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Resolution and contrast sensitivity at low luminances
ylrlon Res. Vol. 12, pp. 825-833. Perganwn Press 1972. Printed in
RESOLUTION
GreatBritain
AND CONTRAST SENSITIVITY AT LOW LUMINANCES A. VANMEETERENand J. J. Vos
Institute for Perception TNO, Soesterberg, The Netherlands (Received 22 July 1971) INTRODUCTION
To GET measurable signals from the image on its surface, the retina must sum the incoming photon flux over some moderately small area. Ricco’s classical law indicates this process. The size of the summation areas affects resolution as well as contrast sensitivity. The effect upon resolution is obvious: details are lost in large summation areas. On the other hand contrast sensitivity is favoured by a large summation area, when vision is limited at low luminances by the statistical fluctuations of the incoming photon flux (DE VRIES, 1943), since these fluctuations are reduced by summation. Resolution and contrast sensitivity can be studied simultaneously by means of the spatial modulation sensitivity functions introduced by SCHADE(1956). Though there is an extensive literature on modulation sensitivity functions of the human eye, reviewed by WESTHEIMER (1965), few measurements have been made at low luminances. New measurements were made therefore to study both, contrast sensitivity and resolution. The results will be discussed in terms of the fluctuation theory. MODULATION
SENSITIVITY
FUNCTIONS
FOR
NATURAL
VISION
The most complete measurements with luminance as a parameter have been made by VAN NES and BOUMAN(1967) with monochromatic lights and an artificial pupil. The measu~ments to be reported here have been made with natural pupils and white light at low luminances. The results may provide information about natural vision in twilight. Modulation sensitivity functions are measured with spatial sine wave patterns. An example of such a pattern is shown in Fig. 1. The modulation, defined as the amplitude divided by the mean luminance, is brought to threshold, and the reciprocal threshold gives the modulation sensitivity. By repeating this for a number of spatial frequencies, a modulation sensitivity function is defined. In fact two such functions were measured simultaneousiy by presenting the patterns in vertical and horizontal orientation. The experimenter presented about ten modulation settings near threshold in random order and in steps of a factor 1.3. The subject’s task was to say “horizontal”, “vertical”, or “no choice” and the transition of “no choice” answers to correct answers was taken as threshold. The subject was not restricted in time, nor forced to make a choice between “vertical” and “horizontal”. As followed from a pilot experiment with the forced choice method, thresholds determined in this simplified way correspond to a detection probability of about 0.75. The forced choice method, though better defined, requires a large number of presentations and is impractical when the interest is mainly in mutual comparison of many modulation sensitivity functions. To study normal free vision in twilight a large sine wave field is required, preferably at a distance of several meters, to avoid accommodation problems. Therefore the sine wave patterns were prepared in the form of slides and projected on a white screen at a distance of 3.5 m from the subject. The field sire was 825
A. VAN MEETEREN AND J. J. Vos
826
17” x 11”. A second projector superimposed a uniform field. Any desired combination of mean luminance and modulation could be adjusted with the aid of neutral density tilters in the light beams and, within limits, by control of the lamp currents. Calibrations were made with a Weston Viscor photocell.
The average results of two subjects and horizontal and vertical pattern orientations are presented in Fig. 2. These modulation sensitivity functions show the general properties, well known from the literature: a maximum modulation sensitivity for medium frequencies, shifting toward lower frequencies and finally disappearing as luminance decreases. The curves of Fig. 2 clearly illustrate the two aspects of modulation sensitivity functions: their spatial bandwidth (IV) represents resolution, whereas their relative height (S) reflects contrast sensitivity. Contrast sensitivity increases with luminance. However, the curves for the 500 . ldLcd/n x 18) * lo-*
200 5 ;i
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100
,x ..? 50
._ z z 20 c .o g 10 2 z
5
2 1
0.5
5 10 20 1 2 spatial frequency w in ppd
50
100
FIG. 2. Modulation sensitivity functions. Spatial frequency is plotted in periods per degree. These measurements refer to normal free vision to a field of 17” x 11”. Average results of two observers and horizontal and vertical pattern orientations. The natural pupil was 5-7 mm in the luminance range concerned.
various luminances cannot be brought to coincide completely by shifts along the sensitivity axis. The curves span a wider range of spatial frequencies as luminance is raised, i.e. the spatial bandwidth increases with luminance. If spatial bandwidth is limited by summation, this implies that the summation area is smaller at higher luminances. Spatial summation in the visual system is known from Ricco’s law. Ricco’s law states that the contrast threshold for small test objects depends on their energy content, as long as they are smaller than “Ricco’s area”. Ricco’s area is only a simplification of more complex spatial interactions. As RATLIFT (1965) proposes, spatial interations, including summation and inhibition, can be characterized by detailed weighting functions. Weighting functions are most straightforwardly determined by a Fourier transformation of the modulation sensitivity functions. This technique was applied by PATEL (1966) to fovea1 modulation sensitivity functions for retinal illuminances in the range of 3-1000 td. Weighting functions for lower retinal illuminances can be computed from our data.
FIG. 1. Spatial sine wave pattern. The luminance is sinusoidally modulated as a function of spatial cobrdinate. The modulation is defined as a/B. The spatial frequency w is the number of periods per unit of angle.
vacing page 826
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In order to use Fourier techrdques the system concerned should be linear. In our case spatial interactions, summation as well as inhibition should be linear. Spatial summation is apparently linear within Ricco’s area. The linearity of Iateral inhibition is recently discussed by THOMAS (1968). Measurable non-linear inhibitions were found when the luminance differences concerned were larger than a factor 2. Thus the possible inhibition in sine wave patterns with low spatial frequencies will be practically linear in threshold measurements where the modulations are always smaller. The calculations of the Fourier transforms were made according to FILON’S(1928) method for numerical evaluation of trigonometric integrals, In preparing the data we had to extrapolate the modulation sensitivity for zero spatial frequency. This extrapolation is easily made for the lowpass curves, but is problematic for the bandpass curves in Fig. 2. However, its effect on the spatial weighting function is only noticeable at large angular distances from the central stimulus. Assuming spatial ~t~~tio~ to be restricted to angular distances of no more than I”, we made the extrapolation such that the weighting function vanishes at that distance. Another extrapolation had to be made at lower retinal illumiuances. Here the experhnental data are restricted to a small frequency range, since the rest of the curves vanish below threshold, as is shown in Fig. 2. This of course does not necessarily imply a small spatial bandwidth. We decided therefore to extrapolate the modulation sensitivity curves to conform the next higher ones, as a better estimate than a truncation.
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3. Weighting functions obtained by Fourier transformation of modulation sensitivity functions, referring to normal free vision and a field of 17” x 11”.
Figure 3 illustrates some of the weighting functions found by Fourier transformation of the modulation sensitivity functions for normal free vision at low retinal illuminance. We prefer retinal illuminance as the relevant parameter here, rather than luminance. The diameters of the natural pupils were measured in the experimental conditions with the aid of an infra-red image tube. Thus luminance could be converted into retinal illuminance. For the sake of clarity weighting functions for intermediate retinal illuminances are omitted but they fit nicely in between. The weighting functions shrink when the retinal illuminance is raised, and reflect lateral inhibition. Their ha~fwidths change from 11.5 to 2-3’ of arc for normal free vision in between O-01 and 100 td. This improvement can only partly be attributed to the contraction of the natural pupil from 7 to 5 mm, resulting in better optics. The halfwidths of the line spread functions for the optics of the human eye, as measured by CAMPBELLand GUBISCH (1966) change from 2.0 to l-5’ of arc in the pupil range concerned.
A. VANMIWIBREN AND J. J. Vos
828
One can state therefore, that the weighting functions for normal free vision, as presented in Fig. 3, are of neural origin mainly. These weighting functions for vision with large fields possibly apply to different retinal positions. FOVEAL
MODULATION
SENSITIVITY
FUNCTIONS
Figure 4 presents modulation sensitivity functions for fovea1 vision. These measurements were made with a small sine wave field of 2.8” x 2*8”, a 3-mm artificial pupil and fixation control. The experimental set up for these measurements is described elsewhere (VAN MEETEREN, Vos and F~OOGAARD,1971). The procedure of the measurements was the same as described above. The results refer to the same subjects. The fovea1 modulation sensitivity 500
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10 frequency
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50 w
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Fio 4. Modulation sensitivity functions. Spatial frequency is plotted in periods per degree. These rn~~rnen~ refer to a foveauy eated field of 2.8” x 2-8” and a 3-mm artificial pupil. Average results of two observers and horizontal and vertical pattern orientations.
differ markedly from the large field results in that sensitivity decreases faster with luminance, whereas high frequency performance tends to be better. They do not show a peaked maximum in the luminance range investigated. Our measurements complement those made by PATEL (1966) at higher retinal illuminances, where the bandpass character appears again. Apparently fovea1 modulation sensitivity functions only change in height, not in shape. This means that the spatial interactions do not change with retinal illuminances below 1 td; above that level lateral inhibition starts to develop (PATEL,1966). Figure 5 gives the weighting functions of the fovea for low retinal illuminance (A) and for 3 td (B). These weighting functions are narrower than those for natural vision, partly due to the 3 mm artificial pupil, but certainly also because of smaller retinal units. functions
INTEGRATED
MODULATION
SENSITIVITY
Mod~ation sensitivity functions represent two aspects of vision: the contrast sensitivity S in the relative heights of the curves, and resolution in the form of the spatial bandwidth
Resolution and Contrast Sensitivityat Low Luminmces
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FIG. 5. Weightingfunctions obtained by Fourier transformation of modulation sensitivity functions, refer&g to vision with a 3-mmartifkial pupil, fixationcontrol and a fovea1field of 2.8” x 2.8”.
W. Contrast sensitivity and spatial bandwidth are not independent of each other, however. Spatial bandwidth can be improved by decreasing the station area but only at the expense of contrast ~nsitivity. The photon fluctuations are relatively greater in a smaller summation area. Let D be the diameter of a summation area and B the luminance. The mean number of photons within the summation area (signal) is proportional to BD2 and the variance (noise) is proportional to (BP)“. The signal to noise ratio thus is proportional to BP/(BP)f = B*D. According to the well known sample-theorem one can substitute D = 1/2W. Further contrast sensitivity at best can be proportional to the signal to noise ratio. Thus, if the eye would follow the limits put by photon fluctuations, one should find a kind of uncertainty relation: s. w :. B”. (1) Due to the bandpass structure it is often difficult to separately define the height Sand the width W of a modulation sensitivity function. Their product S. W on the other hand is well defined; it is the area under the modulation sensitivity function, and can be determined directly. We will call this product the integrated modulation sensitivity. In Fig. 6 the integrated modulation sensitivity is plotted vs. retinal illuminance on logarithmic scales. Curve A refers to the measurements for a large sine wave field of 17” x 11’. The curve to a first approximation follows a square root relation indeed over almost 5 decades. The same holds for curve B which is derived from modulation sensitivity curves measured by VANNES and BOUMAN(1967) with a sine wave field of 4.5” x 8*25”, monochromatic green light, and a 2-mm artificial pupi1.l As argued above this result is in agreement with the hypothesis of DE VRIES(1943), that vision is limited by the statistical fluctuations of the incoming photon flux. Contrast sensi* We have no satisfying expknation of the considerabledif%erence in heightbetweencurve A and curve B of Fig. 6. Compared with the scarce literature on modulation sensitivity for sine wave gratings at lower iuminances curve B seems to be rather high.
830
A.
VAN
MEETERENANDJ.
J.Vos
FIG 6. Integrated mod~ation sensitivity as a function of retinal ilIurninan=. The curves repreaent square root relations. A refers to normal free vision and a large field. The data for B are derived from VANNES and BOUMAN(1967) and refer to monochromatic green light and a 2mm artiikial pupil. The open circles refer to the foveal measurements.
tivity S and spatial bandwidth W (or resolution) may individually deviate from the square root law without coming into conflict with the fluctuation theory, when their interaction is properly realized. Apparently the increase of the signal to noise ratio with luminance can be used either for better contrast sensitivity or for better resolution. The strategy of the visual system in this respect can be studied when S and Ware known separately. However, the separation of S and W is problematic, as remarked before, in view of the bandpass character of the modulation sensitivity functions. We have taken rather arbitrarily for S the maximum values of the modulation sensitivity functions. The value of W then follows from the known product of S and W. This separation is presented for normal free vision in Fig. 7. The increase of information with luminance is at first spent mainly to modulation sensitivity whereas at higher levels some modulation sensitivity is sacrificed for better resohttion. This is about the region of rod-cone transition. No sign of this transition is found in the integrated modulation sensitivity vs. retinal illuminance relation (Fig. 6). Apparently the visual system is able to follow the limit put by the photon fluctuations irrespective of this transition. Or, to put it in a probably more correct perspective, the rod-cone shift is one of the strategic means. It will be clear, that the square root relation gradually changes into Weber’s law at higher luminances: due to neural saturation the signal to noise ratio is frozen at a certain level. At low luminances the absolute threshold sets a limit. This is evident in the fovea1 measurements. Fovea1 integrated modulation sensitivity at low luminances remains clearly
Resolution and Contrast Sensitivity ta Low Luminances
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Fro. 7. Modulation sensitivity S (upper curve, left scale) and spatial bandwidth W (lowest curve, right scale) as a function of retinal illmninance for normal free vision. The increase of the signal to noise ratio with iuminance is at first mainly used for better modulation sensitivity, later also to improve spatial bandwidth.
behind the square root relation and tends to become zero between O-01 and O-1 td. The number of photons detected within the summation area and within the summation time here apparently comes to the absolute threshold. DISCUSSION So far we discussed the product of sensitivity and spatial bandwidth as a function of
luminance on a relative scale. The actual threshold signal to noise ratio within a summation area and within the integration time can be roughly estimated. MORGAN (1965) found surprisingly low values in between O-12 and O-19, starting from contrast sensitivity and resolution data derived from different authors. Modulation sensitivity functions reflect contrast sensitivity and resolution simultaneously and thus allow a more direct estimation. The computation proceeds as follows. With 2.76 x 1Ol8photons of 550 ~~, l-47 X 10q3 W/lm and 1 td equal to O-85 x lo-l3 lm/min* of arc, the number of photons per min’ of arc per see per td is 342. Taking from Fig. 7 a spatial bandwidth of 4-6 ppd at 1 td, the width of the integration area can be estimated at 60/(2 x 4-6) = 65’ of arc, and its area at 43 minz of arc. With an integration time of 0~1set and assuming that only about 12 per cent of the incoming photon flux :is effective (MORGAN, 1965), one arrives at 0.1 x 0.12 x 43 x 342 = 175 photons per integration area-time volume at 1 td. The fluctuations in this number amount to 175f = 13.2. On the other hand, one finds a modulation sensitivity of 86 at 1 td from Fig. 7, indicating a threshold amplitude of 175185= 2.1. Apparently the signal to noise ratio at threshold is Z-1/13+2= O-16. This estimation is as low as Morgan’s. It is impossible to detect a signal with a probability of about 75 per cent when the signal to noise ratio is only 0.16. This low signal to
noise ratio applies to a single summation unit however. The sine wave patterns cover many summation units. It is hypothesized, that several summation units pool their detection
832
A, VAN MEETERENAND J. 3. Vos
probability in some way and that the total probabi~ty of detection of a sine wave pattern builds up over the pattern area. The high angular resolutions connected to vernier acuity and depth perception indicate a similar process. No systematic investigations of probability summation in the detection of spatial sine wave patterns are known to us. Measurements of modulation sensitivity for sine wave or bar patterns, in which the length of the bars and the number of bars involved is varied, might provide information about the characteristics of probability summation. R~HLER and HILZ (1966) found that modulation sensitivity increases with the length of the bars and asymptotes at a bar-length of about I”. COLTMANand ANDERSON(1960) found that modulation sensitivity increases with the number of bars in the pattern and that it asymptotes at 7 bars. Those results indicate that probability summation works to fufl advantage when the testpatterns are larger than 1’ and contain more than 7 bars. HAY and CHESTER(1970) recently argued that the processing of noise might introduce a component in the modulation sensitivity functions that depends on spatial frequency. This component would be absent in transfer functions measured in suprathreshold conditions. DAVIDSON(1968) and CAMPBELL and MAFFEI(1970) compared transfer functions derived from threshold and suprathreshold measurements and found almost no difference. Thus the effect of noise, inherent in threshold measurements, is probably independent of spatial frequency. This conclusion also applies to probability summation. As an alternative to the probability summation hypothesis ROSE(1948) suggests direct summation of photons over the whole (circular) test-object. The resulting signal is compared with the fluctuations in an equal sized area of the background. COLTMAN(1954) extended this hypothesis to bar patterns. There are several objections to this hypothesis. Firstly, as BOUMAN(1961) remarked, it implies that the visual system must know the test stimulus before detecting it. Secondly, the modulation sensitivity for sine wave patterns should be proportional to l/w*, where w is the spatial frequency. However, as mentioned above it is not likely that noise gives rise to a component in the modulation sensitivity function that depends on spatial frequency. Thirdly, the known differences in the size of Ricco’s area at different retinal locations and the corresponding differences in spatial bandwidth could not be explained if the visual system could summate photons over the whole test-stimulus, no matter what its size. REFERENCES BOUMAN, M. A. (1961).History and present status of quantum theory in vision. In Sensory ~omm~~cafion. (edited by R~~ENBLITH,W. A.) New York. BOUMAN, M. A. and VAN DER VELDEN,H. A. (1947). The two-quanta expknation of the dependence of the
threshold values and visual acuity on the visual angle and the time of observation. J. opt. Sac. Am. 37, 908-919. CAMPBELL, F. W. and GUBIKH, R. W. (1966). Optical quality of the human eye. J. Physiol., Land. 186,558578. CAMPBEU, F. W. and MAFFBI,L. (1970). Electrophysiological evidence for the existence of orientation and size detectors in the human visual system. J. Physiol., I&d. 2Q7,635-652. COLTMAN, J. W. (1954). Scintillation limitations to resolving power in imaging devices. J. opt. Sot. Am. 44, 234-237. COLTMAN, J. W. and ANDERSON,A. E. (1960). Noise limitations to resolving power in electronic imaging. Froc. Insrn Radio Engrs 48,858-86X DAVIDSON,M. (1968). Perturbation approach to spatial brightness interaction in human vision. J. opt. Sot. Am. !5&1300-1308. FILON, L. N. G. (1928). On a quadrature formula for trigonometric integrals. Proc. R. Sot. Edit&. 49,38-47. HAY, G. A. and Crmsmas, M. S. (1970). Limiting factors in threshold and suprat~hold vision. Nature, L0m.i.22% 1216-1218.
Resolution
and Contrast Sensitivity at Low Luminances
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VAN MEETEREN,A., Vos, J. J. and E~~GAARD,J. (1971). Das Sehen mit dem Small Starlight Scope, einem Dreistufen-Bildverstr. Optik 32,456467. MORGAN, R. H. (1965). Quantum fluctuation and visual perception. In Diagnostic Radiologic Instrumentation. Charles C. Thomas, Springfield, Ill., 61-91. VAN NIB, F. L. and B~UMAN, M. A. (1967). Spatial modulation transfer in the human eye. J. opt. Sot. Am. 57, 401-406. PATEL,A. S. (1966). Spatial resolution by the human visual system. The effect of mean retinal illuminance. J. opt. Sot . Am. 56,689-964. RATLXFF,F. (1965). Mach Bands: Quantitative Studies on Neural Networks in the Retina. Holden-Day, San Francisco. RBHLER,R. and HILZ, R. (1966). Physical and physiological factors in visual modulation transfer. In Performance of the Eye at Low Lumimnces. pp. 105-117. Excerpta Medica Foundation, Amsterdam. Rosa, A. (1948). The sensitivity performance of the human eye on an absolute scale. J. opt. Sot. Am. 38, 196-208. SCHALW,0. H. (1956). Optical and photo electric analog of the eye. J. opt. Sot. Am. 46,721-739. THOMAS,J. P. (1968). Linearity of spatial integrations involving inhibitory interactions. Vision Res. 8,49-68. DE VRIES,H. (1943). The quantum character of light and its bearing upon threshold of vision, the differential sensitivity and visual acuity of the eye. Physica Eimihoven 10,553-564. WBsrmm.r~~, G. (1965). Visual acuity. A. Rev. Psychol. 16,359-380. Abstract-Modulation sensitivity functions of the eye, measured with spatial sine wave patterns, represent two aspects of vision, its contrast sensitivity (.!?)and its resolution in the form of spatial bandwidth (W). According lo the fluctuation theory of vision, at low luminance the product S. W. should be proportional to the square root of luminance. This is contirmed. According to this relation the increase of the signal to noise ratio with luminance can be used either for better resolution (W) or for better contrast sensitivity (8. The strategy of the eye as to this choice is discussed. R&rtn&Les fonctions de modulation de sensibilite de l’oeil, mesurees avec des mires spatiales sinusoldales, representent deux aspects de la vision, sa sensibilite au contraste (5’) et sa resolution sous forme d’une largeur de bande spatiale( W). Selon la theorie de fluctuations de la vision, aux faibles luminances le produit S. W devrait &re proportionnel a la racine carrel de la luminance, ce qui est con6rme. Selon cette relation, l’augmentation du rapport signal sur bruit avec la luminance peut servir soit A augmenter la resolution (W) soit a accroitre la sensibiliteaucontraste(S). On discute la strategic de l’oeil dans ce choix. ZusannnenfassBBg-Mit Sinusgittem gemessene Kontrasttibertragungsfunktionen des Auges bringen xwei Aspekte der Sebleistung xum Ausdruck: Die Kontrastempfindlichkeit (S) und das AuflosungsvermGgen in Form ortlicher Bandbreite (W). Entsprechend der Fluktuationstheorie sollte bei geringer Leuchtdichte das Produkt SW proportional der Quadratwurzel der Leuchtdichte sein. Das wird be&it&t. Nach dieser Rexiehung kann die Zunahme des Signal-Rausch-Verhahnisses mit der Leuchtdichte entweder xur Verbesserung der Auflosung (W) oder der Kontrastemp8mllichkeit genutzt werden. Es wird besprochen, wie das Auge diese Alternative entscheidet. - @yHKqEiE ¶yBCTBETeJtI.HOCTH ma3a KMO,ny~xmi~~ B3MepBJIBCbCITOMOmUO npOCTpancTBennbIx MTTepHOB, cHHyCO~aBI.HO n3MemnomrixcB H [email protected] BBa acneKTa 3peHaR: er0 KOHTpaCTnyBJnyBcTBETenMBXTb(5) H er0 npOCTpaFtcTBeBTTyIO pa3pemaTOmyro cnoco6~iocr1. J&napemeTOK, COCTOBmBx ~3 ~OJIOCpa3HO2tSaCTOTbI(w). COr~~acno &noKTyamiomioZt ~eopmi 3pemia npri mi3~on ap~ocrn npon3se~emfe s* Wnonxcno 6r.1~5nponopmioHaJtBHOKOpBiOKBanpaTHOMyH3 BpKOCTE.3T0 nOJty¶BBO ITOHTBepXjTenEe.B COOTBeTCTBEE C 3TBM, yBeJni¶eHHe OTHOmenBa CHTHaBaK myMy rtp&i yBe.BBKemiE~KOCTE MOIKIX6bTTb ECIIOJI~~OB~BO ~1x60 AJM nysmero paspemenna (W), ~60 ~nri nymneti xonTpacrnolt ~IlCTBBTeJtBHOCTB (S). Q6cyXnaeTca CTpaTerEB I’JTaXlnpn 3TOMBbI6OpC. PewMe
V.R.12/5--P
RESOLUTION
GreatBritain
AND CONTRAST SENSITIVITY AT LOW LUMINANCES A. VANMEETERENand J. J. Vos
Institute for Perception TNO, Soesterberg, The Netherlands (Received 22 July 1971) INTRODUCTION
To GET measurable signals from the image on its surface, the retina must sum the incoming photon flux over some moderately small area. Ricco’s classical law indicates this process. The size of the summation areas affects resolution as well as contrast sensitivity. The effect upon resolution is obvious: details are lost in large summation areas. On the other hand contrast sensitivity is favoured by a large summation area, when vision is limited at low luminances by the statistical fluctuations of the incoming photon flux (DE VRIES, 1943), since these fluctuations are reduced by summation. Resolution and contrast sensitivity can be studied simultaneously by means of the spatial modulation sensitivity functions introduced by SCHADE(1956). Though there is an extensive literature on modulation sensitivity functions of the human eye, reviewed by WESTHEIMER (1965), few measurements have been made at low luminances. New measurements were made therefore to study both, contrast sensitivity and resolution. The results will be discussed in terms of the fluctuation theory. MODULATION
SENSITIVITY
FUNCTIONS
FOR
NATURAL
VISION
The most complete measurements with luminance as a parameter have been made by VAN NES and BOUMAN(1967) with monochromatic lights and an artificial pupil. The measu~ments to be reported here have been made with natural pupils and white light at low luminances. The results may provide information about natural vision in twilight. Modulation sensitivity functions are measured with spatial sine wave patterns. An example of such a pattern is shown in Fig. 1. The modulation, defined as the amplitude divided by the mean luminance, is brought to threshold, and the reciprocal threshold gives the modulation sensitivity. By repeating this for a number of spatial frequencies, a modulation sensitivity function is defined. In fact two such functions were measured simultaneousiy by presenting the patterns in vertical and horizontal orientation. The experimenter presented about ten modulation settings near threshold in random order and in steps of a factor 1.3. The subject’s task was to say “horizontal”, “vertical”, or “no choice” and the transition of “no choice” answers to correct answers was taken as threshold. The subject was not restricted in time, nor forced to make a choice between “vertical” and “horizontal”. As followed from a pilot experiment with the forced choice method, thresholds determined in this simplified way correspond to a detection probability of about 0.75. The forced choice method, though better defined, requires a large number of presentations and is impractical when the interest is mainly in mutual comparison of many modulation sensitivity functions. To study normal free vision in twilight a large sine wave field is required, preferably at a distance of several meters, to avoid accommodation problems. Therefore the sine wave patterns were prepared in the form of slides and projected on a white screen at a distance of 3.5 m from the subject. The field sire was 825
A. VAN MEETEREN AND J. J. Vos
826
17” x 11”. A second projector superimposed a uniform field. Any desired combination of mean luminance and modulation could be adjusted with the aid of neutral density tilters in the light beams and, within limits, by control of the lamp currents. Calibrations were made with a Weston Viscor photocell.
The average results of two subjects and horizontal and vertical pattern orientations are presented in Fig. 2. These modulation sensitivity functions show the general properties, well known from the literature: a maximum modulation sensitivity for medium frequencies, shifting toward lower frequencies and finally disappearing as luminance decreases. The curves of Fig. 2 clearly illustrate the two aspects of modulation sensitivity functions: their spatial bandwidth (IV) represents resolution, whereas their relative height (S) reflects contrast sensitivity. Contrast sensitivity increases with luminance. However, the curves for the 500 . ldLcd/n x 18) * lo-*
200 5 ;i
+ 16' 0 IO0 q 16'
100
,x ..? 50
._ z z 20 c .o g 10 2 z
5
2 1
0.5
5 10 20 1 2 spatial frequency w in ppd
50
100
FIG. 2. Modulation sensitivity functions. Spatial frequency is plotted in periods per degree. These measurements refer to normal free vision to a field of 17” x 11”. Average results of two observers and horizontal and vertical pattern orientations. The natural pupil was 5-7 mm in the luminance range concerned.
various luminances cannot be brought to coincide completely by shifts along the sensitivity axis. The curves span a wider range of spatial frequencies as luminance is raised, i.e. the spatial bandwidth increases with luminance. If spatial bandwidth is limited by summation, this implies that the summation area is smaller at higher luminances. Spatial summation in the visual system is known from Ricco’s law. Ricco’s law states that the contrast threshold for small test objects depends on their energy content, as long as they are smaller than “Ricco’s area”. Ricco’s area is only a simplification of more complex spatial interactions. As RATLIFT (1965) proposes, spatial interations, including summation and inhibition, can be characterized by detailed weighting functions. Weighting functions are most straightforwardly determined by a Fourier transformation of the modulation sensitivity functions. This technique was applied by PATEL (1966) to fovea1 modulation sensitivity functions for retinal illuminances in the range of 3-1000 td. Weighting functions for lower retinal illuminances can be computed from our data.
FIG. 1. Spatial sine wave pattern. The luminance is sinusoidally modulated as a function of spatial cobrdinate. The modulation is defined as a/B. The spatial frequency w is the number of periods per unit of angle.
vacing page 826
Resolution and Contrast Sensitivity at Low Luminances
827
In order to use Fourier techrdques the system concerned should be linear. In our case spatial interactions, summation as well as inhibition should be linear. Spatial summation is apparently linear within Ricco’s area. The linearity of Iateral inhibition is recently discussed by THOMAS (1968). Measurable non-linear inhibitions were found when the luminance differences concerned were larger than a factor 2. Thus the possible inhibition in sine wave patterns with low spatial frequencies will be practically linear in threshold measurements where the modulations are always smaller. The calculations of the Fourier transforms were made according to FILON’S(1928) method for numerical evaluation of trigonometric integrals, In preparing the data we had to extrapolate the modulation sensitivity for zero spatial frequency. This extrapolation is easily made for the lowpass curves, but is problematic for the bandpass curves in Fig. 2. However, its effect on the spatial weighting function is only noticeable at large angular distances from the central stimulus. Assuming spatial ~t~~tio~ to be restricted to angular distances of no more than I”, we made the extrapolation such that the weighting function vanishes at that distance. Another extrapolation had to be made at lower retinal illumiuances. Here the experhnental data are restricted to a small frequency range, since the rest of the curves vanish below threshold, as is shown in Fig. 2. This of course does not necessarily imply a small spatial bandwidth. We decided therefore to extrapolate the modulation sensitivity curves to conform the next higher ones, as a better estimate than a truncation.
1
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100
8
10 20 angular distance
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40
in min of
OTC
3. Weighting functions obtained by Fourier transformation of modulation sensitivity functions, referring to normal free vision and a field of 17” x 11”.
Figure 3 illustrates some of the weighting functions found by Fourier transformation of the modulation sensitivity functions for normal free vision at low retinal illuminance. We prefer retinal illuminance as the relevant parameter here, rather than luminance. The diameters of the natural pupils were measured in the experimental conditions with the aid of an infra-red image tube. Thus luminance could be converted into retinal illuminance. For the sake of clarity weighting functions for intermediate retinal illuminances are omitted but they fit nicely in between. The weighting functions shrink when the retinal illuminance is raised, and reflect lateral inhibition. Their ha~fwidths change from 11.5 to 2-3’ of arc for normal free vision in between O-01 and 100 td. This improvement can only partly be attributed to the contraction of the natural pupil from 7 to 5 mm, resulting in better optics. The halfwidths of the line spread functions for the optics of the human eye, as measured by CAMPBELLand GUBISCH (1966) change from 2.0 to l-5’ of arc in the pupil range concerned.
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One can state therefore, that the weighting functions for normal free vision, as presented in Fig. 3, are of neural origin mainly. These weighting functions for vision with large fields possibly apply to different retinal positions. FOVEAL
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Figure 4 presents modulation sensitivity functions for fovea1 vision. These measurements were made with a small sine wave field of 2.8” x 2*8”, a 3-mm artificial pupil and fixation control. The experimental set up for these measurements is described elsewhere (VAN MEETEREN, Vos and F~OOGAARD,1971). The procedure of the measurements was the same as described above. The results refer to the same subjects. The fovea1 modulation sensitivity 500
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Fio 4. Modulation sensitivity functions. Spatial frequency is plotted in periods per degree. These rn~~rnen~ refer to a foveauy eated field of 2.8” x 2-8” and a 3-mm artificial pupil. Average results of two observers and horizontal and vertical pattern orientations.
differ markedly from the large field results in that sensitivity decreases faster with luminance, whereas high frequency performance tends to be better. They do not show a peaked maximum in the luminance range investigated. Our measurements complement those made by PATEL (1966) at higher retinal illuminances, where the bandpass character appears again. Apparently fovea1 modulation sensitivity functions only change in height, not in shape. This means that the spatial interactions do not change with retinal illuminances below 1 td; above that level lateral inhibition starts to develop (PATEL,1966). Figure 5 gives the weighting functions of the fovea for low retinal illuminance (A) and for 3 td (B). These weighting functions are narrower than those for natural vision, partly due to the 3 mm artificial pupil, but certainly also because of smaller retinal units. functions
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Mod~ation sensitivity functions represent two aspects of vision: the contrast sensitivity S in the relative heights of the curves, and resolution in the form of the spatial bandwidth
Resolution and Contrast Sensitivityat Low Luminmces
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FIG. 5. Weightingfunctions obtained by Fourier transformation of modulation sensitivity functions, refer&g to vision with a 3-mmartifkial pupil, fixationcontrol and a fovea1field of 2.8” x 2.8”.
W. Contrast sensitivity and spatial bandwidth are not independent of each other, however. Spatial bandwidth can be improved by decreasing the station area but only at the expense of contrast ~nsitivity. The photon fluctuations are relatively greater in a smaller summation area. Let D be the diameter of a summation area and B the luminance. The mean number of photons within the summation area (signal) is proportional to BD2 and the variance (noise) is proportional to (BP)“. The signal to noise ratio thus is proportional to BP/(BP)f = B*D. According to the well known sample-theorem one can substitute D = 1/2W. Further contrast sensitivity at best can be proportional to the signal to noise ratio. Thus, if the eye would follow the limits put by photon fluctuations, one should find a kind of uncertainty relation: s. w :. B”. (1) Due to the bandpass structure it is often difficult to separately define the height Sand the width W of a modulation sensitivity function. Their product S. W on the other hand is well defined; it is the area under the modulation sensitivity function, and can be determined directly. We will call this product the integrated modulation sensitivity. In Fig. 6 the integrated modulation sensitivity is plotted vs. retinal illuminance on logarithmic scales. Curve A refers to the measurements for a large sine wave field of 17” x 11’. The curve to a first approximation follows a square root relation indeed over almost 5 decades. The same holds for curve B which is derived from modulation sensitivity curves measured by VANNES and BOUMAN(1967) with a sine wave field of 4.5” x 8*25”, monochromatic green light, and a 2-mm artificial pupi1.l As argued above this result is in agreement with the hypothesis of DE VRIES(1943), that vision is limited by the statistical fluctuations of the incoming photon flux. Contrast sensi* We have no satisfying expknation of the considerabledif%erence in heightbetweencurve A and curve B of Fig. 6. Compared with the scarce literature on modulation sensitivity for sine wave gratings at lower iuminances curve B seems to be rather high.
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FIG 6. Integrated mod~ation sensitivity as a function of retinal ilIurninan=. The curves repreaent square root relations. A refers to normal free vision and a large field. The data for B are derived from VANNES and BOUMAN(1967) and refer to monochromatic green light and a 2mm artiikial pupil. The open circles refer to the foveal measurements.
tivity S and spatial bandwidth W (or resolution) may individually deviate from the square root law without coming into conflict with the fluctuation theory, when their interaction is properly realized. Apparently the increase of the signal to noise ratio with luminance can be used either for better contrast sensitivity or for better resolution. The strategy of the visual system in this respect can be studied when S and Ware known separately. However, the separation of S and W is problematic, as remarked before, in view of the bandpass character of the modulation sensitivity functions. We have taken rather arbitrarily for S the maximum values of the modulation sensitivity functions. The value of W then follows from the known product of S and W. This separation is presented for normal free vision in Fig. 7. The increase of information with luminance is at first spent mainly to modulation sensitivity whereas at higher levels some modulation sensitivity is sacrificed for better resohttion. This is about the region of rod-cone transition. No sign of this transition is found in the integrated modulation sensitivity vs. retinal illuminance relation (Fig. 6). Apparently the visual system is able to follow the limit put by the photon fluctuations irrespective of this transition. Or, to put it in a probably more correct perspective, the rod-cone shift is one of the strategic means. It will be clear, that the square root relation gradually changes into Weber’s law at higher luminances: due to neural saturation the signal to noise ratio is frozen at a certain level. At low luminances the absolute threshold sets a limit. This is evident in the fovea1 measurements. Fovea1 integrated modulation sensitivity at low luminances remains clearly
Resolution and Contrast Sensitivity ta Low Luminances
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Fro. 7. Modulation sensitivity S (upper curve, left scale) and spatial bandwidth W (lowest curve, right scale) as a function of retinal illmninance for normal free vision. The increase of the signal to noise ratio with iuminance is at first mainly used for better modulation sensitivity, later also to improve spatial bandwidth.
behind the square root relation and tends to become zero between O-01 and O-1 td. The number of photons detected within the summation area and within the summation time here apparently comes to the absolute threshold. DISCUSSION So far we discussed the product of sensitivity and spatial bandwidth as a function of
luminance on a relative scale. The actual threshold signal to noise ratio within a summation area and within the integration time can be roughly estimated. MORGAN (1965) found surprisingly low values in between O-12 and O-19, starting from contrast sensitivity and resolution data derived from different authors. Modulation sensitivity functions reflect contrast sensitivity and resolution simultaneously and thus allow a more direct estimation. The computation proceeds as follows. With 2.76 x 1Ol8photons of 550 ~~, l-47 X 10q3 W/lm and 1 td equal to O-85 x lo-l3 lm/min* of arc, the number of photons per min’ of arc per see per td is 342. Taking from Fig. 7 a spatial bandwidth of 4-6 ppd at 1 td, the width of the integration area can be estimated at 60/(2 x 4-6) = 65’ of arc, and its area at 43 minz of arc. With an integration time of 0~1set and assuming that only about 12 per cent of the incoming photon flux :is effective (MORGAN, 1965), one arrives at 0.1 x 0.12 x 43 x 342 = 175 photons per integration area-time volume at 1 td. The fluctuations in this number amount to 175f = 13.2. On the other hand, one finds a modulation sensitivity of 86 at 1 td from Fig. 7, indicating a threshold amplitude of 175185= 2.1. Apparently the signal to noise ratio at threshold is Z-1/13+2= O-16. This estimation is as low as Morgan’s. It is impossible to detect a signal with a probability of about 75 per cent when the signal to noise ratio is only 0.16. This low signal to
noise ratio applies to a single summation unit however. The sine wave patterns cover many summation units. It is hypothesized, that several summation units pool their detection
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probability in some way and that the total probabi~ty of detection of a sine wave pattern builds up over the pattern area. The high angular resolutions connected to vernier acuity and depth perception indicate a similar process. No systematic investigations of probability summation in the detection of spatial sine wave patterns are known to us. Measurements of modulation sensitivity for sine wave or bar patterns, in which the length of the bars and the number of bars involved is varied, might provide information about the characteristics of probability summation. R~HLER and HILZ (1966) found that modulation sensitivity increases with the length of the bars and asymptotes at a bar-length of about I”. COLTMANand ANDERSON(1960) found that modulation sensitivity increases with the number of bars in the pattern and that it asymptotes at 7 bars. Those results indicate that probability summation works to fufl advantage when the testpatterns are larger than 1’ and contain more than 7 bars. HAY and CHESTER(1970) recently argued that the processing of noise might introduce a component in the modulation sensitivity functions that depends on spatial frequency. This component would be absent in transfer functions measured in suprathreshold conditions. DAVIDSON(1968) and CAMPBELL and MAFFEI(1970) compared transfer functions derived from threshold and suprathreshold measurements and found almost no difference. Thus the effect of noise, inherent in threshold measurements, is probably independent of spatial frequency. This conclusion also applies to probability summation. As an alternative to the probability summation hypothesis ROSE(1948) suggests direct summation of photons over the whole (circular) test-object. The resulting signal is compared with the fluctuations in an equal sized area of the background. COLTMAN(1954) extended this hypothesis to bar patterns. There are several objections to this hypothesis. Firstly, as BOUMAN(1961) remarked, it implies that the visual system must know the test stimulus before detecting it. Secondly, the modulation sensitivity for sine wave patterns should be proportional to l/w*, where w is the spatial frequency. However, as mentioned above it is not likely that noise gives rise to a component in the modulation sensitivity function that depends on spatial frequency. Thirdly, the known differences in the size of Ricco’s area at different retinal locations and the corresponding differences in spatial bandwidth could not be explained if the visual system could summate photons over the whole test-stimulus, no matter what its size. REFERENCES BOUMAN, M. A. (1961).History and present status of quantum theory in vision. In Sensory ~omm~~cafion. (edited by R~~ENBLITH,W. A.) New York. BOUMAN, M. A. and VAN DER VELDEN,H. A. (1947). The two-quanta expknation of the dependence of the
threshold values and visual acuity on the visual angle and the time of observation. J. opt. Sac. Am. 37, 908-919. CAMPBELL, F. W. and GUBIKH, R. W. (1966). Optical quality of the human eye. J. Physiol., Land. 186,558578. CAMPBEU, F. W. and MAFFBI,L. (1970). Electrophysiological evidence for the existence of orientation and size detectors in the human visual system. J. Physiol., I&d. 2Q7,635-652. COLTMAN, J. W. (1954). Scintillation limitations to resolving power in imaging devices. J. opt. Sot. Am. 44, 234-237. COLTMAN, J. W. and ANDERSON,A. E. (1960). Noise limitations to resolving power in electronic imaging. Froc. Insrn Radio Engrs 48,858-86X DAVIDSON,M. (1968). Perturbation approach to spatial brightness interaction in human vision. J. opt. Sot. Am. !5&1300-1308. FILON, L. N. G. (1928). On a quadrature formula for trigonometric integrals. Proc. R. Sot. Edit&. 49,38-47. HAY, G. A. and Crmsmas, M. S. (1970). Limiting factors in threshold and suprat~hold vision. Nature, L0m.i.22% 1216-1218.
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VAN MEETEREN,A., Vos, J. J. and E~~GAARD,J. (1971). Das Sehen mit dem Small Starlight Scope, einem Dreistufen-Bildverstr. Optik 32,456467. MORGAN, R. H. (1965). Quantum fluctuation and visual perception. In Diagnostic Radiologic Instrumentation. Charles C. Thomas, Springfield, Ill., 61-91. VAN NIB, F. L. and B~UMAN, M. A. (1967). Spatial modulation transfer in the human eye. J. opt. Sot. Am. 57, 401-406. PATEL,A. S. (1966). Spatial resolution by the human visual system. The effect of mean retinal illuminance. J. opt. Sot . Am. 56,689-964. RATLXFF,F. (1965). Mach Bands: Quantitative Studies on Neural Networks in the Retina. Holden-Day, San Francisco. RBHLER,R. and HILZ, R. (1966). Physical and physiological factors in visual modulation transfer. In Performance of the Eye at Low Lumimnces. pp. 105-117. Excerpta Medica Foundation, Amsterdam. Rosa, A. (1948). The sensitivity performance of the human eye on an absolute scale. J. opt. Sot. Am. 38, 196-208. SCHALW,0. H. (1956). Optical and photo electric analog of the eye. J. opt. Sot. Am. 46,721-739. THOMAS,J. P. (1968). Linearity of spatial integrations involving inhibitory interactions. Vision Res. 8,49-68. DE VRIES,H. (1943). The quantum character of light and its bearing upon threshold of vision, the differential sensitivity and visual acuity of the eye. Physica Eimihoven 10,553-564. WBsrmm.r~~, G. (1965). Visual acuity. A. Rev. Psychol. 16,359-380. Abstract-Modulation sensitivity functions of the eye, measured with spatial sine wave patterns, represent two aspects of vision, its contrast sensitivity (.!?)and its resolution in the form of spatial bandwidth (W). According lo the fluctuation theory of vision, at low luminance the product S. W. should be proportional to the square root of luminance. This is contirmed. According to this relation the increase of the signal to noise ratio with luminance can be used either for better resolution (W) or for better contrast sensitivity (8. The strategy of the eye as to this choice is discussed. R&rtn&Les fonctions de modulation de sensibilite de l’oeil, mesurees avec des mires spatiales sinusoldales, representent deux aspects de la vision, sa sensibilite au contraste (5’) et sa resolution sous forme d’une largeur de bande spatiale( W). Selon la theorie de fluctuations de la vision, aux faibles luminances le produit S. W devrait &re proportionnel a la racine carrel de la luminance, ce qui est con6rme. Selon cette relation, l’augmentation du rapport signal sur bruit avec la luminance peut servir soit A augmenter la resolution (W) soit a accroitre la sensibiliteaucontraste(S). On discute la strategic de l’oeil dans ce choix. ZusannnenfassBBg-Mit Sinusgittem gemessene Kontrasttibertragungsfunktionen des Auges bringen xwei Aspekte der Sebleistung xum Ausdruck: Die Kontrastempfindlichkeit (S) und das AuflosungsvermGgen in Form ortlicher Bandbreite (W). Entsprechend der Fluktuationstheorie sollte bei geringer Leuchtdichte das Produkt SW proportional der Quadratwurzel der Leuchtdichte sein. Das wird be&it&t. Nach dieser Rexiehung kann die Zunahme des Signal-Rausch-Verhahnisses mit der Leuchtdichte entweder xur Verbesserung der Auflosung (W) oder der Kontrastemp8mllichkeit genutzt werden. Es wird besprochen, wie das Auge diese Alternative entscheidet. - @yHKqEiE ¶yBCTBETeJtI.HOCTH ma3a KMO,ny~xmi~~ B3MepBJIBCbCITOMOmUO npOCTpancTBennbIx MTTepHOB, cHHyCO~aBI.HO n3MemnomrixcB H [email protected] BBa acneKTa 3peHaR: er0 KOHTpaCTnyBJnyBcTBETenMBXTb(5) H er0 npOCTpaFtcTBeBTTyIO pa3pemaTOmyro cnoco6~iocr1. J&napemeTOK, COCTOBmBx ~3 ~OJIOCpa3HO2tSaCTOTbI(w). COr~~acno &noKTyamiomioZt ~eopmi 3pemia npri mi3~on ap~ocrn npon3se~emfe s* Wnonxcno 6r.1~5nponopmioHaJtBHOKOpBiOKBanpaTHOMyH3 BpKOCTE.3T0 nOJty¶BBO ITOHTBepXjTenEe.B COOTBeTCTBEE C 3TBM, yBeJni¶eHHe OTHOmenBa CHTHaBaK myMy rtp&i yBe.BBKemiE~KOCTE MOIKIX6bTTb ECIIOJI~~OB~BO ~1x60 AJM nysmero paspemenna (W), ~60 ~nri nymneti xonTpacrnolt ~IlCTBBTeJtBHOCTB (S). Q6cyXnaeTca CTpaTerEB I’JTaXlnpn 3TOMBbI6OpC. PewMe
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