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Research Article  |   December 2009
Contours in noise: A role for self-cuing?
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Journal of Vision December 2009, Vol.9, 2. doi:10.1167/9.13.2
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      Preeti Verghese; Contours in noise: A role for self-cuing?. Journal of Vision 2009;9(13):2. doi: 10.1167/9.13.2.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

A contour formed of aligned Gabor patches is easily detected amidst a dense background of randomly oriented patches; the longer the string of aligned patches, the more easily the contour is detected (D. J. Field, A. Hayes, & R. F. Hess, 1993). Here we show that a short string of collinear elements acts as a cue pointing to other patches of similar orientation that lie along the path defined by the string. Cueing might increase the gain of similarly oriented elements in the vicinity and/or reduce the number of potential locations (uncertainty) that the observer monitors. To assess the strength of the contour cue, I measured sensitivity to contrast increments on a test patch placed at various offsets relative to the cueing contour. Noise density and the length of cueing contour were also manipulated. Signal detection theory analysis of the psychometric function provided estimates of the gain and uncertainty parameters associated with each condition. In the presence of noise, observers were best at detecting a contrast increment on a patch that was aligned with the cueing string. Gain estimates were high for the aligned condition but decreased with increasing offset from this position. Uncertainty estimates were invariant with offset at a given noise level but increased with increasing noise. Uncertainty also decreased significantly when the length of the cueing string was increased. The high gain for collinear test patches and the ability of the contour to reduce uncertainty at high noise densities is indeed similar to the effect of an explicit cue (e.g., Z.-L. Lu & B. A. Dosher, 1998), suggesting that one part of the contour cues other parts and contributes to the detectability of contours in noise.

Introduction
Amidst the clutter of the natural visual world, human observers effortlessly identify the bounding contours of discrete objects. Neurons in primary visual cortex “view” the world through small apertures that reduce extended contours to segments of various orientations. How does the visual system select the segments that belong together to form a contour? To explore this question, Field, Hess, and Hayes (1993) used a highly reduced representation of natural clutter—a field of Gabor patches, identical except for their randomly selected orientations. In the midst of this orientation noise, they presented a “contour” formed of roughly collinear Gabor patches. Although this contour was identical to the noise in all respects except for the relative orientation of its components, observers could easily detect it provided that the change in orientation and the separation between adjacent components was small. 
These psychophysical constraints on contour detection mirror the slow orientation change in naturally occurring contours. Geisler, Perry, Super, and Gallogly (2001) analyzed contours in natural images and showed that the psychophysical data are well fit by an ideal observer model that is constrained by natural priors, i.e., the likelihood ratio of two orientations falling on a contour (see also Elder & Goldberg, 2002). In Geisler et al.'s (2001) model, a grouping function binds edges together when their probability of being on the same contour exceeds a criterion. Then, a transitivity rule requires that if a first contour segment is connected to a second, and the second is connected to a third, then the first must be connected to the third. 
The natural statistics provide an elegant quantitative account of the likelihood of two oriented fragments lying on a contour given their proximity, local orientations, and their direction. For instance if the two fragments have similar orientation, they are more likely to be on the same contour if the fragments are collinear than if they are parallel. The co-occurrence statistics show that the collinear configuration has a higher likelihood than the parallel configuration. Thus the likelihood ratio seems to reflect the higher visibility of collinear vs. parallel strings in random noise (Field et al., 1993; Geisler et al., 2001; Polat & Norcia, 1996). However, the transitivity rule for grouping multiple contour fragments to form extended contours does not reflect the finding that a longer contour is easier to detect. Because the likelihood ratio only considers pairs of local edge elements, it does not reflect the fact that an edge element is more likely to be on the same contour as an extended co-oriented edge than a short edge at the same distance. In fact, a comparison of Geisler et al.'s (2001) model to their data for detecting contours in noise shows that while the model accounts for human detection of shorter contours, it systematically underestimates human detection of longer contours in noise. This suggests that the pairwise grouping of local edge elements followed by the transitivity rule may not be sufficient to account for the global grouping of contours in noise. 
How does the visual system organize local orientation information into contours? A clue comes from our earlier work with motion trajectories (Verghese & McKee, 2002). There are similarities between motion paths and contours. Due to inertia, objects in motion generally do not change direction abruptly. Contours are in essence static versions of motion trajectories. The bounding contours of natural objects typically change slowly. There is increasing evidence that the human visual system takes advantage of these physical constraints to detect motion along a straight or gently curving path, as well as oriented elements that form a smooth contour (Field et al., 1993; Kovacs & Julesz, 1993; McKee & Welch, 1985; Nakayama & Silverman, 1984; Polat & Sagi, 1993, 1994; Snowden & Braddick, 1989; van Doorn & Koenderink, 1984; Verghese, McKee, & Grzywacz, 2000; Watamaniuk, McKee, & Grzywacz, 1995). 
In the case of motion, the visual system uses a deceptively simple process to combine local motion signals into a smooth motion path: the initial part of the trajectory cues the location and direction of subsequent parts (Verghese & McKee, 2002). I call this process “self-cueing.” Self-cueing explains both the visibility of motion trajectories in noise (Verghese et al., 2000; Watamaniuk et al., 1995) and the improved ability to detect contrast changes at the end compared to the beginning of a motion trajectory (Verghese & McKee, 2002). This paper examines whether the visual system uses self-cueing as a general strategy to organize local information into smooth paths. Specifically I examine whether this self-cueing is involved in building contours out of oriented elements. 
The role of attention
How does a cue select a target, and how can these cueing effects be incorporated into the signal detection theory models? It has been recognized for many decades that cueing enhances detection and discrimination by “drawing attention” to the cued location or feature (Carrasco, Penpeci-Talgar, & Eckstein, 2000; Nakayama & Mackeben, 1989; Posner, 1980). What exactly does “attention” do to the biological representation of the target? Attending to the cued attribute can aid target selection in three ways: (1) by reducing the uncertainty about the target, (2) by increasing the sensitivity of detectors tuned to the cued attribute, and (3) by increasing the selectivity of detectors tuned to the cued attribute (by narrowing the tuning function of the detector). 
A cue can reduce uncertainty by specifying, for example, where the target will occur, or what its orientation will be. The uncued locations are suppressed relative to the cued locations, biasing any competitive interaction between neurons in favor of the cued locations (Desimone & Duncan, 1995; Reynolds, Chelazzi, & Desimone, 1999). This suppression is equivalent to reducing the weighting of neurons feeding into a subsequent stage, without changing their individual properties. Several psychophysical studies have shown that cueing a subset of possible target locations makes location uncertainty equal to the number of cued locations (Burr, Baldassi, Morrone, & Verghese, 2009; Foley & Schwarz, 1998; Palmer, Ames, & Lindsay, 1993; Palmer, Verghese, & Pavel, 2000). 
There is also evidence that the cue enhances the response of individual detectors ( Figure 1A). Physiological studies have shown that cueing a location or a feature has a profound effect on neurons in awake, behaving monkeys. In fact, attending to a feature enhances the responses of a cell when the attended feature matches the cell's preference and suppresses the response when the feature is non-preferred (Martinez-Trujillo & Treue, 2004; Moran & Desimone, 1985; Reynolds et al., 1999; Treue & Martinez-Trujillo, 1999). This modulation in response is evident over the entire tuning function as a change in the gain and is greatest when there are competing stimuli in the visual field (McAdams & Maunsell, 1999; Motter, 1993; Treue & Martinez-Trujillo, 1999). Psychophysical studies have shown enhanced gain at the cued location when the stimuli are clearly visible or embedded in low noise (Carrasco et al., 2000; Dosher & Lu, 2000a). 
Figure 1
 
Effects of attention on the response of a hypothetical neuron as a function of an arbitrary feature value. (A) Attention can enhance the response across the entire tuning curve (solid line), effectively increasing the response at a given feature value by a multiplicative factor. The unattended tuning curve is depicted by a dashed line. (B) Attention can also increase the selectivity of the neuron by restricting the region of feature space over which it responds.
Figure 1
 
Effects of attention on the response of a hypothetical neuron as a function of an arbitrary feature value. (A) Attention can enhance the response across the entire tuning curve (solid line), effectively increasing the response at a given feature value by a multiplicative factor. The unattended tuning curve is depicted by a dashed line. (B) Attention can also increase the selectivity of the neuron by restricting the region of feature space over which it responds.
The evidence that the cue narrows the tuning of detectors ( Figure 1B) is mixed. In psychophysics, observers show a narrower spatial selectivity for the cued location only under conditions of clutter or high noise (Dosher & Lu, 2000b; Yeshurun & Carrasco, 1998). However, the classification image study of Eckstein, Shimozaki, and Abbey (2002) found little evidence for increased spatial selectivity in the presence of a location cue. In physiology, early studies suggested that attending to a stimulus resulted in both an increased gain and a narrower tuning of V4 neurons (Haenny & Schiller, 1988; Spitzer, Desimone, & Moran, 1988). This is in contrast to later studies that measured responses across the entire tuning function and only found evidence for increased gain (McAdams & Maunsell, 2000; Reynolds et al., 1999; Treue & Martinez-Trujillo, 1999; Treue & Maunsell, 1996). 
What implications do the different cueing effects have on signal detection theory models? The two critical variables in SDT are the number of detectors that respond to the signal relative to the total number of detectors responding (uncertainty) and the signal-to-noise ratio in each of these detectors. These two variables have different effects on the psychometric function, which plots the observer's probability of a correct response versus signal strength. A decrease in uncertainty decreases the slope of the psychometric function ( Figure 4A). Any change in signal-to-noise ratio is reflected in a horizontal shift of the psychometric function ( Figure 4B). The signal-to-noise ratio can be altered either by a greater sensitivity (gain) to the signal, or by a more selective filter. When the signal is presented in dense noise, it is not likely that increased gain across the entire bandwidth of the tuning function improves performance because gain would increase both the response to the signal and to the high level of external noise, and thus leave signal-to-noise ratio unchanged (see Verghese, 2001). In our experiments with cluttered backgrounds, it is more likely that performance is improved due to a more selective filter, or due to increased uncertainty. 
The experiments in this paper investigate the role of a contour segment as a cue to nearby segments of similar orientation. To specifically test this cueing hypothesis, I use a contour as a cue to a test patch with a contrast increment and examine whether the improvement in contrast discrimination is due to uncertainty reduction or gain improvement, similar to the analysis of improvement in other explicit cueing paradigms (Carrasco et al., 2000; Foley & Schwarz, 1998; Palmer et al., 1993). 
Methods
The stimuli were generated by a Macintosh G4 with a Radeon graphics card and displayed on a Sony 19′′ monitor. The contour was straight ( Figure 2A) and was made up of 2, 3, 5, or 7 oriented Gabor patches. It was presented at a random location within a 2° box centered on fixation except for one of the conditions in Experiment 3 where it was presented at a fixed location (fixation). As a baseline, I first measured the visibility of contours of various lengths in noise. In these detection experiments, the contour was oriented either ±45° with respect to the vertical, and its spacing was varied between 3 λ and 5.6 λ, where λ is the period of the sine wave in the Gabor patch. The visibility of the contour was measured as a function of the level of noise in the display, for different lengths of the contour, and for different values of spacing between the contour elements. The patches in the contour had a contrast of 0.5, whereas the noise patches had contrasts of 0.25, 0.35, 0.5, 0.71, and 1. These values were chosen to replicate the conditions in the contrast increment experiments (see below). 
Figure 2
 
The test patch is at a location on the contour (as in A), or off the contours (as in B) at the locations marked by outlines that are not present in the actual display. The test patch is always at the same fixed distance from the end of the contour. The surrounding noise elements have been omitted for clarity.
Figure 2
 
The test patch is at a location on the contour (as in A), or off the contours (as in B) at the locations marked by outlines that are not present in the actual display. The test patch is always at the same fixed distance from the end of the contour. The surrounding noise elements have been omitted for clarity.
In the contrast increment experiments, I varied the position of the test patch with respect to the contour in separate blocks of trials, so that the test patch was on or off the contour. The test patch was always in a fixed location with respect to the contour in a block of trials, either aligned with the contour, or placed at 90°, 45°, −45°, and −90° with respect to the extended contour ( Figures 2A and 2B). The spacing between the centers of the patches on the contour was about 4 times λ, the spatial period of the grating. The test patch was either 1 or 2 times this spacing from the lower end of the contour. The patches had a spatial frequency of 8 c/degree and their Gaussian profile has a space constant of 0.067 degrees. 
In the main contrast increment experiments the contour and test patch were presented at one of two orientations, 45° and 135°. The contrast of the contour elements (cue and test) was 0.5. The increment was presented on the test patch in one of the intervals and ranged from 0.52 to 1. The test patch in the other interval had a contrast of 0.5. We used a fixed increment in a block of trials, so observers knew the value of the contrast increment, the two possible orientations of the test patch (45 and 135 deg), their distance and offset relative to the cueing contour and that the test was constrained to fall with the central 2° by 2° region. 
We used a 2IFC procedure with two temporal intervals to measure both contour detection and contrast increment thresholds. Each interval was short (about 200 ms) to minimize eye movements. The location and orientation of the contour was chosen randomly in each interval, subject to the constraints above. In the contrast detection experiments, one of the intervals had a contour, while the other was blank. Observers were asked to choose the interval with the contour. In the contrast increment experiments, both intervals had a contour and a test patch, but in one of the intervals the test patch had a contrast increment relative to the contour. In the other interval the test and contour patches had the same contrast. Observers were asked to choose the interval with the increment. 
Sensitivity to contrast increment thresholds at these positions was measured at various levels of added noise—no added noise, 120, 240, or 480 noise patches in separate blocks. The noise patches were Gabors with one of 12 orientations selected randomly and located at random positions across the display, with the constraint that patches did not come closer than the spacing between the elements in the contour. To prevent observers from doing the contrast increment task by just looking for the brightest patch in the display, we randomized the contrast of the noise patches to one of 5 values that straddled the contrast of the cueing contour (the noise patches had contrasts of 0.25, 0.35, 0.5, 0.71, and 1). The largest value of noise contrast equaled or exceeded the largest contrast increment on the test patch ( Figure 3). 
Figure 3
 
An illustration of the stimulus (cueing contour + test) in noise. The faint box in the center of the display indicated the central 2° square within which the test patch appeared.
Figure 3
 
An illustration of the stimulus (cueing contour + test) in noise. The faint box in the center of the display indicated the central 2° square within which the test patch appeared.
Fitting the signal detection theory model to the data: For each position of the test patch relative to the contour we generated a psychometric function by measuring proportion correct for 100 trials at each of 4 to 5 levels of increment contrast. This function was fit with the uncertainty model outlined in Verghese and McKee (2002), which characterizes a psychometric function by its sensitivity and by its slope (see 1). According to Pelli (1985) the slope is a measure of how many detectors the observer monitors to detect the signal (uncertainty). These detectors may be selective to different regions of space or to different features. In our experiments, uncertainty might arise from the many possible locations within the 2° square box, or the ±45° orientation of the test. Figure 4A shows how a cue can reduce uncertainty about the target and improve thresholds by reducing the steepness of the curve, while Figure 4B shows how a cue can increase sensitivity and improve thresholds by moving or shifting the entire psychometric curve leftward (on a log scale). 
Figure 4
 
Predicted cue effects on the psychometric function. The cued and uncued psychometric functions are shown in black and gray, respectively. The cue can decrease uncertainty M, resulting in a psychometric curve with a shallower slope (left panel). This example shows an uncertainty reduction by a factor of 32. The cue can increase sensitivity k resulting in a leftward shift of the curve, without a change in slope (right panel). The right panel shows a doubling of the gain.
Figure 4
 
Predicted cue effects on the psychometric function. The cued and uncued psychometric functions are shown in black and gray, respectively. The cue can decrease uncertainty M, resulting in a psychometric curve with a shallower slope (left panel). This example shows an uncertainty reduction by a factor of 32. The cue can increase sensitivity k resulting in a leftward shift of the curve, without a change in slope (right panel). The right panel shows a doubling of the gain.
Data for each experimental condition were fit by an iterative procedure to determine the maximum likelihood estimates of the gain parameter k and the uncertainty parameter M. We used a bootstrap technique based on the proportion correct at each contrast to generate confidence limits on the gain and uncertainty parameters. For a given condition and observer, a simulated set of data was generated at each contrast level by taking 100 samples (with replacement) from the observed distribution of proportion correct at that contrast level. The gain and uncertainty parameters were estimated for 100 such simulated sets to obtain an estimate of the spread of these parameters. 
Observers: A total of six observers participated in the experiment, the author and five observers naive as to the purpose of the experiment. 
Experiment 1: Contour detection baseline
This set of experiments replicate previous findings on contour detection (Field et al., 1993) for our stimulus conditions. We measured the visibility of a contour in noise, as a function of noise density, contour length, and the spacing between the patches. This experiment provided a baseline for the contour and noise parameters in our subsequent contour cueing experiments. In this simple detection experiment the contour was present in only one of two intervals and observers were asked to detect the interval with a contour. 
Figures 5A and 5B plot proportion correct as function of the reciprocal of the number of noise patches for two observers. Each color represents a different contour length from 2 to 5 patches. At any given length the contour is easier to detect as the number of noise dots in the display decreases. Moreover, as its length increases, the contour can tolerate a larger amount of noise. This is visible as a leftward shift of the psychometric function with increasing contour length. The increased visibility of longer contours is relevant to how effectively they can cue a test patch (see Experiment 2). 
Figure 5
 
Visibility of a contour in noise as a function of its length (A, B) and spacing between elements (C, D) for two observers. Panels (A) and (B) plot proportion correct as a function of the reciprocal of the number of noise dots at a spacing of 4 λ for contours of various lengths. Noise thresholds corresponding to 82% correct are plotted in (C) and (D) as function of patch spacing for different contour lengths.
Figure 5
 
Visibility of a contour in noise as a function of its length (A, B) and spacing between elements (C, D) for two observers. Panels (A) and (B) plot proportion correct as a function of the reciprocal of the number of noise dots at a spacing of 4 λ for contours of various lengths. Noise thresholds corresponding to 82% correct are plotted in (C) and (D) as function of patch spacing for different contour lengths.
The data in Figures 5A and 5B are for a spacing of 4 λ, where λ is the period of the grating within the Gabor patch. The spacing λ applies to both the patches in the contour and to the closest spacing of the noise. We have similar psychometric functions for spacings of 3, 5, and 5.6 λ (not shown but summarized in Figures 5C and 5D). From each of the psychometric functions, we estimate noise thresholds as the reciprocal of noise dots that yields 82% correct performance. Figures 5C and 5D plot the noise threshold of a contour of a given length as a function of its spacing for the same two observers. As one can see, a contour of a given length is more visible in noise when the spacing between the patches is small. 
So far the results are quite predictable. To relate them to the statistics of natural images, we note that parallel structure is much less probable than collinear structure in the natural world (Elder & Goldberg, 2002; Geisler et al., 2001). Does this lower probability make contours with parallel elements harder to detect? Figure 6 compares the visibility of collinear and parallel contours for the same two observers. The contours were presented in a random location with the constraint that at least a part of the contour fell within a 2° box centered on fixation. It is clear that the contour is much more visible when it is collinear rather than parallel. These data are similar to the data of Field et al. (1993) and Polat and Norcia (1996). The data of Figure 6 show that the advantage of collinear over parallel applies to our stimulus configuration as well. 
Figure 6
 
Proportion correct in a contour detection task as a function of noise level, for two observers. The spacing between the patches was 4λ. When the individual elements are aligned with the contour (filled symbols), detection is significantly higher than when the local elements have an orientation that is orthogonal to the contour (open symbols).
Figure 6
 
Proportion correct in a contour detection task as a function of noise level, for two observers. The spacing between the patches was 4λ. When the individual elements are aligned with the contour (filled symbols), detection is significantly higher than when the local elements have an orientation that is orthogonal to the contour (open symbols).
Experiment 2: Cueing effects of a contour
The effect of noise
According to the association field hypothesis outlined by Field et al. (1993), the detectability of a contour is due to a mechanism “that integrates information across neighboring filters tuned to similar orientations.” This contour integration mechanism may explain the detectability of contours in noise, but it is not well suited to discriminating a small contrast change that occurs only on one element of the contour. According to this hypothesis the contrast increment signal that occurs on one patch would be integrated into the response to the entire contour. Thus as the number of elements in the contour increases, a contrast increment on one element of the contour becomes a smaller fraction of the integrated contour response, leading to poorer contrast discrimination with increasing contour length. The contour cueing hypothesis makes the opposite prediction; an increment on a patch whose orientation and location match that of an extended contour will be easier to detect than an increment on a patch surrounded by random orientations. To test the contour cueing hypothesis, I measured the visibility of contrast increments located along extensions of a contour. 
The increment contrast technique was used in our previous work to show that the detectability of motion trajectories in noise is due to the first part of the trajectory alerting the visual system to any consistent motion in the vicinity (Verghese & McKee, 2002). Analysis of the psychometric functions showed that the threshold improvement at the end of the trajectory was largely due to a significant decrease in uncertainty. This suggests that the initial motion segment of a trajectory in noise cues the location and direction of subsequent motion segments. In this set of experiments I examined the cueing effect of a contour on test patches that are located at an extension of the cueing contour, as well as at various positions off this contour. If the effectiveness of the contour as a cue is completely determined by the visibility of the contour, then a cue of a fixed length at a fixed noise level should yield similar benefits in contrast discrimination, regardless of the offset of the test patch with respect to the cue. This is because conditions are blocked by offset, so the observer knows in advance where the test is with respect to the cue. However, if a contour segment preferentially cues patches located on extensions of the contour, it would indicate that self-cueing is a general strategy that organizes information along smooth paths in space or along smooth trajectories in space time. 
Unless otherwise noted, these experiments were conducted with a contour spacing of 4 λ, with the test patch at a distance of 8 λ (2-element spacing) from the end of the cueing contour segment. The test patch was constrained to fall within a central 2° by 2° in the display. Contrast discrimination was measured for test patches on and off the contour (as in Figure 3), as a function of noise density. The complete data for this set of conditions is summarized in Figure 8, but we first consider a subset of these conditions that illustrate the uncertainty and gain changes described in Figure 4. Figure 7 plots proportion correct as a function of increment contrast along with the fit of the uncertainty model. Uncertainty M and gain k were estimated from individual psychometric functions as described in the Methods section and in 1. The line is the model fit to the data and the associated gain and uncertainty parameters are listed. Figure 7A shows a pair of conditions that yielded a pure uncertainty change, while Figure 7B shows another pair of conditions that yielded a pure gain change. The decrease in uncertainty in Figure 7A was due to a decrease in noise from a high level (480 noise dots) to an intermediate level (240 noise dots) for the aligned condition for observer PV. The change in the visibility of the test patch is largely due to an order-of-magnitude increase in the uncertainty (slope) of the psychometric function corresponding to the high-noise condition. The gain estimates are roughly comparable for the two noise conditions. The uncertainty estimates at other noise levels are summarized in Figures 8C and 8D for observers PV and AJ. 
Figure 7
 
Fitting the uncertainty model to the data to estimate the best fitting values of gain k and uncertainty M. (A) Psychometric functions for the aligned configuration at two levels of noise show a change in uncertainty. (B) Psychometric functions at two values of offset (and the same noise level) show a change in gain.
Figure 7
 
Fitting the uncertainty model to the data to estimate the best fitting values of gain k and uncertainty M. (A) Psychometric functions for the aligned configuration at two levels of noise show a change in uncertainty. (B) Psychometric functions at two values of offset (and the same noise level) show a change in gain.
Figure 8
 
(A, B) Gain estimates versus angular offset of the patch relative to the contour extension for two observers. An offset of 0° represents patches aligned with the contour. The open triangles, circles, and squares represent data under low, moderate, and high noise conditions, respectively. The filled squares represent data for the no-noise condition. (C, D) Uncertainty estimated from psychometric functions versus the noise level. The uncertainty estimates are for the 0° offset condition. Both gain and uncertainty estimates are from psychometric functions for a cue length of 3 and a spacing of 2 elements between cue and test. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 8
 
(A, B) Gain estimates versus angular offset of the patch relative to the contour extension for two observers. An offset of 0° represents patches aligned with the contour. The open triangles, circles, and squares represent data under low, moderate, and high noise conditions, respectively. The filled squares represent data for the no-noise condition. (C, D) Uncertainty estimated from psychometric functions versus the noise level. The uncertainty estimates are for the 0° offset condition. Both gain and uncertainty estimates are from psychometric functions for a cue length of 3 and a spacing of 2 elements between cue and test. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 7B plots proportion correct for a different slice through the data. The two conditions shown here have low noise (120 noise dots), but the test patch was either aligned (0°) or offset 45° from the contour. The improvement in performance due to offset is characterized by a gain change, i.e., a leftward shift of the curve in the aligned condition. Gain estimates appear to be sensitive to spatial configuration, particularly at the low noise level (120 noise dots). It is clear that the 45° offset has a lower gain than the 0° offset. Conversely, the uncertainty as estimated by the slope of the curves was invariant with offset at low noise levels. Thus it appears that under low noise conditions, the gain is higher when the test is aligned with the contour, but under higher levels of noise, it is uncertainty that impairs contrast discrimination. 
Figures 8A and 8B summarize the effect of offset on gain at various noise levels. The error bars on the aligned condition are typical for a noise level and indicate the spread (standard deviation) of gain estimates from our bootstrap procedure. In the absence of noise, the gain was roughly invariant with offset (squares). In the presence of moderate levels of noise (120 to 240 patches), the gain was highest for increments on a patch aligned with the cueing string (offset = 0°) and decreased with increasing offset from the aligned position. At the highest levels of noise (480 noise patches) the estimated gain was so low that the change in gain with offset was not discernable. The decrease in gain with increasing noise is likely due to the normalizing effect of the noise patches (Heeger, 1992). These data indicate that the contour produces an enhanced gain for targets falling on the projected contour line, i.e., zero offset. 
Estimates of uncertainty, however, did not change with offset (not shown); uncertainty remained constant across offset for a given noise level and increased significantly only at the highest noise level. Figures 8C and 8D plot the estimated uncertainty for the aligned condition at the three noise levels. Uncertainty estimates are not significantly different from 1, except for the highest noise level. These data suggest that the cueing contour was able to counteract the noise for densities up to 240 noise patches. Beyond this level of noise, estimates of uncertainty were elevated, indicating that the cue was not effective at this high noise level. 
The effect of contour length
As the visibility of a contour defined by oriented patches increases with length (Geisler et al., 2001), increasing the length of the cueing string should not only increase its visibility but also make it a more effective cue. We verified that the increasing contour length did indeed increase the detectability of the contour in our own stimulus configuration (see Figure 5). We then measured observers' ability to detect increments on a patch that was cued by contours of different lengths. 
Psychometric functions do indeed show better sensitivity to contrast increments as contour length increases. The gain and uncertainty values estimated from these psychometric functions show that the enhanced sensitivity is largely due to changes in uncertainty ( Figure 9) and less due to systematic changes in gain with cue length. In Figures 9A and 9B the different symbols plot the gain estimates for different lengths of cueing contour (2, 3, and 5 for observer PV and 3, 5, and 7 for observer AJ). Gain estimates for contour lengths greater than 2 elements were relatively invariant with cue length. The gain estimates shown in Figures 9A and 9B are for a test patch at a 2-element spacing from the end of the cueing contour and for a noise level of 120 dots (low noise). Similar patterns of sensitivity were obtained at a higher noise level of 240 dots (not shown). Figures 9C and 9D show the effect of contour cue length on uncertainty. Both observers are uncertain in the presence of higher levels of noise, but only when the cueing contour is short (2 to 3 elements long). Because the uncertainty estimates are based on the slope of the psychometric function, which is quite variable, only order of magnitude differences are significant in our bootstrap procedure. These data show that a long contour produces a significant reduction in uncertainty in the presence of high noise. 
Figure 9
 
Gain and uncertainty estimates for contours of various lengths. (A, B) Gain estimates as a function of offset for two observers for a low noise level of 120 dots. (C, D) Uncertainty estimates in the aligned condition with a 3-patch contour cue at noise levels of 120 and 240 dots. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 9
 
Gain and uncertainty estimates for contours of various lengths. (A, B) Gain estimates as a function of offset for two observers for a low noise level of 120 dots. (C, D) Uncertainty estimates in the aligned condition with a 3-patch contour cue at noise levels of 120 and 240 dots. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Experiment 3: Explicitly manipulating uncertainty
In this experiment we specifically address uncertainty reduction. It may be argued that in Experiment 2 uncertainty is an indirect measure that we infer from the psychometric function and that slope estimation is inherently noisy. So here we explicitly manipulate uncertainty and compare contrast increment thresholds in the presence and absence of noise. The cueing string was similar to the earlier experiment and was tilted at ±45° with respect to vertical. In this experiment the test was offset in orientation with respect to the cue at angles of ±30°, ±15°, and 0°. Figure 10 shows the 5 test + cue configurations. For purposes of illustration, the test is shown as vertical, with the cueing string tilted at various angles. In the experiment the cueing string was oblique (either 45° or 135°) and the test was tilted with respect to the string. The test always occurred at the bottom of the cueing string, at a spacing of 1 element. The base contrast of the elements was 50%. The test + cue were presented alone without noise or presented amidst random noise as in Figure 3. The test was constrained to fall within a central 2° by 2° region. We compared increment thresholds with and without noise to determine the contribution of uncertainty. 
Figure 10
 
Cue and test configuration in Experiment 3: The test patch (depicted here with a contrast increment) always occurred at the bottom of the cueing string. For purposes of illustration the test is shown as vertical, but in the experiment it always had an orientation of ±45°. The cue had a tilt of ±30°, ±15°, or 0°, with respect to the test.
Figure 10
 
Cue and test configuration in Experiment 3: The test patch (depicted here with a contrast increment) always occurred at the bottom of the cueing string. For purposes of illustration the test is shown as vertical, but in the experiment it always had an orientation of ±45°. The cue had a tilt of ±30°, ±15°, or 0°, with respect to the test.
The data for symmetric tilts of the test with respect to the cue were similar, so we averaged them. Figure 11 plots contrast thresholds as a function of orientation difference between cue and test for 3 observers. In the absence of noise (green symbols), contrast thresholds were highest when the test was collinear with the cueing contour and decreased as the tilt between cue and test increased. This is similar to the findings of Chen and Tyler (2002) who measured contrast increments with flankers that varied in orientation from collinear to orthogonal. They found that as the tilt between test and cue increased, increment thresholds decreased. So in the absence of noise, the collinear cue appears to mask rather than facilitate increment detection, when the elements are all well above contrast threshold. 
Figure 11
 
Increment thresholds for a test with different relative angles to the cue. The green symbols show thresholds measured without noise and the red symbols show thresholds measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV.
Figure 11
 
Increment thresholds for a test with different relative angles to the cue. The green symbols show thresholds measured without noise and the red symbols show thresholds measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV.
Adding noise to the display caused contrast increment thresholds to increase for all observers, but the trend of thresholds was opposite to what we found in the noise-free display. In the presence of noise (red symbols), thresholds were lowest for the collinear cue and increased with increasing orientation difference between cue and test. This suggests that the collinear cue helps locate the test patch within the central 2° square, and that the effectiveness of this cue decreases as the relative angle between cue and test increased. 
To test this possibility, we repeated the noise condition but added a variant where there was no uncertainty about the test, because it always occurred at fixation. We used the same relative orientations as before but increased the range to include a 45-degree orientation difference between test and cue. 
The filled symbols in Figure 12 plot data for the uncertain condition, while the open symbols plot data for the certain condition (these two sets of data were taken in separate blocks within the same session). When test location was unknown, the data for all observers shows the same trend as in Figure 11, increasing with increasing tilt between cue and test. When test location was known, contrast increments for all three observers are clearly lower than for one in an unknown location. More importantly, contrast thresholds remain roughly unchanged with orientation difference when the test location is known, while thresholds increase with orientation difference under conditions of uncertainty. This experiment demonstrates that one of the primary functions of the adjacent contour elements is to reduce uncertainty about the location of likely additional components of the same contour—a kind of bootstrapping that improves contour visibility. 
Figure 12
 
Increment thresholds in noise for a test tilted at various angles from a cueing contour. All thresholds were measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV. The open symbols show thresholds when the test was in a fixed location and the filled symbols when it was randomly located within a central 2° by 2° square.
Figure 12
 
Increment thresholds in noise for a test tilted at various angles from a cueing contour. All thresholds were measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV. The open symbols show thresholds when the test was in a fixed location and the filled symbols when it was randomly located within a central 2° by 2° square.
Discussion
The results indicate that a contour preferentially cues patches that are located at potential extensions of the contour. Experiment 1 establishes that the visibility of contours increases with contour length and decreases with noise level. Experiment 2 examines the efficacy of these contours when used as cues to a test patch with a contrast increment. Because the location of the test patch is determined only with respect to the cue, it is not surprising that conditions that make the contour cue easier to locate (such as low noise density and increased contour length) also improve performance in the contrast increment task. What is more interesting is that the cue preferentially improves contrast discrimination on test patches that are potential extensions of the cueing contour, particularly under conditions of uncertainty. If the effectiveness of the contour as a cue was completely determined by its visibility in noise, then cues of a fixed length would yield similar benefits in contrast discrimination, regardless of the offset of the test patch with respect to the cue, or its orientation. This is because conditions are blocked by offset and by relative orientation, so the observer knows in advance the location and orientation of the test patch with respect to the cue. However the greatest improvement in contrast discrimination is seen for patches collinear with the cue, either due to increases in gain, or due to decreases in uncertainty for patches collinear with the cue, suggesting an advantage for test patches that lie on extensions of the contour. Experiment 3 shows that the improved contrast discrimination for patches aligned with the contour only occurs under conditions of low visibility: either in the presence of noise as shown here, or for low contrast contours in the absence of noise (Verghese, 2007). For high contrast contours in the absence of noise, sensitivity actually decreases with increasing alignment between contour and test (Figure 11; Chen & Tyler, 2002). 
The changes in uncertainty for patches on extensions of the contour are similar to our results for motion trajectories in noise, where the increased ability to detect a contrast increment on the end as opposed to the beginning of a motion trajectory was due to a substantial reduction in uncertainty (Verghese & McKee, 2002). In the motion study, we saw a hint for increases in gain at lower noise levels. In the present study we find strong evidence that increases in gain account for the improvement in contrast discrimination at low noise, while the improvement is due to changes in both uncertainty and gain at higher noise levels. 
Collinear facilitation and contrast normalization
What class of contour integration model accounts for these effects? One class of model predicts an overall increase in the activity of detectors coding for roughly collinear elements. These include models that implement long-range facilitatory interactions (e.g., Grossberg, Mingolla, & Ross, 1997; Li, 1998) and those that maximize smoothness in contour grouping (Pettet, McKee, & Grzywacz, 1998). The increased activation on the contour explains the increased detectability of contours in noise (as in Field et al., 1993). Such increased activation is not consistent with some aspects of our data. For instance, increased activation and the higher variability associated with higher response levels predicts higher thresholds for increments on the contour, which is not consistent with our results. Furthermore, elements on a contour do not appear brighter: contrast matching experiments show that perceived contrast does not change for patches on the contour as compared to patches off the contour (Hess, Dakin, & Field, 1998; Pettet & Verghese, 1997). This is in contrast to the report that an explicit cue to a location increases the perceived contrast of a Gabor patch at that location (Carrasco, Ling, & Read, 2004). Of course, the conditions in the explicit cueing experiment and the contour-in-noise experiment are quite different. It is possible that in the presence of noise patches, contrast normalization from these patches offsets any increase in perceived contrast. 
Another class of model that might account for increased gain in the collinear configuration is a variant of the contrast normalization models that preferentially weight collinear signals in the normalization pool (Schwartz & Simoncelli, 2001). Gain control does not predict an increase in overall activity but rather increased contrast sensitivity to contrast changes about the mean contrast. If orientations similar to the test were weighted more heavily in the normalization pool, it would explain the higher sensitivity to increments on the test when the cue is collinear with the test. However it does not explain why contours are easily detected in noise. Neither class of model explains both the visibility of contours in noise and the increased sensitivity for increments on a contour in the presence of noise. 
The role of uncertainty
Yu, Klein, and Levi (2002) looked explicitly for uncertainty effects and found only small effects that did not reach significance. The likely reason for the difference between their study and ours is the great difference in stimulus conditions. In our experiments, the test patch was only defined with respect to the cue. So conditions that impaired the visibility of the cue such as high noise and short cue length increased the uncertainty of the test. In contrast, Yu et al. (2002) had a test grating at fixation surrounded by a concentric annulus of orthogonal orientation. They found that the annular surround improved contrast detection of the central test and specifically examined whether this facilitation was due to a reduction in uncertainty (Pelli, 1985). For one of their observers the facilitation was due to an increase in gain, and for the other, the facilitation was due to a reduction in uncertainty, which did not reach significance. We too obtain similar results under conditions of low noise: the improved contrast discrimination for test patches aligned with the contour are largely due to increases in gain, with insignificant decreases in uncertainty. However at larger noise levels, uncertainty reduction contributes significantly to the improvement in contrast discrimination. 
Other studies (Petrov, Verghese, & McKee, 2006; Shani & Sagi, 2006) have examined the role of uncertainty in mediating the improved detection of a Gabor patch in the presence of collinear flankers (Polat & Sagi, 1993, 1994). Both studies (Petrov et al., 2006; Shani & Sagi, 2006) show a decrease in threshold and in the slope of the psychometric function in the presence of collinear flankers, consistent with uncertainty reduction. Shani and Sagi (2006) suggest that this pattern of results can be explained by an alternate mechanism where flankers of similar orientation add to the contrast of the target. While an interaction between collinear mechanisms may indeed facilitate target detection, it is clear that uncertainty plays a role because the presence of any spatial marker improves target detection. For instance, Petrov et al. (2006) showed that a faint thin ring around the target produced a similar improvement in thresholds and slopes as collinear flankers. Yu et al. (2002) showed that orthogonal end flankers that reduce spatial uncertainty produce almost as much facilitation as collinear end flankers that reduce both spatial and feature uncertainty. Shani and Sagi (2006) observed a significantly smaller facilitation with orthogonal than with collinear flankers, but perhaps this is because spatial uncertainty was already low due to the explicit spatial markers present in their experiments. While spatial markers and/or orthogonal flankers produce varying degrees of facilitation, all studies show that collinear end flankers produce the largest facilitation. These results indicate a role for spatial and feature uncertainty reduction, as well as for mechanisms based on collinear interactions (Chen & Tyler, 2002; Pettet et al., 1998; Zenger & Sagi, 1996). 
The role of attention
It is possible that some combination of summation through long-range interactions (Gilbert, 1992; Kapadia, Ito, Gilbert, & Westheimer, 1995) and suppression through contrast-gain control (Schwartz & Simoncelli, 2001) can explain both the increased detectability and cueing strength of collinear contours in noise. Alternately, I propose that a short contour segment acts as a cue to the rest of the contour. Support for this proposal comes from a comparison of Geisler et al.'s (2001) contour visibility data and their model predictions. Recall that according to their model the visibility of a long contour is determined by local grouping according to co-occurrence statistics followed by a transitivity rule. However, an inspection of their Figure 6 shows that the model systematically underestimates human contour detection for long contours. This discrepancy can be remedied by recognizing that a contour segment is an excellent cue to another contour (or patch) that is collinear. As with other explicit cues, the effect of a contour cue may be to increase the sensitivity of the underlying detector and to reduce the number of competing responses. The increase in sensitivity can be achieved through a process akin to gain control and the increase in the visibility of the contour can be achieved through a reduction in the number of competitors (a decrease in uncertainty). 
How can contour segments act as cues? At high contrast levels, any pair of roughly aligned Gabor patches may increase the response of an elongated orientation detector (Jones & Palmer, 1987; Solomon, Watson, & Morgan, 1999) above that generated by the randomly oriented noise. However, a single pair is barely detected in dense noise, and even multiple pairs scattered throughout the noise are not as detectable as an extended contour formed of a comparable number of patches. I am proposing that each pair acts as a selective cue for adjacent patches of similar orientation. Competitive interaction between the cue and noise also suppresses the activity generated by patches of dissimilar orientation (Reynolds et al., 1999). In short, the visual system uses a bootstrapping operation to connect the oriented segments. Identifying a contour is difficult in noisy, cluttered environments where there is considerable uncertainty about which oriented segments belong together. By removing competitors, self-cueing greatly enhances the signal-to-noise ratio of the responses generated by the real meandering contour. In fact Elder and Goldberg (2002), using an entropy measure, show that contour grouping cues such as proximity and good continuation significantly reduce the uncertainty about which oriented segments are to be grouped together. 
The benefits of a contour cue are consistent with explicitly cueing attention. Previous studies show that attending to a stimulus increases gain at low noise levels (Carrasco et al., 2000; Dosher & Lu, 2000a; McAdams & Maunsell, 1999; Treue & Martinez-Trujillo, 1999; Treue & Maunsell, 1996). In our studies, the predominant effect of a contour cue is to increase sensitivity to contrast increments under low noise conditions. This sensitivity increase is most pronounced for test patches aligned with the cue. Under higher noise conditions, the changes in sensitivity are less discernable (Figure 8), but we begin to see an improvement in detecting contrast increments due to reduced uncertainty, likely implemented by suppressing locations and orientations not consistent with the contour cue. Reducing uncertainty is particularly effective under conditions of high noise because it suppresses irrelevant locations. If stimuli are selected based on competitive interactions between detectors, then the non-suppressed locations have a competitive edge. 
Our finding that a contour cue reduces uncertainty at high noise levels is similar to the effect of a motion trajectory in noise. The trajectory acts as a self-cue, pruning out spurious locations and directions and directing attention to the most likely region that contains an extension of the trajectory. Such a self-cueing process can work in concert with mechanisms that favor smoothness, providing the “transitivity rule” to connect locally smooth paths into an extended trajectory or contour. 
Appendix A
Uncertainty model
The model described here is very similar to the model described in Verghese and McKee (2002) and Verghese and Stone (1995). Ideally the observer monitors only the test patch as specified by the cueing contour. However, if the cueing contour is not detected, then the observer might monitor any patch that approximately matches the characteristics of the test patch: an orientation near ±45°, a contrast near 0.5, and a location in the central 2° × 2° area. Uncertainty arises because there may be more than one patch that fits this description. Recall that the noise patches have all possible orientations and one of 5 contrast values centered about 0.5. Thus uncertainty depends on the noise level and the length of the contour cue (see Figures 8 and 9). In our model we assume that the number of patches monitored in each temporal interval is M. This corresponds to M detectors, one for each patch. The detectors have a sensitivity k, and each detector produces a noisy response. The observer finds the largest of these responses in each interval and then chooses the interval with the larger response. Errors arise when the interval without the increment produces a larger response and the probability of error increases with the number of detectors that the observer monitors. 
We assume that the noisy responses are samples from a Gaussian distribution. When the detector is centered on the contrast increment, the response is a sample from a distribution with a mean at c + Δ c, whereas the responses to non-increment contrasts are samples from a distribution centered at c. The value of c in our experiments is 0.5. The variance of this distribution does not represent the variability in response to a single contrast value, but rather the pooled variance across all five noise contrasts. The observer monitors the output of M detectors in each interval and makes a correct choice when the largest response from the increment interval exceeds the largest response from the non-increment interval. Conversely the observer makes an incorrect choice if the non-increment interval produces the larger response. Thus we can write the probability correct as 1-probability that the non-increment interval produces the larger response, i.e.,  
P c o r r e c t ( Δ c ) = P i n c r e m e n t i n t e r v a l l a r g e r = 1 P n o n i n c r e m e n t i n t e r v a l l a r g e r P c o r r e c t ( Δ c ) = 1 M f ( x k c ) F ( x k c ) 2 M 2 F ( x k ( c + Δ c ) ) d x ,
(A1)
where c is the contrast of the trajectory, Δ c is the contrast increment, f( x) is the Gaussian probability density function, and F( x) is the cumulative Gaussian ∫ −∞ x f( x′) dx′. 
The probability that the non-increment interval generates the larger response is given by the terms within the integral. The non-increment interval produces the larger response when one of the M detectors in the non-increment interval has a value x and all the other detectors have a value less than x. The probability that a detector in the non-increment interval produces a response x is given by the Gaussian density f( xkc), where k is the sensitivity parameter and c is the pedestal contrast. The probability that the other detectors generate a response less than x is the product of all their cumulative distributions, i.e., F( xkc) 2 M−2 F( xk( c + Δ c)), where the first term is due to the 2 M − 2 detectors that see the pedestal contrast and the second term is from the one detector in the increment interval that responds to the increment c + Δ c. The exponent 2 M − 2 arises from the M − 1 detectors in the non-increment interval and the M − 1 detectors in the increment interval that see the pedestal contrast c. The expression is multiplied by M because any one of the M detectors in the non-increment interval can produce the largest response. 
Acknowledgments
This work was supported by NSF Grants 0347051 and 0642728 and by Smith Kettlewell Eye Research Institute. 
Commercial relationships: none. 
Corresponding author: Preeti Verghese. 
Email: preeti@ski.org. 
Address: Smith Kettlewell Eye Research Institute, San Francisco, CA, USA. 
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Figure 1
 
Effects of attention on the response of a hypothetical neuron as a function of an arbitrary feature value. (A) Attention can enhance the response across the entire tuning curve (solid line), effectively increasing the response at a given feature value by a multiplicative factor. The unattended tuning curve is depicted by a dashed line. (B) Attention can also increase the selectivity of the neuron by restricting the region of feature space over which it responds.
Figure 1
 
Effects of attention on the response of a hypothetical neuron as a function of an arbitrary feature value. (A) Attention can enhance the response across the entire tuning curve (solid line), effectively increasing the response at a given feature value by a multiplicative factor. The unattended tuning curve is depicted by a dashed line. (B) Attention can also increase the selectivity of the neuron by restricting the region of feature space over which it responds.
Figure 2
 
The test patch is at a location on the contour (as in A), or off the contours (as in B) at the locations marked by outlines that are not present in the actual display. The test patch is always at the same fixed distance from the end of the contour. The surrounding noise elements have been omitted for clarity.
Figure 2
 
The test patch is at a location on the contour (as in A), or off the contours (as in B) at the locations marked by outlines that are not present in the actual display. The test patch is always at the same fixed distance from the end of the contour. The surrounding noise elements have been omitted for clarity.
Figure 3
 
An illustration of the stimulus (cueing contour + test) in noise. The faint box in the center of the display indicated the central 2° square within which the test patch appeared.
Figure 3
 
An illustration of the stimulus (cueing contour + test) in noise. The faint box in the center of the display indicated the central 2° square within which the test patch appeared.
Figure 4
 
Predicted cue effects on the psychometric function. The cued and uncued psychometric functions are shown in black and gray, respectively. The cue can decrease uncertainty M, resulting in a psychometric curve with a shallower slope (left panel). This example shows an uncertainty reduction by a factor of 32. The cue can increase sensitivity k resulting in a leftward shift of the curve, without a change in slope (right panel). The right panel shows a doubling of the gain.
Figure 4
 
Predicted cue effects on the psychometric function. The cued and uncued psychometric functions are shown in black and gray, respectively. The cue can decrease uncertainty M, resulting in a psychometric curve with a shallower slope (left panel). This example shows an uncertainty reduction by a factor of 32. The cue can increase sensitivity k resulting in a leftward shift of the curve, without a change in slope (right panel). The right panel shows a doubling of the gain.
Figure 5
 
Visibility of a contour in noise as a function of its length (A, B) and spacing between elements (C, D) for two observers. Panels (A) and (B) plot proportion correct as a function of the reciprocal of the number of noise dots at a spacing of 4 λ for contours of various lengths. Noise thresholds corresponding to 82% correct are plotted in (C) and (D) as function of patch spacing for different contour lengths.
Figure 5
 
Visibility of a contour in noise as a function of its length (A, B) and spacing between elements (C, D) for two observers. Panels (A) and (B) plot proportion correct as a function of the reciprocal of the number of noise dots at a spacing of 4 λ for contours of various lengths. Noise thresholds corresponding to 82% correct are plotted in (C) and (D) as function of patch spacing for different contour lengths.
Figure 6
 
Proportion correct in a contour detection task as a function of noise level, for two observers. The spacing between the patches was 4λ. When the individual elements are aligned with the contour (filled symbols), detection is significantly higher than when the local elements have an orientation that is orthogonal to the contour (open symbols).
Figure 6
 
Proportion correct in a contour detection task as a function of noise level, for two observers. The spacing between the patches was 4λ. When the individual elements are aligned with the contour (filled symbols), detection is significantly higher than when the local elements have an orientation that is orthogonal to the contour (open symbols).
Figure 7
 
Fitting the uncertainty model to the data to estimate the best fitting values of gain k and uncertainty M. (A) Psychometric functions for the aligned configuration at two levels of noise show a change in uncertainty. (B) Psychometric functions at two values of offset (and the same noise level) show a change in gain.
Figure 7
 
Fitting the uncertainty model to the data to estimate the best fitting values of gain k and uncertainty M. (A) Psychometric functions for the aligned configuration at two levels of noise show a change in uncertainty. (B) Psychometric functions at two values of offset (and the same noise level) show a change in gain.
Figure 8
 
(A, B) Gain estimates versus angular offset of the patch relative to the contour extension for two observers. An offset of 0° represents patches aligned with the contour. The open triangles, circles, and squares represent data under low, moderate, and high noise conditions, respectively. The filled squares represent data for the no-noise condition. (C, D) Uncertainty estimated from psychometric functions versus the noise level. The uncertainty estimates are for the 0° offset condition. Both gain and uncertainty estimates are from psychometric functions for a cue length of 3 and a spacing of 2 elements between cue and test. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 8
 
(A, B) Gain estimates versus angular offset of the patch relative to the contour extension for two observers. An offset of 0° represents patches aligned with the contour. The open triangles, circles, and squares represent data under low, moderate, and high noise conditions, respectively. The filled squares represent data for the no-noise condition. (C, D) Uncertainty estimated from psychometric functions versus the noise level. The uncertainty estimates are for the 0° offset condition. Both gain and uncertainty estimates are from psychometric functions for a cue length of 3 and a spacing of 2 elements between cue and test. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 9
 
Gain and uncertainty estimates for contours of various lengths. (A, B) Gain estimates as a function of offset for two observers for a low noise level of 120 dots. (C, D) Uncertainty estimates in the aligned condition with a 3-patch contour cue at noise levels of 120 and 240 dots. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 9
 
Gain and uncertainty estimates for contours of various lengths. (A, B) Gain estimates as a function of offset for two observers for a low noise level of 120 dots. (C, D) Uncertainty estimates in the aligned condition with a 3-patch contour cue at noise levels of 120 and 240 dots. The error bars represent the spread (standard deviation) of the distribution of gain and uncertainty estimates from the bootstrap procedure.
Figure 10
 
Cue and test configuration in Experiment 3: The test patch (depicted here with a contrast increment) always occurred at the bottom of the cueing string. For purposes of illustration the test is shown as vertical, but in the experiment it always had an orientation of ±45°. The cue had a tilt of ±30°, ±15°, or 0°, with respect to the test.
Figure 10
 
Cue and test configuration in Experiment 3: The test patch (depicted here with a contrast increment) always occurred at the bottom of the cueing string. For purposes of illustration the test is shown as vertical, but in the experiment it always had an orientation of ±45°. The cue had a tilt of ±30°, ±15°, or 0°, with respect to the test.
Figure 11
 
Increment thresholds for a test with different relative angles to the cue. The green symbols show thresholds measured without noise and the red symbols show thresholds measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV.
Figure 11
 
Increment thresholds for a test with different relative angles to the cue. The green symbols show thresholds measured without noise and the red symbols show thresholds measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV.
Figure 12
 
Increment thresholds in noise for a test tilted at various angles from a cueing contour. All thresholds were measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV. The open symbols show thresholds when the test was in a fixed location and the filled symbols when it was randomly located within a central 2° by 2° square.
Figure 12
 
Increment thresholds in noise for a test tilted at various angles from a cueing contour. All thresholds were measured in the presence of noise: 120 noise dots for JB and 240 noise dots for LM and PV. The open symbols show thresholds when the test was in a fixed location and the filled symbols when it was randomly located within a central 2° by 2° square.
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