How well can observers detect the presence of a change in luminance distributions? Performance was measured in three experiments. Observers viewed pairs of grayscale images on a calibrated CRT display. Each image was a checkerboard. All luminances in one image of each pair consisted of random draws from a single probability distribution. For the other image, some patch luminances consisted of random draws from that same distribution, while the rest of the patch luminances (*test patches*) consisted of random draws from a second distribution. The observers' task was to pick the image with luminances drawn from two distributions. The parameters of the second distribution that led to 75% correct performance were determined across manipulations of (1) the number of test patches, (2) the observers' certainty about test patch location, and (3) the geometric structure of the images. Performance improved with number of test patches and location certainty. The geometric manipulations did not affect performance. An ideal observer model with high efficiency fit the data well and a classification image analysis showed a similar use of information by the ideal and human observers, indicating that observers can make effective use of photometric information in our distribution discrimination task.

^{1}The overall size of the 5 × 5 checkerboards was thus 145 × 150 mm (14.75° × 15.26°). The two checkerboards were presented against a dark gray background (4.4 cd/m

^{2}) and were separated horizontally by 48 mm (4.91°). The CIE 1931

*xy*chromaticity of the background and checkerboard patches was held fixed at [0.30, 0.30].

*test checkerboard*and the other the

*standard checkerboard*. The luminances for the 25 patches in the standard checkerboard and almost all of the patches in the test checkerboard were randomly drawn from a truncated Gaussian distribution with a mean of 15.0 cd/m

^{2}, a standard deviation of 5.0 cd/m

^{2}, and a truncation range of [5.0, 50.0] cd/m

^{2}. For the test checkerboard, the luminances of the remaining patches, which we refer to as the

*test patches*, were drawn from a different distribution. The test patch distribution was a truncated Gaussian whose mean and standard deviation were larger than the standard distribution by a multiplicative constant. Across trials, this constant varied between 1 (minimum) and 2.5 (maximum). These corresponded to test patch distributions with a mean of 15.0 cd/m

^{2}, a standard deviation of 5.0 cd/m

^{2}, and a truncation range of [5.0, 50.0] cd/m

^{2}(minimum) and with a mean of 37.5 cd/m

^{2}, a standard deviation of 12.5 cd/m

^{2}, and a truncation range of [12.5, 125.0] cd/m

^{2}(maximum).

*location-known condition*; top panel of Figure 2) or were randomly selected on each trial from the 25 checkerboard patches (

*location-unknown condition*; bottom panel of Figure 2). This resulted in a total of 10 conditions (5 patch numbers × 2 levels of certainty). Conditions were blocked and observers were informed beforehand which condition was being tested. The order of conditions was randomized for each observer. For each condition, observers ran five blocks of 100 trials before moving onto the next condition.

^{2}lower than the location-unknown thresholds.

*mean luminance model*that chooses the checkerboard with the larger mean luminance on each trial, (b) a

*highest luminance model*that chooses the checkerboard containing the highest luminance patch on each trial, and (c) a

*highest range model*that chooses the checkerboard with the highest luminance range on each trial. We constructed variants of these models for the location-known and location-unknown conditions. For the location-known condition, the mean luminance and highest luminance were evaluated only over the known test patch locations in each checkerboard, while the range was obtained by subtracting the lowest non-test patch location luminance from the highest test patch location luminance. For the location-unknown condition, the mean, highest luminance, and luminance range were computed over all the locations in each checkerboard.

**r**is a binary column vector coding responses (left/right) as 1 s and 0 s,

**w**is a row vector with the weights found by the regression, and

**P**is a matrix whose rows were per-trial vectors of luminances obtained from the stimulus. The values in the rows of

**P**were obtained as follows: on each trial, for each checkerboard (left and right), we took the luminances from the test patch locations, the two most luminous non-test patches, and the two least luminous non-test patches. We sorted the luminances for the test and non-test patches in descending order separately, for each checkerboard. Then, we took the differences between the corresponding luminance-ranked patches (i.e., the most intense test patch on the left minus the most intense test patch on the right, the second most intense test patch on the left minus the second most intense test patch on the right, etc.). The regression thus tells us how much weight is assigned to the luminance differences between corresponding rank-ordered test patches across the two checkerboards and to the luminance differences of corresponding rank-ordered non-test patches at the high and low luminance ends of the non-test patch range. We analyzed only the data from the location-known conditions in this way, as we found that our data set did not have sufficient power to provide reliable estimates of the weights when all checkerboard squares were considered.

*F*(5,10) = 10.4,

*p*≤ 0.001. Examination of the plot suggests that this result was driven by the effect of test patch number: Conditions with two test patches led to lower thresholds than conditions with one test patch. This difference is comparable to the difference between the 1- and 2-test patch location-known conditions in Experiment 1, replotted in Figure 7 with X symbols.

*psi*/

*sep*two-patch conditions are greater than those for the

*center*two-patch condition, then those manipulations have a detrimental effect on human performance not accounted for in the model. However, the data in the left panel of Figure 7 show a minimal effect on threshold for those manipulations: A repeated measures ANOVA with observer as a random factor was not significant,

*F*(2,4) = 1.27,

*p*= 0.37. In addition, note that the non-significant trend toward an elevated threshold for the

*sep*condition is small relative to the effect of test patch number.

*F*(2,4) = 4.08,

*p*= 0.68. Simply introducing parallelograms into the image does not appear to affect performance for a single test patch, regardless of its shape.

*F*(2,6) = 0.80,

*p*= 0.49. This pattern suggests that similar strategies were used regardless of scene geometry and that both test patch luminances affected the decision.

Experiment 2: Test patch 2 weights | ||||
---|---|---|---|---|

center | psi | sep | ||

Mean weight | 0.57 | 0.46 | 0.17 | |

SEM | 0.32 | 0.21 | 0.13 | |

Experiment 3: Test patch 2 weights | ||||

bp | fp | m | rp | |

Mean weight | 0.61 | 0.30 | 0.43 | 0.27 |

SEM | 0.20 | 0.14 | 0.16 | 0.06 |

^{2}and a standard deviation of 3.0 cd/m

^{2}, truncated on the interval [1.5, 15] cd/m

^{2}. The test distribution had minimum parameters equal to the standard distribution parameters. Its maximum parameters were a mean of 17.7 cd/m

^{2}, a standard deviation of 8.8 cd/m

^{2}, and a truncation range of [4.4, 44.2] cd/m

^{2}.

*F*(5,10) = 4.4,

*p*≤ 0.02. As with Experiments 1 and 2, thresholds for the 1-test-patch conditions are higher than those for the 2-test-patch conditions.

*F*(3,6) = 0.6,

*p*= 0.66. That is, separating the two test patches in depth did not affect threshold nor was threshold different in the condition where the depth of the two test patches was randomized on every trial.

*F*(3,8) = 1.08,

*p*= 0.41.

**x**represents the luminances of the 50 checkerboard patches presented on a particular trial. If ℓ(

**x**) was greater than 0, the ideal observer indicated that the test was on the left; if ℓ(

**x**) was less than or equal to 0, the ideal observer indicated that the test was on the right. For a given trial, the vector

**x**can be thought of as the concatenation of the vectors

**x**

^{Left}, representing the luminances of the 25 checkerboard patches on the left checkerboard, and

**x**

^{Right}, the luminances of the 25 checkerboard patches on the right checkerboard.

*i*indexes patch location within a single checkerboard (left or right), {

*t*} represents the indices of the test patches within the test checkerboard, and {

*s*} represents the remaining indices within the test checkerboard. The probability

*p*

_{ t }(

*x*) is the probability of observing luminance

*x*at a single patch under the test distribution, and

*p*

_{ s }(

*x*) is the probability of observing luminance

*x*at a single patch under the standard distribution. The corresponding expression when the test is on the right is

*p*

_{ s }(

*x*) was evaluated using the probability density function of a truncated Gaussian distribution with mean

*μ*= 15 cd/m

^{2}and standard deviation

*σ*= 5 cd/m

^{2}(the parameters of the luminance distribution for stimulus patches under the standard illuminant). The Gaussian density function was truncated between the range [5, 50] and renormalized so that the total probability was 1, to match how the stimuli were generated for the experiments.

*p*

_{ t }(

*x*) would be evaluated using the same basic method as described for the standard distribution but with the truncated Gaussian having a mean, variance, and truncation range computed for the test distribution.

*p*

_{ t }(

*x*) was evaluated as a weighted sum of the likelihood for a given set of test distribution parameters, with the weights given by the probability of those parameters. The distribution of the parameters,

*p*(TestDistParam), was estimated by creating a histogram of the parameters used in the observers' experimental runs corresponding to the condition being simulated. With this,

**x**be the vector of eight luminances concatenated from the vectors

**x**

^{Left}, the luminances of patches

*x*

_{1}

^{Left}, …

*x*

_{4}

^{Left}belonging to the left checkerboard, and

**x**

^{Right}, the luminances of the four patches

*x*

_{1}

^{Right}, …

*x*

_{4}

^{Right}belonging to the right checkerboard.

*t*and two

*s*characters, where

*t*in the

*i*th location indicates that the

*i*th patch on the left is drawn from the test distribution,

*p*

_{ t }(

*x*

_{ i }

^{Left}), and

*s*indicates that the

*i*th patch on the left is drawn from the standard distribution,

*p*

_{ s }(

*x*

_{ i }

^{Left}). Let

*C*be a matrix of the set of combinations of

*t*and

*s*in Equation A6 whose rows are

*C*

_{ a }for the six combinations

*a*= 1, 2, … 6. Within a given row

*C*

_{ a }, the columns are indexed by

*i*= 1, 2, 3, 4. We write

*C*

_{ ai }. Equation A6 can then be represented as

*k*test distribution patches displayed at

*n*possible patch locations. In this case, there are

*N*=

*p*(

**x**∣TestOnRight)). The log likelihoods are then compared as in Equation A1.

*p*

_{ t }(

*x*) to model observer uncertainty in the parameters of the test distribution. The weights for signal level were again taken from observers' empirical test distribution parameter probability histograms from the corresponding experimental condition. Because of computational limitations, this simulation was done only for the 1-, 2-, and 3-test-patch cases. As with the corresponding simulations for the location-known case, we found little effect of test level uncertainty and we report results for the case where there was no uncertainty about test illuminant level.