Many models of visual processing assume that visual information is analyzed into separable and independent neural codes, or features. A common psychophysical test of independent features is known as a summation study, which measures performance in a detection, discrimination, or visual search task as the number of proposed features increases. Improvement in human performance with increasing number of available features is typically attributed to the summation, or combination, of information across independent neural coding of the features. In many instances, however, increasing the number of available features also increases the stimulus information in the task, as assessed by an optimal observer that does not include the independent neural codes. In a visual search task with spatial frequency and orientation as the component features, a particular set of stimuli were chosen so that all searches had equivalent stimulus information, regardless of the number of features. In this case, human performance did not improve with increasing number of features, implying that the improvement observed with additional features may be due to stimulus information and not the combination across independent features.

^{1}; Green & Swets, 1974; Barlow, 1978; Burgess et al., 1981; Kersten, 1984), object recognition (Eckstein, Ahumada, & Watson, 1997; Braje et al., 1995; Liu et al., 1995), perceptual learning (Tjan et al., 1995; Gold, Bennett, & Sekuler, 1999), and reading (Abbey, Eckstein, & Shimozaki, 2001). Humans may or may not be able to use parts of this information in their own performance of the task, and, as they are not ideal, can never use all the information. The amount of information used by the human observer can be assessed by comparing the ideal observer performance with human performance (often measured as efficiency, ((Legge, Klitz, & Tjan, 1997’

*d*/Legge, Klitz, & Tjan, 1997’

*d*)

_{ideal}). Several authors have modeled how humans might be suboptimal for a given task, such as the inability to optimally use the signal information (sampling inefficiency), internal or equivalent noise, or intrinsic uncertainty (e.g., Barlow, 1978; Burgess et al., 1981; Pelli, 1985; Solomon & Pelli, 1994; Eckstein et al., 1997).

^{2}illustrates the linear summation model in a single 2-feature trial in Experiment 2. Starting on the left, the model assumes two independent responses at each location, one corresponding to orientation (xFigure 3 for the target location, x

_{t-o}for the distractor locations), and the other corresponding to spatial frequency (x

_{d-o}for the target location, x

_{t-sf}for the distractor locations). These responses are weighted separately by the sensitivity of the observer to that particular feature (

_{d-sf}′

_{o}and

_{d-sf}′

_{sf}), and then summed to give a single combined response for each location (x

*d*for the target location, x

_{t-linear}for the distractor locations). The model then chooses the location with the maximum value for the combined response as the target location.

*d*′ from signal detection theory (Green & Swets, 1974), and linear summation is represented as the vector sum of the sensitivity for each feature. Also, independent features are represented as orthogonal axes for each feature, and the length of the 2-feature vector, which represents performance in the two-feature task, becomes the hypotenuse of the two single-feature vectors.

_{t-o}for the target location, and x

_{d-o}for the distractor locations), and one for spatial frequency (x

_{t-sf}for the target location, and x

_{d-sf}for the distractor locations). The model then chooses the location with the maximal independent featural response (uncombined, unlike the linear summation model) as the target location. In other words, the model chooses the location with the most evidence for target presence along a single feature, amongst the evidence across all features. As the number of features available to perform the task increases, the probability of any of the internal responses to the target (along any one of the available features) taking the maximum value also increases. Thus, this decision rule predicts better performance in the 2-feature visual search task, compared to the single-feature task, similar to the linear summation model. Probability summation, however, is a weaker form of summation than linear summation, and generally predicts a smaller increase in performance in the 2-feature search, relative to the single-feature searches.

^{2}, mean luminance = 24.75 cd/m

^{2}). These stimuli appeared in the center of four static square boxes included to reduce the intrinsic uncertainty in the task (uncertainty in the exact location of the signal, e.g., Burgess & Ghanderharian, 1984; Pelli, 1985; Eckstein et al., 1997). The boxes were 2 deg in length, and were centered 3.44 deg to the right, left, upward, and downward from a central fixation point. A uniform luminance mask of 38.8 cd/m

^{2}appeared for 300 ms immediately following the search display. A high-contrast copy of the target was continuously shown at the bottom of the display. The target and distractor locations were randomized on each trial, and the observer indicated his or her choice of the target’s location for that trial by using a computer mouse.

Experiment 1 (close values) | ||||
---|---|---|---|---|

Condition | Spatial frequency of target | Trials/block | Orientation of target | Octave bandwidth |

Spatial frequency | 2.5 cpd | Vertical | 0.789 | 0.117 |

Orientation | 2.0 cpd | 13/15 deg from verticala^{a} | 1.00 | 0.117 |

2-Feature | 2.5 cpd | 13/15 deg from verticala^{a} | 0.789 | 0.117 |

Experiment 2 (orthogonal values) | ||||

Condition | Spatial frequency of target | Trials/block | Orientation of target | Octave bandwidth |

Spatial frequency | 5.0 cpd | Vertical | 0.387 | 0.0664 |

Orientation | 2.0 cpd | Horizontal | 1.00 | 0.0664 |

2-Feature | 5.0 cpd | Horizontal | 0.387 | 0.0664 |

*d*′) using the standard M-AFC transformation from signal detection theory (Green & Swets, 1974, see ). This index (

*d*′) is the normalized distance between two Gaussian distributions describing the observer’s response to the target and to the distractor over a large number of trials, and typically varies in a 4-AFC task from 0 (chance performance) to about 4 (nearly perfect performance). Predictions of performance in the 2-feature task for independent feature and ideal observer models were derived from the human observers’ performance in the single-feature tasks (see and for details). Analyses of variance were performed at an alpha level of .05 using the statistical package GANOVA (Woodward, Bonett, & Brecht, 1990).

*d*′ for each observer in Experiment 1. A clear effect can be seen, with

*d*′ for the 2-feature search significantly larger than those for the single-feature searches across the observers (

*d*′

_{2f}vs.

*d*′

_{sf}, F(1,21)=43.15, MSE=0.073,

*p*< .0001;

*d*′

_{2f}vs.

*d*′

_{o}, F(1,21)=44.21, MSE=0.058,

*p*<.0001). The

*d*′ for the spatial frequency and orientation searches were nearly equal, as expected from the separate adjustments of the target orientation for each observer.

*d*′

_{2f}/

*d*′

_{sf}on the left, and

*d*′

_{2f}/

*d*′

_{o}on the right. Also included are the predictions for the ideal observer and the two independent feature models. First, it should be noted that all three models predict similar ratios that are greater than one, with the probability summation model predicting a slightly smaller ratio than the other two models. Second, the ratios for the three observers also were significantly greater than one (

*d*′

_{2f}/

*d*′

_{sf,}t(2) = 8.39, standard error = 0.038,

*p*=.0139;

*d*′

_{2f}/

*d*′

_{o,}t(2) = 12.11, standard error = 0.022,

*p*=.0067), reflecting the improvement in performance for the

*d*′s in the 2-feature search. Third, the empirical ratios tended to fall between the predictions of the probability summation model on the low end, and both the ideal observer and the linear summation models on the high end. Across observers, the empirical ratios were significantly smaller than the linear summation predictions (

*d*′

_{2f}/

*d*′

_{sf}, F(1,21) =10.09, MSE=0.021,

*p*=.0045;

*d*′

_{2f}/

*d*′

_{o,}F(1,21)=12.22, MSE=0.022,

*p*=.0022). For both probability summation and the ideal observer, the differences from the empirical ratios across observers approached but did not quite achieve significant levels (probability summation:

*d*′

_{2f}/

*d*′

_{sf,}F(1,21)=4.062, MSE=0.022,

*p*=.0568;

*d*′

_{2f}/

*d*′

_{o,}F(1,21)=2.267, MSE=0.021,

*p*=.1470), (ideal observer:

*d*′

_{2f}/

*d*′

_{sf,}t(2) = 3.848,

*p*=.0614;

*d*′

_{2f}/

*d*′

_{o,}t(2) = 3.664,

*p*=.0681).

*d*′

_{ideal}) for the Experiment 1 in the three conditions can be found in Table 2, calculated as described in . Observer K.F. had slightly higher

*d*′

_{ideal}values for the orientation and 2-feature task, corresponding to the slightly larger orientation differences for her (15 deg), compared to A.P. and S.S. (13 deg). Also note that

*d*′

_{2f,ideal}is about 1.4 times greater than the single-feature searches (spatial frequency and orientation), corresponding to the predictions for the ratios of

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}by the ideal observer.

Single-feature | Orientation | 2-Feature | |
---|---|---|---|

A.P. and S.S. | 4.711 | 4.232 | 6.271 |

K.F. | 4.711 | 4.793 | 6.666 |

*d*′

_{human}/

*d*′

_{ideal})

^{2}) is used to express this comparison (Barlow, 1978; Burgess et al., 1981). Figure 8 depicts the absolute efficiencies of the human observers in Experiment 1. These efficiencies ranged from about 0.12 to 0.22, which coincides to the efficiencies found in other studies of simple detection and discrimination (e.g., Barlow, 1978; Burgess et al., 1981; Burgess & Ghandeharian, 1984; Eckstein et al., 1997). Observers varied slightly in their absolute performance, with A.P. being more efficient than the other observers (A.P. vs. other observers, F(1,21)=64.71,

*p*<.0001). Finally, observers were more efficient in the orientation tasks, relative to the spatial frequency (Experiment 1: F(1,21)=7.434,

*p*=.0126) and 2-feature tasks (Experiment 1: F(1,21)=18.83,

*p*=.0003).

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}greater than one, and the observers’ results were similar to these predictions. As seen from these results, it can be difficult to distinguish the effects of additional stimulus information from the effects of summation across independent features. In the second experiment, a stimulus set was constructed so that the stimulus information was constant across conditions, and therefore predicted equal performance across all three types of searches, regardless of the number of available features.3 As in the first experiment, the independent feature models predicted an increase of performance in the 2-feature search. Therefore, for the stimuli in Experiment 2, the summation models predict ratios of

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}greater than one, whereas the stimulus information predicts ratios equal to one.

*d*′ for all the observers in Experiment 2. Most notably for these stimuli, there was no improvement found in the 2-feature search, as was found in the first experiment. Figure 10 gives the ratios of

*d*′

_{2f}/

*d*′

_{sf}on the left, and

*d*′

_{2f}/

*d*′

_{o}on the right, with the predicted ratios of the linear summation, probability summation, and ideal observer models. As discussed earlier, the ideal observer predicts ratios about equal to one, and the two summation models predict ratios larger than one. The empirical ratios were clearly closer to the predictions of the ideal observer. The empirical ratios across the four observers were not significantly greater than the predicted ratios of the ideal observer (the ratios of

*d*′

_{2f}/

*d*′

_{o}for S.S. were significantly smaller than the ideal observer, t(7) = −9.844.

*p*= <.0001). Conversely, the empirical ratios across the four observers were significantly smaller than the linear summation predictions (

*d*′

_{2f}/

*d*′

_{sf}, F(1,28)=283.7, MSE=0.015,

*p*<.0001;

*d*′

_{2f}/

*d*′

_{o,}F(1,28)=249.6, MSE=0.013,

*p*<.0001), and the probability summation predictions (

*d*′

_{2f}/

*d*′

_{sf,}F(1,28)=92.56, MSE=0.014,

*p*<.0001;

*d*′

_{2f}/

*d*′

_{o,}F(1,28)=73.26, MSE=0.013,

*p*<.0001). As expected, a significant experiment-by-type-of-search interaction was found for the

*d*′s for the three observers common to both experiments (F(2,41)=25.31, MSE=0.056,

*p*<.0001), indicating the different pattern of results across the two experiments. Also, the empirical ratios were significantly different from each other for the three observers across the two experiments, for both

*d*′

_{2f}/

*d*′

_{sf}(F(1,42)=22.34, MSE=0.035,

*p*<.0001) and

*d*′

_{2f}/

*d*′

_{o}(F(1,42)=41.11, MSE=0.045,

*p*<.0001).

*d*′

_{ideal}) for Experiment 2 in the three conditions is listed in Table 3, calculated as described in . Note that

*d*′

_{2f,ideal}for Experiment 2 is nearly equal to the

*d*′s for the single-feature searches, corresponding to predictions for the ratios of

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}by the ideal observer equal to one for Experiment 2. The absolute efficiencies of the human observers for Experiment 2 are shown in Figure 11. The efficiencies ranged from about 0.05 to 0.125, a lower range than in Experiment 1. In fact, the three observers common to both experiments (A.P., K.F., S.S.) were more efficient in Experiment 1 than in Experiment 2 (F(1,42) =173.4,

*p*<.0001), most likely due to the lower contrast necessary for Experiment 2 (with the larger differences in spatial frequency and orientation) leading to increased intrinsic uncertainty (uncertainty about the exact location of the stimulus locations, see Pelli, 1985; Burgess & Ghandeharian, 1984; Eckstein et al., 1997). As in Experiment 1, A.P. was more efficient than the other observers (A.P. vs. other observers, F(1,28)=62.23,

*p*<.0001), and observers were more efficient in the orientation tasks, relative to the spatial frequency (F(1,28)=17.69,

*p*=.0002) and 2-feature tasks (F(1,28)=9.967,

*p*=.0038).

Single-feature | Orientation | 2-Feature | |
---|---|---|---|

All observers | 6.180 | 6.133 | 6.164 |

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}for the ideal observer were compared directly to human performance, and absolute performance of the human observers, as measured by efficiency (see “Results” sections), was several times less than that predicted by the ideal observer. The relevant point is that the ideal observer and the human observers appear to use information from the stimulus similarly. In cases where there is additional featural information, but not additional stimulus information (Experiment 2), the human observers do not show an improvement in the 2-feature task. Therefore, for this task, the human observers’ performance appears to be determined by the stimulus information, and not the featural information. Also, one might consider the consequences of adding internal noise to the ideal observer model,5 equivalent across conditions. By this method, we may more closely match the absolute performance of the human observer with a degraded ideal observer. Regardless of the level of the internal noise added, however, the predicted ratios of of

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}for the degraded ideal observer would be the same.

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}for an observer linearly combining information across the cigar and donuts were estimated to be only slightly less than the predictions for the independent linear combination model. This decrease in ratios across observers was about 0.07 for Experiment 1, and about 0.025 for Experiment 2, small compared to the decrease in the predicted ratios when comparing the probability summation predictions against the independent linear summation predictions (about 0.21).

^{1}Note that others (e.g., Geisler & Davila, 1985) have talked about an ideal observer analysis in terms of an ideal decision after suboptimal processing of the image reflecting the human visual system. In this latter treatment of the ideal observer, the model performance reflects both stimulus information and constraints in the human visual system.

^{2}A common variant of this model is linear summation with equal weighting across the two features (unweighted). The stimulus parameters were chosen to give approximately equal performance in the orientation and spatial frequency searches, and therefore the weighted and unweighted linear summation models gave nearly equivalent predictions for the 2-feature search.

^{3}As described in , the stimulus information in the present task as assessed by the ideal observer depends upon (1) the energies of the target and the distractors, (2) the image noise, and (3) the correlation between the target and the distractors. All the targets in Experiment 2 were chosen to equate these three stimulus parameters across all conditions, including, most importantly, the correlations of the targets with the distractors. This ensures the same stimulus information for each task. Specifically, all the targets were orthogonal to the distractors (correlations = 0).

^{5}The internal noise is added as a scalar to the decision variable of the ideal observer (the dot product of the stimulus with the ideal template, with the ideal template defined as the difference between a template matching the target and a template matching distractor) (see Figure 2 and ).

^{7}Best fits of the predictions of the exact probability summation model to the Quick pooling formula (Graham, 1989; Quick, 1974) (

*d*′

_{2f}= (

*d*′

_{sf}

_{k}+

*d*′

_{o}

_{k})

_{1/k}for the observed data in this study found summation exponents (

*k*) from 2.96 to 3.72. Probability summation has been inferred from exponents ranging from 3 to 5 (Graham, 1989), congruent with the predicted exponents for the exact probability summation model.

*d*′

*d*′ values for an M-AFC procedure (Green & Swets, 1974; MacMillan & Creelman, 1991; Palmer et al., 2000; Eckstein et al., 2000). It is assumed that one response is generated at each of the M locations, which is determined by a univariate Gaussian distribution, with one Gaussian describing the response to the target, and another Gaussian describing the response to the distractor. The distributions have unit variance, with the mean of the distractor distribution equal to zero, and the mean of the target distribution equal to

*d*′. This is a standard set of assumptions for signal detection theory:

_{t}= Gaussian (

*μ*

_{t}=

*d*′,

*σ*

_{t}= 1),

*x*

_{d}= Gaussian (

*μ*

_{d}=

*d*′,

*σ*

_{d}= 1).

*x*

_{t}) and

*M−1*distractor responses (

*x*

_{d}). The maximum response across all locations for each trial is chosen as the target location. Therefore, a correct response is generated when the target response is the maximum value across all locations:

*μ*= 0,

*σ*= 1)

**template**= column vector describing the ideal template

**stimulus**

_{t}= = column vector describing the stimulus at the target location

**stimulus**

_{d}= column vector describing the stimulus at the distractor location

**t**= column vector describing the target

**d**= column vector describing the distractor

**n**= column vector describing the image noise added to the stimuli. The mean of the noise (μ

_{a}) is zero

_{image}= standard deviation of the image noise

**K**= the covariance matrix describing the image noise

*λ*

_{t}= the decision variable for the ideal observer at the target location

*λ*

_{d}= the decision variable for the ideal observer at a distractor location

*σ*

_{λ}= standard deviation for the decision variable

**K**=

*σ*

^{2}

_{image}

**I**, where

**I**is the identity matrix.

*d*′) for all conditions (spatial frequency, orientation,and 2-feature), and the predicted ratios of

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}were calculated from these individual

*d*′s. Notably, the

*d*′ for the ideal observer is a function of the image noise, the energies of the target and the distractors (

**t**

^{t}

**t**,

**d**

^{t}

**d**), and the correlation between the target and the distractor (the dot product,

**d**

^{t}

**t**). In Experiment 2, the targets for all conditions were chosen to be effectively orthogonal to the distractors (correlations equal to zero). Thus,

*d*′

_{ideal}for all conditions in Experiment 2 depend only on the energies of the targets and distractors. The energies were also equalized across conditions, leading to predictions of equal performance across all conditions in Experiment 2.

*d*′ in the 2-feature task.

*x*

_{t-linear},

*x*

_{d-linear}). The weights for the linear combination are the

*d*′s for each task, which are the optimal weightings in this case. On each trial, the maximum value amongst

*x*

_{t-linear}and the three

*x*

_{d-linear}’s is chosen as the target location, where the weighted linear combination of the target responses is

_{2f,linear}for each observer were found from Equation 6. The M-AFC conversion in may be used to convert d′

_{2f,linear}to the predicted percent correct in the 2-feature task.

_{2f, prob sum.}

_{2-feature}for the probability summation model from the

*d*′

_{sf}and the

*d*′

_{o}for each observer.6 The PC

_{2-feature}then was converted to

*d*′

*d*for the probability summation model by using the same conversion to

*d*′ described above in

*d*. Note that this equation is an exact prediction of probability summation (, 2000; Eckstein et al., 1996), and not an approximation, such as the Quick pooling model, which has been a common approximation used by others (Tyler & Chen, 2000; Quick, 1974; Graham & Robson, 1987).Graham, 1989

*d*′

_{2f}/

*d*′

_{sf}and

*d*′

_{2f}/

*d*′

_{o}for the linear summation across cigars and donuts, for the independent linear summation model, and the difference between the predicted ratios. As with the other summation models, the predicted ratios for summation across cigars and donuts were larger than one. Also, the differences in predicted ratios between the independent and cigar/donut linear summation models were small, about 0.07 for Experiment 1 and about 0.025 for Experiment 2. These differences were much less than the difference in predicted ratios for independent linear summation and the probability summation (about 0.21).

Experiment 1 | |||||||
---|---|---|---|---|---|---|---|

Predicted d′_{2f}/d′_{sf}, Linear summation | Predicted d′_{2f}/d′_{o}, Linear summation | ||||||

Observer | Independent | Cigar/donut | Difference | Independent | Cigar/donut | Difference | |

K.F. | 1.383 | 1.308 | 0.072 | 1.455 | 1.376 | 0.068 | |

S.S. | S.S. | 1.351 | 1.337 | 0.071 | 1.422 | 1.407 | 0.070 |

A.P. | 1.359 | 1.330 | 0.071 | 1.430 | 1.399 | 0.069 | |

Experiment 2 | |||||||

Predicted d′_{2f}/d′_{sf}, Linear summation | Predicted d′_{2f}/d′_{o}, Linear summation | ||||||

Observer | Independent | Cigar/donut | Difference | Independent | Cigar/donut | Difference | |

K.F. | 1.430 | 1.354 | 0.025 | 1.454 | 1.377 | 0.023 | |

S.S. | 1.574 | 1.259 | 0.026 | 1.601 | 1.281 | 0.021 | |

A.P. | 1.464 | 1.327 | 0.025 | 1.489 | 1.350 | 0.023 | |

D.V. | 1.425 | 1.358 | 0.024 | 1.449 | 1.382 | 0.023 |