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450 nm and B being 580 nm. To begin with, when the intensities of fields A and B are fairly high, A will look bluish and B yellowish. We then decrease the intensities in A and B equally until both fields are so dark that they have lost their color. During this process A will gradually look brighter than B. At really low luminance levels, field B will disappear while A is still grayish. To restore the subjective impression of equal brightness, the intensity in B must be increased about 100-fold.
In nature this phenomenon can be observed at dusk. As darkness comes at the end of the day, the eye becomes increasingly sensitive to short-wavelength light and less sensitive to long-wavelength red and orange light. If initially, in daylight, a bluegreen flower has the same brightness as a red flower, the blue-green flower will become brighter than the red flower as darkness sets in. This phenomenon was first described by Purkinje (1823), whose name has since been associated with this perceptual shift. The phenomenon can easily be explained by the shift from cone vision in daylight to rod vision at night. The two different spectral sensitivity curves are displayed in Figure 4.8. The scotopic spectral sensitivity curve V0 has the same form as the spectral sensitivity curve of the rods, with a maximum sensitivity at 507 nm. The photopic luminous efficiency curve V is a weighted sum of the M and L sensitivities of the M- and L-cones.
Relative luminous efficiency
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Figure 4.8 Photopic and scotopic spectral luminous efficiency curves for daylight and night vision. The photopic curve, with maximum at 555 nm, is the basis of photometric measurements of light.
Linear and nonlinear response
In a linear detector there is proportionality between stimulus magnitude, I, and response magnitude, R ¼ s I. Irrespective of proportionality factor, s, if the stimulus strength doubles, the response magnitude will also double.
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While there is no such proportionality between light intensity and response of a photoreceptor (its hyperpolarization), the first step in this process, the absorption of light in the receptor’s pigment, is typically a linear function of the intensity. In this first step, light absorption transfers energy from light quanta to the receptor’s pigment, resulting in receptor excitation. After a cascade of intermediate steps, this eventually leads to hyperpolarization of the photoreceptor. When the retina is illuminated by light of a wavelength , the receptor’s excitation, S, is the product of the receptor’s sensitivity, s , for light of this wavelength and the light intensity,
I :
S ¼ s I
For broad-band light this multiplication must be performed for every wavelength (or for small wavelength intervals ) and summed over the spectrum:
S ¼ s I
For a linear detector, the excitation, S, can be directly replaced by the response R ¼ s I. As above, R means output or response, and I is a general symbol that denotes the input (the stimulus magnitude). For the photocell used in a lux-meter to measure illuminance, R is actually the current in milliamperes, but the lux-meter is wavelength-calibrated so that the reading is illuminance, with the unit lux. I is the radiant flux per unit area (irradiance in mW/m2), and s is the constant of proportionality between output and input. The photocell is linear within certain limits, and for the same stimulus (same spectral distribution) the readout is therefore proportional to the power of the incoming radiant flux.
The receptor potential, V, that is eventually generated by the excitation (light absorption), S, is not proportional to S (or I). An increase in light intensity, and the resulting proportional increase in excitation, leads to a change in the receptor potential that depends on the degree of polarization prior to the new excitation. The relationship between light intensity and receptor potential is approximately logarithmic over a large intensity range. This nonlinearity between stimulus intensity and the polarization response is carried over to the next cells in the visual pathway. However, the relationship between the magnitude of the receptor potential and the firing rate of a ganglion cell may be approximately linear.
In a limited sense the eye can be viewed as a transducer of radiant energy, or a detector, like that in Figure 4.9. In cases where the output is nonlinearly related to the input, it is useful to apply an expression of differential sensitivity, or gain, that is valid within a small intensity interval, I, and a response interval, R:
s ¼ R= I
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Figure 4.9 A transducer is a light-sensitive element, such as a photoreceptor in the eye and the detector in a light-measuring instrument that transforms the light input, I, to a different output. For instance, electromagnetic radiation of a certain power in Watts may be transformed to an output response, R, measured in Volts.
This expression for sensitivity (or gain) is more general than that for the sensitivity of a linear system, defined earlier. In a linear system the two definitions give the same result. The difference between the two definitions is made clear in Figure 4.10. For responses that are nonlinear, it is necessary to choose a fixed response criterion, i.e. a constant value of R, when calculating gain. One sees from Figure 4.10 that
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Figure 4.10 The relationship between an input, I, and an output, R, in a linear and a nonlinear system. In both cases, sensitivity (or gain) can be defined as s ¼ R= I.
there is little difference between a linear and a nonlinear intensity–response curve (I–R curve) for small intensities to the left of the figure. Therefore, receptors and other cells can be treated as quasi-linear systems for low intensities, close to threshold.
Figure 4.11 shows schematically a receptor’s polarization as a function of light intensity. In Figure 4.11(a) the dashed I–R curve of Figure 4.10 is drawn for stimuli of different wavelengths (but here on a logarithmic x-axis). This leads to I–R curves for different wavelengths that are displaced parallel to each other on the abscissa. Because they are from the same receptor, all the curves look the same. The parallel shift reflects the receptor’s spectral sensitivity, or the effectiveness of the different wavelengths in exciting it. The fact that all wavelengths can give the same response in
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Figure 4.11 Determining the relative spectral sensitivity (b) from intensity–response curves (I–R curves) of an arbitrary nonlinear detector (a). Relative sensitivity is defined as s ¼ R= I, where a constant response magnitude, R, is set to 1.0 for all wavelengths.
a receptor only by adjusting the intensity of the light is a direct consequence of the univariance principle (see p. 160).
If we want to determine the receptor’s relative spectral sensitivity [as in Figure 4.11(b)] from the I–R curves of Figure 4.11(a), the magnitude of R does not matter. Using the general definition of sensitivity above, we can determine the relative spectral sensitivity, s , as demonstrated in Figure 4.11(b): The relative sensitivity is inversely proportional to the threshold intensity, I, that causes a certain constant change, R (the threshold criterion) in the response.
Spectral sensitivity
Energy-based sensitivity
Until now we have treated the concept of sensitivity in a general way, and the symbol I has been used for the unspecified intensity of an arbitrary physical stimulus. More specifically, for the eye, the spectral sensitivity can be expressed as a function of wavelength as:
seð Þ ¼ R=
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where ¼ (W). is here the radiant flux in Watts (W), and the index ‘e’ indicates that energy units are used rather than quantum units.
In a psychophysical experiment, the threshold response, R, in the expression above is held constant, and the radiant flux, , is changed for each wavelength until the threshold response is reached. The criterion for R may be a 55 percent detection rate for a stimulus presentation, or it might be a judgement of ‘equally bright’ to a comparison field, as in Figure 4.1. The spectral sensitivity is then inversely proportional to the threshold power ¼ that gives the constant criterion responseR. The scotopic luminous spectral sensitivity, V0 , of the human dark-adapted eye was determined from psychophysical experiments such as this.
Quantum-based sensitivity
The spectral absorption curve for rhodopsin, the pigment of the rods, tells us how large a fraction of the incident light of each wavelength is absorbed. Rod excitation corresponds to the number of rhodopsin molecules that are affected, which is proportional to the number of quanta that are absorbed, irrespective of wavelength. A similar reasoning applies to the cone pigments. It has been demonstrated that it is the quantum-based spectral sensitivity derived in psychophysical experiments that corresponds to the spectral pigment absorption in rhodopsin. This confirms that the sensitivity of the eye depends on the quantum absorption in the receptors, which can be estimated from the quantum flux at the cornea after correction for the absorption in the eye media (cornea, lens, vitreous, etc.).
The relationship between spectral sensitivity expressed in inverse energy units, seð Þ, and the sensitivity expressed in inverse quantum units, sqð Þ, is the following:
sqð Þ ¼ hc seð Þ=
Here sqð Þ ¼ R= N, and N ¼ N ðs 1Þ is the number of quanta per second in the light stimulus. The derivation is made by the expression E ¼ h ¼ hc=.
Since hc is constant, the relative quantum sensitivity sqð Þ will be proportional to seð Þ=. Compared with the energy sensitivity seð Þ, this means that the quantumbased spectral sensitivity is twice as large at 400 nm as at 800 nm. The maximum for the quantum-based spectral sensitivity will therefore be shifted towards shorter wavelengths relative to the energy-based sensitivity.
Action spectra of the cones
A spectral sensitivity curve that is derived by varying the intensity of light for each wavelength until the same physiological reaction ð RÞ occurs in the receptor is called an ‘action spectrum’. Action spectra are of great importance in vision science
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Figure 4.12 (a) Relative spectral energy sensitivity for the three cone types of the human retina. L denotes the long-wavelength sensitive cone, M the middle-wavelength sensitive cone, and S the short-wavelength sensitive cone (courtesy of J. H. Wold). (b) Energy-based spectral sensitivity, 1= E, for the dark-adapted human eye. The heavy black curve represents the sensitivity of rods 8 above the fovea, and the thinner gray curve shows the sensitivity of the cones in the fovea. The curves are averages of the results for 22 subjects. The sensitivities are expressed relative to the maximum sensitivity in the fovea. The cone curve has a higher sensitivity than the rods for wavelengths longer than about 650 nm (modified from Pirenne, 1967).
because action spectra that are measured psychophysically can be compared with those measured electrophysiologically.
The three energy-based spectral sensitivities Lð Þ, Mð Þ and Sð Þ of the three types of human cones are shown in Figure 4.12(a). These sensitivities can now be measured by different methods, psychophysically and electrophysiologically, and the correspondence between them is relatively good. The wavelength of maximum sensitivity for L-cones is about 560 nm, 530 nm for the M-cones, and 425 nm for the S-cones, although minor deviations from these values have been found and explored in recent years (Sharp and et al., 1999; see the section ‘Genetics of photopigments’, p. 108).
In Figure 4.12(b), the spectral sensitivity of the rods is compared with that of L-cones in darkness. Surprisingly, L-cones have higher sensitivity than the rods above