The relation between color discrimination and color constancy: when is optimal adaptation task dependent?

Abstract

Color vision supports two distinct visual functions: discrimination and constancy. Discrimination requires that the visual response to distinct objects within a scene be different. Constancy requires that the visual response to any object be the same across scenes. Across changes in scene, adaptation can improve discrimination by optimizing the use of the available response range. Similarly, adaptation can improve constancy by stabilizing the visual response to any fixed object across changes in illumination. Can common mechanisms of adaptation achieve these two goals simultaneously? We develop a theoretical framework for answering this question and present several example calculations. In the examples studied, the answer is largely yes when the change of scene consists of a change in illumination and considerably less so when the change of scene consists of a change in the statistical ensemble of surface reflectances in the environment.

Figure 1

ROC diagram. The ROC (receiver operating characteristic) diagram plots hit rate versus the false alarm rate. An observer can maximize hit rate by responding “same” on every trial. This will lead to a high false alarm rate, and performance will plot at (1,1) in the diagram. An observer can minimize false alarms by responding “different” on every trial and achieve performance at (0,0). Varying criteria between these two extremes produces a trade-off between hits and false alarms. The exact locus traced out by this trade-off depends on the information used at the decision stage. Better information leads to performance curves that tend more toward the upper left of the plot (the solid curve indicates better information than the dashed curve.) The area under the ROC curve, referred to as A′, is a task-specific measure of information that does not depend on criterion. The hatched area is A′, for the dashed ROC curve. The ROC curves shown were computed for two surfaces with reflectances r₁ = 0.15 and r₂ = 0.29 presented in a roving same-different design. The illuminant had intensity e = 100, and the deterministic component of the visual responses was computed from equation 2.3 with g = 0.02 and n = 2. The solid line corresponds to σ_n = 0.05 and A′ = 0.85, and the dashed line corresponds to σ_n = 0.065 and A′ = 0.76. Hit and false alarm rates were computed using the decision rule described in section 2.4.

Figure 2

Effect of gain and noise on discrimination performance. (Top) Histograms of the discrimination measure $A_{i,jk}^{\prime }$ for two values of the gain parameter (solid bars, g = 0.010; hatched bars, g = 0.021). In the calculations, we set σ_n = 0.05, n = 2, μ_r = 0.5, σ_r = 0.3, and e = 100. Calculations were performed for 500 draws from the surface ensemble, and $A_{i,jk}^{\prime }$ was evaluated for all possible 124,750 surface pairs formed from these draws. (Bottom) The mean of the $A_{i,jk}^{\prime }$ (denoted by ) as a function of the gain parameter for four noise levels.

Figure 3

State of model visual system for optimal choice of gain. The top right panel shows the response function for the optimal choice of gain (g = 0.021) when the noise is σ_n = 0.05. Below the response function is a histogram of the light intensities c_i,j reaching the eye, while to the left is a histogram of the resultant visual responses. Calculations were performed for 500 draws from the surface ensemble. Choices of gain less than or greater than the optimum would shift the response function right or left. For these nonoptimal choices, visual responses would tend to cluster nearer to the floor or ceiling of the response range, resulting in poorer discrimination performance.

Figure 4

Effect of illuminant change on discrimination and constancy performance. The filled circles and solid line show how discrimination performance decreases when the illuminant intensity is changed and the adaptation parameters are held constant. Here the x-axis indicates the single scene illuminant intensity used in the calculations for the corresponding point. The open circles and dashed line show how constancy performance decreases when the test illuminant intensity is changed and the adaptation parameters are held fixed across the change. Here the x-axis indicates the test illuminant intensity, with the reference illuminant intensity held fixed at 100. All calculations performed with adaptation parameters held fixed (g = 0.021, n = 2) and for σ_n = 0.05. The surface distribution had parameters μ_r = 0.5 and σ_r = 0.3.

Figure 5

Trade-off between discrimination and constancy. (Left) Each set of connected solid circles shows the trade-off between and for various optimizations of the steepness parameter n (see the text). Each set is for a different noise level (σ_n = 0.01, 0.025, 0.05, 0.075, 0.10), with the set closest to the upper right of the plot corresponding to the lowest noise level. The reference illuminant had intensity e_ref = 100, and the test illuminant had intensity e_test = 160. The surface ensemble was specified by μ_r = 0.5 and σ_r = 0.3 and was common to both the reference and test environments. Both and were evaluated with respect to draws from this surface ensemble. The gain parameter was held fixed at g = 0.02045. The steepness parameter for the reference environment was n = 4.5. Parameters g = 0.02045 and n = 4.5 optimize discrimination performance for the reference environment when σ_n = 0.05. The open circles connected by the dashed line show the performance points that could be obtained for each noise level if there were no trade-off between discrimination and constancy. (Right) Equivalent trade-off noise plotted against visual noise level σ_n. See the discussion in the text.

Figure 6

Adaptation to surface ensemble change for discrimination. Numerical search was used to optimize for two visual environments characterized by a common illuminant but different surface ensembles (surface ensemble 1 and surface ensemble 2). The illuminant intensity was 100. In surface ensemble 1, μ_r = 0.5 and σ_r = 0.3. In surface ensemble 2, μ_r = 0.7 and σ_r = 0.1. The histogram under the graph shows the distributions of reflected light intensities for the two ensembles. The graph shows the resultant visual response function for each case. The solid line corresponds to surface ensemble 1 and the dotted line to surface ensemble 2. The histogram to the left of the graph shows the response distribution for surface ensemble 1 under the surface ensemble 1 response function, while the histogram to the right shows the response distribution for surface ensemble 2 under the surface ensemble 2 response function. All calculations done for σ_n = 0.05 and e = 100. In evaluating for surface ensemble 1, performance was averaged over draws from surface ensemble 1; in evaluating for surface ensemble 2, performance was averaged over draws from surface ensemble 2. The dashed lines show how the visual response to the light intensity reflected from a fixed surface varies with the change in adaptation parameters. (Since the illuminant is held constant, a fixed surface corresponds to a fixed light intensity.)

Figure 7

Trade-off between discrimination and constancy for change in surface ensemble. (Left) The plot shows the trade-off between C versus in the same format as the left panel of Figure 5. When the adaptation parameters are chosen to optimize discrimination () for the test environment (surface ensemble 2), constancy performance () is poor (lower right end of each set of connected dots). When the adaptation parameters are chosen to optimize constancy, discrimination performance is poor (upper left end of each set.) The connected sets of dots show how performance on the two tasks trades off for five noise levels σ_n = 0.01, 0.025, 0.05, 0.075, 0.10. Surface ensemble parameters and illuminant intensty are given in the caption for Figure 6. In evaluating , the adaptation parameters used for computing responses in the reference environment (surface ensemble 1) were held fixed at g = 0.02045 and n = 4.5. These parameters optimize discrimination performance for the reference environment when σ_n = 0.05. (Right) Equivalent trade-off noise plotted against noise level σ_n.

Figure 8

Chromatic example, illuminant change results. Trade-off between discrimination and constancy for illuminant change. Each pair of horizontally aligned panels is in the same format as Figure 5. (Left panels) versus trade-off curves with respect to illuminant changes. The reference environment illuminant was D65; the test environment illuminants were (from top to bottom) the blue, yellow, and red illuminants. The reference and test environment surface ensembles were the baseline surface ensemble in each case. The individual sets of connected points show performance for noise levels σ_n,, = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50. In evaluating , the adaptation parameters for the reference environment were those that optimized discrimination performance in the reference environment. These parameters were optimized separately for each noise level. (Right panels) Equivalent tradeoff noise plotted against noise level σ_n.

Figure 9

Chromatic example, surface ensemble change results. Trade-off between discrimination and constancy for surface ensemble changes. Same format as Figure 8. The reference and test environment illuminants were D65, the reference environment surface ensemble was the baseline esemble, and the test environment surface ensembles were (from top to bottom) the blue, yellow, and red ensembles. The individual sets of connected points in the left panels show performance for noise levels σ_n = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50. In evaluating , the adaptation parameters for the reference environment were those that optimized discrimination performance in the reference environment. These parameters were optimized separately for each noise level.

Figure 10

Summary of equivalent trade-off noise for the chromatic example. The solid black bars show the mean equivalent trade-off noise (+/− one standard deviation) for the three illuminant changes reported in Figure 8. The solid gray bars show the corresponding values for the three surface ensemble changes reported in Figure 9. In each case, the mean and standard deviation were taken over visual noise levels (that is, over the values shown in each of the right-hand panels in Figures 8 and 9.)