**Abstract**:

**Abstract**
**Visual search in real life involves complex displays with a target among multiple types of distracters, but in the laboratory, it is often tested using simple displays with identical distracters. Can complex search be understood in terms of simple searches? This link may not be straightforward if complex search has emergent properties. One such property is linear separability, whereby search is hard when a target cannot be separated from its distracters using a single linear boundary. However, evidence in favor of linear separability is based on testing stimulus configurations in an external parametric space that need not be related to their true perceptual representation. We therefore set out to assess whether linear separability influences complex search at all. Our null hypothesis was that complex search performance depends only on classical factors such as target-distracter similarity and distracter homogeneity, which we measured using simple searches. Across three experiments involving a variety of artificial and natural objects, differences between linearly separable and nonseparable searches were explained using target-distracter similarity and distracter heterogeneity. Further, simple searches accurately predicted complex search regardless of linear separability ( r = 0.91). Our results show that complex search is explained by simple search, refuting the widely held belief that linear separability influences visual search.**

*perceptual*configuration. This problem is compounded by the fact that targets in LS configurations are generally further away from their distracters (resulting in easier search) compared to LNS configurations. Thus, it is not clear whether linear separability influences search difficulty above and beyond that expected from increased target-distracter similarity or distracter heterogeneity alone (Navalpakkam & Itti, 2006).

_{1}and D

_{2}depends on the dissimilarity of T with D

_{1}and D

_{2}the similarity between the D

_{1}and D

_{2}as well as their linear separability. To measure these similarities, we set up simple search displays involving these items and took the similarity between two items A and B to be the average time taken by subjects to search for A among Bs (or vice versa). To assess the impact of linear separability, we asked whether differences between LNS and LS configurations could be explained by target-distracter or distracter-distracter similarities alone. As in previous studies, we sought to identify the impact of linear separability over and above the contributions expected from similarity relations alone.

*perceptually*LNS configurations.

*any*target-distracter configuration (T, D

_{1}, D

_{2}) is LS and used this method to compare LS and LNS configurations. Using this measure, we were able to show that model predictions were equally accurate for LS and LNS target-distracter configurations, suggesting that the linear separability has no impact on visual search.

_{1}and D

_{2}, we also measured search performance across set sizes for T among D

_{1}, T among D

_{2}, and D

_{1}among D

_{2}(or vice versa).

No. | LNS configuration | LS configuration | Observed effect direction (LNS vs. LS) | ||||

T | D_{1} | D_{2} | T | D_{1} | D_{2} | ||

1 | −20° | 0° | −40° | 0° | −20° | −40° | LNS > LS |

2 | −20° | −40° | 20° | −40° | −20° | 20° | LNS > LS |

3 | 0° | 20° | −20° | 20° | 0° | −20° | LNS > LS |

4 | 20° | 40° | −20° | 40° | 20° | −20° | LNS > LS |

5 | 20° | 0° | 40° | 0° | 20° | 40° | LNS > LS |

6 | −20° | −40° | 40° | −40° | −20° | 40° | LNS < LS |

7 | 0° | 20° | −40° | 20° | 0° | −40° | LNS < LS |

8 | 0° | 40° | −40° | 40° | 0° | −40° | LNS < LS |

9 | 0° | −20° | 40° | −20° | 0° | 40° | LNS < LS |

10 | 20° | 40° | −40° | 40° | 20° | −40° | n.s. |

_{1}or more D

_{2}on the target side) were crossed with two possible target locations (left/right) to yield four unique conditions for a given target-distracter triplet (T, D

_{1}, D

_{2}). These conditions were repeated two times to yield a total of eight trials of each triplet, which resulted in 480 complex search trials (20 triads × 8 repetitions × 3 set sizes).

_{1}, D

_{2}) in the complex search task: These were T among D

_{1}, T among D

_{2}, and D

_{1}among D

_{2}or vice versa. Because there were 20 triplets in the complex search task (Table 1) involving only five orientations, there were only 10 unique simple searches (

^{5}C

_{2}). Each of these simple searches was repeated four times each with the target randomly located on the left or right side and two times with each orientation in the pair as target. As a result, there were eight trials at each set size for each image pair (T, D) in which T or D could be the target, giving rise to 240 trials (10 triads × 2 pairs × 4 repetitions × 3 set sizes).

*r*= 0.83,

*p*< 0.00005 for complex search;

*r*= 0.77,

*p*< 0.00005 for simple search). We compared the performance of subjects on LNS versus LS configurations and how this related to their performance on the corresponding simple searches.

*top row*). For the LS search (0°, −20°, −40°), the corresponding simple searches are (0°, −20°), (0°, −40°), and (−20°, −40°) (shown in Figure 1F through 1H,

*bottom row*). According to similarity-based accounts, search should be hard when target-distracter similarity is high or when distracter-distracter similarity is low (i.e., distracters are heterogeneous).

*α*< 0.05, indicated in Table 1).

*t*test,

*p*= 0.12). For the four complex searches in which LNS was easier than LS, the slopes did not differ significantly (mean slopes: 28 ms/item for LNS, 19 ms/item for LS,

*p*= 0.72, paired

*t*test). The resulting RT versus set size plots are shown in Figure 3. Across pairs, we observed effects similar to the example displays in Figure 1, namely that the harder complex search (whether LNS or LS) was always associated with differences in target-distracter similarity relations as measured using simple searches. For the harder complex searches (whether LNS or LS), the target was less similar to one of the distracters—a relatively weak effect that did not reach significance in one group (Figure 3B;

*p*= 0.7 for main effect of LS vs. LNS in an ANOVA with subject, configuration, search triple, and set size as factors) and reached significance in the other group (Figure 3F;

*p*= 0.0001 for main effect of LS vs. LNS as before). This effect, which is in the opposite direction, was outweighed by the fact that the target in the harder search was significantly more similar to one of the distracters in both groups of conditions (LS vs. LNS main effect

*p*< 0.00001; Figure 3C and 3G). The harder complex search was also associated with significantly smaller distracter-distracter similarity in both groups of conditions (LS vs. LNS main effect

*p*< 0.00001; Figure 3D and 3H). Again, two of these simple searches predicted the direction of the effect (whether LNS was harder or easier than LS), and these two appeared to have dominated the third. We conclude that differences between LNS and LS searches, at least for oriented lines, can be accounted for using similarity relations as measured using simple searches.

_{2}in Figure 4) are far smaller than distances between orientations on the same side of the vertical (e.g., 0° and 40°, distance d

_{1}in Figure 4) despite having the same angular separation. This is consistent with the perceptual tendency to confuse mirror-related shapes (Gross & Bornstein, 1978; Wolfe & Friedman-Hill, 1992; Rollenhagen & Olson, 2000). Likewise, the distance d

_{3}between 40° and −40°—an orientation difference of 80°—is slightly smaller than the distance d

_{1}between 0 deg and 40 deg even though the latter involves only a 40° difference (Figure 4).

Condition | No. | LNS configuration | LS configuration | Effect direction (LNS vs. LS) | ||||

T | D_{1} | D_{2} | T | D_{1} | D_{2} | |||

Similar distracters | 1 | T | J | N | T | N | L | LNS > LS |

2 | T | K | O | T | K | M | LNS > LS | |

3 | T | I | M | T | I | O | LNS > LS | |

4 | T | L | P | T | P | N | n.s. | |

Dissimilar distracters | 5 | T | B | F | T | D | F | LNS > LS |

6 | T | D | H | T | F | H | LNS > LS | |

7 | T | A | E | T | A | G | LNS > LS | |

8 | T | C | G | T | C | E | LNS > LS |

*r*= 0.91,

*p*< 0.00005 for complex search;

*r*= 0.94,

*p*< 0.00005 for simple search). Subjects searched for the target faster in LS configurations compared to LNS configurations (

*p*< 0.0005 for the main effect of linear separability, ANOVA on search times with subject, linear separability, similarity, and set size as factors). There were also significant main effects of subject, set size, and similarity (

*p*< 0.00005) but no significant interaction effects. These results are similar to the effects reported by Saumier and Arguin (2003).

*p*= 0.007, paired

*t*test). This difference was also significant for the dissimilar distracter conditions (Figure 6E; mean slopes: 48 ms/item for LNS, 18 ms/item for LS,

*p*= 0.02, paired

*t*test) and approached significance for the similar distracters (Figure 6A; mean slopes: 78 ms/item for LNS and 15 ms/item for LS,

*p*= 0.08, paired

*t*test). These results replicate the findings of previous studies using the same LNS and LS searches. However, the target-distracter and distracter-distracter similarities were never explicitly tested in previous studies.

*p*> 0.1 for main effect of configuration; Figure 6C and 6G). The LNS searches, however, had significantly larger distracter-distracter similarity (

*p*< 0.00001 for the main effect of configuration; Figure 6D and 6H) compared to LS searches. Thus, similarity relations as measured in simple search predict the greater difficulty of the LNS searches compared to the LS searches. We conclude that LNS searches are harder than LS searches because they have greater target-distracter similarity or greater distracter homogeneity. Thus, we again found no evidence of an effect of linear separability over and above that expected out of purely similarity considerations alone.

^{17}C

_{2}) pairs of the shapes used in this experiment using a separate set of 12 subjects with methods identical to that used in Experiment 3. We then performed a multidimensional scaling analysis on the data as before. Distances between stimuli in the multidimensional scaling plot were in close correspondence with the experimentally measured distances (

*r*= 0.88,

*p*< 0.00001). But the original configuration of these stimuli in parametric space (Figure 7A) was strongly distorted in visual search space wherein the stimuli appear to form clusters of thin/thick and curved/flat stimuli (Figure 7B).

^{2}b/c where b was the length of the triangle base, and p and c were the two free parameters, pointiness (which had to be at least 0.5) and curvature (which could range from −1 to 1), respectively. The radius of the arc on the base of the triangle was set to r

_{b}= b/c. The sign of the curvature parameter determined whether the arc was convex (positive values) or concave (negative values). The radii, sides, and curvature then uniquely determined the location of the center of the corresponding circles, and the corresponding arcs were drawn to define the shape.

_{1}on the side of the target) were crossed with two possible target locations (left/right) to yield four trials for a given target-distracter triplet (T, D

_{1}, D

_{2}). Distracter identity at each location was chosen at random in each trial while preserving these constraints. In addition to this, to ensure that the immediate vicinity of the target also contained equal proportions of distracters, four distracters of each type were placed in the 3 × 3 array surrounding the target with locations chosen at random on each trial.

_{1}, D

_{2}) in the complex search task; these were T among D

_{1}, T among D

_{2}, and D

_{1}among D

_{2}(and vice versa). Each simple search was repeated twice with the target either on the left or right side of the array with the result that there were four trials for each image pair in which either could be the target.

_{1}and D

_{2}as (T, D

_{1}, D

_{2}) and the corresponding average reaction time (averaged across trials and across subjects) as RT(T, D

_{1,}D

_{2}). The corresponding simple searches are T among D

_{1}and T among D

_{2}, which are related to target-distracter similarity. We denoted the corresponding average RTs as RT(T, D

_{1}) and RT(T, D

_{2}), respectively. Because the ordering of the distracters was arbitrary, we sorted the data such that RT(T, D

_{1}) was the larger of the two search times. In other words, D

_{1}was always the harder of the two distracters. A third factor that may influence complex search performance is distracter homogeneity, which we measured using the average time taken to search for D

_{1}among D

_{2}or vice versa. Note that, in general, RT(A, B) may not equal RT(B, A) if there is a search asymmetry. In subsequent analyses (see Results) we found search asymmetry to have only a minor impact on complex search.

**y**). The observed simple search data were likewise concatenated to form a 96 × 3 matrix (denoted as

**X**) in which the three columns corresponded to the TD

_{1}, TD

_{2}, and D

_{1}D

_{2}effects (search times or distances). We then sought to find the linear contributions of the three simple search factors (a 3 × 1 vector

**b**) that could predict the complex search. This was done using linear regression (

*regress*function in Matlab, Natick, MA), which finds the best-fitting coefficients as

Model type | # | Model interpretation | Correlation with data (n = 96) | Model coefficients | |||

a | b | c | d | ||||

Models based on search times (RT) | 1 | Complex search time is determined entirely by the search time for the harder distracter. | 0.76 | - | - | - | - |

RT(T, D_{1}D_{2}) = max RT(T, D_{1}), RT(T, D_{2}) | |||||||

2 | Complex search time depends on both the easy and hard distracters. | 0.79 | 0.88 | 0.94 | –0.11 | - | |

RT(T, D_{1}D_{2}) = a × RT(T, D_{1}) + b × RT(T, D_{2}) + c | |||||||

3 | Complex search time depends on the easy and hard distracters as well as on distracter-distracter similarity. | 0.82 | 0.81 | 1.29 | –0.39 | 0.07 | |

RT(T, D_{1}D_{2}) = a × RT(T, D_{1}) + b × RT(T, D_{2}) + c × RT(D_{1}, D_{2}) + d | |||||||

Models based on search distance (1/RT) | 4 | Complex search distance is determined entirely by the hard (nearer) distracter. | 0.84 | - | - | - | - |

d(T, D_{1}D_{2}) = 1/RT(T, D_{1}D_{2}) = min (1/RT(T, D_{1}), 1/RT(T, D_{2})) | |||||||

5 | Complex search distance depends on both the hard and easy distracters. | 0.85 | 0.73 | 0.16 | –0.22 | - | |

d(T, D_{1}D_{2}) = 1/RT(T, D_{1}D_{2}) = a × d(T, D_{1}) + b × d(T, D_{2}) + c | |||||||

6 | Complex search distance depends on hard and easy distracters as well as on distracter-distracter dissimilarity. | 0.91 | 0.60 | 0.45 | –0.35 | –0.05 | |

d(T, D_{1}D_{2}) = 1/RT(T, D_{1}D_{2}) = a × d(T, D_{1}) + b × d(T, D_{2}) + c × d(D_{1}, D_{2}) + d |

**ypred**=

**X**

^{T}

**b**. To compare model predictions with the observed data, we calculated the Pearson's correlation between the two vectors.

*r*= 0.94,

*p*< 0.00001 for complex search;

*r*= 0.79,

*p*< 0.00001 for simple search). Thus, the underlying features and/or task strategies did not vary across subjects. We then set out to investigate whether complex search times could be explained using simple search times.

_{1}and D

_{2}, we selected the corresponding average search times for T among D

_{1}and T among D

_{2}. Note that we did not consider D

_{1}or D

_{2}among T for these calculations, thereby respecting possible search asymmetries that might be present in these displays (however, see below for an analysis of asymmetry). In contrast, the complex search displays contained equal numbers of D

_{1}and D

_{2}, so we took as a measure of distracter homogeneity the average search time for finding D

_{1}among D

_{2}or D

_{2}among D

_{1}. Because distracter identity was arbitrary, we always denoted the harder distracter to be D

_{1}(i.e., the one among which the target was harder to find on average) and the other one as D

_{2}.

*p*< 0.00001, unpaired

*t*test on average RTs for each condition). For these easy and hard complex searches, we calculated the average search times for the corresponding simple searches (Figure 9A). It can be seen that, compared to the hard complex searches, the easy ones had smaller search times for (T, D

_{1}) and (T, D

_{2}) (

*p*< 0.00001, unpaired

*t*test) but larger search times for (D

_{1}, D

_{2}) (

*p*= 0.02, unpaired

*t*test). Thus, just as in Experiments 1 and 2, similarity between the target and either distracter increases complex search time whereas distracter homogeneity reduces complex search time, exactly as predicted by similarity theories of search.

_{1}and D

_{2}is dominated entirely by the time taken to search for the harder of the two distracters. The predictions of this model were positive and significant (

*r*= 0.76,

*p*< 0.00001), but this correlation was relatively small in comparison to the other models. The second model was based on the idea that the time taken for complex search depends on both the time taken to search for the target among the easy as well as among the hard distracters. For this model, the correlation between the predicted and observed search times was higher than Model 1 (

*r*= 0.79,

*p*< 0.00001). Although an increase in quality of fit is expected because this model has an extra free parameter, there is a standard statistical test (partial

*F*test) that estimates the expected increase in the quality of fit due to the extra degrees of freedom and returns the probability that the two models are equivalent. We found that Model 2 predictions were significantly better than Model 1 over and above that expected from the extra degree of freedom,

*F*(2, 3) = 9.47,

*p*= 0.003, partial

*F*test. The third model instantiated the idea that the time taken for complex search depends not only on the time taken to search for the target among the two distracters, but also on the similarity between the distracters themselves. For this model, the correlation between the predicted and observed complex search times was even higher than the previous two models (

*r*= 0.82,

*p*< 0.00001). This increase in quality of fit over Model 2 was significantly greater than expected from the addition of an extra degree of freedom,

*F*(3, 4) = 15.6,

*p*= 0.001, partial

*F*test.

*r*= 0.84,

*p*< 0.00001). The next model (Model 5) was based on the idea that search distances in complex search depend on the distances between the target and the nearest as well as the farthest distracters. The predictions of this model were slightly better (

*r*= 0.85,

*p*< 0.00001), but this increase barely reached statistical significance (

*F*(2, 3) = 3.82,

*p*= 0.053, partial

*F*test).

*r*= 0.91,

*p*< 0.00001, Figure 9B). The improvement in the quality of fit in this model was significantly greater than expected from the extra degree of freedom in Model 6 compared to Model 5,

*F*(3, 4) = 59.7,

*p*< 0.00001, partial

*F*test. The correlation between the observed and predicted data for Model 6 was significantly larger than the correlations for all other models (

*p*< 0.05 for all comparisons, Fisher's

*z*test). Finally, when search distance predictions in Model 6 were converted back into RTs (by taking their reciprocal), they too were strongly correlated with the observed complex search times (

*r*= 0.86,

*p*< 0.00001).

Correlation with data | Model coefficients | ||||

Hard distracter | Easy distracter | Distracter heterogeneity | Constant term | ||

All sets (n = 96) | 0.91**** | 0.60 | 0.45 | −0.35 | −0.05 |

Set 1 (n = 24) | 0.95**** | 0.52 | 0.40 | −0.40 | 0.13 |

Set 2 (n = 24) | 0.89**** | 0.50 | 0.47 | −0.48 | 0.17 |

Set 3 (n = 24) | 0.93**** | 0.69 | 0.45 | −0.26 | −0.29 |

Set 4 (n = 24) | 0.93**** | 0.44 | 0.73 | −0.45 | −0.07 |

*r*= 83%) of the variance in complex search can be accounted for by pair-wise dissimilarities or simple searches. The striking match between complex search and predictions based solely on simple searches places strong limits on the influence of emergent properties such as linear separability. In the subsequent analyses, we present results based on Model 6, and we take up the issue of linear separability in greater detail.

^{2}_{1}and D

_{2}is driven by simple searches T among D

_{1}, T among D

_{2}, and D

_{1}among D

_{2}(or vice versa). As formulated here, this model already takes into account possible visual search asymmetries because it uses, for instance, the search for T among D

_{1}but not D

_{1}among T. However, this framework allowed us to ask, what is the contribution of search asymmetries toward complex search? To address this issue, we set up a variant of the model (Model 6) in which we took the simple search times to be the average of the times taken to search for T among D

_{1}and D

_{1}among T and so on. For each of the 96 complex searches in our data, we also performed a statistical test to determine which of the simple searches (T, D

_{1}) and (T, D

_{2}) were asymmetric (ANOVA with subject and asymmetry as factors, criterion of

*p*< 0.05 for main effect of asymmetry). We separated the complex searches into those in which the simple searches had no asymmetry (

*n*= 46), those in which one simple search (i.e., T among D

_{1}or T among D

_{2}) had an asymmetry (

*n*= 40), and those in which both simple searches had asymmetries (

*n*= 10). For these three groups of searches, we compared the correlation with the observed data of the models with and without asymmetry (Table 5). As expected, there was virtually no difference in the quality of fit of the two variants of Model 6 when neither simple searches had an asymmetry. However, when one of the simple searches had a search asymmetry, the model that incorporated asymmetry had a slightly higher quality of fit (

*r*= 0.84 vs.

*r*= 0.90 for without and with asymmetry), but this difference in correlation was not significant (

*p*= 0.23, Fisher's

*z*test). When both simple searches had asymmetries, the model with asymmetry again outperformed the model without asymmetries (

*r*= 0.79 vs.

*r*= 0.85), but again, the difference in correlations was not significant (

*p*= 0.40, Fisher's

*z*test). However, in both cases, there was a slight increase in the quality of fit when the model incorporated search asymmetry. We conclude that search asymmetries play a relatively minor role in determining complex search.

Type of complex search | Model performance (correlation with data) | |

Without asymmetry | With asymmetry | |

No asymmetric pairs (n = 46) | 0.89***** | 0.90***** |

1 asymmetric pair (n = 40) | 0.84***** | 0.90***** |

2 asymmetric pairs (n = 10) | 0.79* | 0.85** |

_{1}, D

_{2}), the distances between the three stimuli in visual search space can be measured using the reciprocal of the pair-wise search times. In other words, the distance d(TD

_{1}) is the reciprocal of the average time taken to search for T among D

_{1}(and vice versa). For any three points, given the distances d(TD

_{1}), d(TD

_{2}), and d(D

_{1}D

_{2}), we reasoned that if the target T lies along a straight line between D

_{1}and D

_{2}(i.e., it is LNS), then the distance between the distracters d(D

_{1}D

_{2}) must equal the sum of the distances d(TD

_{1}) and d(TD

_{2}); this is illustrated in Figure 10A. The extent of deviation from equality is therefore a measure of how LS this configuration is. For each of the 96 complex search configurations in Experiment 3 (8 subsets × 4 sets × 3 targets per subset), we plotted the sum of d(TD

_{1}) and d(TD

_{2}) against d(D

_{1}D

_{2}) (Figure 10B). In this plot, points close to the unit line (y = x) represent configurations that fall along a line and are, by definition, LNS. Points that fall far away from the unit line represent triangle-like configurations that are clearly LS. Of the 96 complex search configurations, we chose two equal subsets for comparison: the LNS configurations, which were defined as those that fell within 0.41 distance units of the unit line (

*n*= 29), and the LS ones, which were defined as those that fell beyond 1.28 distance units of the unit line (

*n*= 29). We then compared model fits for the LS and LNS configurations (Figure 10C). The correlation between model predictions and the observed complex search distances was slightly lower for the LNS than the LS configurations (

*r*= 0.68 for LNS,

*r*= 0.72 for LS configurations), but this difference did not reach significance (

*p*= 0.76, Fisher's

*z*test). This subtle and insignificant drop in correlation is hardly the impact expected of a linear separability effect. We obtained qualitatively similar results upon varying the threshold distances used to define the two configurations. We therefore conclude that linear separability is unlikely to play a role in complex visual search.

*r*= 0.82,

*p*< 0.05) or LNS from the distracters (

*r*= 0.86,

*p*< 0.005). This difference between correlation was again statistically insignificant (

*p*= 0.81, Fisher's

*z*test). Thus, even for LNS and LS configurations defined independently of our distance criterion, differences in search performance were predicted by the model. We conclude that complex search is largely explained using simple searches and that linear separability has no impact on visual search at least for the shapes tested here.

_{1}and D

_{2}, we characterized its corresponding simple searches, i.e., T among D

_{1}, T among D

_{2}, and D

_{1}among D

_{2}(and D

_{2}among D

_{1}). The first two searches measure the similarity between the target and the distracters. The last one (D

_{1}among D

_{2}or vice versa) measures distracter similarity or homogeneity. The main finding of this study is that, at least for a wide variety of shapes, complex search is quantitatively explained by simple search and that this holds true regardless of the linear separability of the target and distracters. It is consistent with classical similarity theories, which posit that complex search is influenced by target-distracter similarity and distracter homogeneity. This finding however refutes the widely held belief that linear separability influences visual search. We examine below the relationship between these findings and previous studies in this regard.

*r*= 0.33,

*p*= 0.05) and were positively but not significantly correlated with distracter-distracter search slopes (

*r*= 0.18,

*p*= 0.29). These correlations are far lower than those obtained between complex and simple search times (

*r*= 0.91 across all sets, Figure 9), which may reflect the greater noise in search slopes compared to search times. Understanding the relationships between search slopes will require systematic measurements involving fewer conditions but many more trials per condition to obtain reliable slope estimates.

_{1}+ TD

_{2}– D

_{1}D

_{2}. Because this is a linear term, any model containing this term will produce predictions that are identical to a model based on a linear combination of TD

_{1}, TD

_{2}, and D

_{1}D

_{2}. The resulting model will be indistinguishable from a model based on similarity alone. Thus, as several others have correctly argued (Bauer et al., 1996a, 1996b, 1998), any real contribution of linear separability must be substantially larger than predictions based on similarity.

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