Yinon Rer. Vol. 16. pp. 573 to 579. Pergamon Press 1976. Prmvd
QUANTUM
in Great Brain.
FLUCTUATION FOVEAL VISION’
LIMIT
IN
THEODORE E. Cow School of Optometry. University of California. Berkeley. CA 94720. U.S.A. (Received
21 Xocember 1974; in revised form
5 September
1975)
Abstract-A unique test of the hypothesis that only quantum fluctuations limit the detectability of fovea1 luminance change is presented. 10.9’ dia target spots were incremented or decremented in luminance. ROC curves for detection of both luminance increments and decrements were measured and found to conform to theoretical curves predicted on the basis of Poisson distributions of photon arrival. Four different parameters of the ROC curves lead independently to estimates of quantum efficiency with values of about Z x lo-*. The measured quantum efficiency increases by about a factor of 10, to as high as 0018, when the target is reduced to 1.2’ dia. This is consistent with the view that lateral inhibition causes many of the photons caught from a large target to be ineffectual in the task of detecting the luminance change signal.
and the receiving device is said to have a detective quantum efficiency equal to F (Clark-Jones. 1959).
INTRODUCTION
Implicit in equation (2) are the deVries-Rose law (E x B- 1 ‘). Piper’s law (E Y A”‘) and Pieron’s
Quantum fluctuations present a physical limit to luminance detection (Hecht. Shlaer and Pirenne, 1942; Sakitt, 1972) and to luminance change detection (deVries, 1943; Rose, 1948; Lamar, Hecht, Hendley and Shlaer. 1948; Barlow, 1958a; Tanner and Clark-Jones, 1960; Cohn, 1974). At absolute threshold, luminance detection by the human eye appears to match the limit set by quantum fluctuations (Hecht et al., 1942; Sakitt, 1972), but the evidence is not so clear above absolute threshold.
law (E x T”‘). Barlow (1957. 1958a. b) has presented data consistent with these laws but exceptions (Weber’s law, Ricco’s law. Bloch’s law, respectively) cast doubt on the quantum fluctuation interpretation (Cornsweet and Pinkser. 1965; Rushton. 1956; and see Brindley, 1970, for a review). Consider the following measurement (details in Methods). In the fovea, under conditions of low photopic adaptation. a 25:~~ increase of the steady luClassical quantum jluctuation prediction minance of a 10.9’ dia spot lasting 64 msec can just The detectability, d’, of an increment of flux, E be detected (seen 68% of the time with a false alarm (quanta/set DEG’) lasting for duration, T(sec), in the rate of O-30).If the quantum fluctuation limit explains steady background, B (quanta/set DEG’) due to a the detectability of this stimulus equation (2) can be distant, uniform emitting source subtending area, A used to calculate the quantum efficiency. F. For this (DEG’) is given by Tanner and Clark-Jones (1960): measurement and EAT= 15,ooO photons BAT= 60,OW photons so F = 2.5 x lo-‘. In other EAT ,.. . (1) words, 1 in 4000 photons incident at the cornea is d,=(BAT)L!‘. used in this detection task. a value low compared If only a fraction, f, of the available photons are to the rate at which photons incident at the cornea used in the task are likely to be absorbed in fovea1 receptors.’ An alternative to the quantum Huctuation explanation is possible because this value is so low. Suppose. for example, that 1 in 20 photons incident at the cornea is actually used in the detection task. The noisiness ’ Portions first presented at the Fall meeting of the of the stream of received photons can be expressed Optical Society of America, Houston, 1974. by its sipa-to-mean ratio. This would be (devries. * Light calibrations are specified accurate to within 1943) “!BAT+20 + (BAT+ 20) = 0.015. which is &50”/,. Equation (2) uses a Gaussian approximation to too small to explain why a 2%: increment of the Poisson distributions which can cause an underestimate background is only just detectable. Aguilar and Stiles of F by as much as a factor of 2. Losses in the media at 500 nm may be as much as 50% (Ludvigh and IMC- (1954) used this sort of reasoning to argue against Carthy, 1938) but are probably less (Boettner and Wolter, the descriptive value of the quantum fluctuation limit 1962). The Stiles-Crawford effect may account for a loss for scotopic vision. Possibly a source of internal noise. of as much as 25% (LeGrand, 1968, p. 106 assuming 5.5 perhaps Gaussian (Blackwell, 1953). is the cause of mm pupil). Finally. since human cones in the central fovea the poor performance. How can the quantum fluctuahave dimensions comparable to peripheral rods (Yamada, 1969) the inference of Hecht, Shlaer and Pirenne (1942) tion theory, and its alternatives. be further tested! One approach is to seek independent confirmation of a receptor quantum catching efficiency of about 50% that the quantum efficiency. F. is as low as portrayed may be used (although it would be far preferable to know directly quantum catching efficiency of central fovea1 above. This will be elaborated below. Another approach is to find a test of the quantum fluctuation cones). 573
5-4
THEODORE E. COHS
theorq that separates it from other theoriss of the increment threshold. The main purpose of this paper is to present the results of such a test. Further predictions rhror~. In this section
o_f the yrmrum jhrcruuion nso further predictions of the
quantum ~uctuation theory are described and a second computation of quantum efficisncy is presented. Suppose hypothetically that. in a space-time interval confusable with that of the stimulus (Barlow. l958b). the human observer can make use of only 10 (on the average) of the photons caught from the background of 60X00 photons available at the cornea. Consider the performance of this detector in successive tasks where in one an increment and in the other a decrement (each. for purposes of iilustration. a 6p,i modulation of the background) are to be detected. The optimal behavior of the observer is to count photons and to say that the stimulus has occurred if the count exceeds some cutoff. providing the stimulus was the increment (deVries. 1943; Rose. 1948; Tanner and Jones. 1960). The decision rule should be reversed if
90 -
/ /
/
/ /
/
/
/
Foise alarm rote
Fig. 1. Ideal photon counter ROC curves. Theoretical ROC curves for detection of six photon decrement of 10 photon background (upper) and six photon increment of same background (lower). (Probability paper-Keuffel and Esser $478061.) Ordinate: hit rat
QUANTUM
in Great Brain.
FLUCTUATION FOVEAL VISION’
LIMIT
IN
THEODORE E. Cow School of Optometry. University of California. Berkeley. CA 94720. U.S.A. (Received
21 Xocember 1974; in revised form
5 September
1975)
Abstract-A unique test of the hypothesis that only quantum fluctuations limit the detectability of fovea1 luminance change is presented. 10.9’ dia target spots were incremented or decremented in luminance. ROC curves for detection of both luminance increments and decrements were measured and found to conform to theoretical curves predicted on the basis of Poisson distributions of photon arrival. Four different parameters of the ROC curves lead independently to estimates of quantum efficiency with values of about Z x lo-*. The measured quantum efficiency increases by about a factor of 10, to as high as 0018, when the target is reduced to 1.2’ dia. This is consistent with the view that lateral inhibition causes many of the photons caught from a large target to be ineffectual in the task of detecting the luminance change signal.
and the receiving device is said to have a detective quantum efficiency equal to F (Clark-Jones. 1959).
INTRODUCTION
Implicit in equation (2) are the deVries-Rose law (E x B- 1 ‘). Piper’s law (E Y A”‘) and Pieron’s
Quantum fluctuations present a physical limit to luminance detection (Hecht. Shlaer and Pirenne, 1942; Sakitt, 1972) and to luminance change detection (deVries, 1943; Rose, 1948; Lamar, Hecht, Hendley and Shlaer. 1948; Barlow, 1958a; Tanner and Clark-Jones, 1960; Cohn, 1974). At absolute threshold, luminance detection by the human eye appears to match the limit set by quantum fluctuations (Hecht et al., 1942; Sakitt, 1972), but the evidence is not so clear above absolute threshold.
law (E x T”‘). Barlow (1957. 1958a. b) has presented data consistent with these laws but exceptions (Weber’s law, Ricco’s law. Bloch’s law, respectively) cast doubt on the quantum fluctuation interpretation (Cornsweet and Pinkser. 1965; Rushton. 1956; and see Brindley, 1970, for a review). Consider the following measurement (details in Methods). In the fovea, under conditions of low photopic adaptation. a 25:~~ increase of the steady luClassical quantum jluctuation prediction minance of a 10.9’ dia spot lasting 64 msec can just The detectability, d’, of an increment of flux, E be detected (seen 68% of the time with a false alarm (quanta/set DEG’) lasting for duration, T(sec), in the rate of O-30).If the quantum fluctuation limit explains steady background, B (quanta/set DEG’) due to a the detectability of this stimulus equation (2) can be distant, uniform emitting source subtending area, A used to calculate the quantum efficiency. F. For this (DEG’) is given by Tanner and Clark-Jones (1960): measurement and EAT= 15,ooO photons BAT= 60,OW photons so F = 2.5 x lo-‘. In other EAT ,.. . (1) words, 1 in 4000 photons incident at the cornea is d,=(BAT)L!‘. used in this detection task. a value low compared If only a fraction, f, of the available photons are to the rate at which photons incident at the cornea used in the task are likely to be absorbed in fovea1 receptors.’ An alternative to the quantum Huctuation explanation is possible because this value is so low. Suppose. for example, that 1 in 20 photons incident at the cornea is actually used in the detection task. The noisiness ’ Portions first presented at the Fall meeting of the of the stream of received photons can be expressed Optical Society of America, Houston, 1974. by its sipa-to-mean ratio. This would be (devries. * Light calibrations are specified accurate to within 1943) “!BAT+20 + (BAT+ 20) = 0.015. which is &50”/,. Equation (2) uses a Gaussian approximation to too small to explain why a 2%: increment of the Poisson distributions which can cause an underestimate background is only just detectable. Aguilar and Stiles of F by as much as a factor of 2. Losses in the media at 500 nm may be as much as 50% (Ludvigh and IMC- (1954) used this sort of reasoning to argue against Carthy, 1938) but are probably less (Boettner and Wolter, the descriptive value of the quantum fluctuation limit 1962). The Stiles-Crawford effect may account for a loss for scotopic vision. Possibly a source of internal noise. of as much as 25% (LeGrand, 1968, p. 106 assuming 5.5 perhaps Gaussian (Blackwell, 1953). is the cause of mm pupil). Finally. since human cones in the central fovea the poor performance. How can the quantum fluctuahave dimensions comparable to peripheral rods (Yamada, 1969) the inference of Hecht, Shlaer and Pirenne (1942) tion theory, and its alternatives. be further tested! One approach is to seek independent confirmation of a receptor quantum catching efficiency of about 50% that the quantum efficiency. F. is as low as portrayed may be used (although it would be far preferable to know directly quantum catching efficiency of central fovea1 above. This will be elaborated below. Another approach is to find a test of the quantum fluctuation cones). 573
5-4
THEODORE E. COHS
theorq that separates it from other theoriss of the increment threshold. The main purpose of this paper is to present the results of such a test. Further predictions rhror~. In this section
o_f the yrmrum jhrcruuion nso further predictions of the
quantum ~uctuation theory are described and a second computation of quantum efficisncy is presented. Suppose hypothetically that. in a space-time interval confusable with that of the stimulus (Barlow. l958b). the human observer can make use of only 10 (on the average) of the photons caught from the background of 60X00 photons available at the cornea. Consider the performance of this detector in successive tasks where in one an increment and in the other a decrement (each. for purposes of iilustration. a 6p,i modulation of the background) are to be detected. The optimal behavior of the observer is to count photons and to say that the stimulus has occurred if the count exceeds some cutoff. providing the stimulus was the increment (deVries. 1943; Rose. 1948; Tanner and Jones. 1960). The decision rule should be reversed if
90 -
/ /
/
/ /
/
/
/
Foise alarm rote
Fig. 1. Ideal photon counter ROC curves. Theoretical ROC curves for detection of six photon decrement of 10 photon background (upper) and six photon increment of same background (lower). (Probability paper-Keuffel and Esser $478061.) Ordinate: hit rat
