Clin .Vision Sci. Vol. 2, No.3, pp. 187-199, 1988Printed in Great Britain. All rights reserved

0887-6169/88 $3.00 +0.00Copyright © 1988 Pergamon Press pic

THE DESIGN OF A NEW LETTER CHART FORMEASURING CONTRAST SENSITIVITY

D. G. PELLI, 1 J. G. ROBSON 2 and A. J. WILKINS J

'Institute for Sensory Research, Syracuse University, Syracuse, NY 13244-5290, U.S.A.,Physiological Laboratory, Cambridge University, Cambridge CB2 3EG and3Medical Research Council, Applied Psychology Unit, Cambridge, England

(Received 31 August 1987; in revised form 9 October 1987)

Summary-I. A consideration of methods for assessing contrast sensitivity leads to the conclusion that,for a clinical test, letters are more suitable than gratings.

2. A letter chart is described in which letters decrease in contrast but not in size. The letters are arrangedin groups of three; successive groups decrease in contrast by a factor of IfJ2 from a very high contrastdown to a contrast below the threshold of normal observers. A subject's threshold is taken to be the lowestcontrast for which at least two letters in a group are correctly reported.

3. A mathematical model of the observer and the chart-testing procedure has been used to predict howthe accuracy and repeatability of the test score depend on the parameters of the chart and observer. Thisreveals that even a low probability of misreporting supra threshold letters will seriously bias the test scoreif the passing criterion is strict, requiring correct report of all letters in each group, but will have littleeffect if the passing criterion is less strict. This effect of the passing criterion may explain Rubin's [Clin.Vision Sci. 2, No. I (1987)] finding that the new test, which uses a lenient criterion, has excellent test-retestreliability, much higher than the Ginsburg [Am. J. Optom. Physiol. Opt. 61,403-407 (1984)] chart withits strict criterion.

Key words-Contrast sensitivity; letter sensitivity; contrast sensitivity test; clinical testing; charts.

INTRODUCTION

Characterisation of visual function in clinicalpractice is still based primarily upon measure­ment ofletter acuity, supplemented occasionallyby other measures such as perimetry. Whilethese measures are certainly very useful, there isstill a great need for earlier detection of variousdiseases which impair visual function, as wellas for a quick and simple method for character­ising the level of visual function resulting fromstable visual impairment. We believe that thisneed can best be met by supplementing the usualacuity test by a measurement of contrast sensi­tivity for a target of intermediate size. We shalldescribe the design considerations that haveled us to develop a variable-contrast letterchart for the clinical determination of contrast­sensitivity*.

*The Pelli-Robson Leiter Sensitivity Chart will be availablefrom Clement Clarke International Ltd, 15 Wigmore Street,London WIH 9LA, England, and Clement Clarke Inc,3128-D East 17th Ave., Columbus, Ohio 43219, U.S.A.

c.v.s. 23-0

BACKGROUND

Although it has been realised for a long timethat the measurement of increment thresholdsfor such test targets as discs on a uniformbackground can provide useful informationabout the visual system that acuity measure­ments fail to provide, past attempts to introducesuch contrast-sensitivity tests into clinicalpractice (e.g. the George Young Threshold Test,Young, 1918) have had little lasting success.More recently, following the introduction ofcontrast-sensitivity measurements for sinusoidalgratings of different spatial frequencies as ageneral research method for the characterisationof human spatial vision (Schade, 1956; Cam­pbell and Robson, 1968), there has been sub­stantial interest in using such measurements asa basis for clinical visual assessment (e.g. seeWolkstein et al., 1980). While sufficient workhas now been done to suggest that contrast­sensitivity measurements made with gratingswould be useful in the clinic, it seems to usunlikely that the full potential of contrast­sensitivity testing will be realised unless the

187

188 D. G. PELLI et al.

measurement can be made as simple and reliableas the letter-acuity test itself.

In research laboratories, contrast-sensitivitymeasurements for sinusoidal gratings have usu­ally been made using patterns displayed on acathode-ray tube, the electrical signals requiredoften being computer generated and the mea­surement procedure computer controlled. Withsuch equipment, the threshold contrast can bemeasured at as many closely-spaced spatial fre­quencies as desired over the visible range (typi­cally 0.5-40 c/deg), and the result plotted as acontrast-sensitivity function. It is also possible,in a research setting, to use good psychophysicalprocedures (e.g. a two-alternative forced-choicestaircase method with randomly interleavedstimuli of different types) despite the addedcomplexity and extra time required. Measure­ments can certainly be made in this way in aclinical setting (e.g. see Wolkstein et ai., 1980)and apparatus for performing such measure­ments is commercially available (e.g. the NicoletOptronix Vision Tester). However it is not clearthat for routine clinical use, particularly forscreening purposes, it is appropriate to use thesame measures of contrast sensitivity or thesame methods of measurement as have beenadopted in the context of basic vision research.

There are three aspects of the standard re­search procedure whose appropriateness in aclinical context may be questioned. These are(I) the use of cathode-ray-tube displays to gettest patterns of different contrasts, (2) the use ofgratings as test patterns and (3) the measure­ment of contrast sensitivity for test patterns ofseveral different spatial frequencies.

Methods for producing the test patterns ofdifferent contrasts

It is certainly true that cathode-ray-tuberaster-display devices used in conjunction withanalogue or digital electronic waveform gener­ators provide the most satisfactory and con­venient way of producing stimuli of differentlow contrasts. They are particularly satisfactory(in some cases essential) if continuously variablegrey levels or continuous control of contrast isrequired, if very low contrast patterns areneeded (they can provide very uniform back­grounds) or if the contrast must be temporallymodulated or the pattern moving. However,they suffer from being relatively bulky andexpensive, as well as requiring maintenance androutine calibration. These are not importantconsiderations in a research context but in

considering how to provide a routine clinicaltest which might be carried out in every op­tometrist's or physician's office, it is clearlydesirable to find some simpler alternative. Therecan be no doubt that in this context what isneeded is an easily used printed test chart ratherthan a complex apparatus requiring elaborateor time-consuming procedures. The chartshould be durable and inexpensive to produce.Such a chart would display a small number oftest patterns of different contrasts and the sub­jects would be required to identify each testpattern, thereby providing objective evidencethat they could see it.

One obvious method of producing a test chartwith patterns of different contrasts is to photo­graph a cathode-ray-tube display. This is a veryconvenient way of obtaining static demonstra­tion patterns and has been frequently used inrecent years to provide illustrations for researchpublications (e.g. Robson's variable contrastand variable frequency grating reproduced asFig. 4.12 in Ratliff, 1965)where exact renderingof the original is not really necessary. However,the problems encountered in obtaining exactmasters from the original display and in printingmultiple subsequent copies are formidable,whether the printing is by normal half-tonemethods or is photographic. Despite this, twosinusoidal grating test charts for clinical usehave been made photographically (Arden, 1978;Vistech Vision Contrast Test System, Ginsburg,1984) and are commercially available. Photo­graphic methods have also been previously usedto produce other low-contrast test charts inlimited numbers where the requirement forindividual calibration was acceptable (e.g. theLandolt-C chart of Hecht et al., 1949, used fordemonstrating the effects of hypoxia, or theedge charts of Bailey, 1982, or Verbaken andJohnston, 1986).

Another alternative for producing test pat­terns is to use computer graphics methods togenerate half-tones directly. Della Sala et ai.(1985) have described the use of square-wavegratings made up of many closely-spaced paral­lel thin black lines which were varied in theirseparation (and thus their density) to vary themean reflectance of the areas comprising thelight and dark bars of the grating. The authorsdiscuss the simplicity and robustness of thismethod of producing gratings of controlledcontrast using a computer-driven XY plotter toproduce the master patterns.

A variant of this method has now been de-

Measuring contrast sensitivity 189

scribed by Wilkins et al. (1987) whose square­wave gratings* were printed using black dotsproduced by an impact matrix printer (ratherthan pen-plotted lines) to reduce the reflectanceof the dark areas of the test chart. So long as thedots are all equal in size and blackness and donot overlap, the reduction in reflectance (andhence luminance) of a printed area is exactlyproportional to the density (number per unitarea) of the dots. This makes it easy to generategrating patterns with known contrast ratios andprovides masters which can easily and satis­factorily be reproduced by standard printingmethods. The ability to reproduce a test chartusing standard printing methods is important ifproduction costs are to be minimised.

There is a new kind of printing device, muchbetter than impact printers or XY plotters. Pelli(1987) has discussed the use of standard (300dots/inch) xerographic laser printers, as well asphotographic laser typesetters with substan­tially greater resolutions, for the production ofvarious vision test charts. The use of suchdevices is greatly simplified by the availablilityof programming languages such as PostScript,specially designed for controlling them. Thereseems little doubt that programming a lasertypesetter to produce variable-dot-density half­tones is the best method currently available fororiginating test charts which are to be re­produced by standard printing methods.

Although it is not even potentially suitable forrendering test patterns involving continuouslyvarying grey levels, there is one more method ofprinting low-contrast patterns which has beenemployed in producing clinical test charts. Thisrelies on using pale inks to print areas ofconstant reflectance. The George Young Test]used water-based inks diluted by successivelygreater amounts to print progressively palercircular patches randomly placed on succeedingpages of a little book (Young, 1918). Dilutingthe ink by a further factor of two each timeproduced a test patch whose contrast was onehalf that of the previously printed patch.

Much more recently Bailey and Regan haveused grey inks of different reflectances to print(silk-screen) a set of Snellen charts, each withletters of a different low contrast (Bailey, 1982,1987;Regan and Neima, 1983, 1984).While this

"Cambridge Low Contrast Gratings, available fromClement Clarke International Ltd, 15 Wigmore Street,London WIH 9LA, England.

tOriginally published by Raphael's Ltd, London.

is evidently a satisfactory method for the prod­uction of charts with letters having a contrast of0.1 (10%) or greater (as in Regan's charts), itwould be more difficult to use this method toproduce a chart with a much lower contrast orwith many contrasts.

Sinusoidal and other gratings as test patterns

The use of sinusoidal gratings as optical testpatterns was introduced by Selwyn (1948)primarily for studying camera optics andphotographic films. However, since the overallresolution of a photographic system could onlybe understood by taking into account the char­acteristics of the human observer as well asthose of the lens and film, Selwyn also made thefirst direct measurements of the visibility ofsinusoidal gratings. Selwyn's main justificationfor using a sinusoidal grating as a test objectwas based upon the realisation that the lightdistribution in the image of such a target mustnecessarily also be sinusoidal. Selwyn was alsoaware that photographic emulsions as well aslens performance could usefully be examinedusing sinusoidal grating targets and was im­pressed that "the adoption of such ... test ob­jects results in considerable simplification oftheoretical arguments".

A rather similar rationale lay behind Schade's(1956) adoption of sinusoidal gratings for char­acterising human vision in the context of pro­viding a comprehensive description of the wholetelevision chain; each component in the chain-s­camera lens, television pick-up tube, signaltransmission system, cathode-ray-tube displayand human observer-s-could be characterised ina consistent way in terms of its spatial-frequencyresponse function. In many ways this unifiedapproach to an electro-optical system was aspatial counterpart to the characterisation ofelectronic sound-recording and reproducingsystems using pure-tone sounds (sinusoidalpressure fluctuations) and sinusoidally-varyingvoltages.

In both Selwyn's and Schade's work the useof sinusoidal gratings was a practical andtheoretical convenience; it did not depend uponany deep knowledge of the visual mechanismor upon the visual system functioning in anyparticular way; essentially it represented theseauthors' response to a practical need to proceedwith applied work involving vision, in the ab­sence of detailed understanding of the visualsystem. Following Selwyn and Schade therehave been a very large number of subsequent

190 D. G. PELLI et al.

studies of the visual system, both psycho­physical and neurophysiological, which havealso been based upon characterising the visualprocess by an examination of visual perform­ance using sinusoidal gratings. Many of thepsychophysical studies, starting with that ofCampbell and Robson (1968), have concludedthat the visual system operates as a systemof parallel channels or mechanisms tuned todifferent bands of spatial frequencies, whileneurophysiological studies (see Robson, 1983)have shown that neurones at various levels inthe visual system are also more-or-Iess selec­tively responsive to restricted bands of spatialfrequencies (as well as to restricted ranges oforientation). These findings can be taken toimply that sinusoidal gratings play a specialrole in the study of visual function since theyare the elementary signals appropriate to thespatial-frequency domain.

On the other hand, it must be rememberedthat the visual nervous system is comprised ofneurones which, although they may be orien­tation and spatial-frequency selective, havereceptive fields of limited extent. For such neu­rones the most appropriate stimulus will be onewhich is restricted to the area of the neurone'sreceptive field. This suggests that the appropri­ate test stimulus would be a grating patch ofrather small extent, a stimulus which is believedto be visually optimal (Watson et al., 1983).Thedesirability of using stimulus patterns of limitedextent for clinical vision testing may also beargued on the basis of the well-known non­uniformity of acuity and sensitivity over thevisual field which are confounding factors whenlarge area gratings are used. However, while acase could certainly be made for using smallgrating patches as the basis of a standard test ofcontrast sensitivity, there are various reasonswhy, in clinical practice, letters are more desir­able than grating patches. The principal advan­tage of letters is that they lend themselves to theadoption of particularly good psychophysicalprocedures (to be discussed below). But as wellas this, letters are familiar and important ineveryday life. Letters have also been perceivedto have the advantage of including "vertical,horizontal, oblique and curved contours"(Sloan, 1959).

Letter charts

The high-contrast letter-based acuity chartwas first described in very nearly its modern

form by Snellen in 1862. It is a printed chartwhich can be inexpensively reproduced andwhose relevant characteristics (contrast andsize) are very stable. Because letters have dis­tinct forms with which most subjects will be veryfamiliar, it is easy to use correct identification asa criterion for determining whether or not theletters have been seen; this makes the test objec­tive and more reliable than it would be if asubjective detection criterion were used (Vaeganand Halliday, 1982;Higgins et al., 1984). Whilesubjects can be asked to identify the orientationof gratings (as for Ginsburg's chart, 1984), formost subjects this is not as straightforward, oras quick, as naming letters; nor is it possible toprovide as many different, but readily-named,orientations as letters. With a letter chart, thetesting procedure-identifying letters in agroup-is easily explained, easy to score, andvery quick.

A subtle virtue of traditional acuity charts isthat the test is relatively error-tolerant since thesubject is typically required to read only 4 outof 5 letters in each line to pass that line, so thatup to one mistake per line has no effect on thetest result.

Given that the standard letter-based acuitychart is, by its very nature, a most satisfactoryway of determining a subject's high-contrastacuity, and one with which both patients andexaminers are already very familiar, it seemsparticularly desirable to adopt as similar a chartas possible for measuring contrast sensitivity.This argument has been put forward by Reganand Neima (1983, 1984) to support the use ofSnellen charts printed with low-contrast letersto determine low-contrast acuities. As notedearlier, Regan and Neima's charts have, like theregular Snellen chart, letters of various sizes butall of the same contrast. Using a single chart ofthis kind, it is therefore possible to determinefor what size of letter a subject's contrast sensi­tivity has some particular value; by using anumber of charts with different contrasts it ispossible to map out a contrast-sensitivityfunction for letters.

The main reason for clinical contrast sensi­tivity measurements is to determine if subjectswith normal acuity have abnormally low con­trast sensitivity at lower spatial frequencies.Because contrast sensitivity falls steeply both athigh spatial frequencies and at small letter size(Legge et al., 1987), the Regan-Neima low­contrast acuity charts are less sensitive tochanges in contrst sensitivity than is a direct

Measuring contrast sensitivity 191

measure of contrast sensitivrty by a chart In

which fixed-size letters vary in contrast.

How many sizes of letters do we need?

As explained above, the recent interest invisual contrast-sensitivity measurements as aclinical tool has followed from the introductionof the concept of the contrast-sensitivity func­tion (contrast sensitivity for sinusoidal gratingsas a function of spatial frequency) as a funda­mental description of normal visual perform­ance. Initially it was unclear how this functionmight be affected by clinically significant dis­orders, and at that stage it was obviously neces­sary to examine the whole contrast-sensitivityfunction. Now, however, it is fairly clear thatone of four kinds of distortion of the contrast­sensitivity function can occur (Regan andNeima, 1983). There may be a loss of sensitivityat high spatial frequencies only (resulting in areduction in high-contrast acuity), a general lossat all frequencies (resulting in a reduction inhigh-contrast acuity and intermediate­frequency contrast sensitivity), a loss mainly atlow and intermediate spatial frequencies or aloss restricted to intermediate frequencies.While detailed assessment of the visual defect inany particular subject will require the subject'swhole contrast-sensitivity function to be mea­sured, it should be possible to determine if asubject's vision is abnormal by making only twomeasurements. One measurement is required toreveal the existence of reduced acuity at highcontrast and a second to reveal a contrast­sensitivity loss at intermediate spatial fre­quencies. It should be possible to provide thefirst of these measurements using a standardhigh-contrast acuity chart and the second usinga variable-contrast letter chart with letters ofappropriate size.

Ginsburg (\978) has shown that the recog­nition of letters is possible using just the spatialfrequencies of between 1.5 and 2.5 cycles perletter width. Similarly, Legge et al. (\985) haveshown that spatial frequencies higher than twocycles per letter width are unnecessary forachieving optimum reading rates. Since thecontrast energy of letters falls off rapidly withspatial frequency, it is likely that at threshold itis frequency components below about 2.5 cyclesper letter width which provide the visual signalsused by a subject for identifying letters. Thus avariable-contrast letter chart with letters thatsubtend an angle of one half degree at the eyecan be presumed to measure a subject's

contrast-sensinvity at spatial frequencies be­tween 3 and 5 c/deg. This frequency range corre­sponds reasonably well with the optimum fre­quency for normal subjects and is arguably thebest range for determining whether or not asubject has an intermediate spatial-frequencysensitivity loss. The choice of spatial frequencyat which it is best to estimate contrast-sensitivityloss is in many cases probably not very criticalas Pelli et al. (1986, 1988) have shown that theupper part of the contrast-sensitivity function ofnormal and low-vision observers has a fairlystereotyped form whose parameters can be quitewell estimated from measurements of high­contrast acuity and contrast sensitivity for 3 degletters. On the basis of the above discussion webelieve that a single chart with letters all of onesize could be used to obtain as much informa­tion on a subject's contrast sensitivity as wouldbe clinically useful.

A NEW LETTER CHART

We have now produced such a chart withletters of various contrasts (Fig. I) as acomputer-generated variable-dot-density half­tone originated on a phototypsetter (Linotronic300) having a resolution of 2540 dots/inch(\00 dots/mm). Full-size copies of the originalchart have been printed using offset lithography.

The size of the letters on this new chart is suchthat they subtend 0.5 deg at 3 m. While it issupposed that the chart will normally be used ata distance of 3 m, it can be used at much nearerdistances for assessment of low vision.

For the new chart we have chosen to use theSloan letter set, mainly because we suppose thatthis chart will be used in conjunction with astandard high-contrast acuity chart using thesame letter set (National Academy of Sciences­National Research Council, 1980). Althoughthe Sloan letters are said to be "about as nearlyequal in legibility as can be obtained with simplecapital letters" (Sloan, 1959), we do not knowwhether this holds for large, low-contrast let­ters. It is desirable to use letters which areequally well identified and it is possible that analternative letter set or letter forms would bebetter in this respect. We intend to study this.

The chart is read in the conventional wayfrom left to right and from top to bottom. Oneach line of the chart there are two groups, eachof 3 letters. The letters in each group have thesame contrast and the contrast of the letters ineach successive group is less than that of the

D. G. PELLl et al.192

0.05 OSN ZCN 0.20

0.35 SHO CHV 0.50

0.65 KDR ZKD 0.80

0.95 HCD SNO 1.10

1.25 o V S DRH 1.40

1.55 DSN HRK 1.70

1.85 DNZ NVH 2.00

2.15 RDH HKZ 2.30

Fig. 2. Key to the Pelli-Robson chart as used for scoring asubject's performance. The marginal numbers give the logcontrast sensitivity corresponding to the neighbouringgroup of three letters. For example, the number 2.00appears next to a group of letters with a contrast of 1/100

(i.e. 1%) indicating a log contrast sensitivity of 2.00.

letters in the preceding group by a factor of1/.J2. The correct relationship between the con­trasts of the different groups of letters is assuredby the use of the variable-dot-density half-tonetechnique, while the correct absolute contrastlevel is adjusted during printing and checkedafterwards photometrically.

The size of the letters on this new chart is suchthat they sub tend 0.5 deg at 3 m, the distance atwhich we suppose the chart will normally beused. However, in order to detect a loss ofcontrast sensitivity due to wide-angle light­scatter (e.g. from cataract) it is necessary for alarge part of the subject's visual field to beapproximately as bright as the chart itself, sothe chart should be mounted on a white wall. Ifthis cannot be provided, then it would be prefer­able to use a chart with smaller letters and largewhite margins at a near-vision distance. Forassessment of low vision the standard 3 m chartshould be used at a shorter viewing distance, sothat the letters will be well within the acuitylimit.

The subject reads the letters on the chart,starting with those of highest contrast, andcontinues until two or three of the letters in onegroup are incorrectly named. The examinerscores the subject's performance on a key likethat in Fig. 2, which shows all the letters at fullcontrast, and gives the log contrast sensitivitycorresponding to each group. The subject'sscore is then determined by the previous group(the last group in which two or three letters werecorrectly named).

The order in which the letters appear on thechart is randomised but, in order that subjectsshould be aware of the character set being usedin the lower-contrast area of the chart, all ten of

the Sloan letters appear in the top three lines ofthe chart. For most subjects these first high­contrast characters will be readily visible andwill serve not only to familiarise them with thecharacters but also with the nature of the chartand the test procedure.

PARAMETERS OF THE NEW CHART

In deciding on the parameters of the newchart, i.e. the number of letters per line, thedecision rule for finding the subject's thresholdcontrast and the magnitude of the contrastchange from line to line, we developed a statis­tical model for the testing procedure. Thismodel was designed to enable us to choose theoptimum parameters that would maximise theaccuracy of the measurements provided by thechart. The model has two parts, the observerand the chart (with test procedure).

The chart

Each letter on the chart is assumed to be arandom sample from an alphabet of N letters,usually 10. The letters are arranged in groups ofm letters, all of equal contrast. The first groupis assumed to have unit contrast (zero logcontrast; somewhat greater than the maximumcontrast on a printed chart) and each sub­sequent group to be at a lower contrast, the logcontrast being reduced by a constant decrement,b. The subject starts reading at the first groupand continues reading subsequent groups untilfailing a group. The criterion for passing agroup (i.e. not failing) is to name correctly atleast k of the m letters. The raw score assignedto the subject is the log contrast of the lastgroup passed (i.e. for which the subjectidentified at least k of m). If the observer failsthe first group then the raw score is the logcontrast of the first group plus b.

The observer

The goal of the test is to measure theobserver's threshold log contrast, t. We assumethat the observer's probability of correctlynaming a letter at log contrast x is given by apsychometric function, P(x - t)

P(x-t)=(I-f)W(X-t)+fg (I)

where W(x - t) is the probability that the sub­ject will correctly see the letter, c is the prob­ability of the subject making a "misreporting"error, i.e. that the subject will report a randomletter instead of the one actually "seen", and the

VRS DR

Fig. 1. A miniature Pelli-Robson Letter-Sensitivity Chart. It should be noted that the contrast levels of the letters in thisminiature reproduction of the chart are not exactly those on the full-size chart as listed in Fig. 2.

193

Measuring contrast sensitivity 195

guessing rate g is the probability of correctresponse at zero contrast. We set g = l/N,where N is the number of letters in the alphabet.W(x - t) is assumed to be a Weibull (1951)function

W(x - t) = 1 - (1 - g) exp[ _lOJl(x~t)], (2)

where f3is a parameter controlling the steepnessof the Weibull function. The psychometric func­tion is sigmoidal with a lower asymptote g, atransition section with steepness controlled byf3,and an upper asymptote 1 - f. The lowerasymptote is the guessing rate g which we setto l/N.

The calculation

Let the log contrasts of the groups on thechart be xo, x" x2 , • • • , where Xi = X o+ ib is thelog contrast of the ith group, b is the step size,and Xo is the log contrast of a nonexistentzero-th group, i.e. the score assigned whenthe subject fails the first group. The probabilityof passing the ith group is given by the proba­bility of correctly identifying at least k of mletters.

Ppass(i)= f (j)rt»,- t)j[l -'P(x i - t)]m- j

j w k mfor i > 0 (3)

where (~) is the binomial coefficient

(~) =j!(mm~ j)!' (4)

The probability of passing the nonexistentzero-th group is 1

strictly necessary. For a fixed set of parametersone can always print a scoring sheet that assignsmodified scores x; so that the mean modifiedscore equals threshold. However, in practice theexaminer cannot correct for parametricvariation from subject to subject and this is anunnecessary refinement.

Equations (7) and (8) converge quickly, sothat the mean score and variance are easilydetermined by computer calculations. Initialcalculations showed that the variance (1~ andmean error r, - t are remarkably insensitive tot. However, we did consider more than onevalue of threshold, in case it matters whetherthreshold contrast is equal to the contrast of agroup or to an intermediate level between twogroups. Therefore we assumed threshold wasequally likely to take one of two values, a logcontrast of t, = - 2 (i.e. 1% contrast), or a logcontrast of t2 = - 2 + b/2, i.e. half a step higher.The overall mean error, e, and variance, (12, arecomputed from the separate mean errors andvariances

el = rrl - t,

e2 = r'2- t2

e = (e, + e2)/2

(12 = «(1~1 + (1~J/2 + (e] - e2)2/4.

The parameter f3was fixed at 3.5. Increas­ing or reducing f3is equivalent to making areciprocal change in the step size b (and therelative threshold t - Xo, but xo= 0 and theresults depend only weakly on t).

(5) RESULTS

i =0 to OC!

The variance of the score, will be given by

(1~= L (x i-r)2P i • (8)

Obviously the mean score, rr' will dependupon the threshold, t. It might seem that theideal chart would yield rr = t, but that is not

The probability Pi of obtaining score Xi is givenby the product of the probability of passing allgroups up to and including group i and theprobability of failing group i + 1

Pi = [1 - r;» + 1)] 11Ppass(j). (6)j~O to i

Figure 3 shows the overall standard deviation(1 of the test score as a function of the step sizeb for various passing criteria k and numbers mof letters per group with the initial assumptionthat there is no misreporting error, i.e. c = o.

As one would expect, reducing the step sizereduces the standard deviation. However, notethat as the step size is made smaller and smallerthe benefit of further reduction wanes. For mostof the conditions, when the step size is largerthan 0.1 (i.e. a tenth of a log unit) the standarddeviation is smaller than the size of a single step.When the step size is smaller than 0.1 thestandard deviation is mostly larger than a step.Note that the various passing criteria (various kof m) affect standard deviation in the directionone would expect, with more letters per group

(7)rr = L «,r..i=O to 00

The mean score will be

D. G. PELLI et al.

1.00

- -,VI

~x

c .~0uCl'.9c0 0.10+: x 1 of 10

> 0 1 of 2.,• 2 of 31J

~ 0 3 of 40 • 4 of 5

1JC

El/)

-L.LL.LI..l..I.ll0.0110 100 1000

Number of letters on chart

Fig. 4. Standard deviation of estimated log contrast sensi­tivity vs number of letters on the chart, for various passing

criteria, assuming no misreports, £ = o.

• 1 of 1

x 1 of 2

n 2 of 3

o 3 of 4

• 4 of 5

196

1.00

-VI0...E8co.9

c.s 0.10-.~>Q>

1J

'E0

1JC0if,

0.010.Q1

J II II ! ! LLLLJJ

0.1

Step size ( log contrast)

Fig. 3. Standard deviation of estimated log contrast sensi­tivity vs step size, for various passing criteria, assuming no

misreports, £ = o.

Probabi Lity of m isreporting

Fig. 5. Standard deviation of estimated log contrast sensi­tivity vs probability of misreporting, for various group sizesand passing criteria. The alphabet size N is always 10,exceptas indicated. The Ginsburg chart has a I of I passing

criterion and an alphabet size of N = 3.

in the parameters of the observer, or behaviordifferent from that of our model observer. Thereare an unlimited number of ways in which realobservers can deviate from the simple modelassumed here, but we believe that as apractical matter the performance of experiencedobservers is reasonably well modeled by thepsychometric function in equation (1), andthat the main difference between experiencedand inexperienced observers is that the in­experienced observers have a higher (and morevariable) probability c of misreporting theirobservations.

Figure 5 shows the standard deviation of thetest score as a function of the probability e ofmisreporting. A logarithmic step size of 0.15

• 40f5

x 10fl1N=31

+ 20f3(N =31

G 1 of 1

o 1 of 2

• 2 of 3

o 3 of 4

0.01 ..11111111 '1IlII"! ! !!IIwl ! II 111111 'I "'!Ill0.0001 0.001 0.01 0.1

u;~cooCl'

3co

o>Q)

1J

1:'o

1JCoif,

tending to reduce the standard deviation. How­ever, the effect is modest, and two of the condi­tions cross, so that for the smallest step size theI of 2 rule yields the largest standard deviation.

In a contrast-sensitivity chart all the lettersare the same size, so a chart of fixed areawill accommodate a fixed number of letters.Doubling the number of letters per group willrequire halving the number of groups on thechart. The proper comparison then, amongrules with different numbers ofletters per group,is to examine the standard deviation as a func­tion of the total number of letters on the chart.Figure 4 replots the data of Fig. 3 in thisway-standard deviation vs number of letters inthe chart. The number of letters per chart iscalculated as 2m lb.

Surprisingly the lower three conditions arenearly superimposed, showing that, by this mea­sure, it makes no difference whether we havesteps of 2.0/9 = 0.22 among 9 groups of 5 letters(and require 4 of 5 to pass) or steps of2.0/15 = 0.13 among 15 groups of 3 letters (andrequire 2 of 3 to pass). The upper two condi­tions, I of I and I of 2, yield higher standarddeviations over the entire domain (i.e. charts ofless than 10 to more than 100 letters). Thus weshould use at least 3 letters per group to obtainthe highest efficiency, i.e. the lowest standarddeviation for any given number of letters on thechart.

At this point we must consider theappropriateness of our model of human per­formance. It is important that the test be robust,so that the mean and standard deviation of thescore be at most slightly affected by variations

Measuring contrast sensitivity 197

1.0

- 0 1 of 3..~ /. I

xx 2 of 3

c: •0 • 3 of 3 I xo

I01 •2 /•c: / x.~

/0 •'> /Q>'0 • ~'E0'0 /c: /2 •If) ./ "xx-x __L! lllllll • I III!!!!0.1

0.0001 0.001 0.01 0.1

Probability of misreporting

Fig. 6. Standard deviation of estimated log contrast sensi­tivity vs probability of misreporting, for groups of three

letters and various passing criteria.

(i.e. a factor of .)2 in contrast) is assumed; bothour new chart and the Ginsburg chart use thisstep size. An alphabet size, N, of ten letters isassumed for our new chart, and three for theGinsburg chart (gratings of three different ori­entations). The main feature of this graph,which shows the effect of adopting variousdecision rules, is that for each rule the standarddeviation of the test score is essentially un­affected by infrequent misreporting errors, butbeyond some critical probability of misreport­ing the standard deviation rises disastrously.Of the curves relating to a ten-alternative chart,the least satisfactory is clearly that for the 1 of1 rule for which the standard deviation is morethan doubled (to 0.4 log units) if the probabilityof misreporting is only as high as 1%. A similarunsatisfactory behaviour is seen with the three­alternative Ginsburg chart which also uses a 1of 1 rule. The Ginsburg chart differs from ourchart in two ways, a smaller alphabet of 3instead of 10, and a 1 of 1 rule instead of a 2 of3 rule. Figure 5 includes graphs for the 1 of 1and 2 of 3 rules with the smaller alphabet,N = 3, as well as the usual N = 10. Comparingthese graphs shows that reducing the alphabetonly increases the standard deviation somewhat,and does not affect the degree of tolerance tomisreporting. Thus the use of only three testpatterns in the Ginsburg chart is only a minordrawback, increasing the standard deviationslightly, but the use of a strict 1 or 1 criterionis a major flaw, resulting in high susceptibility tomisreporting errors. This difference in sus­ceptibility to misreporting errors may explainwhy Rubin (1988) found poor test-retest re-

liability using the Ginsburg chart (52% intra­class correlation for normal subjects and 60%for patients) and high test-retest reliability usingthe Pelli-Robson chart (98% and 86%).

We suspect that even experienced observershave a probability of misreporting of around1% and that inexperienced observers may havea higher probability, perhaps occasionally ashigh as 5%. Note that the left-to-right orderingof the curves is in order of leniency; the moreforgiving the rule, the more misreporting can betolerated without loss of accuracy. It is clearthat the 2 of 3 rule compares favorably with theother rules, tolerating more misreporting thanall but the 1 of 2 rule, but having nearly halfthe standard deviation of that rule when mis­reporting is rare.

Figure 6 compares the 2 of 3 rule with other3-letter rules: 1 of 3 and 3 of 3. As in Fig. 5, themore lenient rules provide more tolerance formisreporting. However, the 2 of 3 rule is agood compromise, having the lowest standarddeviation for misreporting rates in the range0.007-2%.

Figure 7 shows that the misreporting ratealso affects the mean score. This is becausemisreporting may cause a subject to fail one ofthe many groups before reaching threshold. Theplotted bias is the difference between the meanscore at a particular misreporting rate and themean score when the misreporting rate is zero.The graph is qualitatively similar to Fig. 6, butthe critical misreporting rates are higher, so thatthis bias is not likely to be important in practice,as the standard deviation with high rates ofmisreporting would already be prohibitivelylarge. However, it is worth pointing out that

1.0

0.8o 1 of 3u;x 20f 3 •e

Ic: 0.6 • 30f 30o01

2 0.4 -'" /0

iii0.2 •/

/-

0.0 LL.J.JGI4~~,l..a..ay~1=t'l..or;iiI:·_-'-..LJ.J.WJJ0.0001 0.001 0.01 0.1 1

Probability of misreporting

Fig. 7. Bias in the estimated log contrast sensitivity vsprobability of misreporting, for various passing criteria.

198 D. G. PELLI et al.

the graphs of standard deviation eventually fall(when the bias rises) at high misreporting rates(Fig. 5). This is because the misreporting rateseventually become so high that the observerregularly fails the first group. Thus empiricalevidence of a consistently high test score doesnot necessarily imply accurate measurement ofa high threshold, as it could also be the result ofa high misreporting rate.

Conclusions from modelling

Statistical modelling of the testing processindicates that for minimum standard deviationof the final threshold estimate there should be atleast 3 letters per group. To minimise the effectof misreporting it is essential that the rule belenient. A 2 of 3 rule is essentially immune tomisreporting at rates up to about 1%. If mis­reporting rates are higher than 2% then an evenmore lenient rule should be used. A I of 3 ruleis immune up to about 10%, but the cost is thatat low misreporting rates its standard deviationis nearly twice that of the 2 of 3 rule. Theseconclusions have been incorporated in thedesign of the new chart described earlier.

The poor test-retest reliability of the Gin­sburg chart (Rubin, 1988) may be due to its 1of 1 criterion, partly because having only oneletter per group is inefficient (see Fig. 4), butmostly because a strict criterion makes the testextremely vulnerable to misreporting errors.Figure 5 indicates that if the observer misreportsonly 1% of the trials the standard deviation willbe increased from 0.2 to 0.5 log units.

GENERAL DISCUSSION

Clinical contrast-sensitivity measurementspromise to detect and assess visual diseaseswhich do not affect visual acuity. However,none of the previously available methods seemto be sufficiently quick and reliable for it to beexpected that they will be widely used. It iscertainly possible to make measurements, usingcurrently available methods, for assessing theprogress and disabling effect of known disease,though we believe that use of our new chartcould make this easier. One potential benefit ofcontrast-sensitivity measurement is the earlydetection of such disorders as diabetic reti­nopathy and glaucoma. As Hyvarinen et al.(1983) have written of glaucoma, "The largeinterindividual and possible interocular vari­ations in contrast sensitivity make it difficult toassess the effect of glaucoma until we have

routine measurements of contrast sensitivity ina standardised situation throughout life."

We hope that the new chart whose design hasbeen described and explained here may makethe routine measurement of contrast sensitivitya real possibility.

Acknowledgements-This research was supported by Na­tional Eye Institute grant EY04432 to DGP.

REFERENCES

Arden G. B. (1978) The importance of measuring contrastsensitivity in cases of visual disturbance. Br. J. Ophthal.62, 198-209.

Bailey I. L. (1982) Simplifying contrast sensitivity testing.Am. J. Optom. Physiol. Opt. 59, 12P.

Bailey I. L. (1987) Mobility and visual performance underdim illumination. In Night Vision: Current Research andFuture Directions (Symposium Proceedings). NationalAcademy Press, Washington, DC.

Campbell F. W. and Robson J. G. (1968) Application ofFourier analysis to the visibility of gratings. J. Physiol.,Lond. 197, 551-566.

Della Sala S., Bertoni G., Somazzi L., Stubbe F. andWilkins A. J. (1985) Impaired contrast sensitivity indiabetic patients with and without retinopathy: a newtechnique for rapid assessment. Br. J. Ophthal. 69,136142.

Ginsburg A. P. (1978) Visual information processing basedupon spatial filters constrained by biological data. Ph.D.dissertation, Cambridge Univ. Reprinted as AFAMRLTech. Rep. 78-129, Library of Congress 79-600156.

Ginsburg A. P. (1984) A new contrast sensitivity vision testchart. Am. J. Optom. Physiol. Opt. 61, 403-407.

Hecht S., Hendley C. D., Frank S. and Schlaer S. (1949)Contrast discrimination charts for demonstrating theeffect of anoxia on vision. J. opt. Soc. Am. 39, 922-923.

Higgins K. E., Jaffe M. J., Coletta N. J., Caruso R. C. anddeMonasterio F. M. (1984) Spatial contrast sensitivity:importance of controlling the patient's visibility criterion.Arch. Ophthal. 61, 403-407.

Hyviirinen L., Rovamo J., Laurinen P., Saarinen J. andNasanen R. (1983) Contrast sensitivity in monocularglaucoma. Acta ophthal. 61, 742-750.

Legge G. E., Pelli D. G., Rubin G. S. and Schleske M. M.(1985) Psychophysics of reading-I. Normal vision.Vision Res. 25, 239-252.

Legge G. E., Rubin G. S. and Leubker A. (1987) Psycho­physics of reading-V. The role of contrast in normalvision. Vision Res 27,1165-1178.

National Academy of Sciences-National Research Council(1980) Recommended standard procedures for the clinicalmeasurement and specification of visual acuity. Report ofworking Group 39. Adv. Ophthal. 41, 103-148.

Pelli D. G. (1987) Programming in PostScript: imaging onpaper from a mathematical description. BYTE 12(5),185-202.

Pelli D. G., Rubin G. S. and Legge G. E. (1986) Predictingthe contrast sensitivity of low-vision observers. J. opt.Soc. Am. A3, P56.

Pelli D. G., Legge G. E. and Rubin G. S. (1988) Clinicalmeasurement of contrast sensitivity. Invest. Ophthal. vi­sual Sci. (Submitted).

Measuring contrast sensitivity 199

Ratliff F. (1965) Mach Bands: Quantitative Studies on NeuralNetworks in the Retina. Holden-Day, San Francisco,Calif.

Regan D. and Neima D. (1983) Low-contrast letter chartsas a test of visual function. Ophthalmology 90, 1192~ 1200.

Regan D. and Neima D. (1984) Low-contrast letter chartsin early diabetic retinopathy, ocular hypertension, glau­coma and Parkinson's disease. Br. J. Opthal. 68, 885~889.

Robson J. G. (1983) Frequency domain visual processing.In Physical and Biological Processing of Images, edited byBraddick O. J. and Sleigh A. C. Springer, Berlin.

Rubin G. S. (1988) Reliability and sensitivity of clinicalcontrast sensitivity tests. C/in. Vision Sci. 2(3), 169-177.

Schade O. (1956) Optical and photoelectric analog of theeye. J. opt. Soc. Am. 46, 721~739.

Selwyn E. W. H. (1948) The photographic and visualresolving power of lenses. Photographic J. 88B, 6-12.

Sloan L. (1959) New test charts for the measurement ofvisual acuity at far and near distances. Am. J. Ophthal. 48,807-813.

Snellen H. (1862) Scala tipographia per mesurare i/ nisus.Utrecht.

Vaegan and Halliday B. L. (1982) A forced-choice testimproves clinical contrast sensitivity testing. Br. J. Oph­thai. 66, 477-491.

Verba ken J. J. and Johnston A. W. (1986) Population normsfor edge contrast sensitivity. Am J. Optom. Physiol. Opt.63, 724-732.

Watson A. B., Barlow H. B. and Robson J. G. (1983) Whatdoes the eye see best? Nature, Lond. 302,419-422.

Weibull W. (1951) A statistical distribution function of wideapplicability. J. appl. Mech. 18, 293~297.

Wilkins A. J., Della Sala S., Somazzi L. and Nimmo-SmithI. (1988) Age-related norms for the Cambridge LowContrast Gratings, including details concerning theirdesign and use. C/in. Vision Sci. 2(3), 201-212.

Wolkstein M., Atkin A. and Bodis-Wollner I. (1980) Con­trast sensitivity in retinal disease. Ophthalmology 87,1140--1149.

Young G. (1918) Threshold tests. Br. J. Ophthal. 2, 384-392.