Intensity Discrimination Intensity discrimination is the process of distinguishing one stimulus intensity from another Two types: Difference thresholds – the two stimuli are physically separate Increment thresholds – the two stimuli are immediately adjacent or superimposed Fig. 1.1 Increment threshold stimuli (edges of stimuli touch each other) Difference threshold (separated stimuli) A B L T = L + L L T = L - L Fig. 1.2 L T L L T L = L + L T L u m in a n c e + L L - L L T = L- L 0 L Theory and Practice Theory: Quantum fluctuations provide a theoretical lower limit for intensity discrimination by an “ideal” observer Fig. 2.7 Mean = 16 From Dr. Kraft’s course – Hecht, Shlaer & Pirenne Mean = 8 Photon emission follows a Poisson distribution Mean = 4 Probability that the numbers of quanta on the x-axis will occur in any given flash Mean = 2 0.4 Mean = 1 0.0 0 5 10 15 20 25 30 Number of Quanta in a Flash To distinguish a flash with a mean of 8 from a flash with a mean of 9 quanta is impossible! The distributions overlap almost completely Mean of 8, vs. mean of 9 0.15 0.1 Series1 0.05 36 31 26 21 16 11 0 6 Series2 1 If present the flash with a mean of 8 photons many times and a flash with a mean of 9 photons presented many times, there will be many times that the 9 photon flash will have more photons than the 8 photon flash S1 Mean of 8, vs. mean of 12 0.15 0.1 Series1 0.05 36 31 26 21 16 11 6 0 1 Series2 S1 Mean of 8, vs. mean of 16 0.15 0.1 Series1 0.05 36 31 26 21 16 11 6 0 1 Series2 S1 Mean of 8, vs. mean of 20 0.15 0.1 Series1 0.05 36 31 26 21 16 11 6 0 1 Series2 S1 In a Poisson distribution, the variance is equal to the mean. The standard deviation (SD) is the square root of the mean. In a two-alternative forced-choice task, to reach threshold (75% correct), LT must differ from L by 0.95 SD. (e.g., L = 0.95 SD) 0.15 0.1 Series1 0.05 36 31 26 21 16 11 6 0 1 Series2 S1 Moreover, as L increases, the minimum threshold L also increases with the L because the variance in a Poisson distribution equals the mean, so the SD changes with the square root of the mean 0.15 0.1 Series1 0.05 36 31 26 21 16 11 6 0 1 Series2 S1 As a result, an “ideal” observer would follow the deVries-Rose Law (deVries, 1943;Rose, 1948): L L=K where L is the threshold luminance difference, L is the background, or reference, luminance and K is a constant. Theory and Practice In practice: at low background intensities, human observers behave as an ideal detector (follow the deVries-Rose Law) L (milliLamberts) 10000 1000 Fig. 3.1 100 10 1 0.1 0.01 0.001 0.0001 0.00001 0 1 0 00 1 .00 1 0 .0 1 0 0 0 0. 0.0 0.1 1 10 L (milliLamberts) 10 0 1 00 0 10 00 0 At higher intensity levels, the intensity discrimination threshold is higher than expected from an ideal detector (e.g., Weber’s Law holds) The constant proportional relationship between the increment threshold and the reference or background level is called Weber’s law, which is mathematically expressed as: L L = K where L is the threshold luminance difference, L is the background, or reference, luminance and K is a constant. L (milliLamberts) 10000 1000 Fig. 3.1 100 10 1 0.1 0.01 0.001 0.0001 0.00001 00 1 .0 00 1 0.00 1 0 .0 1 0 0 0 0. 0.1 1 10 L (milliLamberts) 10 0 1 00 0 10 00 0 the fraction: threshold L divided by the reference luminance, L is called the “Weber fraction” (threshold L / L) You always can determine the Weber fraction, even when Weber’s Law does not hold L/L Fig. 3.2 1.0 0.8 Weber’s Law does NOT hold (L/ L rises as L decreases) 0.6 Weber’s Law holds 0.4 0.2 0.0 0.00001 0.0001 0.001 0.01 0.1 1 10 L (milliLamberts) 100 1000 10000 100000 Both the deVries-Rose and Weber’s laws fail to account for thresholds at high light intensities Fig. 3.3 Log Increment Threshold, L 6 4 2 0 -2 -4 Predicted by DeVries Rose Law Predicted by Weber's Law -6 -6 -4 -2 0 2 4 6 Log Background Intensity, L (cd/m2) The increment threshold data of a rod monochromat (circles) plotted along with the theoretical lower limit (deVries-Rose, dotted line) and the predictions of Weber’s Law (solid line). Luminance values are in cd/m 2. (Redrawn from Hess et al. (1990) More practical issues: How changes in other stimulus dimensions affect the Weber fraction The Weber Fraction is affected by stimulus size, duration, wavelength, and retinal location (eccentricity from the fovea) #1 Stimulus size: the Weber fraction is lower (smaller) for larger test stimuli Log Weber Fraction, L/L Fig. 3.4 2 Test Field Diameter 4' 10' 18' 55' 121' 1 0 -1 -2 -3 -7 -6 -5 -4 -3 -2 -1 0 Log Background Intensity, L (cd/m2) 1 2 3 More practical issues: Is a target visible under certain conditions? Log Weber Fraction, L/L 2 This is the target’s Weber fraction. It is NOT a threshold Test Field Diameter 4' 1 121' 0 Is a spot with a particular -1 relative to luminance, background, visible? It depends on its size. -2 If the target is 121’, it is visible If 4’, it is not visible -3 -7 -6 -5 -4 -3 -2 -1 0 1 2 Log Background Intensity, L (cd/m) 2 3 Need to distinguish between the Weber fraction of a target vs. the threshold of a viewer. For a subject or patient viewing a target, if the subject’s Weber fraction is below a line, then the subject’s threshold is better (smaller). If the Weber fraction of a target is below the line, the target is NOT visible to someone whose threshold is on the line. The smaller the threshold L, the smaller is the value of the Weber fraction for a given background L, (only the numerator changes) and the more sensitive the visual system is to differences in light intensity. A B L T = L + L L T = L - L Fig. 1.2 L T L L T L = L + L T L u m in a n c e + L L - L L T = L- L 0 L The “dinner plate” example: Plate with luminance of 0.0102 footlamberts. Background is 0.01 footlamberts L is thus 0.0102– 0.01 = 0.0002. L/L = 0.0002/0.01 = 0.02 (plate is 2% more intense) From Figure 3-4, can learn that this is not visible. Log Weber Fraction, L/L 2 Test Field Diameter 4' 10' 18' 55' 121' 1 0 -1 . -2 -3 -7 -6 -5 -4 -3 -2 -1 0 Log Background Intensity, L (cd/m2) 1 2 3 More practical issues: Is a target visible under certain conditions? Log Weber Fraction, L/L 2 Test Field Diameter 4' 1 Is a spot with a particular luminance, relative to 0 background, visible? It depends on its size. 121' -1 -2 Plate’s Weber fraction -3 -7 -6 -5 -4 -3 -2 -1 0 1 2 Log Background Intensity, L (cd/m) 2 3 Continuing: How changes in other stimulus dimensions affect the Weber fraction #2 Short-duration flashes are harder to see (are less discriminable) than long-duration flashes That is, the threshold L increases as flash duration becomes shorter. Sensitivity = 1/threshold Continuing: How changes in other stimulus dimensions affect the Weber fraction #3 Threshold L varies with eccentricity from the fovea At low luminance levels, threshold is lowest (sensitivity is highest) about 15-20 degrees from fovea and the fovea is “blind” At high luminance levels, threshold is lowest at the fovea Log Increment Threshold, L (Apostilbs) This is the basis for visual field tests -4 -3 Background luminance, L 0 0.001 0.01 0.1 1 10 100 1000 -2 Fig. 3.5 -1 0 1 2 3 -60 -45 -30 -15 0 15 30 45 60 75 90 Angular Distance From Fixation (deg) Increment threshold as a function of eccentricity from the fovea for several luminance levels. The top line shows the threshold when the background luminance (L) is very low (0 apostilbs). The bottom line shows the threshold for a background L of 1000 apostilbs. Note, on the Y-axis, that lower thresholds (higher sensitivities) are upwards on the graph. (Modified from (Lynn, Felman & Starita, 1996).) Intensity discrimination can be limited at many places within the visual system Sensory Magnitude Scales Revisited Using the “just noticeable difference” (jnd) to create a scale for sensory magnitude vs. stimulus magnitude L + threshold L = LT LT is one jnd more intense than L. LT + threshold L = LT2 LT2 is one jnd more intense than LT And so on… Sensory Magnitude 12 10 8 6 4 L 2 0 0 50 100 Stimulus Luminance, L (cd/m2) 150 200 Sensory Magnitude 12 10 L + threshold L = LT 8 LT is one “just noticeable difference” (jnd) more intense than L. 6 LT 4 L 2 0 0 50 100 Stimulus Luminance, L (cd/m2) 150 200 Sensory Magnitude 12 LT + threshold L = LT2 10 LT2 is one jnd more intense than LT 8 LT2 6 and 2 jnd’s larger than L LT 4 L 2 0 0 50 100 Stimulus Luminance, L (cd/m2) 150 200 Sensory Magnitude 12 10 LTn+1 LTn 8 When Weber’s Law holds, the threshold Ls keep getting larger, so 1 jnd is a larger increase in stimulus luminance LT2 6 LT 4 L 2 0 0 50 100 Stimulus Luminance, L (cd/m2) 150 200 Fechner’s Law Sensory Magnitude 12 10 8 6 4 2 Fechner's Law: Log(L) 0 0 50 100 Stimulus Luminance, L (cd/m2) 150 200 Fechner’s Law relates the magnitude of sensation to the increment threshold Fechner’s law: = k log( ) where is sensory magnitude, is an arbitrary constant determining the scale unit, and is the stimulus magnitude Comparing Fechner’s Law with Stevens’ Power Law Sensory Magnitude 12 Fig. 3.6 10 8 Stevens’ Power Law resembles Fechner’s Law when the exponent is <1 6 4 2 Stevens' Law: L0.15 Fechner's Law: Log(L) 0 0 50 100 Stimulus Luminance, L (cd/m2) 150 200