Tilted frames of reference have similar effects on the perception of gravitational vertical and the planning of vertical saccadic eye movements

We investigated the effects of a tilted reference frame (i.e., allocentric visual context) on the perception of the gravitational vertical and saccadic eye movements along a planned egocentric vertical path. Participants (n = 5) in a darkened room fixated a point in the center of a circle on an LCD display and decided which of two sequentially presented dots was closer to the unmarked ‘6 o’clock’ position on that circle (i.e., straight down toward their feet). The slope of their perceptual psychometric functions showed that participants were able to locate which dot was nearer the vertical with a precision of 1°–2°. For three of the participants, a square frame centered at fixation and tilted (in the roll direction) 5.6° from the vertical caused a strong perceptual bias, manifest as a shift in the psychometric function, in the direction of the traditional ‘rod-and-frame’ effect, without affecting precision. The other two participants showed negligible or no equivalent biases. The same subjects participated in the saccade version of the task, in which they were instructed to shift their gaze to the 6 o’clock position as soon as the central fixation point disappeared. The participants who showed perceptual biases showed biases of similar magnitude in their saccadic endpoints, with a strong correlation between perceptual and saccadic biases across all subjects. Tilting of the head 5.6° reduced both perceptual and saccadic biases in all but one observer, who developed a strong saccadic bias. Otherwise, the overall pattern and significant correlations between results remained the same. We conclude that our observers’ saccades-to-vertical were dominated by perceptual input, which outweighed any gravitational or head-centered input.

Rapid eye movements to a virtual target are biased by illusory context in the Poggendorff figure

In order to determine the influence of perceptual input upon oculomotor responses, we examined rapid saccadic eye movements made by healthy human observers to a virtual target defined by the extrapolated intersection of a pointer with a distant landing line. While corresponding perceptual judgments showed no evidence of systematic bias, eye movements showed a strong bias, in the direction of assimilation of the saccade trajectory to the shortest path between the end of the pointer and the landing line. Adding an abutting vertical inducing line to make an angle of 45 deg with the pointer led to a larger bias in the same direction as the classical Poggendorff illusion. This additional Poggendorff effect was similar in direction and magnitude for the eye movements and the perceptual responses. Latency and dynamics of the eye movements were closely similar to those recorded for a control task in which observers made a saccade from the start fixation to an explicit target on the landing line. Further experiments with inducing lines presented briefly at various times during the saccade latency period showed that the magnitude of the saccade bias was affected by inducer presentation during the saccade planning process, but not during the saccade itself. We conclude that the neural mechanisms for extrapolation can feed into the control of eye movements without obvious penalties in timing and accuracy and that this information can instantaneously modify motor response throughout the planning phase, suggesting close association between perceptual and motor mechanisms in the process of visuo-spatial extrapolation.

Awareness is the key to attraction: dissociating the tilt illusions via conscious perception

The tilt illusion is a compelling example of contextual influence exerted by an oriented surround on a target's perceived orientation. A vertical target appears to be tilted away from a 15° oriented surround but appears to be tilted toward a 75° tilted surround. We tested the claim that these biases result from distinct sensory processes: a low-level repulsive process and a higher-level attractive process. If this claim were correct, then surround visibility would be a requirement for attraction, but it would not necessarily be a requirement for repulsion. Indeed, Motoyoshi and Hayakawa (2010) have demonstrated that repulsion can survive removal of the surround from phenomenal awareness using adaptation-induced blindness. Here we sought to test this prediction by measuring the orientation biases in a parafoveally presented Gabor patch surrounded by tilted gratings after 20-s adaptation. The adapting stimulus was an annularly windowed plaid composed of vertical and horizontal jittering gratings. Observers were instructed to maintain fixation throughout the trial and report whether the Gabor appeared to be tilted clockwise or anticlockwise of vertical. They also had to indicate whether the surround was visible after adaptation. Postadaptation biases were then compared with those obtained in a control experiment without dynamic adaptation. We found large repulsive biases induced by 15° oriented surrounds, but no attractive biases were induced by 75° tilted surrounds. This result shows that attractive effects do require visual awareness and thereby provides robust evidence for the existence of two separate mechanisms mediating the phenomenology of the tilt illusions.

Four theorems on the psychometric function

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, . This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, , can be approximated by , where is the of the Weibull function that fits best to the cumulative noise distribution, and depends on the transducer. We derive general expressions for and , from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when , . We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4–0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.

Aftereffect of perceived regularity

Regularity is a ubiquitous feature of the visual world. We demonstrate that regularity is an adaptable visual dimension: The perceived regularity of a pattern is reduced following adaptation to a pattern with a similar or greater degree of regularity. Stimuli consisted of 7×7 element arrays arranged on square grids presented in a circular aperture. The position of each element was randomly jittered from its baseline position by an amount that determined its degree of irregularity. The elements of the pattern consisted of dark Gaussian blobs (GBs), difference of Gaussians (DOGs), or random binary patterns (RBPs). Observers adapted for 60 s to either a single pattern or a pair of patterns with particular regularities, and the perceived regularities of subsequently presented test patterns were measured using a conventional staircase matching procedure. We found that the regularity aftereffect (RAE) was unidirectional: Adaptation only caused test patterns to appear less regular. We also found that RAEs transferred from GB adaptors to both DOG and RBP test patterns and from DOG and RBP adaptors to GB patterns. We suggest that regularity is coded by the peakedness in the distribution of spatial-frequency channel responses across scale, and that the RAE is a result of a flattening of this distribution by adaptation. Thus, the RAE may be a consequence of contrast normalization, and an example of norm-based coding where irregularity is the norm.

Recognition criteria vary with fluctuating uncertainty

In distinct experiments we examined memories for orientation and size. After viewing a randomly oriented Gabor patch (or a plain white disk of random size), observers were given unlimited time to reproduce as faithfully as possible the orientation (or size) of that standard stimulus with an adjustable Gabor patch (or disk). Then, with this match stimulus still in view, a recognition probe was presented. On half the trials, this probe was identical to the standard. We expected observers to classify the probe (a same/different task) on the basis of its difference from the match, which should have served as an explicit memory of the standard. Observers did better than that. Larger differences were classified as "same" when probe and standard were indeed identical. In some cases, recognition performance exceeded that of a simulated observer subject to the same matching errors, but forced to adopt the single most advantageous criterion difference between the probe and match. Recognition must have used information that was not or could not be exploited in the reproduction phase. One possible source for that information is observers' confidence in their reproduction (e.g., in their memory of the standard). Simulations confirm the enhancement of recognition performance when decision criteria are adjusted trial-by-trial, on the basis of the observer's estimated reproduction error.

Perceived pattern regularity computed as a summary statistic: implications for camouflage

Why do the equally spaced dots in figure 1 appear regularly spaced? The answer 'because they are' is naive and ignores the existence of sensory noise, which is known to limit the accuracy of positional localization. Actually, all the dots in figure 1 have been physically perturbed, but in the case of the apparently regular patterns to an extent that is below threshold for reliable detection. Only when retinal pathology causes severe distortions do regular grids appear perturbed. Here, we present evidence that low-level sensory noise does indeed corrupt the encoding of relative spatial position, and limits the accuracy with which observers can detect real distortions. The noise is equivalent to a Gaussian random variable with a standard deviation of approximately 5 per cent of the inter-element spacing. The just-noticeable difference in positional distortion between two patterns is smallest when neither of them is perfectly regular. The computation of variance is statistically inefficient, typically using only five or six of the available dots.

Efficiencies for the statistics of size discrimination

Different laboratories have achieved a consensus regarding how well human observers can estimate the average orientation in a set of N objects. Such estimates are not only limited by visual noise, which perturbs the visual signal of each object's orientation, they are also inefficient: Observers effectively use only √N objects in their estimates (e.g., S. C. Dakin, 2001; J. A. Solomon, 2010). More controversial is the efficiency with which observers can estimate the average size in an array of circles (e.g., D. Ariely, 2001, 2008; S. C. Chong, S. J. Joo, T.-A. Emmanouil, & A. Treisman, 2008; K. Myczek & D. J. Simons, 2008). Of course, there are some important differences between orientation and size; nonetheless, it seemed sensible to compare the two types of estimate against the same ideal observer. Indeed, quantitative evaluation of statistical efficiency requires this sort of comparison (R. A. Fisher, 1925). Our first step was to measure the noise that limits size estimates when only two circles are compared. Our results (Weber fractions between 0.07 and 0.14 were necessary for 84% correct 2AFC performance) are consistent with the visual system adding the same amount of Gaussian noise to all logarithmically transduced circle diameters. We exaggerated this visual noise by randomly varying the diameters in (uncrowded) arrays of 1, 2, 4, and 8 circles and measured its effect on discrimination between mean sizes. Efficiencies inferred from all four observers significantly exceed 25% and, in two cases, approach 100%. More consistent are our measurements of just-noticeable differences in size variance. These latter results suggest between 62 and 75% efficiency for variance discriminations. Although our observers were no more efficient comparing size variances than they were at comparing mean sizes, they were significantly more precise. In other words, our results contain evidence for a non-negligible source of late noise that limits mean discriminations but not variance discriminations.

Evidence for a subtractive component in motion adaptation

Adaptation to a moving stimulus changes the perception of a stationary grating and also reduces contrast sensitivity to the adaptor. We determined whether the first effect could be predicted from the second. The contrast discrimination (T vs. C) function for a drifting 7.5 Hz grating test stimulus was determined when observers were adapted to a low contrast (0.075) grating of the same spatial and temporal frequency, moving in either the same or the opposite direction as the test. The effect of an adaptor moving in the same direction was to move the T vs. C function upwards and to the right, in a manner consistent with an increase in divisive inhibition. We also measured the effect of adaptation on the motion-null point for a counterphasing grating containing two components, one moving in the same direction as the adaptor and the other in the opposite direction. Adaptation increased the amount of contrast of the adapted component required to achieve the motion-null point. However, this shift could not be predicted from the effects of adaptation on contrast sensitivity. In particular, the balance point was shifted in gratings of high contrast where there was no effect of adaptation on contrast discrimination. We suggest that adaptation has a subtractive (recalibration) effect in addition to its effects on the contrast transduction function, and that this subtractive effect may explain the movement after-effect seen with stationary tests.

Visual discrimination of orientation statistics in crowded and uncrowded arrays

When required to identify the orientation of an item outside the center of the visual field, the mean orientation predicts performance better than the orientation of any individual item in that region. Here I examine whether the visual system also preserves the variance of orientations in these so-called "crowded" displays. In Experiment 1, I determined the separation between items necessary to prevent neighbors from interfering with discrimination between different orientations in a single, target item. In Experiment 2, I used this separation and measured the effect of orientation variance on discrimination between mean orientations in these consequently uncrowded displays. In Experiment 3, I measured the relationship between the just-noticeable difference in variance and the smaller of two orientation variances in uncrowded displays. Finally, in Experiments 4 and 5, I reduced the separation between items and measured the effect of crowding on mean and variance discriminations. When considered together, the results of all these experiments imply that the visual system computes orientation variances with both more efficiency and greater precision than it computes orientation means. Although crowding made it difficult for some observers to discriminate between small amounts of orientation variance, it had no other significant effect on visual estimates mean orientation and orientation variance.

Cortical distance determines whether flankers cause crowding or the tilt illusion

Differences between target and flanker orientations become exaggerated in the tilt illusion. However, small differences sometimes go unnoticed. This small-angle assimilation shares many similarities with other types of visual crowding but is typically found only with small and/or hard-to-see stimuli. In Experiment 1, we investigated the effect of stimulus visibility on orientation bias using relatively large stimuli. The introduction of visual noise increased the perceived similarity of target and flanker orientations at retinal eccentricities of 4° and 10°; however, small-angle assimilation was found only at 10°. The effects of eccentricity were reduced in Experiment 2, when our stimuli were "M-scaled" for equal cortical coverage. Further support for a cortical substrate was obtained in Experiment 3, in which the effects of target-flanker separation were measured. When biases from all three experiments are expressed as a fraction of the inducing flankers' angle, and plotted as a function of the approximate cortical separation between the target and its closest flanker, they form a curve like the cross-section of half a Mexican hat. We conclude that the tilt illusion and small-angle assimilation reflect opponent influences on orientation perception. The strength of each influence increases with cortical proximity and stimulus visibility, but the one responsible for assimilation has a lesser extent.

Orientation uncertainty reduces perceived obliquity

The influence of prejudice on perception should be greatest when certainty about stimulus identity is least. We exploited this relationship to reveal visual biases for the cardinal orientations: vertical and horizontal. Specifically, when we increased the variance of orientations in an array of grating patches, estimates of the mean became less oblique. This result is consistent with a stable prior, or prejudice, for those orientations most prevalent in natural scenes.

Strong tilt illusions always reduce orientation acuity

The apparent spatial orientation of an object can differ from its physical orientation when differently oriented objects surround it. This is the "tilt illusion". Previously [Solomon, J. A., & Morgan, M. J. (2006). Stochastic re-calibration: Contextual effects on perceived tilt. Proceedings of the Royal Society of London. Series B, Biological Sciences, 273, 2681-2686], we reported a loss of orientation acuity whenever a large physical tilt was required to compensate for the tilt illusion and make a target appear horizontal. Since all of those targets appeared to be at least approximately horizontal, we concluded that orientation acuity was not wholly determined by the target's apparent orientation. In the present study, we used oblique (i.e. neither horizontal nor vertical) reference orientations to more directly examine the effect of perceived orientation on orientation acuity. The results show that when surround and reference were parallel, there was no tilt illusion and acuity was high. Acuity suffered whenever the tilt illusion caused a large discrepancy between the target's physical and perceived tilts. Since this was true even for tilted references, context-induced acuity loss cannot be simply an "oblique effect" of the target's physical orientation.

A 'dipper' function for texture discrimination based on orientation variance

We measured the just-noticeable difference (JND) in orientation variance between two textures (Figure 1) as we varied the baseline (pedestal) variance present in both textures. JND's first fell as pedestal variance increased and then rose, producing a 'dipper' function similar to those previously reported for contrast, blur, and orientation-contrast discriminations. A dipper function (both facilitation and masking) is predicted on purely statistical grounds by a noisy variance-discrimination mechanism. However, for two out of three observers, the dipper function was significantly better fit when the mechanism was made incapable of discriminating between small sample variances. We speculate that a threshold nonlinearity like this prevents the visual system from including its intrinsic noise in texture representations and suggest that similar thresholds prevent the visibility of other artifacts that sensory coding would otherwise introduce, such as blur.

A single "stopwatch" for duration estimation, a single "ruler" for size

Although observers can discriminate visual targets with long exposures from otherwise-identical targets with shorter exposures, temporally overlapping distracters with an intermediate exposure can produce a striking degradation in performance. This new finding suggests that observers can only estimate one duration at a time. Discrimination on the basis of size, rather than duration, did not degrade as rapidly with the number of distracters but was still worse than predicted by unlimited-capacity models. The critical difference between estimates of temporal length and estimates of spatial length seems to be that the former can only be made at the end of an exposure, while the latter can be made at any time during an exposure. When sizes varied throughout the trial and decisions were based on terminal sizes, the set-size effect was as large as that obtained for duration discrimination. We conclude that when textural filters are not available for segregating a target from distracters, efficient estimates of size or duration require the serial examination of individual display items.

Contrast discrimination: second responses reveal the relationship between the mean and variance of visual signals

To explain the relationship between first- and second-response accuracies in a detection experiment, Swets, Tanner, and Birdsall [Swets, J., Tanner, W. P., Jr., & Birdsall, T. G. (1961). Decision processes in perception. Psychological Review, 68, 301–340] proposed that the variance of visual signals increased with their means. However, both a low threshold and intrinsic uncertainty produce similar relationships. I measured the relationship between first- and second-response accuracies for suprathreshold contrast discrimination, which is thought to be unaffected by sensory thresholds and intrinsic uncertainty. The results are consistent with a slowly increasing variance.

Full-wave and half-wave processes in second-order motion and texture

A theory of human second-order motion perception is proposed and further applied to the discrimination of texture slant. The computational algorithms for deriving the direction of left-right motion from a sequence of images are equivalent to the algorithms for deriving the direction of slant (e.g. from top left to bottom right or top right to bottom left) in a single 2D image. There is a broad range of phenomena for which Fourier analysis of the image plus a few simple rules gives a good account of human perception. The problem with this first-order analysis is that there exists a broad class of 'microbalanced' stimuli in which the motion or slant is completely obvious to human subjects but is invisible to first-order analysis. Microbalanced stimuli require second-order analysis which consists of non-linear preprocessing (spatiotemporal filtering followed by rectification of the input signal) before standard motion or slant analysis. To determine whether the second-order rectification is half-wave or full-wave, we construct two special microbalanced stimulus types: 'half-wave stimuli' whose motion (or texture slant) is interpretable by a half-wave rectifying system but not by full-wave or a first-order (Fourier) analysis and 'full-wave stimuli' which are interpretable only after full-wave rectification. Such experiments show that second-order texture-slant perception utilizes both half-wave and full-wave processes, second-order motion-direction discrimination depends predominantly on full-wave rectification and second-order spatial interactions such as lateral contrast-contrast inhibition and second-order Mach bands are exclusively full-wave.

Intrinsic uncertainty explains second responses

In the simplest form of signal-detection theory (SDT), all stimuli give rise to equal-variance Gaussian probability density functions (PDFs) of sensation, with means proportional to stimulus intensity. As this simple SDT cannot accurately describe psychometric functions for two-alternative forced-choice (2AFC) detection experiments, it is commonly modified in one of two ways: with a non-linear transducer or intrinsic uncertainty. Most results can adequately be explained by either modification, but Swets et al.'s (1961) two-response 4AFC (2R4AFC) detection experiment is an exception. Simple SDT cannot predict the relationship between first- and second-response accuracies and non-linear transduction does not help. A previously unacknowledged facet of intrinsic uncertainty is that the same uncertainty required to fit 2AFC psychometric functions also produces an excellent fit to Swets et al.'s 2R4AFC results, without requiring any additional assumptions. This result is derived within the context of a primer on SDT.

The relationship between search efficiency and crowding

The role of target salience in crowding has remained controversial largely because salience usually escapes objective measurement. Here we address this problem using search efficiency as a measure of target salience. In separate experiments observers determined whether parafoveal arrays of vertical Gabor patterns contained targets having a unique colour, a unique direction of motion, and a unique temporal frequency. We analysed search efficiency in the conventional manner using reaction-time gradients (in seconds per item). We also considered accuracy gradients (in percent-correct per item). Crowding is typically quantified by comparing the acuity for a target within an array to the acuity for a target presented alone. We measured orientation acuity for determining whether a slightly tilted target was clockwise or anticlockwise of vertical. Targets with a unique colour or direction of motion were found to pop out, ie (with one exception) reaction-time and accuracy gradients were insignificantly different from zero. Acuity for these targets was significantly greater than acuity for targets whose neighbours had the same colour and direction of motion. Manipulation of temporal frequency produced a wide range of search efficiencies. For three of four observers we found a linear relationship between acuity and the accuracy gradient, shallow gradients being associated with high acuity. In general, we find that crowding is weakened when observers can find a parafoveally presented target quickly and accurately.

Stochastic re-calibration: contextual effects on perceived tilt

The human visual system exaggerates the difference between the tilts of adjacent lines or grating patches. In addition to this tilt illusion, we found that oblique flanks reduced acuity for small changes of tilt in the centre of the visual field. However, no flanks--regardless of their tilts--decreased sensitivity to contrast. Thus, the foveal tilt illusion should not be attributed to orientation-selective lateral inhibition. Nor is it similar to conventional crowding, which typically does not impair letter recognition in the fovea. Our observers behaved as though the reference orientation (horizontal) had a small tilt in the direction of the flanks. We suggest that the extent of this re-calibration varies randomly over trials, and we demonstrate that this stochastic re-calibration can explain flank-induced acuity loss in the fovea.

Predicting the motion after-effect from sensitivity loss

The widely accepted disinhibition theory of the motion after-effect (MAE) proposes that the balance point of an opponent mechanism is changed by directional adaptation. To see if the post-adaptation balance point could be predicted from contrast adaptation, we measured threshold-vs-contrast (i.e., T-vs-C or dipper) functions, before and after adaptation to moving gratings. For test stimuli moving in the same direction, adaptation shifted the point of maximum facilitation (i.e., the dip) upwards and rightwards. For tests moving in the opposite direction, adaptation produced a similar, but smaller, shift. These shifts are consistent with a change in divisive gain control. They are also consistent with subtractive inhibition followed by half-wave rectification. We attempted to use transducer functions derived from these data to predict the strength of the MAE. When combined, gratings moving in the adapted and opposite directions appeared perfectly balanced (i.e., counterphasing) when the latter was given approximately 2% more contrast than was predicted on the basis of the derived transducers. This small under-prediction may be indicative of sensory recalibration. Finally, we found that adaptation did not alter the fact that low-contrast stimuli could be detected and their direction identified with similar accuracy. We conclude that both static and dynamic forms of MAE are primarily caused by a decreased sensitivity in directionally tuned mechanisms, as proposed by the disinhibition theory.

Attentional capacity limit for visual search causes spatial neglect in normal observers

When observers simultaneously monitor several positions in the visual field, distracting stimuli have a devastating effect on the ability to discriminate between similar shapes. For example, the minimum tilt necessary for an observer to discriminate between a clockwise and anticlockwise tilt has been shown to increase with the square root of the number of untilted distractors. Here we show that these rapid visual searches remain inefficient even with extended practice. Moreover, each of our observers performed particularly poorly when uncued targets appeared in certain idiosyncratic positions, as though he or she neglected to process part of the visual field. This type of neglect is not commensurate with the popular 'max rule' strategy, in which observers simply report the direction of the largest apparent tilt. Nor is it consistent with tilt averaging. It is, however, consistent with an attentional effect in which both the signal and the noise from neglected positions are decreased, leaving the local signal/noise ratio constant. We show that our data can be well fit by models in which discriminations are based on a combination of these locally weighted, noisy signals.

Monocular texture segmentation and proto-rivalry

When the right eye's target is the left eye's distracter and vice versa, orientation-defined search is impossible unless, as we show here, the elements are close together. More than 1s was required to find inverse-cyclopean texture boundaries when elements were arranged on a 16 x 16 grid. Less than 250 ms was required for a 24 x 24 grid covering the same area. The conventional view is that binocular rivalry requires at least 200 ms to develop, but our results suggest a more rapid access to monocular signals. We call this rapid form of access "proto-rivalry."

Stimulus contrast and the Reichardt detector

The direction of a drifting grating can become more easily identified when a stationary, flickering grating, with the same spatial and temporal frequencies, is added to it. This amplification has been accepted as evidence that motion perception depends on the product of visual signals elicited before and after a target changes position, as computed by a Reichardt detector. However, amplification is also consistent with a model in which direction identification depends on the product of detection probabilities before and after the position shift. In this paper, we compare the Reichardt detector with a model of Probability Multiplication. For 2-frame sequences, similar results are predicted by Probability Multiplication and a Reichardt model, in which the performance-limiting noise is early (i.e. it is added prior to signal multiplication). Many new and previously published results are consistent with these predictions. Other results are documented in which the amplification is too large to be consistent with Probability Multiplication. To explain these latter results, Reichardt detectors must have both early and late noises.