Reports 513

Wasielewski P. Establishment of lens epithelial cell 10. Dziedzic DC, Reddan JR, Hartzer MK, Misra IC. Estab- lines from Emory and cataract resistant mice and their lishment of cell lines from cryopreserved lens-capsule response to hydrogen peroxide. Lens Eye Tox Res. epithelial preparations and whole lenses. Invest Oph- 1989;6:687-701. thalmol Vis Sci. 1992; 33:1038.

Masking Potency and Whiteness of have previously shown5'6 that when the of white image is constant, contrast sensitivity in Noise at Various Noise Check Sizes noise is independent of viewing distance, eccentricity, Heljd Kukkonen* Jyrki Rovamo,^ retinal illuminance, and exposure time as long as con- and Risto Nasanen% trast sensitivity is lower with spatial noise than without, that is, the effect of external noise exceeds the effect of Purpose. The masking effect of spatial noise can be in- internal noise. The spectral density of external spatial creased by increasing either the rms contrast or check noise equivalent to the internal neural noise2'4 is small- size of noise. In this study, the authors investigated the est at medium spatial and increases to- largest noise check size that still mimics the effect of ward low and high spatial frequencies.7 Therefore, in grating detection and how it depends on only high spectral densities of external spatial noise the bandwidth and spatial of a grating. will exceed the effect of internal noise at low and high Methods. The authors measured contrast energy thresh- spatial frequencies. olds, E, for vertical cosine gratings at various spatial frequencies and bandwidths. Gratings were embedded The effect of spatial noise on detection threshold in two-dimensional spatial noise. The side length of the is determined by its spectral density within the spatial square noise checks was varied in the experiments. The frequency range of the stimulus or, rather, the detec- 2 8 9 spectral density, N(0,0), of white spatial noise at zero tion filter used by the visual system. ' ' However, the frequency was calculated by multiplying the noise check spectral density of white noise has a constant value area by the rms contrast of noise squared. at all spatial frequencies, and its masking effect can 2 Results. The physical signal-to-noise ratio at threshold therefore be described by a single number. For spatial [E/N(0,0)]°l5 was initially constant but then started to noise, generated by adding a random number to each decrease. The largest noise check that still produced a noise check, spectral density is constant at low spatial constant physical signal-to-noise ratio at threshold was frequencies but dies out oscillating at high spatial fre- directly proportional to the spatial frequency. When quencies. Thus, such spatial noise always has a limited expressed as a fraction of grating cycle, the largest noise bandwidth, and there is a spatial frequency above check size depended only on stimulus bandwidth. The which noise is no more white. smallest number of noise checks per grating cycle To increase the spectral density of spatial noise needed to mimic the effect of white noise decreased consisting of checks, we can either increase the con- from 4.2 to 2.6 when the number of grating cycles in- creased from 1 to 64. trast of noise or the size of checks. When noise con- trast cannot be increased further, the only way to in- Conclusion. Spatial noise can be regarded as white in crease the spectral density is to enlarge noise checks. grating detection if there are at least four square noise However, an increase in noise check size also reduces checks per grating cycle at all spatial frequencies. Invest the bandwidth within which noise can be considered Ophthalmol Vis Sci. 1995;36:513-518. white. Therefore, it is important to know how large checks can be safely used without compromising the When external is strong in comparison whiteness of noise within the detection filter used by to the internal neural noise of the visual system, it the visual system. A common estimate for the band- reduces human contrast detection performance.1"4 width of the detection filter is 1 to 2 octaves,2'9 but it The reduction is determined by the signal-to-noise ra- depends on both the bandwidth of the stimulus8 and tio, which is constant at the detection threshold.2'4 We the characteristics of the noise.2 These results imply that there is no single estimate for the bandwidth

From the *Department of Communication and Neurosdence, University of Keek, within which noise is collected. Therefore, the largest Staffordshire, United Kingdom; the ^Department of Vision Sciences, University of noise check size that still mimics the effect of white Aston, Birmingham, United Kingdom; and the \Department of Physiology, University of Helsinki, Helsinki, Finland. noise in grating detection has to be determined exper- Supported in part by The Finnish Ministry of Education, the Association of Finnish imentally. Ophthalmic Opticians, the Information Centre of Optics Business, the Finnish Culture Foundation, and the National Agency for Health and Welfare. RN is In this report, we used various noise check sizes, supported by The Academy of Finland. grating areas, and spatial frequencies, and we investi- Submitted far publication December 29, 1993; revised September 7, 1994; accepted September 9, 1994. gated at which square check sizes contrast energy Proprietary interest category: N. threshold in noise is maximal and still proportional Reprint requests: Jyrki Rovamo, Department of Vision Sciences, University of Aston, Aston Triangle, Birmingham B4 7ET, UK. to the spectral density of spatial noise, calculated by

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multiplying check area by the square of the rms con- trast of noise.

METHODS. Contrast, Check Size, and Spectrum of Spatial Noise. The two-dimensional spectral density function3 of spatial noise is given by

sin (nfxnx) sin (nfyny) xnycj nfxn; irfyny

where n^ and Vy are the horizontal and vertical noise check side lengths, c,, is the rms contrast of noise, and f* and fy are the horizontal and vertical spatial frequencies. At low spatial frequencies, where [sin (nfxnx)/ Spatial frequency (c/deg) 2 2 (itfxnx)] [sin (ftfyiiy)/(irfyTiy)] «* 1, the spectrum is flat and, according to equation (1), the spectral density FIGURE l. The spectral density of two-dimensional spatial of two-dimensional noise can be calculated by multi- noise plotted as a function of spatial frequency along the horizontal frequency axis. The broken line refers to a noise plying the square of the rms contrast of noise c^ by with the noise check side length of 0.1 deg and rms contrast the check area n^riy: of 0.3. The solid line refers to noise with the noise check side length of 0:2 deg and rms contrast of 0.3. The arrows N(0,0)= (2) indicate the cut-off frequency of noise equal to the inverse l of twice the noise check side length, fc = (2n)~ . Hence, noise contrast and noise check size both affect the spectral density of noise. Noise cut-off frequency, f , on the other hand, c noise contrast at each spatial frequency. This variance depends only on the size of the noise check n in the 2 is reflected in the response of the detection mecha- following way : nism, leading to an uncertainty of the presence of the signal and, therefore, a reduction in the human visual (3) performance. Apparatus. The apparatus is described in detail in an earlier study.6 Therefore, only its main features are Along the horizontal (fy = 0) and vertical (fx = provided here. 0) frequency axes of the Fourier space, the spectral Stimuli were generated under computer control density of two-dimensional noise, calculated by equa- on a 16-inch, high-resolution color monitor driven at tion (7), has decreased to 41 % of its maximum value 60 Hz by a VGA graphics board that generated 640 X at the theoretical cut-off frequency of noise. 480 pixels. The pixel size was 0.42 X 0.42 mm2. The Two examples of the spectrum of two-dimensional display was used in a white mode. The average phot- spatial noise, calculated by equation (i), are shown in opic luminance of the display was 50 cd/m2. The lumi- Figure 1. They demonstrate the effect of noise check nance response of the display was linearized by using size on the spectral density function. The side length the inverse function of its nonlinear luminance re- of noise check is doubled, whereas noise contrast is sponse when computing the stimulus images. kept constant. As expected, noise spectral density is To obtain a monochrome signal of 1,024 intensity increased fourfold at low spatial frequencies, and the levels selected from a monochrome palette of 65,536 cut-off frequency of noise, calculated according to intensity levels, we used a video summation device and equation (2), is halved due to the doubling of noise a periodic dithering signal. The contrasts of the grat- check side length. The same effect on the spectral ing stimuli displayed were found to be independent density of noise at low spatial frequencies can be ob- of spatial frequency and orientation up to 2 c/cm. At tained by doubling the noise contrast. This, of course, 4 c/cm contrast decreased by 0.12 log)0 units. does not affect the cut-off frequency of noise. Stimuli. Vertical cosine gratings were created, and The spectral density function of noise represents the experiments were run using a software developed the mean power spectrum of an infinite number of by one of the authors (RN). The gratings were embed- noise images. The power spectrum of a single noise ded in spatial noise produced by adding to each noise image varies greatly because of the random luminance check within the grating area a random number variation of each check. In fact, the spectral density drawn independently from a Gaussian distribution function is directly proportional to the variance of with zero mean. The distribution was truncated at

Downloaded from iovs.arvojournals.org on 09/30/2021 Reports 515 ±2.5 SD units. The rms contrast of spatial noise was varied by changing the standard deviation of the Gaussian luminance distribution. The side length of the square noise checks varied from 1 to 64 pixels. The luminances of the neighboring noise checks were uncorrelated. Thus, noise was white at low spatial fre- quencies (equation (7) and Fig. 1). The contrast energy of an image is calculated by numerically integrating the square of contrast wave- form c(x,y):

E = I2,<*{x,y)p*, (4)

where p is the side length of the square pixel. The contrast waveform is c(x,y) = [L(x,y) — L^/LQ, where L(x,y) is luminance at location (x,y) on the screen and LQ is the average screen luminance. Figure 2 shows examples of the grating stimuli consisting of four cycles. Gratings are embedded in noise consisting of noise checks of increasing size. The rms contrasts of the grating and noise are constant in all frames. When the size of each noise check increases from 1 X 1 to 8 X 8 pixels, the masking effect of noise increases in Figure 2. However, when the noise check size increases further to 64 X 64 pixels, the masking effect decreases because grating becomes visible inside each noise check. Procedures. The contrast energy thresholds were determined by a standard two-alternative forced- choice algorithm. Grating contrast was changed in steps of 0.1 logio units. The contrast required for the probability of 0.84 correct was estimated as the arith- metic average of the last eight reversal contrasts.6 Each data point is die geometric mean of at least three threshold estimates. The experiments were performed in a dark room. The head of the subject was stabilized by using a chin rest. The only light source was the computer screen. The stimuli were viewed binocularly with natural pupils of 5 to 6 mm in diameter. Thus, the average retinal illuminance produced by our dis- play was about 1,200 phot.td. Subjects were asked to fixate the center of the stimulus. No fixation point was used. Subjects. Three subjects, aged 25, 27, and 28 years, served as observers. The study followed the prin- ciples of the Declaration of Helsinki. Informed con- sent was obtained from all subjects after the nature of the experimental procedures had been fully ex- plained. KT had corrected, nonastigmatic myopia (OD -6.00 DS/OS -4.00 DS); OU had corrected,

FIGURE 2. Cosine gratings comprising 4 cycles and embed- ded in spatial noise consisting of square checks. Noise check size from the top to bottom is 1 X 1, 8 X 8, and 64 X 64 pixels. The rms contrasts of the gratings and noise are con- Downloaded from iovs.arvojournals.org on 09/30/2021 stant in all three frames. 516 Investigative Ophthalmology & Visual Science, February 1995, Vol. 36, No. 2

nonastigmatic anisotropia (OD +0.75 DS/OS -0.75 R = [E/N(0,0)]os, and plotted as a function of the DS); and HK had uncorrected hyperopia (+0.5 DS side length of noise checks (n) expressed in terms of OA). Their binocular Snellen acuities were 6/4. a fraction of grating cycle. The number of grating cycles within the square stimulus area was 1, 4, 16, or RESULTS. In the experiments depicted in Figure 64 corresponding to the bandwidth of 1.56, 0.43, 0.11, 3, contrast energy threshold, E, was measured as a or 0.03 octaves at half-height across the Fourier spec- function of increasing noise check side length for grat- trum of the stimulus. ings embedded in two-dimensional spatial noise. As Figure 3 shows, the physical signal-to-noise ra- Noise contrast was constant, but the side length of tio at threshold, that is, the square root of the ratio each square noise check increased from 1 to 16 or 64 between contrast energy threshold and the spectral pixels. The spectral density of noise at zero frequency, density of noise at zero frequency, was initially con- N(0,0), varied with noise check size from 2.82 X 10"4 stant but then decreased with increasing noise check to 7.21 X 10~2 deg2 in Figure 3A, from 3.38 X 10~6 side length at all stimulus bandwidths and spatial fre- to 0.220 deg2 in Figure 3B, from 2.40 X 10"5 to 1.57 quencies. As the dashed lines in Figure 3 reveal, con- deg2 in Figure 3C, and from 1.05 X 10"5 to 2.68 X trast energy threshold started to decrease at the same 10~3 deg2 in Figure 3D. The results are expressed in noise check side length. The constant physical signal- terms of the physical signal-to-noise ratio at threshold, to-noise ratio at threshold indicates that contrast en-

10 • 1 cycle A 4 cyclea B

,01. tlO 3 o H—£ H I'* oonstant contrast o \NN^ energy threshold 1

10" '; • LScAhg Signal- A A Sc/d.J A ScJttog \ • ecMag i 97%

16 cycles 64 cyclei

Noise check side length (cycles) Noise check side length (cycles) FIGURE3. The physical signal-to-noise ratio at threshold, R= [E/N(0,0)]05, for cosine gratings in two-dimensional spatial noise plotted as a function of noise check side length expressed in terms of the fraction of a grating cycle. Noise check side length increased from 1 to 16 pixels in (A, D) and from 1 to 64 pixels in (B, C). The side length of one pixel was 0.042 cm. The rms contrast of noise was constant at 0.30 in (A, D) and 0.35 in (B, C). On the screen, the spectral density of noise at zero frequency, N(0,0), increased from 1.59 X 10~4 to 4.06 X 10~2 cm2 in A and D, and from 2.16 X 10"4 to 0.885 cm2 in B and C. The stimulus was either 0.77 c/cm grating within a stimulus window of 1.3 X 1.3 cm2 (A), 0.75 c/cm grating within a stimulus window of 5.3 X 5.3 cm2 (B), 2 c/cm grating within a stimulus window of 8 X 8 cm2 (C), or 4.0 c/cm grating within a stimulus window of 16 X 16 cm2 (D). To obtain the spatial frequencies of 1.5 to 6 c/deg, we used viewing distances of 223 cm in A, 115 to 458 cm in B, 43 to 172 cm in C, and 43 cm in D. Open symbols in A and D refer to subject OU and inB and C to subject KT. Closed symbols refer to subject HK. The dashed lines in A to D indicate signal-to-noise ratio if the contrast energy threshold remained constant as a function of check side length. Percentages refer to variances ex- plained by the solid lines.

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ergy threshold increased in proportion to the spectral density of spatial noise calculated by equation (2). At these noise check sizes, the detection threshold was determined by the spectral density of noise, and the noise mimicked the effect of white spatial noise in grating detection. The decrease of contrast energy threshold with increasing check side length, on the other hand, implies that the average spectral density of noise in the vicinity of the spatial frequency of the grating decreased although the spectral density of noise at zero frequency increased. The initial increase and subsequent decrease in the contrast energy threshold with increasing noise check size is in agreement with the perceptual phenomenon shown O 10° in Figure 2. The critical side length of the noise check refers Number of grating cycles to the largest size that still mimics the effect of white FIGURE 4. The critical number of noise checks per grating noise in grating detection, It is here indicated by the cycle plotted as a function of the number of graung cycles transition from constant physical signal-to-noise ratio within the square stimulus area. The percentage refers to at threshold to decrease in the signal-to-noise ratio. the variance explained by the solid line. Horizontal lines correspond to the respective averages of the horizontal part of the data in each frame. Least creasing number of cycles, the number of noise checks squares lines of the form, logR = a log(n) + b, were needed to mimic the effect of white noise started to fitted to the decreasing part of the data in each frame. decrease, obeying the least squares equation bc — 5.56 The variances explained by the solid lines in each IfOA'j -pne variance explained by the solid line in Fig- frame were calculated by ure 4, calculated by equation (5), was 97%. DISCUSSION. Our results showed that when I (log R - log R )' 2 esl noise checks were below the critical size, contrast en- r = 100 1 - (5) I(log R - Ramf ergy thresholds for cosine gratings were direcdy pro- portional to the spectral density of noise at zero fre- where R refers to the data, R,,st to the estimates, and quency, N(0,0). The physical signal-to-noise ratio thus -ftave = ni~'Z log R where m is the number of data remained constant at threshold. However, when noise points. We used logR instead of R because Ris plotted check exceeded the critical size, contrast energy on a logarithmic scale. The explained variances were threshold as well as the physical signal-to-noise ratio 96% to 99%, as indicated in Figure 3. at threshold started to decrease with increasing noise As Figure 3 shows, the critical noise check side length, check size. The result means that spatial noise con- nc, expressed in terms of the fraction of a cycle, was sisting of square noise checks mimics the effect of constant for each grating bandwidth irrespective of white noise in grating detection only when the side the spatial frequency. The value of nQ was found to be length of square noise checks is below the critical size. 0.246, 0.230, 0.319, and 0.390 for gratings comprising An obvious reason for the decrease in contrast 1,4,16, and 64 cycles, respectively. Its inverse indicates energy threshold after the noise check side length

the critical number of noise checks per grating cycle, bc, which exceeded the critical size is the reduction of the spec- refers to the minimum number of noise checks per tral density of noise in the vicinity of the spatial fre- grating cycle needed to mimic the effect of white spa- quency of the grating. An additional contributing fac-

tial noise in grating detection. The value of bc was tor could be the block structure of noise (see Fig. 2): found to be 4.07, 4:35, 3.13, and 2.56 for gratings When noise check size increases, the block structure comprising 1 to 64 grating cycles. The cut-off fre- of noise becomes clearly visible, and its masking effect

quency of noise;' fc = (2nc)~', corresponding to the diminishes because the number of edges in the stimu- critical check size, was 2.0, 2.2, 1.6, and 1.3 times lus declines and the visual system starts to process higher than the spatial frequency of the grating con- each block as a separate stimulus window so that less sisting of 1 to 64 cycles. attention is paid to the edges. Figure 4 shows the critical number of noise checks The critical noise check size was inversely propor- per grating cycle as a function of the number of grat- tional to spatial frequency and thus independent of ing cycles within the stimulus area. For the stimuli spatial frequency when expressed in terms of the frac- consisting of a smail number of cycles (k < 4), the tion of a grating cycle. On the other hand, the nar- critical number of noise checks per grating cycle was rower the bandwidth of the stimulus, the greater the approximately constant at 4.2. However, with the in- critical noise check size in terms of the fraction of

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grating cycle. Consequently, fewer noise checks per for the grating comprising 64 cycles seems low. In this grating cycle were needed to mimic the effect of white case, the spectral density of noise has at the nominal noise with decreasing bandwidth of the grating. The frequency of the grating decreased to 59% of its maxi- critical number of noise checks per grating cycle was mum value at zero frequency. This percentage also found to be constant at about 4 for the gratings con- seems to be low to produce the same effect on contrast sisting of 1 to 4 cycles, but it decreased to 2.6 when energy -threshold as white noise. One explanation the number of grating cycles increased to 64. Accord- could be that the bandwidth of the detection mecha- ] ingly, the cut-off frequency of noise fc = (2nc)~ corre- nism increases toward the spatial frequencies below sponding to critical check size was about twice the the nominal spatial frequency of the grating, when spatial frequency of a grating consisting of 1 to 4 cy- the bandwidth of the stimulus decreases below a cer- cles, but it was only 1.3 times the spatial frequency of tain limit. Other possible explanations have already ; the grating consisting of 64 cycles. been discussed. The critical number of noise checks per grating We can conclude that in grating detection, the cycle still mimicking the effect of white noise was masking potency of noise with square noise checks found to decrease with decreasing bandwidth for grat- and constant rms contrast can be increased by increas- ings comprising more than 4 cycles. This means that ing the size of the noise check up to the critical size. the gap between the cut-off frequency of the noise In addition, such a noise mimics the effect of white and the nominal spatial frequency of the grating de- noise up to the same critical size in the sense that creased. The result suggests that the bandwidth of contrast energy threshold is proportional to the spec- the detection mechanism, at least above the nominal tral density of noise,.calculated by multiplying check spatial frequency of the grating, decreases with de- area by the square of the rms contrast of noise. creasing bandwidth of the stimulus, in accordance 8 with the result of Carter and Henning. However, the Key Words detection mechanism seems to be unable to adjust its bandwidth when the number of cycles increases from white noise, spatial noise, critical noise check size, noise spectral density 1 to 4. If we assumed, therefore, that the visual system 9 employs detection filters of constant bandwidth, our References result above 4 cycles would suggest the use of a detec- tion filter at, or shifted to lower spatial frequencies. 1. Burgess AE, Wagner RF, Jennings RJ, Barlow HB. Ef- This means off-frequency looking2 in the detection of ficiency of human visual signal discrimination. Sci- gratings comprising more than 4 cycles. However, in ence. 1981;214:93-94. 2. Pelli D. Effects of Visual Noise. Cambridge: Cambridge our case, off-frequency looking would produce a lower University; 1981. Thesis, i signal-to-noise ratio and poorer performance at 3. Legge GE, Kersten D, Burgess AE. Contrast discrimi- threshold, which is opposite to the original idea about 2 nation in noise. / Opt Soc Am. 1987;A4:391-404. the purpose of off-frequency looking. In addition, the . 4. Pelli DG. The quantum efficiency of vision. In: Blake- 10 results of Henning et al imply that data could be more C, ed. Vision: Coding and Efficiency. Cambridge: explained by several models differing in parameters, Cambridge University Press; 1990:3-24. such as location and bandwidth of detection filters 5. Rovamo j, Franssila R, Nasahen R. Contrast sensitivity (adjustable or fixed), summation rule of filter outputs, as a function of spatial frequency, viewing distance, and subsequent detector (for example, peak-to- and eccentricity with and without spatial noise. Vision through or energy). Res. 1992; 32:631-637. 6. Rovamo J, Kukkonen H, Tiippana K, Nasanen R. Ef- Figure 4 suggests that the gap between the cut- fects of luminance and exposure time on contrast sen- off frequency of spatial noise and the nominal spatial sitivity in spatial noise. Vision Res. 1993;33:1123-1129. frequency of the grating would decrease monotoni- 7. Luntinen O, Rovamo J, Nasanen R. Contrast sensitivity cally. It is, however, improbable that the critical num- as a function of grating area and spectral density of ber of noise checks per grating cycle would reach or external spatial noise. Vision Res. In press. decrease to less than 2, even if the number of cycles 8. Carter BE, Henning GB. The detection~of gratings in increased to more than 64, because two noise checks narrow-band visual noise. J. Physiol '(Land). 1971; 219:355-365. per grating cycle means that the theoretical cut-off 1 9. Stromeyer CF, Julesz B. Spatial frequency masking in frequency of noise, fc = {2nc)~ , is equal to the spatial vision: Critical bands and spread of masking. J Opt Soc frequency of the grating. According to equation (7), Am. 1972:62:1221-1^32. the spectral density of noise along the horizontal fre- 10. Henning GB, Hertz BG, HintonJL. Effects of different quency ixis has at the cut-off frequency decreased to hypothetical detection mechanisms on the shape of 41% of its maximum, N(0,0), at zero frequency. Even spatial-frequency filters inferred from masking experi- the value of 2.56 noise checks per grating cycle found ments: I: Noise masks. J Opt Soc Am. 1981;71:574-581.

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