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13.7. Combined eye aberrations, diffraction    ▐     13.9. Eye spectral response

13.8. Eye intensity response, contrast sensitivity

Eye light-intensity response

Human eye is capable of responding to an enormous range of light intensity, exceeding 10 units on logarithmic scale (i.e. minimum-to-maximum intensity variation of over 10-billion-fold). Inevitably, eye response to the signal intensity, which determines its apparent intensity, or brightness, is not linear. That is, it is not determined by the nominal change in physical stimulus (light energy), rather by its change relative to its initial level.

In general, there is a minimum required change in signal intensity needed to produce change in sensation, and the latter is not necessarily proportional to the former. It was the father of photometry, Pierre Bouguer, who in 1760 first noted that the threshold visibility of a shadow on illuminated background is not determined by the nominal differential in their illumination level, but on the ratio between the two intensities. In other words, that eye brightness response is not proportional to light's nominal (physical) intensity, but proportional to its intensity level. This threshold ratio, which he found to be 1/64 (around 1.5%) did not change with the change of intensity level.

In 1834, German physiologist Ernst H. Weber, based on measurements for a number of different physiological responses, established a general empirical law stating that the minimum noticeable change in input intensity ΔI - so called increment threshold - is in a constant relationship with the intensity level I, i.e.

ΔI/I=constant         (161)

The value of ΔI/I, called Weber fraction, constant, or ratio, varied with the type of physiological response; for brightness response, the initial values varied somewhat from one experiment to another (1/64 Bouguer 1760, 1/100 Weber 1834, 1/38 Steinheil 1837,  1/100 Fechner 1858), but it appeared constant, and the value variations were ascribed to different techniques used and/or to variations in individual sensitivities. More recent experiments came to the ratio values specific for the two main types of retinal photoreceptors, ~0.14 and 0.015-0.03 for the cones and rods, respectively (FIG. 237).

FIGURE 237: If Weber law strictly holds, brightness increment threshold, either alone or as a ratio vs. intensity, always changes linearly with the change in intensity. When both scales of a graph are linear, and the vertical scale is the increment threshold, the plot vs. intensity has slope equal to Weber ratio (top left). When both scales are logarithmic, the plot slope equals 1 (i.e. 45, top right). With the ratio on the vertical scale, the plot is always a horizontal line, since the ratio value is a constant (bottom). As will be addressed ahead, actual data do not support this simple concept. Similarly, the values found experimentally are for the specific stimulus/background/ambient forms, and are not necessarily valid to quantify a general rode/cone response. For instance, the 0.14 Weber ratio for cones was established based on the judgment of observers looking at a pair of bright disks briefly flashed at the same time (Cornsweet and Pinsker, 1965). Just a change in a single factor - for instance, size of the background relative to the stimulus - can significantly alter the increment threshold.

Gustav T. Fechner, German physicist, Weber's student, expanded onto Weber's empirical law: assuming that just noticeable difference in sensation ΔS, corresponding to the threshold signal change ΔI, is a unit change in sensation, defined sensation as S=k(ΔI/I), where k is numerical constant, an integer. Assuming that for small values of ΔS and ΔI, dI/I=dS/k integrates to S=klnI+C, with C being an arbitrary constant. This converts into:

S=2.3klog10I+C        (162)

which is the general form of Fechner's law or, alternately, Weber-Fechner law. Fechner called it psychophysical law, believing it applies to all senses (1860). The choice of integer k affects shape of the curve, which becomes more open (i.e. less steep for small values of I, and steeper for larger I values) as k increases. The choice of C merely affects plot's vertical position.

In its basic form (Weber law), this implies that eye response to object luminance, as brightness discrimination, is not proportional to its actual (physical) intensity level; rather, that it changes with the intensity level, remaining nearly constant relative to it. This, in turn, under assumption that the relative value of just noticeable difference in brightness sensation is a unit of the sensation change, means that the perceived object brightness changes with the logarithm of object's actual brightness.

Fechner was aware that empirical data does not support this simple concept as a general law. But he diminished the magnitude of discrepancy, hypothesizing that it becomes significant only at the extremes of perceived brightness, due to the retinal signal noise: the lowest neural noise level (dark light) at the low end, and saturation at the high end. In order to account for the former, he modified the law into S=kΔI/(I+I0), with I0 being the neural noise level at near-zero illumination.

However, in 1924 Selig Hecht pointed out that empirical data clearly indicates that eye brightness response does not follow Weber-Fechner law over significant portion of its response range (FIG. 238).

FIGURE 238: The solid blue line tightly fits the experimental data by Aubert (1865), Koenig and Brodhun (1889) and Blanchard (1918). It indicates that the Weber-Fechner law - according to which the smallest perceptible change in intensity ΔI vs. intensity level I is constant, thus forms a straight horizontal line when plotted over the range of luminance that the eye can adapt to - applies only to a portion of eye's photopic response, approx, 1-100 millilamberts (1mL=10/π cd/m2). Since the luminance is on a log scale, the plot is greatly compressed horizontally, resulting in an upward swing of ΔI/I toward mesopic and, particularly, scotopic illumination level. Actual nominal value of ΔI diminishes with the illumination level, but at a slower rate than luminance (dashed). When log ΔI is plotted against log(I), strict adherence to Weber-Fechner law would require it to form a straight line with the slope 1 (measured as a ratio of the height of the vertical intercept vs. corresponding length at the horizontal scale, thus the slope is 1 for a 45 line). As the plot shows, log(ΔI) increasingly deviates from research data at very high and, particularly so, low and very low illumination levels. Only in the 1-100 mL range (approximately) the plot nearly conforms to the Weber-Fechner law. Note that the range of intensity is as given by Hecht, with 0.00001 mL at the low end; more recent sources extend it to 0.000001 mL, for the Imax/Imin ratio of about 1010.

 Later experiments confirmed this discrepancy. Based on the available data, Stanley S. Stevens proposed a new general law for human perceptual response, in the form S=kIa, where k is an arbitrary constant and a the exponent (power). Change in the intercept k affects plot height height in a coordinate system, thus changes with the choice of unit for I and/or S; it also changes with the adaptation level (i.e. luminance intensity level). The exponent a equals the plot's slope on log-log coordinates.

The log-log plot on FIG. 238 indicates that between the two extremes - dark light (neural noise preventing any change in the sensation at retinal illuminations below certain level) at the rods threshold, and saturation at the cone limit - there are approximately four different sub-ranges, each with somewhat different response function of the general form S=kIa. For the scotopic range, the exponent a~0.5, thus the smallest perceptible change in intensity changes with the square root of it. Throughout mesopic range, a~2/3, while in the photopic range it gradually shifts from a~1 in its lower portion - where it, therefore, conforms to the Weber law - to a~1.3 at the higher luminance levels.

Interestingly, as it was well known to Hecht, the neural response to light intensity - measured by its photochemical reaction - for the most part does conform to the Weber law. As an empirical fit to it, he used the relation CI=x2/(a-x), with a and x being the concentration of original photochemical before it reacted with light, and that of photochemical changed by interaction with light, respectively; C was a constant, estimated at ~100 for the rods, and ~0.25 for the cones and I the retinal illumination. Taking a=1, and 0≤x≤1, the relation can be written as CI=x2/(1-x); for x as a function of I, it produces sigmoid (S-curve), with most of its length between the floor (where the curve asymptotically approaches zero) and ceiling (asymptotically approaching its maximum, when all of the original photochemical is being converted) closely enough approximated by a straight line. However, empirical data clearly indicated that brightness perception does not mirror the underlying photochemical reaction.

More recently (Rushton and Naka, 1966), a simpler empirical relation was found to be well approximating the magnitude of neural response N of both, rods and cones, in terms of stimulus intensity I and semi-saturating constant σ, as N=I/(I+σ) which, with steady background I0 added to test stimulus becomes N=σI/(I+σ')σ', with σ'=σ+I0. The semi-saturating constant σ is the light flash that raises neural response N to half its maximum value, thus when I=σ, N=0.5. Starting with this relation - known as hyperbolic formula, or H-function - eye brightness response to more complex fields can be described, and the agreement with experimental data is, as mentioned, good.

According to Rashton, σ for human rods is about 1,000 quanta (photons) absorbed per rod per flash. The absolute rod threshold is about 100,000 times smaller, or 1 photon absorbed per 100 rods. This physiological fact supports Rashton's conclusion that rod response cannot be explained simply by bleaching of rhodopsin on the rod level, but that a higher-level processing, such as that at the level of rode community, i.e. conglomerate of rods, is required. Cone response is similar, only their threshold and semi-saturating constant are both higher. The multi-level processing of the light interacting with photoreceptors should, at least in part, help explain why eye brightness response does not mirror photochemical reaction at the level of photoreceptor cell.

As noted, graph on FIG. 238 and the corresponding figures are based on older data, obtained generally by testing the whole eye, with all retinal photoreceptors actively participating. More recent experiments focused on determining the isolated response of the specific photoreceptor types (FIG. 239). 

FIGURE 239: While the older eye brightness response research generally did not selectively interfere with the retinal function, more recent experiments went after isolating particular type of retinal photoreceptors, in order to determine their specific response. Three examples at left are for isolated rod response. Units for all are scotopic trolands (sTd), which represent luminance falling onto the pupil given in cd/m2, multiplied with the pupil area in mm2 (for instance, 5 sTd at 4mm pupil diameter indicates 0.4 cd/m2 background luminance; with 1 sTd retinal illumination corresponding to approx. 0.1 cd/m2 background luminance, the conventional upper limit to rods' activity (1-10 cd/m2) is between 10 and 100 sTd, or 1 and 2 on log scale. All three examples used some form of Stiles' rod isolation technique, with deep red background w/centered in it green test (stimulus), projected onto extrafoveal area (the cones, being more sensitive to deep red, are effectively blinded to the stimulus by the background). TOP: A commonly referred to plot showing that rods' response to brightness for the most part follows Weber law, with the increment threshold to background luminance ratio remaining constant  (i.e. plot slope close to 1, or 45 on log-log coordinates, and is nearly horizontal for Weber's fraction expressed as a ratio number). At a very low background luminance, it is neural noise (dark light) that prevents response; above this level, the response nearly follows square root (DeVries-Rose) law, as a brief transition into Weber law. High background luminance causes rod saturation. MIDDLE: Plots interpolated into the actual data for 6 observers. Unlike Aguilar and Stiles, who used significantly larger stimulus and steady background, the 13 arcmin 1.5 milliseconds stimulus here is temporally coincident with the 18-degree background it is centered on, both projected at 18 from fovea (i.e. they are a simultaneous 1.5 msec flash, except the orange plot, which is for 1 msec flash). All but one have slope significantly below 1, indicated by FIG. 238, closer to 0.5. Individual differences with respect to the scotopic threshold level can be significant, nearing hundredfold. Replacing coincidental flashing background with a steady one had little effect on the plot shape, but it shifted it 1.5-2 logarithmic units (30-100 times) to the left. Similarly, prolonging test duration to 0.1 sec (100msec) shifted a plot nearly as much to the left and downward (i.e. both test and background are considerably fainter at the threshold level). Note that the vertical scale here is normalized to a threshold intensity unit. BOTTOM: A 30 msec 4.5 test (square) centered on 11 background, projected 12 from fovea. With steady background, the conditions are similar to Aguilar and Stiles', but the plot section between threshold and saturation onset has weaker slope. The corresponding Weber fraction plot (dashed blue) has similar overall shape but, consequently, deviates more from the horizontal in this range (unlike Aguilar and Stiles, plot indicates desaturation point at the high end of background luminance, which could be due to cone activation). With flashed background, rods saturate at a significantly faster rate, the shorter delay of a stimulus vs. background, the more so (open circles are for zero delay, i.e. 0.4 seconds background starting simultaneously with 0.03sec test, crosses for 1.4 sec background preceding test by 1 sec). The corresponding weber fraction quickly skyrockets (colid blue), and the sub-range nearly conforming to the Weber law is relatively small. Curiously, flashed background plots shift to the left of the steady one, opposite to Hallett. If steady preadapting field is added, the increment threshold plots shift to the right and up.

In general, it is considered that cones respond to light in a similar fashion as rods, only at higher luminance levels. This, however, doesn't give us a clear picture of what that response is. As the above examples indicate, eye brightness response is too complex to be accurately described with a simple concept, such as Weber-Fechner law. At best, it is applicable to limited sub-ranges of eye brightness response and some specific stimulus/background forms.

Similar conclusion is suggested for eye brightness response to sine-wave grating (conventional-MTF-like pattern). Experimental data du not support the concept of square root law transition from the dark light level into the Weber law domain (FIG. 239, top). This concept may be relatively close approximation only for a narrow range of spatial frequencies, but the model of contrast sensitivity change that fits experimental data best does not agree with it being generally applicable (FIG. 240).

FIGURE 240: Fitting eye contrast sensitivity function (CSF) into experimental data for sine-wave grating at low illumination levels, based on DeVries-Rose/Weber concept (top) and the actual best fit (bottom). Contrast sensitivity scale is in a unit of (modulation threshold)-1, with the contrast modulation threshold being 1, or 100% (so, for instance, contrast sensitivity 100 means that the minimum contrast detectable by eye is 100-1, or 1%). TOP: contrast sensitivity plots satisfying three simultaneous constraints: (1) the empirical contrast vs. acuity curve, (2) empirical CSF ceiling at 2000 sTd illuminance, and (3) 0.5 slope CSF (square root law) until it reaches the CSF ceiling (contrast sensitivity increases with the luminance level because the increment threshold, according to the square root law, increases at such rate; after transition into Weber law domain, the increment threshold remains constant vs. luminance, and the contrast sensitivity, consequently, remaining unchanged, with zero slope). At mid-to-higher frequencies (4 and 16 cycles/degree), these plots fall far short of reaching the empirical CSF ceiling at 2000 cTd. They also produce discontinuity in the acuity line (sensitivity vs. frequency projection on 3-D CSF surface), not supported by empirical observations. BOTTOM: CSF plots that satisfy the three constraints, and also produce continuous acuity line, have to be non-linear, of the type shown. For the lowest frequencies, CSF is nearly stagnant over the range of illuminance, nearly conforming to Weber law. Over a very-low-frequency sub-range, CSF has significantly weaker slope than 0.5, while over low frequencies it comes close to the DeVries/Weber law concept. At mid and high frequencies, however, much steeper slope than 0.5, which agrees with most of the empirical data, and curvilinear plot are required, (Garcia-Perez, 2005).

By its derivation, Steven's power law is more comprehensive than Weber-Fechner's law. Like the latter, it starts with the assumption that ΔI/I=constant=c1, but expands by assuming that the corresponding smallest noticeable change in sensation vs. sensation level is also a constant, not necessarily equal to c1: or ΔS/S=constant=c2, c1≠c2. Taking c1/c2=a gives ΔI/I=ΔS/aS which, after differentiating, leads to:

S=kIa         (163)

where, unlike k in the Weber-Fechner law, which is an integer by the starting assumption, here it can vary continuously. For a=1, thus with the plot slope equaling 1 (45) too, power law coincides with Weber law, since c1=c2 and ΔS=ΔI. Based on his experiments, Stevens came to the following exponent values for eye brightness response:
uniform 5 stimulus in dark point source brief flash flashed point source
0.33 0.5 0.5 1

As mentioned, the actual values of constants in either law vary with the choice of unit and, with the power law, k is also allowed to vary with the adaptation level. Assuming identical apparent brightness unit, and neglecting changes with the adaptation level (i.e. with the level of luminous intensity), some basic characteristic of the two laws are illustrated on FIG. 241.

FIGURE 241: Plots based on Weber-Fechner (logarithmic) and Steven's (power) laws of psychophysical (sensual) response applied to luminous intensity. If placed at approximately the same (zero intensity) origin, the logarithmic curve (blue) indicates significantly faster initial rate of response to the increasing luminosity than the power curve (red, illustrating rods function, with 0.33 exponent, and green, illustrating cones function, with 0.5 exponent), but slower rate of response at the higher intensity levels. The two curves deviate significantly over the range of intensities; however, farther from the origin, somewhat different logarithmic curve, displaced from the origin (gray), can be constructed to nearly coincide in rate with the power curve over a portion of intensity range (gray plot is left somewhat higher for clarity; it can be lowered simply with a small numerical increase of its constant C, -5). In other words, the rate of eye response to changes in luminous intensity over a limited range of intensities farther from origin can be, in general, closely enough described by either logarithmic or power response.

As mentioned, the above is only an illustration, far from an accurate description of eye brightness response. For instance, with intensity plotted on the linear scale, the range of scotopic intensities (<0.001mL, or 0.01/π cd/m2) is an infinitesimally small fraction of the entire range of intensities (up to 10,000mL) that the eye can adapt to. For that reason, it is preferred to plot intensity on some kind of logarithmic scale, as it is done on the graphs depicting more accurately eye brightness response (FIG. 242).

FIGURE 242: Eye brightness response over the range of intensities, from photopic threshold to the level of discomfort (based on Brightness function: Effects of adaptation, Stevens and Stevens, 1963). Luminance is given in decibels (dB), with 0 dB set at 10-7 mL, or 0.31 μcd/m2 (since the value in decibels is given by 10log(I/It), where I is some arbitrary intensity equal to or larger than the threshold intensity It (any 10 dB differential implies a 10-fold change in intensity or, for x as the dB differential, the corresponding intensities ratio is 100.1x ). Apparent brightness is given in brils (bril is a unit of psychological scale introduced by S.S. Stevens, defined as apparent brightness resulting from a 5-degree white patch of 40dB - equaling 0.001mL, or 0.000314 cd/m2 - luminance seen by dark-adapted eye in a brief exposure). Each individual plot is a form of B=k(I-It)a power function - so logB=alog(I-It)+logk) - with the constants k and a varying with the change in luminance, so that the curve fits experimental data for given level of adaptation (i.e. luminance level). Adding threshold intensity It to the power function results in the straight line of a power function plot on a log-log graph quickly turning down when approaching the threshold level. As the luminance increases, the intercept k decreases (from 10 at fully dark-adapted eye), resulting in lowering of the straight portion of the plot; at the same time, the exponent a increases, resulting in steeper slope of the straight portion (from 0.33 near the rods threshold, to 0.44 for 84dB threshold. While sufficient change in luminance intensity inevitably causes shift in the adaptation level, with the corresponding change in the threshold level, graph suggests that any given luminous intensity will appear brighter the lower level of initial adaptation, but the rate of increase in apparent brightness with the intensity is higher for higher level of adaptation. Interpolating through the points of apparent brightness for each adaptation level plot forms a non-power curve that describes eye brightness response over an extended range of luminous intensities (dotted red).

The above results by Stevens and Stevens are obtained using a uniformly illuminated patch (object, target, stimulus)  on a larger background, whose luminance was determining the adaptation level. For more complex images, specifically photographic images, Bartleson and Breneman found different response, with the best-fit function being not a power function, but one producing a curved, not straight, plot farther from the threshold on log-log coordinates (FIG. 243).

FIGURE 243: Eye response to photographic image under varying image and surround luminance (based on Brightness perception in complex fields, Bartleson and Breneman, 1967). Similarities vs. response with a uniform patch are: (1) perceived brightness for any luminance level increases with lowering the adaptation level, and (2) the rate of increase in perceived brightness is higher for higher level of adaptation. However, unlike the illuminated patch, where eye brightness response is well described with a simple power function, the best fit function here has more complex form:
logB=2.037+0.1401logI-a exp(b logI)
where additional parameters a and b are constants varying with the luminance level. Resulting log-log plot away from threshold is not a straight line, but takes a moderately convex form.

According to it, perceived brightness of a complex picture is a function of its luminance (horizontal scale) and surround (ambient) luminance. For given picture luminance, perceived brightness, expectedly, diminishes as the surround luminance increases (vertical plot position), while the rate of change in the perceived brightness with the change of image luminance (i.e. plot slope) increases with the surround luminance. Over the extended range of surround luminance, perceived brightness increases with picture luminance when it is of similar level (green), or consistently higher (orange) than surround luminance. For picture luminance consistently lower than that of the surround (blue), perceived brightness increases with the surround luminance up to a point, after which it nearly stagnates and, possibly, starts decreasing at high surround luminance levels.

Unlike Stevens' graph for a uniform patch, where the visual response (brightness) is measured against up to a several times larger background, determining the adaptation level, brightness of a complex picture in Bertleson and Breneman is evaluated under varying ambient (surround) luminance levels, which is here the main determinant of eye adaptation level. The constant 0.1 bril threshold over the range of luminance should be due to the method of determining brightness of a complex image, by matching brightness of image elements to that of a previously scaled series of neutral stimuli. Effectively, the brightness scale for any given luminance level here is comparative (relative) in its nature: this comparative brightness is different than nominal brightness, and we use a different word for it: lightness

Considering similarities between the plots for simple and complex stimulus, it didn't come as a surprise that further analyses found that the latter can be expressed as a decline from Stevens-type power function, that is itself a power function of the surround luminance (Choi, 1994). In other words, Bartleson/Breneman function describing eye brightness response to a complex image is only a more complex power-function form, still within Steven's general postulate, which was that the human sensual response follows power law.

But, as mentioned, unlike simple, uniform stimulus, brightness is neither the only, nor most important attribute of a complex image. What is usually more important is the level of contrast between its components. This aspect of eye response can be called visual tone reproduction of that in the original image. It is not dependant directly on the image brightness, rather on the lightness of image elements, which itself depends not only on the nominal brightness of image elements, but also on the image luminance relative to that of the surround. While both, lightness and brightness are part of visual perception, they refer to two different perceptional modalities. As opposed to brightness, which is the attribute of a visual sensation resulting from a given magnitude of light emission (luminance), lightness is defined as the brightness of an area judged relative to the brightness of a similarly illuminated area that appears to be white or highly transmitting (FIG. 244).

FIGURE 244: Neural processing produces lightness perception that deviates from the actual intensity pattern and corresponding nominal brightness. Change of background luminance changes perceived lightness (A, lightness induction or simultaneous contrast; squares and lines are all of identical intensity). So does combining areas of different intensities (B, and C, the latter randomly turning white circles into black).

Lightness can be thought of as relative brightness, but it is not formally, since it does not depend on the actual intensities of picture elements alone, rather on their visual appearance relative to each other; elements of identical emission intensities - thus of given constant brightness when projected against neutral background - can have different lightnesses under different surround luminance levels. Brightness ranges from bright to dim, while lightness ranges from light to dark.

Lightness of different areas of a complex image is key element of its perceived contrast, which is in turn key for tone reproduction. As such, eye response function for complex images - i.e. visual tone reproduction - is directly related to contrast transfer. When colors are present, hue and saturation are also important elements of image tone reproduction. As the plots on FIG. 243 indicate, image brightness (lightness) for any given image luminance level increases inversely to surround luminance. However, the increased lightness of its elements due to darkened surrounding results in decreased image contrast, at the rate proportional to the change in the plots' slope. For instance, Bartleson and Breneman found that image reproduced on a transparency (slide) and projected in dark surroundings needs to have inherent contrast higher by 1.5 on log-log coordinates (~32 times) than the original for the optimum tone reproduction; that corresponds to the increase of a power function exponent (i.e. slope) by a factor of 1.5. On the other hand, printed image viewed in illuminated surroundings only need to have identical inherent contrast to the original for the optimum tone (contrast) reproduction.

All this only scratches the surface of the complexities of eye response to light intensity, but should illustrate well that the common notion of it being described as simply logarithmic is oversimplification, to say the least. Available empirical data is based on the limited range of stimuli form and duration, mainly brief flashes combined with some form of background. On the other hand, telescopic object range from point-like like stars, to complex extended images, like planetary surfaces, and are generally a continuous signal, less than perfectly stabilized on the retina. While the logarithmic eye response to brightness certainly can be applicable in some cases, it won't be in the others. In general, power low covers more ground, which makes it more appropriate as a general law usable for describing eye response to light intensity.

Also, distinction needs to be made with respect to eye brightness response within any adaptation level vs. response over an extended range of luminosities connecting a number of successive adaptation levels. The two respective response curves are different, and the former may be, in general, adhering closer to a logarithmic function - although not necessarily similar between different adaptation levels - than the latter. On the other hand, relatively short span of successive adaptation levels is nearly straight on log log coordinates, hence should be also well fitted with a logarithmic function.

Amateur astronomers have simple, effective means of excluding ambient (surround) light from the equation, in which case eye response is mainly determined by the stimulus (observed object) and its background (telescopic field of view). But it is not unusual for the ambient luminance to be significant factor. Hence, all scenarios are possible. To make it more complicated, the retina is capable of selective local adaptation, i.e. its different portions can be in different adaptation modes.

Above considerations suggest very clearly that there is no simple way to present the complexities of eye response to light intensity. Less so for the telescopic eye, which operates under conditions generally different than a typical "laboratory eye". There is no research specific to the telescopic image of any type that is known to me. At best, any single graph can illustrate some basic characteristics of that response, which have only very generalized implications with respect to how human eye responds to light intensity when coupled up with a telescope (FIG. 245).

FIGURE 245: Eye brightness response graph for complex image (scene) indicated by tests under laboratory conditions (based on Gonzalez and Woods, 1992). The horizontal scale, unlike the vertical, is logarithmic, which makes the graph greatly compressed horizontally: going from left to right, every plot section corresponding to a whole logarithm would be ten times longer than it is, if plotted on linear scale. That would make both, scotopic and photopic plots much more elongated, and especially the latter; they would be, more or less, similar to the plots on FIG. 173 or, for that matter, the "transitional" plots on FIG. 174/175, with the transition from exclusively rod function (scotopic) to exclusively cone function (photopic) much smoother than what it appears like to be here. Likewise, the plot is "decompressed" if the vertical scale is also put in logarithmic form, as shown in the top inset. Compressing the plot horizontally, however, makes it easier to illustrate an important property of eye brightness response, which is that it cannot, at any given moment, cover the whole range of adaptations to light intensity it is capable of. Rather, at any given adaptation level, it responds to a significantly smaller range of luminosity, with the light signal appearing black at the threshold level, and glaringly white at its limit. Any prolonged exposure to light intensities closer to the threshold, or the limit of such adaptation level, causes eye to shift to another level. The graph indicates that such adaptation level spans nearly four log units of the luminosity range, and only about 0.2 log unit on the brightness scale (the latter fairly in agreement with FIG. 174/175 indicate). Within an adaptation level, brightness response is likely to be nearly logarithmic (i.e. following nearly straight lime on log-log coordinates) over a good portion of mesopic and photopic range. It may deviate significantly from logarithmic toward scotopic and higher photopic sub-range, but the specific response can vary significantly with the stimulus type, background and surroundings. 

The entire range of brightness corresponding to the range of eye adaptation between scotopic threshold and photopic limit is, as FIG. 242/243 imply, about three logarithmic units, or 1,000 bril. At any given adaptation level, between the absolute threshold and discomfort level, the available brightness range is significantly narrower than that, with a stimulus appearing black if bellow level's threshold, and glaringly white if above its limit. As FIG. 242 suggests, this adaptation-level brightness range is the narrowest in the scotopic mode, with the max-to-min brightness ratio in the low single digits while the widest in the full-blown photopic mode, when it can approach a 100-fold.

For the typical amateur, who generally observes adapted to light conditions, it is eye brightness response within the adaptation level that is more relevant than adaptation plot constructed over an extended range of light intensity. If we stay away from the threshold and limit of such an adaptation level, eye brightness response is most likely to be nearly logarithmic - i.e. forming straight line on log-log plot - for both, simple stimulus (FIG. 242) and (nearly so) for complex image (FIG. 243). That is probably as much as most of the amateurs need to know. More specific information is hard to find, since most of research was not conducted with "telescopic eye", with the typical astronomical objects, nor under typical field conditions. Most of the information available is only partly relevant.

But for any accurate measurement, it is necessary to know exactly - or as close to it as possible - what is the eye brightness response. For instance, if human eye response to stellar brightness is not logarithmic, but rather a power function, what would be the consequence? The difference may not be negligible (FIG. 246).

FIGURE 246: The familiar concept of stellar magnitude is based on the assumption of logarithmic eye response to point-source brightness. For simplicity, brighter magnitude here is assumed to be a larger positive number, contrary to the conventional designation. In order to illustrate the degree of possible deviation of this concept from the actual eye response in case it follows power law instead, the two functions - the conventional logarithmic and a possible power function - are plotted together. The power function simply satisfies the condition that it yields 0 for (L/L0)=1 and 5 for (L/L0)=100, i.e. that a 100 times more luminous star appears to be 5 times brighter to the eye. It uses ~0.39 power exponent, which is between Steven's original exponents for 5 patch in dark (0.33) and point-source (unspecified conditions, 0.5). The point-source nearly certainly wasn't a star - and for a visual star it would be different than for telescopic stars - so this power function should be sufficiently close for the purpose (the constant doesn't change shape of the plot, it merely affects its vertical position). It is immediately noticeable that the two functions differ at every other point except for the luminosity ratio L/L0 equaling 1 and 100, L being the luminosity of a brighter star, and L0 that of an arbitrary magnitude fainter star. The logarithmic function consistently assigns brighter magnitude to a given luminosity ratio than the power function, and the difference can be significant. For instance, a six times more luminous star would be classified as nearly two magnitudes brighter in the logarithmic concept, and only about a single magnitude brighter in the power law concept.

The bottom line is that eye response to light intensity - or the sensation called brightness - is not linear, i.e. proportional to the light input. Rather, it is, in general, well approximated by some form of a power function of light intensity, and in many cases it can be also described as logarithmic. In other words, it is not the nominal change in the light flux entering the eye that determines the change in brightness; it is the change in the flux relative to the initial flux magnitude.

While the exact eye brightness response is mainly important in the professional and academic circles, the principle can be helpful to the average amateur as well. For instance, it helps understand why the telescopic image of the Moon appears so much brighter than looking at the Moon directly, despite having much lower nominal surface brightness. At 100x magnification, Moon's image area is 10,000 times larger and, consequently, its surface brightness is, neglecting light transmission losses, as much lower. However, to the eye, surface brightness is only 4 times lower, if the response is logarithmic, and ~20 times lower if the response is a simple Stevens-type power function with 0.33 exponent. In the first case, the telescopic Moon would have 2,500 times greater total brightness (8.5 times brighter perceived integrated magnitude), and still as many as 500 times (6.8 magnitudes) greater in the second.

Eye brightness response is related to already mentioned eye contrast sensitivity (FIG. 240), which defines its resolving limit as a function of luminance level and object's (sine-wave grating) angular size (spatial frequency).

Eye contrast sensitivity

MTF analysis of the image formed by a telescope objective is not a "finished product". To some degree, it will be changed by eyepiece aberrations and, when finally projected onto the retina, it is a subject to the effect of physiological processes. As a result, perceived contrast and resolution limit will depend not only on those inherent to the image, but also on its brightness level and angular size on the retina. The specifics of it are described with the aid of eye Contrast Sensitivity Function (CSF), a plot interpolated into empirical data, as shown on FIG. 247.

FIGURE 247: Minimum contrast needed by the eye to resolve MTF-like (sine-wave) pattern, varies with its illumination level and angular size on the retina. Both, spatial frequency (in cycles/degree) and contrast level are given in logarithmic form, to "magnify" the effect at the level of a fraction of the percent in the contrast scale, as well as the effect in the range of large details. Detail size on the retina is given in cycles/degree; 60 cycles per degree is the conventional limit to eye resolution of 1 arc minute. Contrast sensitivity, as a function of detail size and retinal illuminance, is defined by the minimum contrast level at which the image remains resolved. For instance, 10 cycles/degree (6 arc minutes) image size requires 0.6-0.7% minimum contrast in photopic (bright-light) conditions,1-2% in average mesopic conditions, and 10-15% in average scotopic (low light) conditions. Contrast sensitivity peaks for ~9' detail size in photopic conditions, shifting toward larger details in mesopic and scotopic conditions. At the same time, maximum contrast sensitivity diminishes from nearly 0.6% (photopic) to nearly 2% (scotopic). Limiting resolution, at 100% contrast level (along the horizontal scale), also diminishes noticeably with the decrease in illumination being, as expected, the highest in bright-light conditions, and lowest in low-light conditions.

The significance of the CSF for astronomical observing is in helping to determine optimum magnification level for details and objects of different luminosity levels. Like other eye properties, contrast sensitivity can vary widely individually. It is not determined by the quality of the eyesight; an individual with poor eyesight can have better than average contrast sensitivity, and the other way around.

13.7. Combined eye aberrations, diffraction    ▐     13.9. Eye spectral response  

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