**Previous work on the straddle effect in contrast perception (Foley, 2011; Graham & Wolfson, 2007; Wolfson & Graham, 2007, 2009) has used visual patterns and observer tasks of the type known as spatially second-order. After adaptation of about 1 s to a grid of Gabor patches all at one contrast, a second-order test pattern composed of two different test contrasts can be easy or difficult to perceive correctly. When the two test contrasts are both a bit less (or both a bit greater) than the adapt contrast, observers perform very well. However, when the two test contrasts straddle the adapt contrast (i.e., one of the test contrasts is greater than the adapt contrast and the other is less), performance drops dramatically. To explain this drop in performance—the straddle effect—we have suggested a contrast-comparison process. We began to wonder: Are second-order patterns necessary for the straddle effect? Here we show that the answer is “no”. We demonstrate the straddle effect using spatially first-order visual patterns and several different observer tasks. We also see the effect of contrast normalization using first-order visual patterns here, analogous to our prior findings with second-order visual patterns. We did find one difference between first- and second-order tasks: Performance in the first-order tasks was slightly lower. This slightly lower performance may be due to slightly greater memory load. For many visual scenes, the important quantity in human contrast processing may not be monotonic with physical contrast but may be something more like the unsigned difference between current contrast and recent average contrast.**

*straddle effect*. We have continued to study this effect (Graham, 2011; Graham & Wolfson, 2013; Wolfson & Graham, 2009). Foley (2011) replicated the straddle effect and studied it further. Kachinsky, Smith, and Pokorny (2003) have presented results we think may be closely related to the straddle effect (see also Pokorny, 2011). The straddle effect's relationship to previously reported effects in adaptation and masking literature is discussed by Wolfson and Graham (2009) and Foley (2011).

*test pattern*contains four Gabor patches arranged in a 2 × 2 grid; two are at one contrast (C1) and two are at another (C2). Test contrasts C1 and C2 vary from trial to trial. The test pattern is presented between the

*adapt pattern*and the

*posttest pattern*. The adapt and posttest patterns are entirely identical. The Gabor patches in the adapt and posttest patterns are always at 50% contrast. All the Gabor patches throughout a single trial (all the patches in the adapt, test, and posttest patterns) have the same spatial frequency and orientation and occupy the same spatial positions.

*above*the adapt contrast. We will refer to this case as an Above test pattern. In the middle row, the two test contrasts

*straddle*the adapt contrast; that is, one test contrast is above and the other is below the adapt contrast. We refer to this case as a Straddle test pattern. In the bottom row, both the test contrasts are

*below*the adapt contrast, and we refer to this case as a Below test pattern.

*constant-difference series*.

*d*′ values rather than percentage correct. We show these

*d*′ values here (which we have not done in our previous publications on the straddle effect) in order to better compare the results in the second-order orientation-identification experiment with results from, for example, the first-order experiments presented later in this article.

*d*′ values, there is a problem if performance is 100% or 0% (i.e., the proportion is either 1 or 0): In these cases, the

*d*′ value computed straightforwardly from the experimental results will be infinite. To correct for this problem we used a rule to truncate the computed infinite values to finite values that can be plotted. The truncation rule we used allows sample size (the number of trials entering into any plotted point) to affect the truncated value. Truncated data points in this and subsequent figures are shown as open symbols. See Appendix B for details of the

*d*′ computations.

*butterfly curve*(Hochberg, 1978, p. 240, although in a very different perceptual-adaptation context, involving bathwater temperature). The terms

*contrast normalization*and

*contrast comparison*that label regions of the curve are given to the processes discussed in this subsection.

- The comparison level equals the recent (much less than 1 s) weighted average of contrast at that spatial position.
- The comparison level is subtracted from the current input contrast.
- The magnitude of the difference is sent upstream, but information about the sign of that difference is lost or at least degraded—that is, a full-wave or partial rectification occurs.

- The later stimulus has a contrast 3 arbitrary units above that of the earlier stimulus. Then (see Figure 6) the contrast-comparison process's output to this later stimulus will be +3 arbitrary output units.
- The later stimulus has a contrast 3 arbitrary units below that of the earlier stimulus. Then (see Figure 6), the contrast-comparison process's output to this later stimulus will also be +3 arbitrary output units.

*d*′ computations are discussed.

*greater*for the test contrast that is

*lower*. Thus, if the observers choose the greater magnitude of change to answer the question of “Which of the two intervals has the higher test contrast?,” they will choose the lower test contrast and be systematically wrong.

*d*′ computed in two different ways. We wanted to use

*d*′ values to compensate for possible kinds of response bias and thus allow comparison of this first-order (same–different two-interval) experiment to the second-order (orientation-identification) experiment. Any

*d*′ computation is the application of a detection-process model in order to calculate from experimental results to a quantity one might call true sensitivity without contamination by quantities one might call response biases. One can never be certain that the detection-process model is perfect. Indeed, one can usually be sure that it is not. Fortunately, experience has shown that the detection-process models do not need to be perfect for the calculated

*d*′ values to be quite informative and prevent errors of interpretation.

*d*′ measures:

*d*′

*-conservative*and

*d*′

*-unconfounded*. Appendix B describes these measure in detail; here we will describe them tersely. The middle column of Figure 9 shows the measure we call

*d*′-conservative because it is conservative with respect to our major conclusions here. In particular,

*d*′-conservative underestimates the depth of the notch corresponding to the straddle effect. The right column shows a measure we think better, which we call

*d*′-unconfounded because it corrects for a substantial confound. The difference between these two

*d*′ measures is in how the false-alarm rate is estimated (and indicated cryptically on the vertical axis labels). For these

*d*′ values—as for the ones in Figure 4—we had to truncate

*d*′ values to avoid the problems caused by performances of 0% or 100% correct. Cases where truncation occurred are shown by open symbols in the figure.

*d*′ values. We used the traditional

*d*′ computation for the second-order orientation-identification results (Figure 4), the values shown here as gray stars. The first-order results are the

*d*′-unconfounded values shown in Figures 9 and 10 (same symbols as in the earlier figures). The results for test-contrast differences |C1 − C2| equal to 5%, 10%, and 20% are shown in the three columns.

*d*′ values from the second-order experiment (gray stars) are substantially higher than those from the first-order experiment (colored symbols).

*not*confined to second-order spatial pathways but also occurs in first-order pathways. This is the major novel result of the present article. The accompanying Weber-law behavior occurs in first-order as well as second-order pathways. This result is less surprising, as Weber-law behavior has been shown in many situations.

*d*′-conservative and

*d*′-unconfounded—are shown, along with percentage “diff,” in Figures C1–C3.) The horizontal axes show average test contrast. The value of the adapt contrast is indicated by a red asterisk on each horizontal axis and is labeled at the right end of the row. The position of the straddle-effect notch moves as the adapt contrast moves, following the adapt contrast so that the notch is always centered where the average test contrast equals the adapt contrast. The first-order results share this general and important property with second-order results.

*d*′-conservative and

*d*′-unconfounded are in Appendix C (Figures C6, C10, and C11).

*all*abrupt changes in contrast, there is a rectification process whose result is that increases and decreases in contrast act similarly and are confusable. We suspect now that there is a shifting, rectifying contrast-comparison process acting on all visual patterns, not just gratings and textures.

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- Same–different two intervals—single Gabor patch—Figures C4 and C5
- Same–different two intervals—single big disk—Figures C8 and C9

- A 2 × 2 grid of horizontal or vertical Gabor patches (all of the same orientation; horizontal is shown in Figure 13), with fixation at the center of the four patches.
- A single horizontal or vertical Gabor patch (horizontal is shown in Figure 13), with fixation at the center of the patch.
- A single big disk, with fixation at the center of the disk.

^{2}. The single Gabor patch or the 2 × 2 grid of Gabor patches was centered within a 16° × 16° (1,024 × 1,024 pixel) gray square at 55 cd/m

^{2}. Therefore, the mean luminance of the whole pattern was also 55 cd/m

^{2}. (Further details of the Gabor-patch patterns appear in the subsection More about the adapt, test, and posttest patterns, and their relation to one another.)

^{2}. This 55 cd/m

^{2}will be referred to as the background luminance for the big-disk pattern, as the luminance of the disk was in general greater than 55 cd/m

^{2}and thus the mean luminance of the full big-disk pattern was in general greater than 55 cd/m

^{2}.

*not*present during the test pattern. Alternately, the test pattern here could be described as the adapt (background) pattern

*plus*an added (or subtracted) probe pattern. The relationship of these experiments here to earlier experiments in the literature using the background–probe terminology is discussed in Wolfson and Graham (2009), at the end of the Procedure subsection under Methods.

*average test contrast*is the mean of C1 and C2—that is, (C1 + C2)/2.

*test-contrast difference*.

*constant-difference series*.

- A fixation “point,” namely a small square filled with a lighter gray than the mean luminance as shown to scale in, for example, Figure 1. This was used for all the 2 × 2 Gabor-patch grid experiments and the big-disk experiment.
- An outline square, shown to scale in Figure C4. This was used for the single-Gabor-patch experiment.

^{2}—that is, equal to the mean luminance in the case of Gabor-patch experiments and to the background luminance in the case of the big-disk experiment. Note that a 0% contrast adapt or test pattern is exactly the same as a gray screen.

*without*replacement across trials to make the numbers of trials of each the same. But the nonsubstantive characteristics were sampled

*with*replacement. The nonsubstantive characteristics include whether Gabor-patch orientation was horizontal or vertical; whether the contrast-defined stripes were horizontal or vertical; which interval of a two-interval trial which test pattern came in; and which position of the two-position same–different experiment had which test contrast.

^{2}. The room was dark except for the CRT screen and a dim lamp. Each CRT's lookup table was linearized. Stimuli were generated and presented using MathWorks' MATLAB with the Psychophysics Toolbox extensions (Brainard, 1997; Pelli, 1997). The actual duration of each stage (adapt, test, etc.) of each trial was checked for accuracy. In the event of inaccuracy—typically an extra refresh of the CRT (1/85 of a second)—the trial was excluded from subsequent analysis. Such inaccuracies occurred on less than 1% of the trials.

*d′*values

*d*′ in this article are not standard because the experimental and theoretical frameworks here are somewhat different from most standard cases. The reader already familiar with signal-detection theory can skip ahead to Three ways of calculating

*d*′ values for all our same–different experiments.

*d*′ will be defined as the difference between the mean of the signal and noise distributions divided by the standard deviation of either distribution. For this equal-variance Gaussian model, the value of

*d*′ is then easily computed (e.g., Macmillan & Creelman, 2005, pg. 8, equation 1.5) as

- z(x) is the
*z*score characterizing the standard normal distribution that is associated with the cumulative probability*x*. - H (for “hit”) is the conditional probability that an observer responds “yes” given that a signal stimulus has occurred on a trial. It is called the
*hit rate*. - F (for “false alarm”) is the conditional probability that an observer responds “yes” given that a noise stimulus has occurred on a trial. It is called the
*false-alarm rate*.

*d*′ values in the three tasks follows. First, however, a general point that applies to all our graphs of

*d*′ values, no matter how they were computed.

*d*′ values to deal with proportions of 1 and 0

*d*′ values from becoming infinite. More specifically, we used the following simple rule to truncate the values of

*d*′:

*d*′ for the second-order orientation-identification experiment

*symmetric criterion*, because the line representing the criterion falls exactly in the middle of the two distributions, making the whole drawing symmetric around that criterion line.

*d*′ from an equal-variance Gaussian model with a symmetric criterion (symmetric bias) reduces to (Macmillan & Creelman, 2005, p. 9, equation 1.7):

*d*′ values plotted in Figure 4, right column.

*d*′ values for all our same–different experiments

*d*′ becomes:

*d*′ values. The three different estimates of the true

*d*′ value came from three different ways of calculating the false-alarm rate. They are described in the following, after two general points:

*d*′, the important general results from these experiments remain true, in particular: A straddle effect exists—that is there is a notch in the graph of performance versus average test contrast centered at the point where the average test contrast equals the adapt contrast; and there is maximal performance for patterns containing both test contrasts somewhat below (or both somewhat above) the adapt contrast. Once both the test contrasts become far from the adapt contrast, performance is below maximum.

*d*′ values are used for our figures here (e.g., Figure 9, middle and right columns). Why the first way is not shown is described in the next subsection.

*d*′ values (a way that is biased in this context)

*d*′. It assumes that the false-alarm rate is the same for all the trials in a given experiment no matter how the test-contrast values differ from trial to trial. This single false-alarm rate is estimated as the proportion of all Same trials on which the observer responded “diff.” In our figures here, we do not show the

*d*′ values calculated this first way for two reasons, one practical and one theoretical.

*d*′ uses the same estimated false-alarm rate for every test stimulus. Thus, the calculated

*d*′ values for a set of stimuli will be monotonic with the hit rates for those stimuli. These hit rates are plotted in all our relevant figures in the left column (the large colored symbols in the left column of, e.g., Figure 9). In particular, the

*d*′ values calculated this first way will necessarily have all the peaks and valleys that the hit rates in the left column show.

*d*′—that the false-alarm rate on every trial is identical—is easily shown to disagree with the observers' performances. First note that all “diff” answers on Same trials are, by definition, false alarms. In our relevant figures these “diff” answers on Same trials are plotted as the bottom curve in the left column (small black dots connected by a black line in, for example, Figure 9). If the assumption underlying this first way of calculating

*d*′ were true, then these bottom curves would be straight horizontal lines except for statistical variability. In our data, however, these bottom curves are systematically bent away from a straight line, showing a dip when the average test contrast equals the adapt contrast (marked with a red asterisk)—that is, for the Same-Straddle stimulus.

*d*′ values:

*d*′-conservative

*d*′ produces values shown in the middle column of results figures here, for example, Figure 9. We call values calculated this way

*d*′

*-conservative*for reasons described later. This second way of calculating

*d*′ assumes that the false-alarm rate varies from trial to trial because it is determined by the average test contrast on each trial.

*d*′ (

*d*′-conservative). The bottom half illustrates the third way of calculating

*d*′ (

*d*′-unconfounded), which will be described later. The numbers used in these examples are based on observer MC's performance, shown in the top row of Figure 9. The results of these numerical examples are summarized in Figure B3.

*d*′-conservative, the false-alarm rate on all trials of this Diff-Above pattern is determined by the average test contrast of 65%. So we estimate the false-alarm rate F by using the measured performance for the Same test pattern having both its test contrasts equal to 65% (small black dots). The observer's measured false-alarm rate on that pattern was 10.1%, thus producing z(F) = −1.28. Thus the estimated

*d*′-conservative for this Diff-Above pattern = z(H) − z(F) = 1.28 − (−1.28) = 2.56.

*d*′-conservative is the probability of an observer's responding “diff” to the Same-Straddle pattern (i.e., the pattern having both test contrasts, and average test contrast, equal to the adapt contrast of 50%). The measured performance F in this example was 12.7%, leading to a z(F) = −1.14. Thus the estimated

*d*′-conservative for this Diff-Straddle pattern = z(H) − z(F) = −0.72 − (−1.14) = 0.42.

*d*′ values:

*d*′-unconfounded

*d*′-unconfounded and

*d*′-conservative is in the way the false-alarm rate is calculated. The hit rate is the same for both calculations. We use the term

*d*′-unconfounded for reasons that will be described later.

*d*′-unconfounded is identical to that for calculating

*d*′-conservative (90%), and thus z(H) = 1.28. The false-alarm rate F for

*d*′-unconfounded is calculated from the pool of all Same trials in which the test contrasts both equaled 55% (called Same-55%) or both equaled 75% (called Same-75%). The proportion of times the observer responded “diff” (gave a false alarm) on Same-55% trials was 21.3%. For Same-75% trials the false-alarm rate was 5%. Since the numbers of trials of Same-55% and Same-75% were identical, we can average the two false-alarm rates (21.3% and 5.0%) to get the false-alarm rate F on the whole pool (13.1%). Thus, z(F) = −1.12. So the estimated

*d*′-unconfounded for the Diff-Above case is z(H) − z(F) = 1.28 − (−1.12) = 2.40. This

*d*′-unconfounded estimate is quite similar to the

*d*′-conservative estimate for this pattern.

*d*′-unconfounded is identical to that for calculating

*d*′-conservative (23.7%), and accordingly z(H) = −0.72. The false-alarm rate F for

*d*′-unconfounded is calculated from the pool of all Same trials in which the test contrasts both equaled 40% or both equaled 60%. The proportion of times the observer said “diff” (gave a false alarm) on Same-40% trials was 17.7%. For Same-60% trials the false-alarm rate was 24.4%. Since the numbers of trials of Same-40% and Same-60% were identical, we can average the two false-alarm rates (17.7% and 24.4%) to get the false-alarm rate F on the whole pool (21.0%). Thus, z(F) = −0.08. So the estimated

*d*′-unconfounded for the Diff-Straddle case is z(H) − z(F) = −0.72 − (−0.08) = 0.09. This

*d*′-unconfounded estimate is very different from the

*d*′-conservative estimate for this pattern.

*d*′-unconfounded before we ran the experiments reported here. Thus, for some experiments we did not use the Same patterns that would be necessary to compute

*d*′-unconfounded exactly as above. Where we had not used a necessary Same pattern, we estimated the performance on that missing one from the performance on the Same pattern that was nearest (in test contrast) to the missing one. These exceptions occurred in two places in our experiments:

*not*constant across average test contrast. In particular, it shows a dip precisely where the dip occurs in the curves for the Diff patterns—that is, where the average test contrast equals the adapt contrast. At first we were puzzled by this dip in the curve for Same trials. But then we realized there was a unique feature that the observer could use to almost always correctly identify the Same-Straddle trials. Notice that in five of the six kinds of trials illustrated in Figure 8 (e.g., the two kinds in the top row of the figure), there are four changes in contrast during each interval: at the start of the adapt pattern, at the start of the test pattern, at the end of the test pattern, and at the end of the posttest pattern. Thus, during each two-interval trial, there are 2 × 4 = 8 transitions.

*d*′-conservative systematically uses smaller false-alarm rate estimates z(F) for the Straddle pattern than for the other patterns because it uses the false-alarm rate from the Same-Straddle pattern. This elevates the

*d*′-conservative estimates for the Straddle patterns—by subtracting a smaller z(F)—relative to the other two ways of calculating

*d*′. And thus

*d*′-conservative plots almost always show a shallower (more conservative) straddle-effect notch than

*d*′-unconfounded plots.

*d*′-conservative is not actually conservative with respect to reality. This argument would make sense if the assumption used in calculating

*d*′-conservative were true—that is, if the false-alarm rate operating on a given trial were determined by the average test contrast on that trial. However, as described in the previous subsection, we are reasonably certain that Same-Straddle trials are in a class by themselves and do not reflect the false-alarm rate on any other trial.

*d*′-unconfounded calculations show deeper straddle-effect notches for the following reason: They do

*not*allow the confounding effect of the special nature of the Same-Straddle case (the unusual number of transitions) to completely dominate the calculation of

*d*′ at the center of the notch.

*d*′-unconfounded is the most valid of our three ways of calculating

*d*′ and that it most closely reflects the similarity dimension. It is this similarity dimension that can show the effects of the contrast-comparison and contrast-normalization processes, and it is these spatial processes that we are studying here.

*d*′ produces totally unconfounded values, but it seemed to be the best term we could think of.

*More detail for interested readers*: A person might suggest that it would be even better (less confounded) to prevent the Same-Straddle case from ever entering into any

*d*′ estimates. Calculating

*d*′-unconfounded does allow it to enter into the

*d*′ calculation for Diff patterns that have one test contrast equal to the adapt contrast. But even there its effect is diluted by the patterns that have the other test contrast (not equal to the adapt contrast). We did some calculations excluding the Same-Straddle pattern entirely and it made little difference except to make the straddle effect notch even deeper occasionally.