428 J. Opt. Soc. Am. A/Vol. 8, No. 2/February 1991

Effects of chromatic and luminance contrast on reading

Kenneth Knoblauch, Aries Arditi, and Janet Szlyk

Vision Research Laboratory, The Lighthouse, Inc., 111 East 59th Street, New York, New York 10022

Received August 21, 1989; accepted August 24, 1990

Reading performance was measured for drifting text defined by chromatic contrast with various amounts ofluminance contrast present. With 0.12 luminance contrast added, reading performance was unaffected by thepresence of chromatic contrast over a range of character sizes varying 30-fold. When luminance contrast wasreduced to near the threshold for reading, chromatic contrast sustained reading rates of nearly 300 words perminute, almost as high as those found with high luminance contrasts. Low-pass filtering of chromatic text had aproportionately greater effect on small characters, as would be predicted from the lower bandwidth of chromaticvisual channels. Arguments are presented suggesting that reading rates for equiluminant text are sustained byluminance transients introduced by transverse chromatic aberrations of the eye.

The influence of contrast in reading is important not onlybecause text of a wide range of contrasts is encountered inthe environment but also because many ocular conditionslower the effective contrast of the reading stimulus. Moststudies of the role of contrast in reading, however, havetreated only the luminance dimension. In general, readingis found to be fastest when the luminance difference be-tween text and background is maximal. For example, usinghigh-contrast edges, Legge and Rubin' demonstrated thatfor observers with normal vision the luminance contrast andthe background luminance (or the text luminance when thebackground was dark) determine reading rate regardless ofthe color of the text. These results are consistent withthreshold studies that show that the color of a grating doesnot affect luminance-contrast sensitivity 2 for frequenciesremoved from the diffraction limit.3 4 In many situationsoutside the laboratory, however, reading involves text of onecolor with a surround of another color, in other words, withchromatic as well as luminance contrasts.

While studies have examined the interaction of chromaticand luminance contrasts on low-contrast stimuli,5-8 few haveexplored the suprathreshold domain. Tinker and Paterson9

found the legibility of colored inks on differently coloredpapers to depend primarily on the luminance differencebetween the text and the background, but the range of con-ditions that they could examine was limited by the nature oftheir stimulus medium. Lippert' 0 reported that legibility ofbriefly presented digits depended on the color differencebetween the digits and the background. While Lippert sys-tematically varied luminance and chromatic differences be-tween text and background, he used a range of charactersizes that was near the acuity limit of the chromatic-con-trast-sensitivity function. Legge et al." have recently re-ported extensive measurements of the influence of lumi-nance and chromatic contrasts on reading, but they restrict-ed their investigation to the long-wave limb of the CIEchromaticity diagram and examined only relatively largesizes.

To explore the relationship between chromatic-contrastsensitivity and reading, we examined reading performancefor text stimuli defined by various combinations of lumi-

nance and chromatic contrasts and for characters of differ-ent sizes. We also briefly examined the influence of blurringon chromatic text. Blurring the text served two functions.First, it minimized luminance transients that might occur inthe retinal image of the text owing to transverse chromaticaberrations. Second, it permitted us to assess whether thebandwidth criterion for reading chromatic text was similarto that found for luminance text. An analysis presented inAppendix A suggests that chromatic aberrations could pro-duce substantial luminance transients under certain condi-tions.

METHOD

Apparatus and CalibrationAll stimuli were generated by an Amiga 1000 computer anddisplayed on a 13-in. (33-cm) color video monitor (AmigaModel 1080). Luminances of foreground and backgroundcolors were measured directly from the monitor with a digi-tal photometer (Spectra). The chromaticity coordinates ofall stimuli were measured with a spectrophotometer withtristimulus filters (Photo Research), which was also used tocheck the luminance, and were subsequently checked with aspectroradiometer (EG&G).

StimuliIn the following discussion X and At will be assumed to repre-sent lights of different spectral composition that, additional-ly, map into distinct coordinates in the CIE chromaticitydiagram. The luminance contrast of all text stimuli will bespecified by using Michelson's formula,12 regardless of thecolor difference between text and background. The pres-ence of chromatic contrast in a text-background pair will bedefined as there being a difference in their chromaticitycoordinates, but no further attempt will be made to quantifythis contrast since it depends in a complex manner on boththe location and the distance between the two points in thespace. Quantifying the chromatic contrast within a uniformchromaticity diagram would also be of limited use, as suchdiagrams that are available are uniform only over local re-

0740-3232/91/020428-12$05.00 © 1991 Optical Society of America

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Vol. 8, No. 2/February 1991/J. Opt. Soc. Am. A 429

L

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At

1

C

Al

I II II II II I

I2 I

2

L

A

3

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I II

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POSITION

Fig. 1. Luminance profiles of contours of characters illustratingvarious combinations of luminance and chromatic contrasts. 1, Adark contour embedded in a background of spectral composition t

illustrates maximum luminance contrast (L). 2, A contour of spec-tral composition X is matched in luminance to a background ofdifferent spectral composition and of different chromaticity, illus-trating chromatic contrast (C) with no luminance contrast. 3, Acontour of the same spectral composition as the background illus-trates an intermediate value of luminance contrast with no chromat-ic contrast. 4, A contour matched in luminance contrast with case 3but differing in chromatic contrast illustrates a stimulus with lumi-nance and chromatic contrast (L + C).

gions of the space. Instead we utilized an indirect approachto assess the influence of chromatic contrast on readingperformance.

The conditions of the experiments described below arerepresented in Fig. 1 as luminance profiles of a contourembedded in a background. Luminance profiles 1 and 3, inwhich the spectral composition of the background u is iden-tical (or metameric) to that of the text stimuli will be re-ferred to as having only luminance contrast or achromaticcontrast. The two profiles differ in the size of the luminancecontrast present and in their mean luminance, although thebackground luminances are equal. Profile 1 displays themaximum attainable luminance contrast, and profile 3 dis-plays a lower contrast. In profile 2 the luminance contrast iszero, but, because there is a difference in chromaticity be-tween the text and the background (text of spectral composi-tion X, background g), chromatic contrast is present. Inprofile 4 the contours of the text stimuli are defined in termsof both a luminance and a chromaticity difference. Profiles3 and 4, then, which are matched in luminance contrast,mean luminance, and background luminance, differ only intheir chromatic contrast. Differences in reading rate be-tween these two conditions then would be due to the pres-ence of the chromatic contrast.

For all conditions, the background was set to a luminanceof 48 cd/M2 with chromaticity coordinates x = 0.31, y = 0.35,represented on the chromaticity diagram in Fig. 2 as pointW. All observers described the color appearance of thebackground as white. This point also represents the chro-maticity coordinates of the text used for conditions consid-ered achromatic. The chromaticity coordinates of the lightsused for text stimuli in the chromatic conditions, indicatedby points Y, BG, and M in Fig. 2, are given in columns x andy of Table 1. In our experimental conditions the hues ofthese lights were described by normal observers as yellow,blue-green, and magenta or purple, respectively.' 5

Observers were tested with achromatic text at three con-

trast levels, 0.02, 0.12, and 0.96. The text was defined as aluminance decrement in all cases. The lowest contrast wasproduced by placing a beam splitter at a 45-deg angle tocover the text window and by optically superimposing a lightbox, suitably filtered to maintain the correct backgroundluminance and color, as a veiling source. The Amiga's colorresolution is only 4 bits per gun. Within this limited colorgamut, we could not produce equiluminant pairs of lightsbut had to be satisfied with what we will refer to as near-equiluminant pairs. Observers were tested with such pairswith residual luminance contrasts of 0.010 (Y), 0.009 (BG),and 0.007 (M). Finally, observers were tested with lumi-nance- and chromatic-contrast text of approximately 0.12contrast. Luminance and contrasts for all text backgroundpairs are indicated in Table 1.

We used a fixed-space, sans-serif font. Center-to-centerspacing of a pair of the capital letter 0 was 1.1 cm, and theheight was 2 cm. Reading material was obtained from stan-

1.0

Y 0.5

0.0 L0.0 0.5 1.0

XFig. 2. Color specification of the text and background stimuli usedin this study represented in the CIE chromaticity diagram. Thesolid triangle indicates the color gamut measured on the video dis-play. The points indicate nominal colors W (white), Y (yellow), GB(blue-green), and M (magenta), whose chromaticity coordinates andluminances are indicated in Table 1.

Table 1. Summary of Chromaticity, Luminance, andContrast Characteristics of Text Stimulia

Luminance(cd/m2 ) Contrast

Stimulus x Y Yhigh Ylow Chigh Clow

W 0.31 0.35 37.4 1.06 0.956 0.128Y 0.41 0.55 47.4 35.3 0.157 0.010BG 0.25 0.35 47.5 36.5 0.140 0.009M 0.32 0.26 49.1 37.7 0.124 0.007

a All values above describe only the text characters. Contrasts were com-puted with respect to a white background of 48.4 cd/M2 . The Yhigh lumin-ances were used for computing the C10 _ contrasts, and the YI0w luminanceswere used for computing the Chigh contrasts.

Knoblauch et al.

430 J. Opt. Soc. Am. A/Vol. 8, No. 2/February 1991

dardized reading-comprehension tests and corresponded toninth-grade reading levels.

The influence of chromatic aberration was examined byblurring the text with a sheet of ground glass (1.4-mm thick-ness, Edmond Scientific) interposed between the observerand the display. Ground glass has been shown in a numberof contexts13 "16"17 to function as a spatially isotropic, low-passfilter with an approximately Gaussian shape. The distancebetween the ground glass and the display was chosen so thatthe bandwidth of the filter (measured by the point at whichits modulation transfer function had declined to lie) was 2cycles per character. Over a wide range of character sizes,Legge et al.13 found this value to be the minimum-spatial-frequency bandwidth required for nearly optimal reading.For our apparatus, this distance was determined by thefollowing procedure. First, a vertical sine-wave grating of1.82 cycles per centimeter (or 2 cycles per width of an indi-vidual character) was presented to an observer on a cathode-ray tube (Joyce Electronics, P4 phosphor). The mean lumi-nance was set at 50 cd/M2 , nearly the same as it was in themain experiment. In principle, the spatial frequency atwhich these measurements were made can be arbitrary aslong as it is resolvable. The observer sat at a distance so thatthe grating subtended 3 cycles per degree, as this frequencycorresponds roughly to the peak of the contrast-sensitivityfunction. While the grating was presented continuously,the observer made ten threshold settings. The contrast ofthe grating was then raised to a value e times the mean of theten threshold values. At this point, the ground glass wasintroduced between the display and the observer, and thedistance determined from the display that reduced the grat-ing again to the threshold (6 mm). This distance was used inexperiments in which it was desired that the text be blurredwith a spatial filter of bandwidth 2 cycles per character.

The spectral-filtering characteristics of the ground glass

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were checked by having an observer make heterochromaticflicker-photometric matches on a color display (Electro-home, G09-401) with and without the ground glass inter-posed between the observer and the display. The stimuluswas a disk of 1-deg diameter presented on a dark surround ata luminance of 50 cd/M2 . Two of the guns were alternatedtemporally in square-wave fashion at 12.5 cycles per second.The presence of the ground glass was not found to produce asignificant difference in the photometric settings for any ofthe pairings of the guns of the display.

ObserversTwo of the authors served as observers for all experiments.Each had normal color vision and normal or corrected-to-normal visual acuity. The uncorrected distance acuities ofthe subjects were 20/10 for observer AA and 20/20 for JS.The main results were replicated on naive observers.

ProcedureObservers received extensive practice on the reading taskduring collection of pilot data for this and other projects.Observers were light adapted to the screen luminance for 5min before each session. One color contrast was tested in agiven session, and the size of the text was randomly variedwithin sessions. Color contrast was varied randomly acrosssessions.

Each experimental trial consisted of a presentation of aline of 80 characters of left- and right-justified text (spacesincluded) drifted at fixed velocity from right to left througha window of 6-character width located in the middle of thescreen. The reduced window size confined the text to arelatively homogeneous portion of the screen. Legge et al.'3

have shown that optimal reading performance with driftingtext is obtained when the window is larger than a 4-characterwidth. Preliminary data collected with a 45-character win-

10 0.1 1 10

CHARACTER WIDTH (degrees)Fig. 3. Reading rate for achromatic text as a function of character width for three luminance contrast levels for two observers. Error bars indi-cate standard deviations about the geometric means.

I I I " I I I'I I I I I II 'I~~~~~~~~~~~~

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0.96 -

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Vol. 8, No. 2/February 1991/J. Opt. Soc. Am. A 431

1 10 0.1

I I I I 1 1I

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CHARACTER WIDTH (degrees)Fig. 4. Reading rate as a function of character width for text that is approximately matched in luminance contrast but differs in chromaticcontrast. Error bars indicate standard deviations about the geometric means.

dow yielded results similar to those presented here.18 Theobserver tracked the text monocularly and read it aloud.Text velocity was increased until the observer could not readthe text without making errors. The minimum speed forwhich this was true, corrected for the number of errors, wastaken as an estimate of reading rate. Three measurementsof reading rate were obtained for each character size withineach session.

Character size was varied by changing the viewing dis-tance. Trial lenses aided accommodation at close dis-tances.' 9 Character width varied between 0.2 and 6.0 deg ofvisual angle.

RESULTS

Figure 3 shows data from two observers, measuring readingrate as a function of character width for text that varied onlyin luminance contrast. Each point represents the geometricmean of three measurements. At high contrast, reading ratevaried only slightly for character widths between 0.2 and 6deg (filled circles). Reading rate peaked at character widthsbetween 0.2 and 1 deg and fell off gradually from this peakout to character widths as large as 6 deg. Lowering theluminance contrast to 0.12, approximately a factor of 7 (opencircles), reduced reading rates by an average of 0.1 log unitfor the two observers. This small difference, however, wasnot statistically significant (F,,l = 29.13; p = 0.12). Lower-ing the contrast to 0.02 reduced reading rates more dramati-cally, with the largest effects at the smallest characterwidths (filled squares). The peak reading rate shifted tocharacter widths between 2 and 4 deg for low contrasts. Inaccord with the results of Legge et al.'4 we found that read-ing performance in observers with normal vision was im-mune to luminance-contrast reduction over a large range.

Figure 4 shows, for two observers, a comparison of reading

rates in the presence and the absence of chromatic contrastfor conditions in which luminance contrast is approximatelymatched. The results for the achromatic text are replottedfrom Fig. 3 as open circles and represent performance for aluminance contrast of 0.12. The nominal hue of the chro-matic text is designated in the figure for each symbol type.Each had a similar luminance contrast (see Table 1). Forobserver AA, the presence of chromatic contrast produced asmall but systematic improvement in reading rates at thethree largest character widths tested, but no similar effectwas evident in the data of JS. The logarithm of the ratio ofreading rates between chromatic-plus-achromatic text andachromatic text is plotted in Fig. 5 as a function of charactersize. Points above the baseline indicate improvement ofreading performance due to the presence of chromatic con-trast. Statistical tests did not reveal a significant effect ofchromatic contrast on reading performance, either when thefour conditions were compared as in Fig. 4 (F3,3 = 2.57; p =0.23) or when the difference from the achromatic conditionwas evaluated as in Fig. 5 (F2,2 = 2.50; p = 0.28).

Figure 6 examines reading performance when the textcontrast was near equiluminant. Recall that the residualluminance contrast for each of the chromatic curves was lessthan or equal to 0.01 (see Table 1). For purposes of compar-ison, the achromatic-contrast curves from Fig. 3 for 0.96 and0.02 contrast have been reproduced as filled circles andsquares, respectively. All the near-equiluminant curves fellbetween these two curves. Although at the largest characterwidth some colors (Y for JS and Y and BG for AA) producedreading nearly as fast as high-luminance-contrast text, read-ing performance with near-equiluminant text was not betterthan high-contrast luminance text at any character width(F2 ,2 = 31.12; p = 0.03). Except for three points in the dataof observer AA, the near-equiluminant curves were all abovethe low-contrast achromatic curve. Since there was less

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432 J. Opt. Soc. Am. A/Vol. 8, No. 2/February 1991

luminance contrast present in the near-equiluminant curvesthan in the low-contrast achromatic curve and since theresidual luminance contrast, if present alone, would be near-ly at or below threshold, the increment in reading perfor-mance must have been due to the presence of the chromaticcontrast. Interestingly, the greatest increases in perfor-mance for the chromatic conditions were at the smallestcharacter widths. For both observers, the ordering of read-ing performance from worst to best was M, Y, and BG.

Knoblauch et al.

Figure 7 shows, for observer AA only, reading rates as afunction of character size for near-equiluminant text passedthrough a spatial filter of 2 cycles per character bandwidth.Reading rate declined sharply for characters smaller than0.7 deg. For larger characters, the relative change in readingperformance was consistent with that shown in Fig. 6 for thisrange. The ordering of the colors was the same as thatshown in Fig. 6 except for the reversal between Y and BG at 3deg. The filtering by the ground glass produced the most

I I I I IIII

Obs: AA A yellowmagenta

- blue-green

l l l l lI

1 10 0.1 1 10

CHARACTER WIDTH (degrees)Fig. 5. Logarithm of the ratio of reading rates for text with and without chromatic contrastbars indicate 95% confidence intervals about each point.

plotted as a function of character width. Error

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CHARACTER WIDTH (degrees)Fig. 6. Reading rate for text defined by nearly equiluminant chromatic contrast plotted as a function of character size. The reading functionsfor achromatic contrast text of 0.96 and 0.02 from Fig. 3 have been replotted for comparison as filled circles and filled squares, respectively. Er-ror bars indicate standard deviations about the geometric means.

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Vol. 8, No. 2/February 1991/J. Opt. Soc. Am. A 433

1000

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Fig. 7. Reading rate for text passed through a spatial filter of 2cycles per character bandwidth plotted as a function of charactersize. Error bars indicate standard deviations about the geometricmeans.

severe effects on the M text, reducing reading performancefor the larger character sizes by a factor of 5. The maineffect of the spatial filtering on near-equiluminant text wasto attenuate reading performance severely at small charac-ters.

DISCUSSION

The use of oral reading for assessing the capacity of thevisual system for processing information raises the questionof whether speech was the limiting factor in these measure-ments. While it is difficult to prove otherwise for charactersizes at the peak of the high-contrast achromatic function,for which reading speeds reached 300-500 words per minute,speech cannot be the limiting factor off the peak to eitherside for which reading speed declines. It should be empha-sized that at no time did observers ever receive a 300-wordtrial, as each 80-character line averaged approximately 12words in length. The short line length permitted observersto recite a portion of the trial from memory. Our prelimi-nary data'8 and that of others"",13 who have utilized silentcontrols indicate that somewhat higher reading rates can beobtained with silent reading, but the forms and relations ofthe functions do not change. Unfortunately, with silentreading, error rates are not easily assessed. Higher readingrates have also been obtained by using shorter lines.20

Our reading rates are not exceptional for these conditions.References 11 and 13, with similar conditions, regularly re-ported peak reading rates of at least 300 words per minute,and Legge et al." have recently reported an observer whoread at rates as high as 670 words per minute. Anecdotally,observers often perceived themselves as speech limited whenthey were not. For example, for 2-deg high-contrast achro-matic letters, observer AA complained that his readingspeeds were limited by his ability to speak fast, but, when

character size was reduced to 0.3 deg, his reading perfor-mance increased 67%.

Our results show that, for observers with normal spatialand color vision, the effect of chromatic contrast on readingperformance depends on the level of luminance contrastpresent. For luminance contrasts that produced only amodest decrement in reading performance, the introductionof chromatic contrast produced no systematic effect on read-ing rate. For example, we did not see an improvement inreading performance with chromatic contrast added to aluminance contrast of 0.12. We emphasize that these re-sults are not likely to reflect a ceiling effect on oral perfor-mance at character sizes off the peak, where reading speedhas declined. These results are consistent with those ofLegge et al.," who found luminance and chromatic contraststo combine independently in determining reading speed.

When luminance contrast was sufficiently lowered, effectsof chromatic contrast were evident. In general, chromatictext with luminance contrast less than 0.01 produced higherreading rates than achromatic text with 0.02 contrast. Theimprovement in performance was greatest for the smallestcharacter sizes. Reading performance with near-equilu-minant text was quite high, reaching nearly 300 words perminute, though not better than that for luminance text ofhigh contrast at any character width. It is important topoint out, however, that higher chromatic contrasts couldhave been generated by using a background other than W(e.g., Y against B). Thus rates higher than those measuredhere might be attainable.1'

Lippert 0 found that legibility of digits as measured with areaction-time task could be predicted with a color-differencemetric based on a modification of CIE 1976 (L*, u*, v*)

1.5

1.0

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0.5

0L _400 500 600 700

WAVELENGTHFig. 8. Schematic diagram of the support of the spatial and spec-tral distribution of a Y bar in a W surround. The shaded areaindicates a region of zero luminance. The dotted curves indicatethe spatial skewing of the spectral distribution that would result inthe retinal image as a result of lateral chromatic aberrations. Theposition scale in this diagram is arbitrary.

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434 J. Opt. Soc. Am. A/Vol. 8, No. 2/February 1991

uniform color space. He presented a nomogram for assess-ing the legibility of character-background pairs for charac-ters in the size range 0.36-0.55 deg. To evaluate Lippert'sresult and our own, we must consider the constraints placedon reading rate by contrast sensitivity with respect to char-acter size.

Recent estimates of the contrast sensitivity for chromaticgratings place the high-frequency cutoff at between 10 and12 cycles per degree.2 ' Assuming letters that are the dimen-sion of standard Sloan optotypes used in acuity testing (5times as wide and high as the stroke width), these values seta limit of 0.21 to 0.25 deg on the minimum size for resolvingthe chromatic spatial variation of letters. The font used inthe present study was approximately twice as high as it waswide so that observers might be expected to have a some-what lower letter-size limit. Nevertheless, reading rate fornear-equiluminant text began to fall for character widthsbelow approximately 0.3 deg.

Legge et al.'3 have shown that a minimum bandwidth of 2cycles per character is necessary for sustaining the maxi-mum reading rate of text defined by luminance contours.This value is based on the le decay point of low-pass spatialcharacteristics that these investigators used for filteringtheir text stimuli. From Mullen's data2l for chromatic con-trast sensitivity, the le bandwidth is approximately 3 cycles

3.0-deg letters 0.7-deg letters

per degree for both R-G and Y-B chromatic-contrast-sensi-tivity functions. By this criterion, reading rate should de-cline for equiluminant text with characters smaller than 0.67deg. This argument depends on the assumptions that thechromatic-contrast-sensitivity function behaves as a low-pass filter and that, as with luminance text, observers re-quire at least 2 cycles per character for reading chromatictext, optimally. By the above arguments, the size range thatLippert'0 examined would seem to be below what is optimalfor mechanisms sensitive to chromatic contrast. We exam-ined characters in the range 0.2-6.0-deg wide and found noevidence to suggest a rolloff in performance until well below0.67 deg. Thus, at first glance, reading performances withnear-equiluminant text seem to be superior to those thatwould be predicted by the constraints of chromatic contrastsensitivity.

We can think of two possible explanations for the highreading rates for small characters defined by near-equilu-minant contrast. Throughout this study luminance wasdefined in terms of photometer readings based on the CIEstandard observer. Individual deviations from the standardobserver may have introduced luminance cues that providedsufficient contrast in order to support reading in the pres-ence of unresolved chromatic contours. The coarsely sam-pled color gamut of the Amiga color space, unfortunately,

0.2-deg letters

-0.5 0 0.5 1.0 -O.5 0 0.5 1.0

POSITION (deg)-0.5 0 0.5 1.0

yellow/white

magenta/white

blue-green/white

Fig. 9. Computed horizontal luminance profiles of stroke of a character X nearly equiluminant with the background u after the introduction oftransverse chromatic aberrations for characters of 3.0-, 0.7-, and 0.2-deg width for each of 3 chromatic-contrast conditions. The characterstroke is one-fiftl of tle widt of te character. Te edges of the stroke are indicated only in the top, left-hand figure by dotted lines. Lateralchromatic aberration results in luminance transients on each side of the stroke. A magnified detail of the transients is presented in each of theright-hand plots. The offset of the optic from the visual axis is 5.0 deg.

6

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Knoblauch et al. Vol. 8, No. 2/February 1991/J. Opt. Soc. Am. A 435

3.0-deg letters

-0.5

0.7-deg letters 0.2-deg letters

0 0.5 1.0 -0.5 0 0.5 1.0 0.5 0 0.5 1.0

POSITION (deg)Fig. 10. Same as Fig. 9 except that the offset of the optic from the visual axis is 2.0 deg.

prevented us from using individual corrections. Neverthe-less, both observers performed similarly with respect to allconditions examined, as have several other, naive subjectsused in pilot experiments.

A second possible source of luminance cues is chromaticaberrations of our observers' eyes. Transverse (or lateral)

chromatic aberrations can produce a wavelength-dependentspatial shift in the retinal image. This shift is a prismaticeffect that results from the lack of correspondence betweenthe optical and visual axes of the eye. The effect of the shift

on luminance edges defined by broadband light is to reduce

the contrast of high-spatial-frequency components. In-

deed, this phenomenon has been held to be the primary

factor limiting luminance-contrast sensitivity near the acu-

ity limit in broadband light.2 2 With chromatic stimuli, how-

ever, spatial luminance transients are usually introducedthrough the spatial skewing of the spectral distributions on

the two sides of an edge (Fig. 8).Figures 9 and 10 show predicted luminance profiles after

the introduction of transverse chromatic aberrations for

character strokes from various width fonts. (See Appendix

A for details of calculations.) The stroke in each case was

defined by a spectral distribution X taken from one of thethree near-equiluminant text conditions of the currentstudy in a surround of W light, , (symbols and edge locations

are indicated only in the top left-hand boxes of Figs. 9 and10). The skewing of X with respect to ,4 results in a decre-

mental transient on one edge and an incremental one at theother. These spatial transients would outline each charac-

ter, increasing its effective luminance contrast. The blur-ring effects from axial chromatic aberrations and the linespread of the eye have not been included in these simula-tions in order to emphasize specifically the prismatic effects.While the line spread of the eye would be expected to lowerthe effective luminance contrast, our calculations (AppendixA and Table 2) indicate these transients to be more effectivethan threshold stimuli. The figures show that, as strokewidth is reduced, the distance between the luminance tran-sients decreases. At a sufficiently small character width,the stroke would be defined almost entirely in terms ofluminance contrast.

Recently Thibos, et al.2 3 have measured an average valueof 2.0 deg for the average offset between the optic and visual

Table 2. Characteristics of Luminance TransientsGenerated by Lateral Chromatic Aberrations

Half- PeakWidth at Deviation Estimated

Half-Height from Mean Visibility(sec) (cd/M2) Factor

Eccentricity (deg) 5.0 2.0 5.0 2.0 5.0 2.0Stimulus

Y/W 19.0 7.8 1.8 1.8 2.4 1.1M/W 9.4 3.8 6.5 6.5 4.9 1.9BG/W 10.3 4.2 12.9 12.9 12.2 4.8R/G 10.0 4.0 5.0 5.0 21.6 8.6

6

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0

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DIj

2

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0~~~~~~~~~~~~~~~~~~~5

0~~~~~~~~~~~~~~~~~~~~~~~~~~

-0.02 0.02

0

20 - _ ~~~~~~~~~~~~~~~6040 - 6

0~~~~~~~~~~~~~~~~~~~4

-0.02 0.02

.0

.0

20~ ~~~~~~~~~~~~5 i i0 I , I . I . I I . I . I . I I ' I ' I

30-0.02 00

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yellow/white

magenta/white

blue-green/white4

I

v

436 J. Opt. Soc. Am. A/Vol. 8, No. 2/February 1991

axes. For the simulations in Figs. 9 and 10, however, weused values of 5.0 and 2.0 deg, respectively, because previousmeasures 24 gave a value of approximately 5.0 deg. With thebroadband sources produced by our display, the transient atthe edges of each letter would be a combination of the wave-length spectra of the text and background smeared acrossthe transient and weighted individually by the luminancesat each wavelength. The acuity limit of the visual systemwould not be expected to permit resolution of the spectrumwithin the transient, except perhaps as a fringe at the edge.Under natural viewing conditions, however, it is thoughtthat the fringes are not typically perceived as colored, owingto adaptation.2 5

We specifically avoided using an achromatizing lens thatmight lessen such aberrations because such a lens would notbe present for observers in the real world for situations inwhich they encountered text defined by color contrast. Ad-ditionally, the task requirement that observers track drift-ing text while responding orally would have made the controlof chromatic aberrations with such a lens or an artificialpupil impractical. However, to evaluate the potential influ-ence of chromatic aberrations in our data, we examinedreading performance with text blurred by a spatial filter of 2cycles per character bandwidth. While such a filter does notcompletely eliminate the luminance contrast in the retinalimage, it attenuates the high spatial frequencies in the stim-ulus that would produce the highest-contrast luminancecomponents. (See Appendix B.) Without these high spa-tial frequencies contributing to luminance artifact, readingrate appears to be constrained by the bandwidth of thechromatic-contrast-sensitivity function as evidenced by thedrop in reading performance for characters smaller then 0.67deg, indicated by the dashed line in Fig. 7. The lower overallrate may reflect the true performance of the chromatic sys-tems for reading. Thus, on one hand, the sharp decline inreading performance for spatially filtered, near-equilumin-ant text supports the notion that chromatic-contrast sensi-tivity constrains reading of equiluminant text according tothe same principles as luminance-contrast sensitivity con-strains reading of luminance text. On the other hand, thehigh reading rates for unfiltered, near-equiluminant smallcharacters raise the intriguing possibility that in practicalsituations chromatic aberrations may have the beneficialeffect of enhancing the visibility of some contours thatwould otherwise not be resolvable.

APPENDIX A: TRANSVERSE CHROMATICABERRATIONS OF BROADBAND APERIODICSTIMULI

The purpose of this appendix is to evaluate computationally,for the stimuli that were used in the present study, whetherthe luminance transients at chromatic edges in the retinalimage generated by transverse chromatic aberrations of theeye were sufficiently visible to influence reading perfor-mance. Only the effects of transverse chromatic aberra-tions are considered here. Diffraction and other aberrationshave been excluded from the analysis, although longitudinalchromatic aberrations, as well, may introduce frequency-dependent luminance components in equiluminant stimu-li.26 Our analysis considers only the effects on edges that areorthogonal to the line between the optical and visual axes,

which should be nearly vertical, even though smaller effectscan be expected from oblique edges as well.

Thibos2 2 presented a method based on a reduced-eyemodel for computing the luminance distributions resultingfrom transverse chromatic aberrations in the retinal imageof a broadband (or polychromatic) sine-wave grating. Thegrating is computed to be the integral over wavelength of theluminance distribution with a phase-shift term that dependson wavelength. The method can be adapted for computingthe luminance distribution of aperiodic images of arbitraryspectral composition.

The horizontal luminance profile L of a two-color imagecan be considered to be the integral over the wavelength ofthe sum of foreground (Fg), background (Bg), spatial (x) andspectral (X) luminance distributions:

r700L(x) = j Fg(x, X) + Bg(x, X)dX. (Al)

J400

The effect of transverse chromatic aberrations is to convolvethe foreground and background distributions with a deltafunction whose spatial offset in degrees is wavelength depen-dent:

J700L(x) = [Fg(x, X) + Bg(x, X)] * 6[x- (X)]dX.

J400(A2)

The function ¢(X) in the model of Thibos22 is given by

0(X) = - a + 3, (A3)

where e in the present circumstances is the angular differ-ence between the optical and visual axes

a = sin'1[3.98 sin(e)/5.55],

= sin-'[sin(a)/n]

and n is taken to be the index of refraction of the model eyebased on Cornu's formula for water:

n = 1.31848 + 6.662/(X - 129.2),

where is measured in nanometers. (Consult Thibos22 for adetailed discussion of the parameters a, fi, etc.) The wave-length dependence of 0 results from the formula for n. Thesymbol * represents the convolution operation.

The effect of transverse chromatic aberrations can be vi-sualized as a skewing of the spectral distribution of lumi-nance in the -x plane. The results of such a skewing areshown schematically as the dotted curves in Fig. 8 for a Y baron a W background. Since luminance at a given position iscomputed by integrating across the wavelength, the lumi-nance in the vicinity of a chromatic edge is composed ofcomponents of the spectral distributions from both sides ofthe edge. If the foreground and the background are set to beequiluminant in the object plane, then incremental or decre-mental luminance transients will result at the edge in theretinal image.

Horizontal luminance profiles for foveally viewed letterstrokes that include transients generated by the transversecomponent of chromatic aberrations are shown in Figs. 9 and10 for letter sizes and color contrasts corresponding to thoseused in the present study, under two different assumptionsabout the angular difference between the visual and opticalaxes. These profiles were computed as follows. The spec-tral energy distributions of the R, G, and B guns of our

Knoblauch et al.

Vol. 8, No. 2/February 1991/J. Opt. Soc. Am. A 437

display were measured with an EG&G spectroradiometer.The chromaticity coordinates of these were calculated. Giv-en the chromaticity coordinates and luminance of each ofthe colors that we wished to check (Table 1), the tristimulusvalues were computed, and the quantities of the gun prima-ries (kR, kG, kB) for producing each of these colors werecalculated from the system of equations

XR XG XB1 kR1 [XCi

YR YG YB _jkG = YCI, (A4)

ZR ZG ZB kBJ ZC

where the columns of the matrix are the tristimulus valuesfor each of the guns, R, G, and B, and the vector on the right-hand side of the equation represents the tristimulus valuesfor an arbitrary light C. The coefficient vector of k's wasused to calculate a linear combination of the spectral energydistributions of the gun primaries for generating the spectraldistribution of C. This spectral energy distribution wasthen weighted by Judd's modified V(X) function and nor-malized to sum to the appropriate luminance

LC(X) = tV(X)[kRR(X) + kGG(X) + kBB(X)], (A5)

where t = Y,/ExV(X)[kRR(X) + kGG(X) + kBB(X)] will becalled the spectral luminance distribution of the light C.

The skewing function (X) was computed for eccentrici-ties of 5 and 2 deg at intervals of 10 nm. While earlierstudies of chromatic aberration yielded values for the eccen-tricity of the optic axis of approximately 5 deg,24 more recentstudies23' 27 indicate a smaller value, perhaps as small as 2deg. For convenience, the skewing function was normalizedto produce zero shift at 700 nm by subtracting k(700) from it.

If LL(X) and LR(X) are the spectral luminance distribu-tions on the left-hand and right-hand sides of an edge atposition xe, then the luminance, L(x), at position x can becomputed with the algorithm

if [X < Xe + O(X)],

then L(x) = L(x) + LL(),

else L(x) = L(x) + LR(X) (A6)

evaluated iteratively over X.28

Figures 9 and 10 show the luminance profiles resultingfrom our computations for 5- and 2-deg eccentricities, re-

spectively, of the optical axis. The luminance transientsappear as narrow spikes, incremental on one edge and decre-mental on the adjacent one. They vary in peak deviationfrom the mean luminance and width at half-height accord-ing to the color pairs utilized (see Table 2). Reducing thesize of the letter simply brings the opposite edges closer

together and, in so doing, brings the transients into closerproximity. It is possible that for the smallest characters the

proximity of the transients permits them to interact to pro-duce effectively an even greater local luminance contrast.Reducing the eccentricity of the optical axis from 5.0 to 2.0deg reduces the half-width at half-height of the transientsby, on average, a factor of 2.4 but leaves the peak deviationunchanged.

An additional computation included in Table 2, in the

R/G row, is for a R character stroke against a G background

at 6.0 cd/M2. This computation was performed simply byusing the R and G guns alone at equal luminance for fore-ground and background, respectively. This condition wasincluded because it corresponds to one used by Legge et al."

in their study of color contrast and reading. While strongeffects of chromatic aberrations are usually associated withjuxtaposition of extreme portions of the spectrum, it is note-worthy that the R/G combination produces a luminancetransient with a peak deviation of greater than 80% of themean luminance. The reason for this extreme effect is un-doubtedly the narrow-band line at approximately 620 nmthat is typical of the spectral energy distribution of the Rprimary of television displays. Thus, while this calculationwas based on the R and G primaries from the current study,the conclusions are likely to hold for other display systems aswell. Legge et al." assessed the axial chromatic aberrationof their stimuli and found it to be minimal, but they did notevaluate the transverse component.

Visibility of these transients was estimated by comparisonwith data from Hecht and Mintz, 29 who measured the mini-mum width of dark lines visible as a function of luminancelevel. They demonstrated that resolution of thin lines de-pends on intensity discrimination rather than on acuity.Under optimal conditions, observers were found to be able todetect lines of 0.5-sec width and 3.0-deg length. We cancompare their results with our decremental transients be-cause the transients are sufficiently narrow that, owing tothe line spread of the eye, their retinal distribution should beindependent of size.

Hecht and Mintz found that observers could resolve linesof approximately 0.5- and 0.75-sec width, respectively, at 50and 6 cd/M2. We assume that the background had thespectral distribution of illuminant A.30 The luminance pro-file of each of these line widths was computed by assumingan eccentricity of either 5.0 or 2.0 deg from the optic axis,from the second equation of Note 28, and the areas of thedecremental profiles were computed. Similarly, the areas ofthe decremental profiles of the decremental transients ofFigs. 9 and 10 were calculated. The ratios of the areas of thetransients from Figs. 9 and 10 to the corresponding areasfrom the line-resolution data were taken as estimates of thefactor above threshold (if greater than 1.0) of the decremen-tal luminance transients. These factors are displayed in theEstimated Visibility Factor column of Table 2.

All the factors are greater than 1.0, indicating that thetransients are all above threshold. Even with the assump-tion that the difference between the optic and visual axes isas small as 2.0 deg, the luminance transients arising at thechromatic contour are predicted to be substantially abovethreshold for some of the color pairs. Among the stimuliused in the current study, the predicted effectiveness of thetransient from the BG/W condition may account for thesubstantial advantage for reading that this condition dis-played over the other near-equiluminant conditions in Fig.6. Why the Y/W condition should be superior to the M/Wcondition, which displays more salient luminance transientsby the current criterion, is not known. It should be indicat-ed that the Y/W stimulus has a greater colorimetric purity(88.2% versus 42.2%) and a greater residual luminance con-trast (0.01 versus 0.007) than the M/W stimulus. We cannotexclude these or other factors as the possible sources of thesuperiority of the Y/W over the M/W condition.

Knoblauch et al.

438 J. Opt. Soc. Am. A/Vol. 8, No. 2/February 1991

APPENDIX B: SPATIAL-FREQUENCYCHARACTERISTICS OF TRANSIENTS FROMTRANSVERSE CHROMATIC ABERRATIONS

The luminance transients in the retinal image that resultfrom lateral chromatic aberrations contribute most of theirenergy in the high spatial frequencies and outside the mini-mum bandwidth required for reading. The first part of thisassertion can be seen from considering that the transientsare localized to edges of the contours. However, the argu-ments for both parts can be made quantitative. Considerthat the horizontal luminance profile L(x) of the pair oftransients that define the two edges of an isolated contour ofan equiluminant character stroke of width x0 is approxi-mately of the form

L(x) = f(X/w) * [6(x + x/2) - 6(x - xo/2)], (BI1)

where f(x) is the luminance profile of the transient at a singleedge, w is a scaling factor that determines the width of thetransient, * represents the convolution operation, and wehave suppressed a term for the mean luminance level. TheFourier amplitude spectrum of the above expression is

L(v) = IwIF(wv)sin(7rx 0v), (B2)

where F is the Fourier transform of f and is spatial frequen-cy.

Given unimodal f and values of w in the range 0.003-0.006deg (see Table 2 and Figs. 9 and 10), we can assume that Fwill be approximately constant over the visible range offrequencies. Thus the frequency dependence of £ is sinus-oidal. is zero at v = 0 and peaks at multiples of 1/(2xo).

Recall from the Discussion that the reading speed of lumi-nance text is undisturbed if a bandwidth of at least 2 cyclesper character is provided.' 3 In terms of standard Sloanoptotypes, the minimum bandwidth would be 2/(5xo), or20% below the peak of the first mode of L. If the bandwidthrequirements for reading chromatic text are similar to thatof luminance text, then, clearly, a considerable portion of thespectrum of L can be removed by low-pass filtering withoutdisturbing the minimal information present in the characterfor optimal reading. Note also that L scales with letter size,so that one cannot argue that chromatic aberrations will beless of a problem with large letters in the way in which it canbe so argued for low spatial frequencies. This point is evi-dent from Figs. 9 and 10 also.

ACKNOWLEDGMENTS

This research was supported by National Institutes ofHealth (NIH) grant EY07747 to Kenneth Knoblauch andNIH grant AG06551 and National Aeronautics and SpaceAdministration, Ames Research Center, grant NCC2-541 toAries Arditi. We thank Alan Lewis for help with spectro-photometric calibrations and Larry Thibos, James Gordon,Laurence Maloney, and Gordon Legge for helpful discus-sions and comments.

REFERENCES AND NOTES

1. G. E. Legge and G. S. Rubin, "Psychophysics of reading. IV.Wavelength effects in normal and low vision," J. Opt. Soc. Am.A 3, 40-51 (1986).

2. M. A. Nelson and R. L. Halberg, "Visual contrast sensitivityobtained with colored and achromatic gratings," Hum. Factors21, 225-228 (1979).

3. J. Pokorny, C. H. Graham, and R. N. Lanson, "Effect of wave-length on foveal grating acuity," J. Opt. Soc. Am. 58, 1410-1414(1968).

4. F. L. Van Ness and M. A. Bouman, "Spatial modulation transferin the human eye," J. Opt. Soc. Am. 57, 401-406 (1967).

5. K. Kranda and P. E. King-Smith, "Detection of colored stimuliby independent linear systems," Vision Res. 19,733-745 (1979).

6. D. H. Kelly, "Spatiotemporal variation of chromatic and achro-matic contrast thresholds," J. Opt. Soc. Am. 73, 742-750 (1983).

7. D. Jameson, "Opponent-colours theory in the light of physiolog-ical findings," in Central and Peripheral Mechanisms of ColourVision, D. Ottoson and S. Zeki, eds. (Macmillan, London, 1985),pp. 83-102.

8. K. T. Mullen, "Spatial influences on chromatic opponent con-tributions to pattern detection," Vision Res. 27,829-839 (1987).

9. M. A. Tinker and D. G. Patterson, "Studies of typographicalfactors influencing speed of reading VII. Variations in color ofprint and background," J. Appl. Psych. 45, 471-479 (1931).

10. T. M. Lippert, "Color-difference prediction of legibility perfor-mance for CRT raster imagery," in Digest of the Society forInformation Display (Society for Information Display, PlayaDel Rey, Calif., 1986), Vol. 17, pp. 86-89.

11. G. E. Legge, D. Parish, A. Luebker, and L. Wurm, "Psychophys-ics of reading. XI. Comparing color contrast and luminancecontrast," J. Opt. Soc. Am. A 7, 2002-2010 (1990).

12. Michelson's formula specifies contrast as (Lmax - Lmin)/(Lmax +Lmin), where Lmax is the maximum luminance and Lmin is theminimum. While this formula is appropriately applied to peri-odic stimuli, it also has been used in other contexts for specifica-tion of reading stimuli (e.g., Refs. 11, 13, and 14). We utilize ithere for convenience and so that our results can be easily com-pared with other reading studies that have specified their stim-uli similarly. Strictly speaking, the contrasts specified here arenot directly comparable with those given for periodic stimuli,such as sine-wave gratings.

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

14. G. E. Legge, G. S. Rubin, and A. Luebker, "Psychophysics ofreading. V. The role of contrast in normal vision," Vision Res.27, 1165-1177 (1987).

15. Given the maximum luminance available from each of the gunsof our display, these lights were located at the limits of theequiluminant color gamut at this luminance level. The colori-metric purities of Y, BG, and M were estimated to be, respec-tively, 82.2%, 27.0%, and 42.2%, based on W. Dominant wave-lengths were estimated as, respectively, 567 nm, 493 nm, and551c.

16. G. Stigmar, "Blurred visual stimuli II. The effect of blurredvisual stimuli in vernier and stereo acuity," Acta Ophthalmol.49, 364-379 (1971).

17. G. Westheimer and S. P. McKee, "Stereoscopic acuity withdefocused and spatially filtered retinal images," J. Opt. Soc.Am. 70, 772-778 (1980).

18. K. Knoblauch, A. Arditi, and J. Szlyk, "Reading rate depends onluminance contrast but not chromatic contrast," Invest. Oph-thalmol. Vis. Sci. Suppl. 29, 44 (1988).

19. For distances closer than 2 m, observers were fitted with a triallens with dioptric power equal to the reciprocal of the desireddistance to the display. Observers were then positioned so thatthe lens was one focal length from the display. Such an ar-rangement produced a unit magnification independent of thedistance to the observer's eye behind the lens. Because theobserver's head position was free, the distance to the displaywas checked frequently throughout an experimental session,and trials were discarded if the distance of the subject to thedisplay was found to have changed.

20. A. Arditi, K. Knoblauch, and I. Grunwald, "Reading with fixedand variable character pitch," J. Opt. Soc. Am. A 7, 2011-2015(1990).

21. K. T. Mullen, "The contrast sensitivity of human color vision tored-green and blue-yellow chromatic gratings," J. Physiol. 359,381-400 (1985).

Knoblauch et al.

Vol. 8, No. 2/February 1991/J. Opt. Soc. Am. A 439Knoblauch et al.

22. L. N. Thibos, "Calculation of the influence of lateral chromaticaberration on image quality across the visual field," J. Opt. Soc.Am. A 4, 1673-1680 (1987).

23. L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, and P. A.Howarth, "Theory and measurement of ocular chromatic aber-ration," Vision Res. 30, 33-50 (1990).

24. Y. Le Grand, Form and Space Vision, M. Millodot and G. G.Heath, trans. (Indiana U. Press, Bloomington, Ind., 1967).

25. R. Held, "The rediscovery of adaptability in the visual system:effects of extrinsic and intrinsic chromatic dispersion," in Visu-al Coding and Adaptability, C. S. Harris, ed. (Erlbaum, Hills-dale, N.J., 1980).

26. D. I. Flitcroft, "The interactions between chromatic aberration,defocus and stimulus chromaticity: implications for visualphysiology and colorimetry," Vision Res. 3, 349-360 (1989).

27. P. Simonet and C. W. Campbell, "The optical transverse chro-matic aberration on the fovea of the human eye," Vision Res. 30,187-206 (1990).

28. Note that this algorithm will produce spurious results if thewidth of a contour is smaller than the variation of A, owing to theuse of discrete wavelength intervals. This was not the case forthe character sizes considered here. However, if it were, onecould utilize the formula

e0(x w) 700 6(w)L(x) = Bg(X)d X+ Fg(X)dX,

f4 00 f(x) fJ (X )

where 0(x) = -'(X), the inverse of 0, and w is the width of thecontour. It is convenient to sample the spectral luminancedistributions and the function 0(x) more densely with an inter-polating function, such as a cubic spline. The additional accu-racy of the above equation was not found to produce significantchanges in the luminance profiles calculated for characterstrokes reported here. If Fg(X) is zero, then the above equationreduces to

700 r0()L(x) = I Bg(X)d- W Bg(X)dX.

400 80(x-w)

29. S. Hecht and E. U. Mintz, "The visibility of single lines atvarious illuminations and the retinal basis of visual resolution,"J. Gen. Physiol. 22, 593-612 (1939).

30. G. Wyszecki and W. S. Stiles, Color Science: Concepts andMethods, Quantitative Data and Formulae, 2nd ed. (Wiley,New York, 1982).