Binding of features helps object recognition in contour integration but hinders it in crowding. In contour integration, aligned adjacent objects group together to form a path. In crowding, flanking objects make the target unidentifiable. However, to date, the two tasks have only been studied separately. K. A. May and R. F. Hess (2007) suggested that the same binding mediates both tasks. To test this idea, we ask observers to perform two different tasks with the same stimulus. We present oriented grating patches that form a “snake letter” in the periphery. Observers report either the identity of the whole letter (contour integration task) or the phase of one of the grating patches (crowding task). We manipulate the strength of binding between gratings by varying the alignment between them, i.e., the Gestalt goodness of continuation, measured as “wiggle.” We find that better alignment strengthens binding, which improves contour integration and worsens crowding. Observers show equal sensitivity to alignment in these two very different tasks, suggesting that the same binding mechanism underlies both phenomena. It has been claimed that grouping among flankers reduces their crowding of the target. Instead, we find that these published cases of weak crowding are due to weak binding resulting from target–flanker misalignment. We conclude that crowding is mediated solely by the grouping of flankers with the target and is independent of grouping among flankers.

*Critical spacing*is the center-to-center distance between the target and the flankers beyond which the flankers do not affect target identification. Critical spacing is proportional to eccentricity (Bouma, 1970; Pelli & Tillman, 2008; Toet & Levi, 1992). Most theories of crowding hold that the difficulty in identification arises from excess binding that inappropriately combines flanker features with those of the target (Levi, 2008; Levi, Hariharan, & Klein, 2002; Parkes, Lund, Angelucci, Solomon, & Morgan, 2001; Pelli, Palomares, & Majaj, 2004). That is, the features of the target and the distracters are bound together if the separation between them is less than the observer's critical spacing for that eccentricity. We noted that alignment matters in contour integration. Does it matter in crowding as well?

*contour integration task*asks them to identify the letter. The

*crowding task*asks them to identify the phase of the central gabor (i.e., to indicate if the light half of the gabor was on the right or left side). The relative orientation between gabors is changed to vary the goodness of continuation (alignment) between them. One advantage of using a letter identification task instead of the usual “snake in the grass” detection is that, by testing identification rather than detection, we can draw conclusions about object recognition. The contour integration task employed here is similar to the Pelli et al. (2009) “snake letter” identification task, which measured the effect of alignment on object recognition.

^{2}. Each gabor is a sinewave grating vignetted by a Gaussian envelope. The contrast function for a vertical gabor at fixation is

*ψ*is phase offset,

*f*= 1 c/deg is spatial frequency,

*x*and

*y*are horizontal and vertical position in deg, and

*λ*= 0.37 deg is the space constant of the envelope. We will later refer to the

*dark-to-light transition*of this function as the line in

*x*–

*y*space at which the argument of sin( ) is zero. (When we fit a sine to this, to measure “wiggle,” the contact is at the point along the transition line where the envelope exp( ) is maximum.) The luminance function for vertical gabors at

*n*locations is

*x*

_{ i },

*y*

_{ i }is the position of the

*i*th gabor. The gabors are presented in a grid, 3 horizontally and 5 vertically, to form a letter. The center of the grid is 10 deg to the right of fixation. The center-to-center separation between adjacent gabors is 1.5 deg.

*θ*= 0, 20, 40, 60, or 90 deg) with respect to the letter path. However, the direction of the tilt alternates between adjacent gabors (±

*θ*with respect to the letter path). An extra orientation jitter, randomly chosen from a uniform distribution (−5 to 5 deg), is applied to each gabor on each trial. The variation in relative orientation of adjacent gabors controls the goodness of continuation (alignment) between them. Following Pelli et al. (2009), we quantified the misalignment as

*wiggle*. The smaller the wiggle, the better the alignment. To calculate a letter's wiggle, we identify straight chains of gabors in the letter. Within each chain, for each pair of gabors adjacent to each other, we fit a one-half sinewave to make tangential contact (at each end) with the dark-to-light transition of each gabor (Figure 3). At the point of contact, the dark side of the transition is always to the right of the sine as the sin( ) argument increases.

*wiggle*. We repeat this for each pair of adjacent gabors. We then average the wiggles of all gabor pairs in all the straight chains in the letter, to obtain the letter's

*wiggle*(Figure 3). This procedure is a generalization of the Pelli et al. (2009) method of measuring the wiggle of periodic stimuli to include some non-periodic stimuli.

*crowding*and

*contour integration*. Each observer participated in both. The stimulus parameters and procedure are the same in both, except where noted. Each task has five conditions, which differ only in the orientation

*θ*of the component gabors relative to the letter path. Wiggle increases with

*θ*. We ran four blocks of 40 trials per condition. The order of blocks (task type by wiggle angle) is randomized for each observer. Each block begins with a press of the space bar. A black fixation square appears at the center of the screen throughout the whole block. Viewing is binocular. The many-gabor stimulus is presented in the right field for 150 ms. The stimulus is the capital letter I in the crowding task and one of the capital letters C, E, F, H, I, L, O, P, T, or U in the contour integration task. (This difference in letters—“I” versus others—between tasks is inconsequential, as shown in the Results section.) The observer then indicates his or her response with a key press. The crowding task is to report the phase of the target gabor—the gabor in the center of the grid (i.e., the center of the letter I)—in other words, to indicate the left/right location of the light half of the gabor. The phase of this target gabor is random (

*ψ*= 0° or 180°) on each trial. The rest of the gabors have zero phase. The observer indicates the target's phase, after the stimulus is taken away, by pressing either the left (or up) or the right (or down) arrow key. The left and up arrow keys are equivalent, and the right and down arrow keys are equivalent. The contour integration task is to report the identity of the letter, which is selected independently, randomly, for each trial. In this task, all the gabors have zero phase. A correct response is rewarded with a low-frequency beep, whereas an incorrect response is indicated by a high-frequency beep. The next trial is presented after an intertrial interval of 1 s.

*θ*of 0, 20, 40, 60, and 90 deg. In the crowding task, the phase of the target gabor is random on each trial: either 0 or 180 deg. That is, the target is phase-aligned with the flankers in the same-target-phase trials (target phase 0 deg) and misaligned in the opposite-target-phase trials (target phase 180 deg). The phase difference results in different amounts of wiggle in these two types of trials, so we analyze the same- and opposite-target-phase trials separately and plot the results at their respective wiggles.

*θ,*there are two kinds of trials: trials with target phase 0 deg and trials with target phase 180 deg. These two types of trials have different wiggles (Figure 3). Since we are interested in the effect of wiggle on crowding, we analyze data from the trials with different target phases separately. The crowding task was to identify the phase of the target, so the 0 and 180 deg target-phase trials must be interleaved. Therefore, to obtain contrast thresholds for each phase, we adopted the following procedure. We collected all the trials from all blocks at each

*θ*. We then separated these trials into two groups: 0 and 180 deg phases. Each group had roughly 80 trials. For each group, we obtained two contrast thresholds by randomly picking half the trials and using QUEST to obtain a threshold for each half. Thus, for the crowding task, we report the average of two log contrast thresholds per wiggle.

*B*that depends solely on the amount of wiggle. We suppose that at zero wiggle all trials are bound,

*B*= 1, and that at 60 deg wiggle all trials are unbound,

*B*= 0. We call the amount of wiggle that produces 50% probability of binding (

*B*= 0.5) the wiggle threshold

*W*

_{50}. If both tasks involve the same binding, then they must have the same wiggle threshold. The following six steps explain how we arrive at and test this prediction. First, the expected proportion correct for each task depends on contrast

*c*and on whether the trial is bound or unbound. Lacking complete models for how the two tasks are performed, we cannot say how much difference the binding should make, and the dependence of the probability of binding on wiggle may be non-linear and even non-monotonic. However, the expected fraction of trials that are bound is the probability of binding

*B,*so the measured proportion correct

*P*(including a mixture of bound and unbound trials) will be a linear interpolation (0 to 100%) between the proportions correct at 0 and 60 deg wiggle,

*P*= (

*P*

_{0°}−

*P*

_{60°})

*B*+

*P*

_{60°}. Thus, proportion correct

*P*is linearly related to probability of binding

*B*. Second, the

*P*vs. log

*c*psychometric function, as a whole, is non-linear (sigmoidal), but it is smooth, so we suppose that, over a modest range, proportion correct is linearly related to log contrast. Third, the cascade of several linear relations is itself a linear relation, so, provided

*P*is near some criterion value (e.g., 0.82), log contrast threshold is linearly related to probability of binding. Fourth, the results in Figure 4 are all collected at a fixed threshold criterion

*P,*satisfying the third step's proviso. Fifth, although the linear relation is task dependent, the linearity itself implies that the midway point of binding probability,

*B*= 0.5, corresponds to the midway point of log contrast threshold, log

*c*= (log

*c*

_{0°}+ log

*c*

_{60°}) / 2. Thus, sixth, if the same binding process mediates both tasks, we predict that the wiggle threshold

*W*

_{50}will be the same for both tasks.

*W*

_{50}for each task and observer. First, we fit quadratic polynomials to each observer's results (log contrast threshold vs. wiggle, 0 to 60 deg, shown as dashed curves in Figure 4). A single polynomial is fitted to same- and opposite-target-phase thresholds in the phase discrimination task, as they trace out the same curve. Then, to estimate the observer's

*W*

_{50}, we find the amount of wiggle at which that curve's log contrast is midway between the log contrast thresholds at wiggles of 0 and 60 deg. The predicted equality of wiggle thresholds across tasks rests on linearity. As explained in the previous paragraph, all tasks for which log

*c*is linearly related to wiggle will have the same wiggle threshold: 30 deg. Thus, it is an essential feature of our test that we allow for a non-linear relation by making a quadratic polynomial fit. Figure 5 is a scatter plot of each observer's wiggle threshold

*W*

_{50}for the two tasks: contour integration vs. crowding. As predicted, all the data points lie close to equality (diagonal line) and close to 30 deg.

*F*(1,16) = 89.2,

*p*< 0.0001; LH:

*F*(1,16) = 39.4,

*p*< 0.0001; CRK:

*F*(1,19) = 96.4,

*p*< 0.0001]. That is, it was easier to identify a letter made of gabors than it was to indicate the phase of a single gabor among flankers. Importantly for our hypothesis, the interaction between task and wiggle was highly significant in all three observers [SBR:

*F*(3,16) = 12.8,

*p*< 0.0005; LH:

*F*(3,16) = 12.7,

*p*< 0.0005; CRK:

*F*(3,19) = 13.2,

*p*< 0.0005].

*W*

_{50}and its confidence interval. We thank the anonymous reviewers for suggesting measurements at 90 deg wiggle and more precise citation of the ideas of May and Hess (2007) and for provoking us to develop a rigorous comparison of wiggle thresholds across tasks (Figure 5). We also thank Brian Keane, Sarah Rosen, Elizabeth Segal, and Katharine Tillman for helpful suggestions. This research was supported by U.S. National Institutes of Health Grant R01-EY04432 to Denis Pelli.