with very low speeds,working up from zero. In the case of any small amplitude disc inwhich the transition from darker to lighter is gradual, it is obviousthat if the speed is only low enough' the. eye by continually

262

I,


adapting will not appreciate the change of intensity, whereas thesame amplitude of fluctuation if more rapid will be perceived.There should, therefore, for the sine curve discs, be a speedbelow which the sensation of flicker is not produced. This waseasily found experimentally to be the case. With the 1/8 and 3/16amplitude discs turning over once every three or four seconds,no fluctuations of intensity were visible. As the speeds increased,a point was reached where flicker began, then when the speed wasmuch further increased, flicker disappeared once more, at thepoints already determined. The amplitude-critical speed curveis therefore completed not by continuing the curve through thelarge amplitude points to the origin, but by turning back, asshown, through the low speed beginning-of-flicker points towardthe maximum amplitude axis. We may summarize these smallamplitude phenomena by the observation that here we clearlyhave to do with rate of change of amplitude, whereas for the largeamplitudes and with abrupt transitions it is possible that we areconcerned primarily with the magnitude of the amplitude.

Critical Speeds of the E Series.-These are exhibited in Fig. 4to the right, the abscissae being openings (in degrees and infractional parts) the ordinates critical speeds. In making these,several different wedge values were used, always below the breakin the straight line relation and chosen to keep the mean bright-ness fairly constant. This saved the observational discomfortof working with unnecessarily low intensities for the small open-ings, but, as Fig. 3 shows, would not affect the values of thereadings.

The critical speeds are highest for the smallest openings and therelationship between speed and opening is in general logarithmic.This relationship is in marked contrast to the high intensitybehavior,4 where the speeds are lowest for the largest and smallestopenings, passing through a maximum at opening 2.

The Transition from Low to High Intensity.-An interestingquestion that arises in considering the flicker phenomena at highand at low intensities is whether these are one phenomenonwhich passes through a change at some critical condition ofbrightness or are two entirely separate effects due to two different

May, 19223 263


processes. This comes down to the question whether the abruptbreaks in the inclined straight lines of Fig. 3 occur at (A), adefinite peak brightness, (B), a definite average brightness, or(C) with no relation to the brightness of the observed field.

At once by reference to the data for 30 degrees (E1 , 0 = 1/12)and 330 degrees (E7, k= 11/12) opening it is seen that the breakin direction does not occur at a definite peak value since this isthe same for both at the same wedge value, and the wedge valuesdiffer by over 5 units (log difference =1.17) corresponding tomore than a ten-fold change in brightness. Referring next to thedata for discs D1 and D5, we find a wedge scale difference of threeunits for the break point, although here the mean brightness is thesame at the same wedge values. There is thus no connectionbetween brightness and the transition from one relationship tothe other. The data of Fig. 3 appear in fact to show that thesloped log I-critical frequency lines, occurring in the region ofcolor vision, and the horizontal ones occurring in the region ofcolorless vision, belong to two quite separate coexisting processes.The critical speed for any brightness is roughly that correspond-ing to the process demanding the higher speed.

EMPIRICAL EXPRESSION OF RESULTS

Discussion of the theoretical aspects of these results is de-ferred to a subsequent communication and the present paperwill be concluded by pointing out an empirical expression whichhas been found to represent the main series of observations justdescribed with remarkable accuracy in terms of the Fourieranalyses of the wave-form used. In order to make clear thestatement of this relation, it is necessary first to assemble togetherthe Fourier series expansions of the disc contours used. Byreference to any comprehensive text on heat conduction we findthe following expressions:

For the D1 (sine curve discs), the variation of intensity with timeis given by

I1 2 +la sinct (1)

where I, is the instantaneous intensity, I is the intensity with the

264


disc removed, a is the amplitude and the frequency in cyclesper second.For the C series (saw-tooth symmetrical)

I=: .+81a (sin t-1 sin 3 ot+-sin 5 t-. . ) (2)2 ?r2 9 25

For the B series (saw-tooth with one abrupt transition)

I=1+ 2 1 a(sin cwt+- sin 2 cot+I sin 3 t+. . .) (3)2 7r 2 3

the plus or minus sign applying to different directions of motionof the disc.For the E series:

4 _ 1It Ia+ 4 a(sin r cos t +- sin 27r cos 2 cot+. .) (4)

or 2If as is the case in our experiments a = 2 and 4, is the fractionalopening, the average value is l+, and (4) becomes

+21 1Is=I4+2- (sin irk Cos wt+- sin 27r cos 2 cot+. .) (5)or 2

The A series constitute a special case of the E series for which= 2, and sin ro = 1, so that

I 4ITa~csct 11h=-+ (cos o+- cos 3 t+- cos 5 ct+. . ) (6)2 7r 3 5

Now the expression which has been found to represent theexperimental data of the main series of observations with con-siderable accuracy is a simple function of the ratio of the coeffi-cient of the first periodic term of the Fourier expansion to theconstant term, or the average value. If we put

2X coefficient of 1st periodic term

constant termwe find that all the experimental points of this main series with theexception of the low amplitude sine curve values are given by theexpression

co =c log 2W (7)a

in which c is the critical speed, c is a scale constant, and a is asmall number of the order of a few hundredths. The constants

7 The constant 2 is introduced in order to have an expression in terms of the rangeof fluctuation, according to the diffusion theory.

May, 1922] 265


c and a are the same for all wave shapes, c being 8.07, and 5=.04.Substituting in (7) we have,

for the A discs W=c log (16 a) (8)

for the B discs w=c log (8 a) (9)

for the C discs W=c log (72 ) (10)

for the D discs cW=c log (4 a) (11)(a)

for the E discs X=c log (4 sin 7ro) (12)(irka)

The agreement of these formulae with the data is shown by thefull curves drawn through the points in Fig. 4 and for the A1,B1, C1, D1, discs by the data in Table 1.8

It will be seen that all the "square topped" stimuli points fallaccurately on the curves given by the formulae; and that theformulae hold for all wave-forms for large amplitudes. Thepoints corresponding to sine-wave forms of small amplitude lieentirely away from the (full) line indicated by the formula due tothe absence of any critical speed for the lowest amplitudes. It isprobable that the symmetrical saw-tooth wave-forms (C discs)would also give points lying on a curve distorted at low ampli-tudes from the curve of the formula, although if the explanationadvanced for the peculiarity of the sine-wave form points iscorrect both saw-tooth forms must yield points which approachzero speed at zero amplitude, as do the square topped forms.The general formula (7) cannot, in view of these shown andsuspected deviations from experimental fact, claim to be com-plete. It does, however, represent the more important lowintensity critical speed relations with sufficient approximation tosuggest that it must be very close to the true complete formula.

8 The points shown for discs B1 and C,, in Fig. 4 which were not included in themain series, are extrapolated from the Al and Di values by utilizing the well determinedratio previously obtained.

266


It will perhaps make this formula more intelligible if a possiblephysical interpretation is put on it. Let us suppose that to aperiodic stimulus Asinwt at the surface of incidence there corre-sponds at a certain depth in a conducting medium the periodic

reaction A e sin cot. This is a degradation in amplitudesimilar to that occurring in heat conduction, according to theFourier diffusion law, except that in the latter case the amplitude.

is reduced by the factor e 2K, where x is the depth, and Kthe diffusivity.9 On this assumption we have, corresponding tothe stimulus

-+1 a sin wt (13)2

-the reaction -+I a e sin wt--) (14)2

In this the range of fluctuation is

2 a e c (15)

The part this is of the whole reaction isco co

2Iae C C

4 a e (16)

2

If now we take as.the criterion for the disappearance of flickerthat the fractional range must fall below some definite value,8, we have, for the critical condition

_~ c)(17)4 ae a

9 The use of this factor, which is called for by the "diffusion" theory (see ref. 4)leads to formulae in which the first periodic term figures as a square. Actually,due to the short frequency range in which all the observations fall, the formulaeinvolving the square fit the data nearly as well as (7). They demand, however, inorder to fit, a value of a of about .001. This is so far below the very large values of theFechner fraction which hold at low intensities as to force the conclusion that thediffusion theory must be modified if it is to cover this illumination region.

May, 1922] 267


4a 2Wor &)=c log - = c log (18)

where W is the quantity used in the empirical formula.In the case of the more complicated wave-forms, the factor

involving c exponentially will enter with higher values of c inthe successive terms of the expansion, making them so smallas to be negligible, so that formula (18) holds for all cases.

SUMMARY

1. At low intensities, with blue light, critical speed of disappear-ance of flicker becomes independent of the intensity, butdifferent for each wave-form of the stimulus.

2. The relationship between critical speed and wave-form isapproximately represented by the equation

co = c log 2W

where W is the coefficient of the first periodic term of the Fourierexpansion representing the wave-form, divided by the meanvalue.

*RESEARCH LABORATORIESTE AMERICAN TELEPHONE & TELEGRAPH COMPANY

AND TEE WESTERiN ELECTRIC COMPANY INC., NEW YORK.JUNE 24, 1921.

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critical frequency relations in scotopic vision

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