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Article  |   October 2012
Local contextual interactions can result in global shape misperception
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Journal of Vision October 2012, Vol.12, 3. doi:10.1167/12.11.3
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      J. Edwin Dickinson, Clare Harman, Olivia Tan, Renita A. Almeida, David R. Badcock; Local contextual interactions can result in global shape misperception. Journal of Vision 2012;12(11):3. doi: 10.1167/12.11.3.

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Abstract
Abstract
Abstract:

Abstract  Adaptation in the visual system frequently results in properties of subsequently presented stimuli being repelled along identifiable axes. Adaptation to radial frequency (RF) patterns, patterns deformed from circular by a sinusoidal modulation of radius, results in a circle taking on the appearance of having modulation in opposite phase. Here we used paths of spatially localized gratings (Gabor patches) to examine the role of local orientation adaptation in this shape aftereffect. By applying the tilt aftereffect (TAE) as a function of the local orientation difference between adaptor and test, concomitant with adjustment of local position to accommodate the orientation change and preserve path continuity (Euler's method), we show that a TAE field can account for this misperception of shape. Spatial modulation is also observed spontaneously in a circular path of Gabor patches when the local patch orientations are rotated from tangential to the path. This illusory path modulation is consistent with the path orientation being attracted to the orientation of the patches. This consistent local rule implies a local explanation for the global effect and is consistent with a known illusion with a local cause, the Fraser illusion (FI). A similar analysis to that used for the TAE shows that the Fraser illusion can account for this particular alteration of perceived shape. A model which proposes that local orientations are encoded after considering the activation in a population of neurons with differing orientation tuning can accommodate both effects. It is proposed that these distinct processes rely on the same neural architecture.

Introduction
Visual information in a scene is sampled by a mosaic of light receptors in the retina. The function of the visual system of the brain is to provide an interpretation of this information in terms of objects or motion flow fields to which meaning can be ascribed. Processing within the system is parallel and hierarchical (Felleman & Van Essen, 1991), with the processing of form information predominantly constrained to a cortical pathway projecting from the primary visual cortex, V1, to the inferior-temporal (IT) cortex (Young, 1992). Neurons of V1 act as localized spatial filters (De Valois, Albrecht, & Thorell, 1982; De Valois, Yund, & Hepler, 1982; Hubel & Wiesel, 1959, 1968; Maffei & Fiorentini, 1973), assembling their oriented contrast sensitivity profiles from the temporally correlated output of linearly arranged but circularly symmetric receptive fields of retinal ganglion cells (Reid & Alonso, 1995). Many neurons of the earliest cortical stage of visual processing, therefore, give particularly strong responses to appropriately oriented lines or edges. Although proximity of the receptive fields of activated neurons is sufficient to bind adjacent responses into a path (Watt, Ledgeway, & Dakin, 2008), paths that activate adjacent neurons with approximately collinear receptive fields are much more salient and robust to noise (Field & Hayes, 2004; Field, Hayes, & Hess, 1993; Li & Gilbert, 2002). Such contours often represent the profiles, or spatial boundaries, of objects and, therefore, serve to circumscribe objects in the visual field. 
Evidence that the lateral occipital cortex (LOC), a cortical area higher in the hierarchy and implicated in object processing, produces a similar fMRI response to grey scale photographs and line drawings of the same scene (Kourtzi & Kanwisher, 2000) implies that the boundaries alone are sufficient to allow the parsing of the visual field into objects. This result provides neurophysiological support for the same argument made on theoretical grounds by Attneave (1954), who showed that the information content of images is concentrated at the boundaries of objects, but it still leaves open the question of how objects are recognized. Attneave, however, also proposed that the points of maximal information content are the points of maximum curvature in such boundaries and showed psychophysically that observers were inclined to refer to such points when analyzing shape. These arguments imply that the shape of an object is critical to its identification and that the points of maximum curvature on a shape feature strongly in its analysis. Consequently, models of object processing have been devised that encode shape using the polar positions of the points of maximum curvature relative to the object center (Pasupathy & Connor, 2001, 2002; Poirier & Wilson, 2006), although other features such as sides have also been shown to provide significant cues to shape (Poirier & Wilson, 2007). While there is some debate about the mechanisms of shape processing it is clear that global processing of shape occurs via the assimilation of local position and orientation into increasingly more complex cues. 
Much of the analysis performed by the visual system is, however, based on contrast in stimulus properties rather than absolute measures, reflecting the necessity to discriminate between similar stimuli. Accuracy in the estimation of absolute measures is often sacrificed for an enhancement in the capacity to notice difference (Badcock & Westheimer, 1985b) and an enhancement in ability to discriminate between oriented lines has been reported after adaptation to similarly (Clifford, Ma-Wyatt, Arnold, Smith, & Wenderoth, 2001; Regan & Beverley, 1985) and/or orthogonally (Clifford et al., 2001; Dragoi, Sharma, Miller, & Sur, 2002) oriented lines. The results are, however, somewhat equivocal (see Kohn, 2007, for a review). In the context of these studies, adaptation might render stimuli more discriminable from each other by an increase in the precision of the mechanisms responsible for the discrimination. Adaptation, though, can also result in an exaggeration in the perceived difference between an adaptor and a subsequently experienced stimulus without the necessity for an improvement in resolution. Exposure to one stimulus results in the perceptual experience of a subsequently presented stimulus of a similar type being repelled along the perceptual dimension, or dimensions, that characterize that stimulus type. For example, after exposure to a line of a particular orientation, a subsequently presented line with a similar but not identical orientation appears rotated away from the orientation of the adapting line, a phenomenon known as the tilt aftereffect (TAE) (Gibson & Radner, 1937; Kohler & Wallach, 1944; Mitchell & Muir, 1976). The orientations of the successively presented lines are perceived as being more different to the orientation of the adaptor than they actually are. Even conceding this exaggeration of difference, however, line orientations can be discriminated to much greater precision than the orientation tuning bandwidth of individual neurons of V1 (Regan & Beverley, 1985; Westheimer, Shimamura, & McKee, 1976). This suggests that the orientation of a line is encoded in the activation of a population of cells with a range of preferred orientations (Coltheart, 1971; Gilbert & Wiesel, 1990). Each cell represents a filter tuned to a particular orientation and carries a label for that orientation. Gilbert and Wiesel (1990) suggested that the response of each filter could be represented by a vector with a magnitude proportional to activity in the cell and an orientation parallel to the line and that the perceived orientation of a line is given by the vector sum of the responses of the population. Exposure to an adapting line diminishes the sensitivity of the filters sensitive to the orientation of that line, resulting in a rotation of the vector sum due to a test line with similar but not identical orientation to the adapting line away from the orientation of the adapting line. The difference in orientation between the adapting and test lines is thus exaggerated. 
The logic of the TAE example is often extrapolated to more complex shapes, with the adaptation of shape processing mechanisms assumed to be responsible for the perception of a shape being moved along axes which characterize that shape. However, in this paper we explore an alternative explanation for the apparently higher-level shape adaptation effects. We propose that displacement of perception of shape along dimensions that characterize an object might result from local TAEs. Obviously, accommodation of the TAEs in the stimuli requires that the positions of specific features are often misperceived but there is substantial evidence that for the TAE and other contextual interactions this is the case (Badcock & Westheimer, 1985a, 1985b; Ganz, 1964; Kohler & Wallach, 1944). Models already exist which predict the redistribution of features of complex stimuli to accommodate changes in local perceived orientation due to adaptation (Meese & Georgeson, 1996). 
It has recently been shown that aftereffects of adaptation to complex shapes and faces can be accounted for by the systematic application of local TAE across the stimuli in a TAE field (Dickinson, Almeida, Bell, & Badcock, 2010). However, that study relied upon data derived using adaption times measured in seconds and the time-course of adaptation to extended two dimensional shapes (Dickinson, Han, Bell, & Badcock, 2010; Leopold, Rhodes, Muller, & Jeffery, 2005; Suzuki & Cavanagh, 1998) has been shown to be rapid, developing in fractions of a second. This is consistent with the results of Sekuler and Littlejohn (1974) pertaining to the TAE, who showed that measurable aftereffects were produced after as little as 10 ms of adaptation. The TAE is retinotopically constrained (Knapen, Rolfs, Wexler, & Cavanagh, 2010) and, therefore, the aftereffects perceived after adaptation to complex shapes that might be attributed to a TAE field should also be approximately retinotopic. There is increasing evidence that a large proportion of such aftereffects are retinotopic (A. Afraz & Cavanagh, 2009; S. R. Afraz & Cavanagh, 2008). Indeed, face identity aftereffects of opposite directions can be induced simultaneously in different regions of the visual field (S. R. Afraz & Cavanagh, 2008). Any positionally invariant face or shape aftereffects might, then, be attributed to persistent retinotopic aftereffects aggregated across multiple eye movements. Dickinson, Mighall, Almeida, Bell, and Badcock (2012) argued for this position, demonstrating a rapid inception of shape and face aftereffects and showing them to be retinotopic in nature. That paper did not, however, explicitly model the effects of the TAE field (Dickinson, Almeida, et al., 2010) for such brief adaptation periods. While we will argue for the TAE field here it must be noted that different mechanisms of adaptation might apply to produce the TAE at these very short timescales (Kohn, 2007). 
A TAE field has the potential to render similar, successively experienced, complex stimuli more discriminable without modifying the selectivity of the analyzers of these complex stimuli (this point, of course, concedes the existence of higher level analyzers of shape and makes no predictions about what adaptation in such mechanisms might look like). The TAE can account for adaptation to curves (Blakemore & Over, 1974), shapes, and faces (Dickinson, Almeida, et al., 2010) for adaptation times of a few seconds. The current study examines whether the TAE field explanation for aftereffects of adaptation to complex shapes can be extrapolated to the brief adaptation durations used in Dickinson et al. (2012) and also to adaptation to orientations cues that do not form a path. The concept of the TAE field also raises the question of whether other local contextual interactions might also result in effects perceived as global when applied systematically over extended stimuli. This possibility is explored using the Fraser, or twisted cord, illusion (Fraser, 1908) applied to closed paths modulated in radius and/or orientation contained within the path. 
In a recent study, Day and Loffler (2009) manipulated position and orientation information independently by sampling a path and modifying the local orientation information with respect to the tangent to the path. The path was defined using Gaussian windowed cosine gratings, known as Gabor patches. The shapes used were radial frequency (RF) patterns, shapes distorted from circular by a sinusoidal modulation of radius around the pattern (Loffler, Wilson, & Wilkinson, 2003; Wilkinson, Wilson, & Habak, 1998). It has been shown that as cycles of specific radial frequency modulation are added to a pattern, the threshold for detection of deformation decreases at a rate that cannot be accounted for by the increasing probability of detection of a single cycle of modulation (probability summation). This demonstrates that the RF patterns are processed globally in the sense that the detection of shape relies on information accumulated across the stimulus (Bell & Badcock, 2008; Dickinson, Han, et al., 2010; Loffler et al., 2003; Wang & Hess, 2005). Two stimulus types were used in Day and Loffler (2009). In the first the path of patches was modulated in radius and the axes of the patches aligned tangential to the path. In the second the path was circular but the axes of the patches were aligned as though tangential to a path modulated in radius. A circular path with modulated orientations appears distorted towards the shape that the orientations were drawn from. Day and Loffler (2009) refer to this situation as having conflicting orientation and positional cues. When compared with patterns where the positional and orientation cues were consistent, the perceived distortion of a pattern with conflicting cues was slightly more than 60% of that of a pattern with consistent cues. They concluded that orientation incompletely captures the position of the elements. As the modulation information has been shown to integrate around a RF pattern (Bell & Badcock, 2008; Loffler et al., 2003), their suggestion that a weighted combination of orientation and position information determines the percept implies that both position and orientation information on the path are encoded globally (see also Wang & Hess, 2005). The processing of modulation information in RF patterns has previously been shown to be restricted to low frequencies of RF modulation (Loffler et al., 2003) but invariant to size of the pattern (Wilkinson et al., 1998). These properties point to an object-based processing where the features of the object are encoded relative to a pattern center (Bell, Dickinson, & Badcock, 2008; Poirier & Wilson, 2006). However, Day and Loffler (2009) also show that a proportion of the effect persists for straight-line stimuli that cannot circumscribe an object with a center. This suggests that, rather than a global measure of shape based on orientation information influencing the perceived shape, and hence local positions within the pattern, local discrepancies between the orientation of the axes of the patches and the tangent to the path might cause local displacements in perceived positions which are then integrated into a global shape. The Gabor patches provide local samples of orientation which are not necessarily tangential to the path, and the path itself provides a second order local orientation signal. We propose that the Fraser illusion causes the orientation of the path to be misperceived and that the locus of the path is modified to accommodate this misperception (Meese & Georgeson, 1996). In the RF pattern with conflicting orientation and position cues, the orientation difference varies systematically around the path and so locally the path resembles the twisted cord or Fraser illusion (Fraser, 1908) but the direction of twist in the cord reverses periodically around the path (see Panel C of Figure 1). Again this interpretation does not deny the existence of global mechanisms for shape processing but suggests that the global mechanisms might use the misperceived positions and orientations in their analysis of shape. 
Figure 1
 
Adaptor stimuli used in Experiment 1. (A) A radius and orientation adaptor stimulus with radius and orientation modulation. (B) A radius only adaptor with radius modulation only. (C) A circular orientation only adaptor with orientation modulation only. (D) An adaptor with radius and orientation modulation but with the radii of each of the patches permuted (permuted). The amplitude of modulation, A of Equations 1 and 2, for the adaptors was 0.1 when applied and zero otherwise. Test stimuli were RF patterns that varied in amplitude around zero and had three cycles of modulation in 2π radians.
Figure 1
 
Adaptor stimuli used in Experiment 1. (A) A radius and orientation adaptor stimulus with radius and orientation modulation. (B) A radius only adaptor with radius modulation only. (C) A circular orientation only adaptor with orientation modulation only. (D) An adaptor with radius and orientation modulation but with the radii of each of the patches permuted (permuted). The amplitude of modulation, A of Equations 1 and 2, for the adaptors was 0.1 when applied and zero otherwise. Test stimuli were RF patterns that varied in amplitude around zero and had three cycles of modulation in 2π radians.
In Experiment 1 of this series of experiments we used sampled closed paths, modulated in local position and/or orientation as adaptors, to show that the aftereffects of adaptation are always in the direction of increasing local orientation difference. In this experiment this was achieved by nulling the aftereffect using conventional sampled RF pattern test stimuli (modulated in radius with the orientation of patches tangential to the path). These patterns required the same phase but lower amplitude of modulation than the adaptor to achieve perceptual circularity. This might be interpreted as an opposite phase global aftereffect but it is also consistent with the exaggeration of local orientation difference. Bell and Kingdom (2009) have shown that a test pattern with the same phase but larger amplitude than the adaptor is perceived as of greater amplitude still, but in the same phase. They describe this as a bidirectional RF aftereffect, which implies that it is a global effect, but it is also consistent with the application of the TAE across the visual field in a TAE field. 
Experiment 2 derived a function that describes the TAE versus local orientation difference between adaptor and test patterns. This function was then used to demonstrate that the aftereffects of adaptation to sampled RF patterns modulated in radius, orientation, or both radius and orientation are predictable from a TAE field. 
Experiment 3 measured the threshold for discrimination of unmodulated patterns from those perceived to be deformed due to modulation of radius and/or orientation and showed that the effects of modulation of radius and orientation sum linearly. 
Experiment 4 derived the function that describes the perceived orientation misperception of a twisted cord comprised of three linearly arranged patches, as the orientation difference between the axes of the patches and the axis of the cord was varied. This function was then used to demonstrate that the discrimination thresholds for RF patterns modulated in orientation can be accounted for by the perceived positional displacement introduced locally by the Fraser illusion. 
General methods
Apparatus
Stimuli were created using Matlab (Version 6.5, Mathworks, Natick, MA, USA) and presented on a Sony G420 monitor from the frame buffer of a Cambridge Research Systems 2/5 video card installed on a PC. The monitor had a resolution of 1024 × 768 pixels (17.1° × 12.8°) and a refresh rate of 100 Hz. Luminance calibration was performed using an Optical OP200-E photometer (head model number 265) and associated software (Cambridge Research Systems). Observers were seated in a darkened room (<1 cd/m2 ambient luminance) at a distance of 115 cm from the monitor, stabilized by a chinrest. At this observing distance the pixel dimensions, vertically and horizontally, were 1 min (1′) of visual angle. The observers signaled their responses using a computer mouse. 
Observers
Four experienced psychophysical observers participated. CH and ED are authors; HM and VB were naïve to the purpose of the experiments. All had normal or corrected to normal visual acuity. All observers participated in Experiments 1 and 3. Observers ED and VB participated in Experiments 2 and 4. The study was approved by the University of Western Australia ethics committee and was, therefore, conducted in accordance with the Declaration of Helsinki. 
Experiment 1: Brief adaptation to a path of Gabor elements modulated in radius and/or orientation results in shape aftereffects
Introduction
After adaptation to a RF pattern, a subsequently presented test circle has the appearance of being modulated in opposite phase to the adaptor. A subsequently presented test RF pattern with the same phase and higher amplitude appears to have higher amplitude still. This pattern of results has been described as a bidirectional global shape aftereffect (Bell & Kingdom, 2009). An alternative explanation, exaggeration of local orientation differences with concomitant distortion of the path, was proposed by Dickinson, Almeida, et al. (2010), who showed that the direction and magnitude of shape aftereffects perceived after adapting to RF patterns were consistent with the systematic local application of the TAE across the test stimulus in a TAE field. The TAE at a point is defined by the orientation difference between the oriented feature in the test pattern and the oriented feature in the adapting pattern at that point. A TAE field represents the TAE expected to be experienced by any oriented feature at any point within a test image across the visual field. Some spatial and temporal averaging must be expected but simple, smoothly varying, and similar but not identical adaptor and test pairs produce simple and smoothly varying TAE fields (Dickinson, Almeida, et al., 2010). Experiment 1 was devised to show that a shape aftereffect can be induced by brief adaptation to patterns modulated in radius and/or orientation, as would be predicted by a TAE field explanation. This explanation for the aftereffect is intrinsically local but its effects could give the impression of a bidirectional aftereffect along an axis encoding a RF pattern in a particular phase. Indeed, Day and Loffler (2009) showed that when orientation and position cues in sampled RF patterns were in conflict they interfered and proposed that analysis of shape required the global integration of local position and orientation cues. Experiment 1, therefore, measured and compared the magnitudes of aftereffects after adaptation to paths of Gabor patches modulated in radius (position), orientation, or both. An additional condition measured the magnitude of adaptation to a disrupted path stimulus created by permuting the radii of the set of patches of a path modulated in radius and orientation. The patches of this stimulus were, therefore, constrained to an annulus with a width of twice the amplitude of modulation and had the same distribution of orientation as a function of theta but were not constrained to a path. The hypothesis that the aftereffects are of local origin predicts substantial aftereffects for all conditions. However, if the aftereffects were due to global shape interactions the disrupted path would be expected to induce little or no aftereffect because the disparate pieces of information would not be assembled into a path (Field et al., 1993; Hayes, 2000; Loffler et al., 2003). This, of course, is not a test for the existence or otherwise of a global shape processing mechanism but for the locus of the adaptation responsible for the aftereffect. 
Methods
Stimuli
All stimuli were composed of Gabor patches, localized one dimensional gratings. The grating of each patch was in cosine phase with respect to the center of the Gaussian contrast envelope. The gratings had a spatial frequency of 8 c/° and the contrast envelope had a full width at half height of 9.4 min of visual angle. Maximum luminance within a patch was 90 cd/m2, a Weber contrast of one to the background of 45 cd/m2. The patches were used as elements of a path. Thirty-six patches formed a closed path with a mean radius of 2° of visual angle. The path was manipulated by independently modulating the radius of the path and/or orientation cues within the path. The null, or unmodulated, pattern was circular with all of the patches aligned perpendicular to the local radius. The radius, r, of a pattern modulated in radius was given by where θ is the polar angle, r0 the mean radius, A the amplitude of modulation, and φ the phase of the sinusoid (φ = 0 for all patterns used in Experiment 1). The pattern so formed is known as a RF3 pattern, which has three cycles of modulation in 2π radians. For a pattern modulated in orientation the angle made by the axis of each patch with the perpendicular to the local radius, α, was given by where r(θ) is defined by Equation 1. When conforming to this function the patches were oriented as though they were tangent to a closed path with a modulation of radius defined by Equation 1. The patches of a pattern with the same amplitude of modulation, A, in the equations that describe radius and orientation as a function of θ would, therefore, be arranged tangential to the path. Henceforth a RF3 pattern modulated only in radius will be referred to as radius only and a pattern modulated only in α orientation only. 
Individual test patterns had the same amplitude of modulation of radius and orientation and so the patches were always tangential to the path. A test pattern with a positive (negative) modulation amplitude would have the appearance of a rounded triangle resting on an apex (base). Four types of adaptor patterns were used: 
  •  
    A pattern with radius and orientation modulation (radius & orientation).
  •  
    A pattern modulated only in radius (radius only).
  •  
    A pattern modulated only in orientation (orientation only).
  •  
    A pattern with radius and orientation modulation applied but with the radii of the patches randomly permuted to create a disrupted contour with the same range of radii as in the other radius and orientation and radius only adaptors employed (permuted).
The four adaptor conditions are displayed in Figure 1. A no-adaptor condition was also tested where the adaptor was replaced with a blank interval. 
Procedure
The task of the observers was to report whether a test pattern was perceived as having a positive or negative amplitude of modulation, that is, whether the pattern appears to be resting on an apex or a side respectively. A single trial comprised an adaptor presented for 160 ms, a blank interstimulus interval (ISI) of 640 ms and then a test stimulus presented for 160 ms. Each condition was tested in three blocks of 180 trials. Nine amplitudes of modulation were used to sample each psychometric function with the 540 trials divided equally across the amplitudes. The probability of responding that the pattern was perceived as having a positive amplitude was calculated for each amplitude and a cumulative normal distribution fitted to the data describing probability as a function of amplitude increasing in the positive direction. The mean of the distribution yielded a measure of the amplitude at the point of subjective equality (PSE), the amplitude at which the observer was equally likely to respond that the amplitude was positive or negative. The amplitude at the PSE is also the amplitude required to null the illusory deformation due to the aftereffect and is, therefore, a measure of the magnitude of the aftereffect. 
Results
The amplitudes at the PSE for each of the conditions tested are displayed in Figure 2
Figure 2
 
Aftereffects of adaptation to the four conditions of Experiment 1. The four conditions are grouped, with the labels for the conditions corresponding to the labels for the adaptors displayed in Figure 1; (A) radius and orientation modulation; (B) radius modulation only; (C) orientation modulation only; (D) radius and orientation modulation with permuted radii (permuted). Error bars are 95% confidence intervals. A substantial aftereffect is experienced after adaptation to all four adaptor types.
Figure 2
 
Aftereffects of adaptation to the four conditions of Experiment 1. The four conditions are grouped, with the labels for the conditions corresponding to the labels for the adaptors displayed in Figure 1; (A) radius and orientation modulation; (B) radius modulation only; (C) orientation modulation only; (D) radius and orientation modulation with permuted radii (permuted). Error bars are 95% confidence intervals. A substantial aftereffect is experienced after adaptation to all four adaptor types.
In the absence of an adaptor a small but consistent bias is seen across all observers to perceive a pattern with a positive amplitude of modulation as circular. This might be a persistent effect of adaptation as within testing sessions, the no-adaptor blocks of conditions were interleaved with the blocks of conditions with adaptors, all of which had positive amplitudes of modulation. The aftereffects of all conditions with adaptors were significantly larger than this effect. 
A repeated measure analysis of variance was performed to compare the effects of the adaptors. The means were shown to be different, F(4) = 46.37, p < 0.0001. A Newman-Keuls multiple comparison test showed that all the adaptor conditions were different from the unadapted condition (p < 0.001). Condition D (permuted), a disrupted path of patches modulated in orientation, is at least as effective as an adaptor as that of Condition C (orientation only), a circular path of patches modulated in orientation. The aftereffect is larger for Condition D (permuted) for each observer and the Newman-Keuls comparison of the pair of conditions showed that the means were significantly different (q = 4.000, p < 0.05). This result demonstrates that the adaptation occurs in the absence of a path of collinear elements. Comparing Conditions B (radius only) and D (permuted) we find that the effectiveness of the adaptors does not differ significantly (p > 0.05). The aftereffect of adaptation to Condition A (radius & orientation), the condition with modulation of radius and orientation, is the largest (for Conditions B–D the size of the aftereffects are 78%, 62%, and 71%, respectively, of the aftereffect for Condition A). Movie 1 is an example trial from Condition D (permuted). The test pattern is unmodulated yet after adaptation it appears to have a negative modulation amplitude. 
Discussion
Given that the aftereffects experienced for Condition D (permuted) are comparable in size to those for Conditions B (radius only) and C (orientation only) it is not parsimonious to surmise that the aftereffects are due to adaptation of mechanisms analyzing global shape because the path, and hence global shape, has been disrupted by permuting the patch radii (Loffler et al., 2003). We propose that the aftereffects are due to the systematic application of local tilt aftereffects (TAEs) across the stimuli in all cases. The concept of the TAE field was developed in Dickinson, Almeida, et al. (2010) to formalize this concept. In Experiment 1 of this study the adaptors were presented for only 160 ms rather than 2 s, and the amplitudes of the aftereffects at the PSE were smaller than those of the previous study. However, the amplitude at the PSE for Condition A (radius & orientation), for which the patches of the adapting stimuli are aligned tangential to the path, is comparable to those reported in an earlier experiment using RF patterns defined by a path of Gabor patches (Dickinson, Han, et al., 2010). In this earlier study aftereffects were seen for adaptation times of as little as 40 ms and it was also shown that the adaptation was persistent, perhaps accounting for the misperception of shape in the no-adaptor condition. 
 
Movie 1.
 
An example trial from Condition D (permuted) of Experiment 1. An adaptor incorporating modulation of radius and orientation but with the radii of the patches permuted is presented for 160 ms. After an interstimulus interval of 640 ms the test pattern is presented for 160 ms. The test pattern has no modulation but appears to be modulated in opposite phase to the adaptor. This is despite the fact that the radial modulation cue has been removed and the path is disrupted leaving only local orientation modulation.
Experiment 2 measures the TAE as a function of the orientation difference between the adaptor and test for an adaptation time of 160 ms. This function is then used to predict the aftereffect expected due to a TAE field. 
Experiment 2: A TAE field can account for the aftereffects of adaptation to shapes defined by sinusoidal modulation of radius and/or orientation
Introduction
The concept of a TAE field has been proposed as an explanation for the aftereffects of adaptation to complex patterns (Dickinson, Almeida, et al., 2010). A TAE field is essentially a spatially continuous representation of the TAE experienced locally on a particular pattern due to adaptation to previously experienced patterns. The perceived shape of any pattern under observation is systematically modified by the influence of the extant TAE field. At any point the local TAE is derived from the orientation difference between the adaptor (or successively experienced adaptors, given the persistence of the effect) and the test stimulus at or adjacent to that point. The function describing the variation of the TAE with orientation difference has a characteristic shape. Dickinson, Almeida, et al. (2010) used the first derivative of a Gaussian to represent the function, fitting data from Clifford, Wenderoth, and Spehar (2000). However, the duration of adaptation used in Experiment 1 is short in comparison to conventional TAE experiments; therefore in this experiment we have sampled the function for an adaptor duration of 160 ms. That aftereffects were observed in Experiment 1 despite the fact that the patches of the adaptor and test patterns were not necessarily strictly coincident implies some measure of spatial flexibility in the TAE. Some degree of spatial averaging of orientation might, therefore, be expected. In this experiment, therefore, we used groups of three patches in the adaptor and test presented such that individual patches were not necessarily spatially coincident across adaptor and test, and also to facilitate comparison with Experiment 4, which used three linearly arranged patches to examine the equivalent function describing the Fraser illusion. The range of the TAE for the stimuli used in this experiment was measured in a corollary investigation. 
Methods
Stimuli
Adapting and test stimuli were groups of three Gabor patches (see Movie 2 and Figure 3). The patches were placed at random at the intersections of three radial lines and three concentric circles (the radii and circles were not represented explicitly in the stimuli). The implicit concentric circles had radii of 20, 16.3, and 11.5 min of visual angle. The radial lines were spaced equally at 2π/3 radians (120°) but phase was randomized across stimuli. The circle radii were selected so that the patches did not noticeably overlap and so that the average distance between the centers of the patches, at 27.8′, was not dissimilar to the average distance between any three patches of the adaptor conditions of Experiment 1. The axes of the gratings of the patches were parallel in all stimuli of this experiment but typically not collinear. The centers of the adaptor and test stimuli were coincident but no constraint was made on the relative spatial arrangement across the adaptor and test. 
 
Movie 2.
 
An example trial from Experiment 2: The adapting and test stimuli are presented for 160 ms each with a 640 ms ISI interposed. The gratings of the patches in the test stimulus are vertical but appear repelled in orientation from those of the adaptor.
Figure 3
 
An illustration of the timecourse of a trial in Experiment 2. An adaptor is presented for 160 ms followed by a test interval with the same duration after an interstimulus interval of 640 ms comprising the grey background.
Figure 3
 
An illustration of the timecourse of a trial in Experiment 2. An adaptor is presented for 160 ms followed by a test interval with the same duration after an interstimulus interval of 640 ms comprising the grey background.
Procedure
Each trial comprised an adapting stimulus presented for 160 ms, a 640 ms ISI, and a test stimulus presented for 160 ms. The task of the observer was to report whether the gratings of the patches of the test stimuli appeared to be oriented clockwise or anticlockwise of vertical. Figure 3 is an illustration of a trial and Movie 2 is an example trial. 
The orientation of the patches of the adapting stimuli varied across the whole range of possible orientations with respect to the vertical, thereby sampling the whole function describing the variation of the TAE with respect to the orientation difference between adaptor and test. The conditions were blocked with three blocks of 180 trials per condition (540 trials for each condition). A range of nine orientations of the test stimulus was used to sample the psychometric function and the probability of responding that the test patches were oriented clockwise of vertical calculated for each of the orientations. A cumulative normal distribution was fitted to the data, with the mean of the distribution yielding the PSE, which in this case was the test orientation that would be perceived as vertical. The function describing the variation of the TAE with respect to the orientation difference between adaptor and test was sampled, first with the adapting and test stimuli at fixation and then again at an eccentricity of 2° to the right of fixation (because the patterns used in Experiment 1 had a mean radius of 2° and therefore central fixation would place the contour at an eccentricity of 2°). In the second set of conditions a fixation marker was present 2° to the left of the position at which the adaptor and test appeared. 
Results
The data describing the variation of the TAE with orientation difference between test and adaptor is shown in Figure 4. The convention adopted here is the same as that used in Dickinson, Almeida, et al. (2010). If the mathematical convention of θ increasing in an anticlockwise direction from the positive x-axis is followed and the adaptor is clockwise of the test then orientation change between adaptor and test is in the positive direction. Adaptation causes the perceived orientation of the test to be repelled in an anticlockwise or positive direction. Conversely if the adaptor is anticlockwise of the test then the orientation change between adaptor and test is negative and the perceived orientation after adaptation will be repelled in a clockwise or negative direction. The illusory orientation change always serves to exaggerate the orientation difference between adaptor and test. Two sets of data are displayed. The blue open circle data points represent the TAE at fixation and the red squares the TAE at an eccentricity of 2°. First derivative of Gaussian (D1) functions are fitted to the datasets. The maximum TAE for the function derived at fixation (blue curve) can be different (it is for ED but not for VB) to that at an eccentricity of 2° (dashed red curve) but the standard deviations of the two functions are very similar within each observer (19.74 ± 2.02 (95% CI) and 20.32 ± 2.57, respectively, for ED and 25.25 ± 3.22 and 26.77 ± 5.29, respectively, for VB). For modeling purposes the red curve was used as adaptation to the path in Experiment 1 occurred at an eccentricity of 2°. The form of the functions approximate those conventionally found for the TAE (Clifford et al., 2000). 
Figure 4
 
The tilt aftereffect (TAE) as a function of the difference in orientation (Δ orientation) between the adapting and test patches. A positive (negative) value for Δ orientation indicates that the orientation of the test patches is anticlockwise (clockwise) of the adapting patches. A positive (negative) TAE indicates repulsion of orientation of the test patches in an anticlockwise (clockwise) direction from the orientation of the adapting patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1). The colored bar is used to associate the size of the TAE to a color for use in displaying the local TAE in the modeling of the TAE across the stimulus as a scalar field.
Figure 4
 
The tilt aftereffect (TAE) as a function of the difference in orientation (Δ orientation) between the adapting and test patches. A positive (negative) value for Δ orientation indicates that the orientation of the test patches is anticlockwise (clockwise) of the adapting patches. A positive (negative) TAE indicates repulsion of orientation of the test patches in an anticlockwise (clockwise) direction from the orientation of the adapting patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1). The colored bar is used to associate the size of the TAE to a color for use in displaying the local TAE in the modeling of the TAE across the stimulus as a scalar field.
Modeling
Experiment 1 showed that an adaptor that had the local orientations appropriate for a path with a sinusoidal modulation of radius and orientation, but with the radii of individual patches permuted (permuted) was as and more effective, respectively, as paths modulated in radius (radius only) or orientation (orientation only). The orientation defined by the axes of the patches was modulated in Conditions C (orientation only) and D (permuted) while the orientation of the path was modulated in the adaptor of Condition B (radius only) by virtue of the modulation of the radius (while the patch orientations remained concentric). In the adaptor of Condition A (radius & orientation) the orientations of both the patches and the path were modulated, perhaps explaining its increased efficacy as an adaptor. 
Dickinson, Almeida, et al. (2010) introduced the concept of a TAE field to demonstrate how seemingly global aftereffects can often be accounted for by the systematic application of local aftereffects across the visual field. The effects of adaptation to RF patterns on the perceived shape of subsequently presented circles are easily modeled as a function of the polar angle θ used in the definition of the RF patterns (see Equation 1). The local orientation difference between the adaptor and test patterns (nominally a circle) is derived as a function of θ (using Equation 2), and then the TAE this implies is calculated for each sample of θ in the field. In the modeling performed here the TAE at each point is calculated using the function described by the dashed red lines in Figure 4, which describes the relationship between the TAE and the orientation difference between adaptor and test locally. The details of this procedure are reported in Dickinson, Almeida, et al. (2010). The TAE is, however, retinotopically constrained (Knapen et al., 2010) and therefore the magnitude of the effect would be expected to fall off with increasing distance introduced locally between the adapting and test stimuli due to the modulation of radius. In order to determine the rate of decline with distance for the stimuli used here, the same procedure used to determine the TAE as a function of the orientation difference between adaptor and test at an eccentricity of 2° was adapted to measure the effect as the adaptor and test were progressively moved apart on the vertical axis. An orientation difference of 20° was used for all distances. The results are displayed in Figure 5
Figure 5
 
Decline of the magnitude of the TAE with distance in the visual field at an eccentricity of approximately 2° for the stimuli used in Experiment 2. The value of the TAE when the adaptor and test were spatially coincident was derived from the fit to the data in Figure 4 pertaining to an eccentricity of 2° of visual angle (dashed red line). The orientation of the adaptor was 20° clockwise of the test orientation. Increasing distance between the centers of the adaptor and test clusters of patches is represented along the x-axis and the TAE represented on the y-axis. A Gaussian function is fitted to the data. The function is constrained to have an amplitude equal to the TAE at 0° distance (at an eccentricity of 2°) and a mean of zero degrees separation. The function, therefore, only has one free parameter, the standard deviation. Error bars are 95% confidence intervals.
Figure 5
 
Decline of the magnitude of the TAE with distance in the visual field at an eccentricity of approximately 2° for the stimuli used in Experiment 2. The value of the TAE when the adaptor and test were spatially coincident was derived from the fit to the data in Figure 4 pertaining to an eccentricity of 2° of visual angle (dashed red line). The orientation of the adaptor was 20° clockwise of the test orientation. Increasing distance between the centers of the adaptor and test clusters of patches is represented along the x-axis and the TAE represented on the y-axis. A Gaussian function is fitted to the data. The function is constrained to have an amplitude equal to the TAE at 0° distance (at an eccentricity of 2°) and a mean of zero degrees separation. The function, therefore, only has one free parameter, the standard deviation. Error bars are 95% confidence intervals.
The fall-off in magnitude of the TAE is adequately represented by a Gaussian function constrained to a mean of zero and an amplitude equal to the TAE when the adaptor and test are spatially coincident at an eccentricity of 2° (derived from the D1 function fitted to the TAE vs. orientation difference data). The standard deviation of the Gaussian is 0.60° ± 0.13° (95% confidence interval) of visual angle for ED and 0.52° ± 0.48° for VB. This Gaussian function is used to constrain the TAE field to a range of radii centered locally on the path describing the adapting RF pattern. The TAE field for each observer is represented in Figure 6. Yellow represents a TAE of 5° in the positive (anticlockwise) sense and red a TAE of 5° in the negative (clockwise) sense (see color bar in Figure 4). Orange represents the absence of a TAE. 
Figure 6
 
The perceived shape (black line) of a circle after adaptation to a RF pattern with an amplitude of sinusoidal modulation of 0.1 in zero phase, as predicted by a TAE field. Panel A shows the prediction for observer ED and B the prediction for observer VB.
Figure 6
 
The perceived shape (black line) of a circle after adaptation to a RF pattern with an amplitude of sinusoidal modulation of 0.1 in zero phase, as predicted by a TAE field. Panel A shows the prediction for observer ED and B the prediction for observer VB.
In order to derive a representation of the perceived shape of a circle following adaptation, a line is projected upwards from a point on the positive x-axis 2° of visual angle from the center of the pattern, which is where the adaptor and test patterns would intersect the x-axis. The line advances on a notionally circular trajectory but respects the local TAE as it proceeds (Dickinson, Almeida, et al., 2010). The integrity of the path is preserved and so accommodation of the TAE necessitates a misperception of position. The path so formed returns to its starting position after traversing 2π radians (360°). The amplitude of modulation of the path created for observer ED is 0.032 and for observer VB it is 0.016. If we make the assumption that positions within the path are modified to accommodate the change in local orientation yet preserve colinearity then we can compare these predictions with the measurements made in Experiment 1 of aftereffects of adaptation to a modulated path where the axes of the patches are tangent to the path (adaptor Condition A—radius & orientation). These aftereffects are 0.023 (±0.001 95% confidence intervals) and 0.017 (±0.002) for observers ED and VB, respectively (the offset seen in the no adaptor condition has been removed and the errors added in quadrature). The predicted aftereffect is large enough to account for the whole of the measured aftereffect (the average measured aftereffect is 83% of the average predicted aftereffect). 
Discussion
The modeling shows that a TAE field could account for the aftereffects of adaptation to sampled RF patterns. Further evidence that the aftereffect is not due to adaptation within a global shape processing mechanism is provided by the demonstration of adaptation to orientation in the absence of an explicit path. The measured aftereffect for this condition is as large as, and larger than, those for adaptation to a modulated path with concentrically oriented patches (radius only) and a circle with modulated patch orientation (orientation only) respectively. 
Experiment 3: Sensitivity to radius and orientation modulation adds linearly
Introduction
The results of Experiments 1 and 2 demonstrate that the TAE, when applied continuously across a stimulus in a TAE field, can account for adaptation to radius and orientation modulation in RF patterns. Misperception of shape is, however, also produced by modulation of orientation in the adapting patterns used in Experiment 1. Inspection of Panel C of Figure 1 shows that modulation of the orientation of Gabor patches arranged in a circular path results in a percept of modulation of the radius of the path (Day & Loffler, 2009; Loffler, 2008). Conversely the perceived modulation of the radius of a path of Gabor patches whose orientations are perpendicular to the local radius (Panel B of Figure 1) is of somewhat lower amplitude than the perceived modulation of a path with patches oriented tangential to the path (Panel A of Figure 1). The effect of the modulation of the patches, therefore, has the effect of introducing a modulation of the path in the same phase. The effects of the modulation of orientation and radius are in the same direction. This experiment investigates the relative contribution of the local orientation and position cues to the perceived modulation of an RF pattern. 
Methods
Stimuli
The stimuli for this experiment were modulated closed paths of Gabor patches. The test stimuli of this experiment were formed in the same way as the adapting stimuli of Experiment 1 (see Equations 1 and 2) but had lower amplitudes of modulation. The frequency of modulation was the same as that applied to the adaptor and test stimuli of Experiment 1, that is, RF3 modulation, or three cycles of modulation in 2π radians (360°), but the patterns were presented in random phase (φ of Equations 1 and 2 was randomized across stimuli). 
Procedure
Thresholds for the detection of modulation in radius (radius only), orientation (orientation only), or both radius and orientation (radius & orientation) were measured using a two interval forced choice (2IFC) task. One interval contained an unmodulated reference pattern (a circular path of Gabor patches aligned tangential to the path) and the other a test pattern with a random phase of modulation (φ was selected at random from the range 0 to 2π radians [0 to 360°]). The test and reference patterns were both presented for 160 ms, and a blank 640 ms interstimulus interval was interposed between the two. The order of presentation of the two patterns was randomized, and the task of the observer was to report which of the two intervals contained the test pattern. The method of constant stimuli (MOCS) was employed using nine amplitudes of modulation in the test stimuli. Trials of the different conditions were blocked, with 180 trials performed in each block. For each condition three blocks of trials were completed by each observer, a total of 540 trials per condition. The responses were binned for amplitude and the probability of correct response calculated for each amplitude of modulation. A cumulative normal distribution was fitted to the data (running from a chance probability of correct response [50% correct] to a probability of 1 [100% correct], the mean yielding the threshold for 75% correct response). All other experimental details are the same as for Experiment 1. Movie 3 is an example trial. 
 
Movie 3.
 
An example trial from Experiment 3. A test and a reference stimulus are presented for 160 ms each prior to and after an interstimulus interval of 640 ms. The test stimulus is modulated in orientation. The order of presentation is randomized and the task of the observer is to indicate the interval that contained the test stimulus.
Results
Thresholds for the detection of modulation in RF patterns are displayed in Figure 7
Figure 7
 
Thresholds for the detection of modulation within a path of Gabor patches. Thresholds are lowest for patterns with modulation in both radius and orientation, slightly higher for patterns solely with modulation of radius, and substantially higher for modulation of orientation alone. Error bars represent 95% confidence intervals.
Figure 7
 
Thresholds for the detection of modulation within a path of Gabor patches. Thresholds are lowest for patterns with modulation in both radius and orientation, slightly higher for patterns solely with modulation of radius, and substantially higher for modulation of orientation alone. Error bars represent 95% confidence intervals.
Paired t tests (one tailed) show that the thresholds for the detection of modulation in the pattern are significantly higher when solely radius (radius only) modulation (t(3) = 4.353, p = 0.0112) or solely orientation (orientation only) modulation (t[3] = 8.127, p = 0.0019) is applied in comparison with thresholds for patterns with radius and orientation modulation (radius & orientation). Thresholds are also significantly larger for orientation modulation in comparison with radius modulation (t[3] = 7.491, p = 0.0025) demonstrating that the percept of shape due to radius modulation is more salient than that of orientation modulation. This pattern of results is consistent with the subjective appearance of the stimuli in Figure 1 (and the results of Experiment 1). By taking the reciprocal of the average thresholds we can compare sensitivities. Sensitivity (the reciprocal of threshold) to orientation modulation is 48 ± 14 (95% CI.), radius modulation 91 ± 10, and both orientation and radius modulation 135 ± 16 (the error is the 95% CI in the mean of the individual reciprocals). It is immediately evident that sensitivity to both of the cues combined is the sum of the sensitivities of the two cues individually. If the radius modulation and orientation modulation were analyzed by independent global detectors one might expect the sensitivities to sum in quadrature. The predicted sensitivity to both cues under these circumstances would be 103 ± 21. A t test shows this prediction to be significantly different from the measured sensitivity (t[6] = 2.684, p = 0.0364). The results support a linear summation of the effects of the radius and orientation modulation. 
Discussion
The results of this experiment demonstrate sensitivity to radius modulation and orientation modulation. Sensitivity to both cues presented together is shown to be a linear sum of sensitivity to the two cues independently. A plausible explanation for this result is that the two cues sum at the local level. This requires that the illusory modulation of position due to the orientation modulation adds locally to the genuine modulation of local position and that the global pattern analyzers use the perceived local positions rather than the veridical positions. The alternative, favored by Day and Loffler (2009), is a global integration of local orientation and position information within a mechanism sensitive to both. This means that both orientation and position information is encoded as a function of polar angle. The direction of illusory displacement is, however, consistent with another orientation-dependent local spatial effect, the Fraser (or twisted cord) illusion. The Fraser illusion will, therefore, produce an illusory shape distortion and if this distortion is sufficient to account for the sensitivity to the orientation modulated patterns the global explanation that orientation information is encoded globally is redundant. Experiment 4, therefore, was designed to measure the size of the Fraser illusion as a function of the angle between the orientation of the patches and the local orientation of the tangent to the path using a stimulus appropriate for modeling the illusory modulation. This function is then used to model the Fraser illusion in a RF pattern modulated in orientation. Should the effect of the orientation modulation be shown to be simply the Fraser illusion, the interpretation of the orientation modulation in the stimuli would have to be changed. The stimuli said to be modulated in position and orientation would not be subject to the Fraser illusion because the patches of the path would all be tangential to the path. The stimuli said to be modulated solely in position would have a subtractive influence of the Fraser illusion on modulation. 
Experiment 4: The Fraser illusion can account for the illusory deformation of a circular path of Gabor patches modulated in orientation
Introduction
Examination of Figure 1 reveals that the perceived amplitude of modulation of the adaptor of Condition A (radius and orientation) is larger than that of Conditions B (radius only) and C (orientation only). The adaptor of Condition C has no modulation of radius but is perceived as distorted from circular while the adaptor of Condition B is as distorted as Condition A (has the same amplitude of modulation of radius), but appears slightly less so. Inspection of the local orientation context around the path representing the adaptor of Condition A reveals that the sign of the illusory displacement of the path is consistent with the Fraser, or twisted cord, illusion. Experiment 4 examines the illusion of tilt in a path defined by Gabor patches as a function of the angle between the axis of the Gabor patches and the path. These data are then used to model the illusory distortion of the path to test whether the Fraser illusion is an adequate explanation for this perceived distortion. 
Methods
Stimuli
The test stimuli were short, linear paths of three Gabor patches. The patches were separated, center to center, by 21 min of visual angle, approximating the separation of the patches in the RF pattern stimuli. The test path was oriented approximately vertically and the orientation of the patches was varied systematically relative to the axis of the path. All parameters associated with the patches and apparatus were the same as for the previous experiments. 
Procedure
Each trial comprised a test stimulus presented for 160 ms. The task of the observer was to report whether the axis of the linear path of Gabor patches was oriented clockwise or anticlockwise from vertical. The orientation of the patches comprising the path varied across the whole range of possible orientations with respect to the axis of the path. The whole function describing the size and direction of the Fraser illusion, with respect to the orientation difference between the axis of the path and the orientation of the patches, was sampled. Conditions were blocked, with three blocks of 180 trials per condition (540 trials for each condition) performed. A range of nine orientations of the axis of the path with respect to the vertical were used to sample the psychometric function and the probability of responding that the axis of the path was oriented clockwise of vertical calculated for each orientation. A cumulative normal distribution was fitted to the data with the mean of the distribution yielding the PSE, the orientation of the path that would be perceived as vertical. The function describing the variation of the Fraser illusion with respect to the orientation difference between the orientation of the patches and the axis of the path was sampled, first with the stimulus at fixation and then again at an eccentricity of 2° to the right of fixation. In the second set of conditions a fixation marker was present 2° to the left of the position at which the stimulus appeared. Movie 4 is an example trial from Experiment 4
 
Movie 4.
 
An example trial from Experiment 4. The test path of patches is presented in a single interval of 160 ms duration. The patches within the path are oriented 20° clockwise of vertical. The path is vertical but appears oriented clockwise of vertical.
Results
Figure 8 displays data describing the Fraser illusion as a function of the orientation difference between the orientation of the patches in the path and the axis of the path. 
Figure 8
 
The Fraser illusion as a function of the orientation difference between the orientation of the patches and the axis of the path of patches. A positive orientation difference indicates that the axis of the path is anticlockwise of the orientation of the patches that comprise the path. The Fraser illusion for a positive (negative) orientation difference is negative (positive) indicating that the perceived orientation of the axis of the path has been drawn towards the orientation of the patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1).
Figure 8
 
The Fraser illusion as a function of the orientation difference between the orientation of the patches and the axis of the path of patches. A positive orientation difference indicates that the axis of the path is anticlockwise of the orientation of the patches that comprise the path. The Fraser illusion for a positive (negative) orientation difference is negative (positive) indicating that the perceived orientation of the axis of the path has been drawn towards the orientation of the patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1).
A D1 function is again fitted to the data of the two observers. The Fraser illusion acts in the opposite direction to the TAE (cf. Figure 4). Again the standard deviation of the fitted functions was similar for the data at fixation and at 2° eccentricity (22.18 ± 5.62 (95% CI) and 23.14 ± 2.08 for ED and 30.43 ± 7.38 and 27.14 ± 5.67 for VB). Again, because the paths of Gabor patches modulated in orientation describing the RF patterns used in Experiment 3 were at approximately 2° from fixation, the D1 functions fitted to the data collected at an eccentricity of 2° were used in the modeling of the Fraser illusion in these paths. 
Modeling
In the TAE, adaptation to a line of a particular orientation causes the orientation of a subsequently presented line to be repelled from that of the adaptor. In the Fraser illusion, the orientation of a line is drawn towards the orientation of elements that comprise the line. Experiment 2 showed that systematic application of the local TAE could account for the illusory modulation in a circular path of Gabor patches after adaptation to a path of Gabor patches with a sinusoidally modulated radius. Here we used the data displayed in Figure 8, describing the Fraser illusion induced in a line of Gabor patches as a function of the orientation difference between the orientation of the patches and the orientation of the line. The procedure is analogous to that employed to model aftereffects in Dickinson, Almeida, et al. (2010) and in Experiment 2 of this paper to model the illusory shape in a circle of Gabor patches modulated from circular orientation. The local orientation difference between the orientation of the patches and the orientation of the local tangent to the path of Gabor patches is derived as a function of θ around the path using Equation 2 (with φ set to zero to allow direct comparison with the modeling of the aftereffect in Experiment 2) and then the Fraser illusion this implies is calculated for each sample of θ in the field. The Fraser illusion at each point on the path is calculated using the function described by the dashed red lines in Figure 8. The modeling in Experiment 2 represented the TAE as a distributed field. The Fraser illusion is obviously constrained to the line itself but the process of deriving a representation of the perceived shape of a circle distorted by the Fraser illusion is, however, similar. Again a line is projected upwards from 2° on the positive x-axis (assuming the origin is at the center of the pattern). The line advances on a nominally circular trajectory but respects the local Fraser illusion as it proceeds. For illustrative purposes the perceived shapes of patterns with amplitudes of orientation modulation of 0.1 are calculated for the two observers ED and VB. The patterns produced are illustrated in Figure 9. The illusory tilt is color coded green for anticlockwise local rotation and blue for clockwise (8° at the maximum of the color range). The effect is constrained to be within the path of Gabor patches. 
Figure 9
 
The predicted perceived distortion of a circular path due to the Fraser illusion in a path of orientation modulated Gabor patches. The amplitude of orientation modulation was 0.1. Panel A is the prediction for observer ED and Panel B the prediction for VB.
Figure 9
 
The predicted perceived distortion of a circular path due to the Fraser illusion in a path of orientation modulated Gabor patches. The amplitude of orientation modulation was 0.1. Panel A is the prediction for observer ED and Panel B the prediction for VB.
The predicted perceived modulation of the patterns is in the same phase as the orientation modulation, as predicted by the Fraser illusion (and the predicted pattern looks somewhat like Panel C of Figure 1—a circular path of patches with an orientation modulation of 0.1). The amplitude of illusory radius modulation of the path created for observer ED is 0.044 and for observer VB it is 0.037 (or 44% and 37% of the orientation modulation amplitude). In order to determine if the Fraser illusion can account for the thresholds for detection of modulation in orientation modulated patterns we can introduce the measured thresholds for detection of orientation modulation (see Figure 7) into the model to predict the perceived positional modulation induced by the Fraser illusion and then compare this with the threshold for radius modulation. The measured threshold for the detection of orientation modulation is used to create a Fraser illusion field (FI field) for that amplitude of orientation modulation using the function describing the Fraser illusion (FI) versus local orientation difference derived in Experiment 4. A nominally circular path is then projected through that field as a succession of straight line segments, the orientations of which are adjusted to accommodate the local FI at their starting points (Euler's method). The error associated with using the FI at the start of each straight line segment can be made arbitrarily small by reducing the length of each segment. The line ultimately arrives back at its starting point and the predicted amplitude of perceived radial modulation of the pattern produced can be ascertained as the maximum excursion from a circular path (a Matlab m file implementing the model for the two observers is included as Supplementary Information). An orientation modulation of 0.0181 (0.0238) for observer ED (VB) gives an illusory radius modulation (peak illusory positional displacement of the path) of 0.0089 (0.0093). The thresholds for detection of genuine radius modulation in the absence of an influence of an orientation context (patches aligned tangential to the path) are 0.0063 ± 0.0007 95% CI (0.0074 ± 0.0014). The predicted magnitudes of perceived modulation can, therefore, accommodate the detection thresholds for genuine modulation of the path. 
Discussion
The results of this experiment demonstrate that the illusory shape of circular paths of Gabor patches modulated in orientation can be accounted for by a systematic application of the Fraser (or twisted cord) illusion around the pattern. 
General discussion
Adaptation to RF patterns causes subsequently presented circles to appear as though they are modulated in opposite phase to the adaptors (Anderson, Habak, Wilkinson, & Wilson, 2007). The aftereffects of such adaptation have conventionally been attributed to modification of the response of mechanisms encoding shape. Recently, Dickinson, Almeida, et al. (2010) showed that the aftereffects can be accounted for by systematic application of the TAE across the stimulus in a TAE field. This explanation, though, was only demonstrated for adaptation times of a few seconds and it has been shown that the adaptation can be rapidly acquired (Dickinson, Han, et al., 2010). The TAE can, however, also be rapidly acquired (Sekuler & Littlejohn, 1974) and so one might expect that a TAE field might serve as an explanation for rapid adaptation in RF patterns. Experiments 1 and 2 confirm that this could be the case. The concept of a TAE field is, of course, not specific to RF patterns and so can be generalized to all adaptors and test patterns to provide a mechanism for enhancing the perceived difference between successively presented similar stimuli. 
Day and Loffler (2009) investigated a misperception of shape in a circular path of Gabor patches similar to that produced by adaptation but induced by a modulation in local orientation information around the path (Loffler, 2008). They attributed this percept to a global shape processing mechanism sensitive to modulation in orientation. Inspection of the path, however, reveals that the perceived local positional displacement of the path is in the same direction as that predicted by the Fraser, or twisted cord illusion (Fraser, 1908). Experiment 3 measured thresholds for detection of modulation in radius and/or orientation and showed that sensitivity summed linearly. Experiment 4 showed that the Fraser illusion can induce a perceived positional modulation sufficient to account for the thresholds for detection of modulation in the patterns modulated in orientation. 
As suggested in the Introduction, a role for the TAE field readily presents itself in the exaggeration of difference in successively presented stimuli. The utility of the analogous process of systematic application of the Fraser illusion seems less obvious. Collinear facilitation in adjacent receptive fields has, though, been proposed as a mechanism by which partially occluded boundaries are bound into a path (Field et al., 1993; Li & Gilbert, 2002). Local orientation cues tangential to a path might serve to enhance the sensitivity of larger scale oriented filters tuned to the path orientation. Indeed, Purkinje (1819) observed that, after prolonged viewing, long straight lines became deformed into a succession of differently oriented line segments. This has been taken to indicate that lines are processed on more than one spatial scale (Tyler & Nakayama, 1984), the premise being that adaptation desensitizes receptors most sensitive to the orientation of the line resulting in discrepancies in the local population responses of activated neurons. Tyler and Nakayama (1984) broke a line on an oscilloscope into segments and manipulated the angle of these segments with respect to the axis of the line. The function that described the perceived orientation of the line with respect to the orientation of the segments was observed to be defined by the competing influences of the Fraser illusion, an illusory tilt in the direction of the local elements, and the Zöllner illusion, an illusory tilt in the opposite direction to the local elements. Tyler and Nakayama (1984) attribute the assimilative effect of the Fraser illusion to excitatory crosstalk between populations of cells with the same spectrum of orientation preferences but with different sized receptive fields. The Zöllner illusion might be considered as due to an analogous inhibitory interaction. Connections between approximately collinear receptive fields might be expected to be excitatory to facilitate the rapid binding of collinear elements into a path. It has been argued that connections between receptive fields with larger differences in preferred orientation are inhibitory and serve to exaggerate the angle between lines (Blakemore, Carpenter, & Georgeson, 1970). Such competition between excitatory and inhibitory influences might have been expected in this study but the results of our Experiment 4 showed no evidence of the repulsive effect of the Zöllner illusion. However, Skillen, Whitaker, Popple, and McGraw (2002) showed that the relative magnitudes of the excitatory and inhibitory effects were dependent upon the ratio between the spatial scales of the receptive fields responsible for the analysis of the grating and the path. In the experiments of this study the receptive fields of the cells responsible for the perception of the path can be expected to be large in comparison with those responsible for the analysis of the orientation of the gratings. For such conditions, Skillen et al. (2002) report no inhibitory effect. 
The similarity of the form of the functions derived in Experiments 2 and 4, which describe the TAE and Fraser illusion as the angular difference between the test orientation and the context is varied, suggests that the underlying mechanism responsible for the two effects might be common. The model illustrated in Figure 10 shows how the illusions might be understood. The column of panels on the left shows a potential mechanism for the TAE and the column on the right the Fraser illusion. Figure 10A denotes the sensitivity of a bank of orientation selective channels as a function of orientation. The channels are spaced at intervals of 10° and therefore 18 channels are required to populate the whole spectrum of possible orientations. This arrangement was adapted from the model of Gilbert and Wiesel (1990), which in turn was based on electro-physiological recordings of neurons in monkey striate cortex (Hubel & Wiesel, 1968). The sensitivity profile of each channel is Gaussian, with a standard deviation of 20°. The channels at the left and right of the figure (peaking at −80° and 90°, respectively) are, of course, contiguous. Therefore, we must imagine a closed loop of orientation selectivity. 
Figure 10
 
Proposed analogous mechanisms for the TAE (left column) and Fraser illusion (right column). A bank of orientation selective channels (blue Gaussian functions) forms a closed loop representing the full spectrum of possible orientations. In Panel A (D) an adaptor (context) orientation activates the channels sensitive to that orientation to a degree that is proportional to the height of the Gaussian sensitivity profile at that orientation. The activation of each channel is represented by the height of the red lines (this height is the height of the Gaussian at the point it is intersected by the adaptor or context orientation). The sensitivities of the bank of channels represented in Panel B (E) are modified by the activation shown in Panel A (D). For the tilt aftereffect these may well be the same bank of channels and the depth of adaptation is proportional to the prior activation. For the Fraser illusion we postulate that the banks of channels are different, perhaps with discrete spatial frequency sensitivity, and that the sensitivity of the second bank of channels is increased in proportion to the activation of the first by the context orientation. The activation due to the test line is again indicated by red bars. The perceived orientation of the test line (dashed red line) is given by the orientation of the resultant of the vectors describing the activity across the bank of channels.
Figure 10
 
Proposed analogous mechanisms for the TAE (left column) and Fraser illusion (right column). A bank of orientation selective channels (blue Gaussian functions) forms a closed loop representing the full spectrum of possible orientations. In Panel A (D) an adaptor (context) orientation activates the channels sensitive to that orientation to a degree that is proportional to the height of the Gaussian sensitivity profile at that orientation. The activation of each channel is represented by the height of the red lines (this height is the height of the Gaussian at the point it is intersected by the adaptor or context orientation). The sensitivities of the bank of channels represented in Panel B (E) are modified by the activation shown in Panel A (D). For the tilt aftereffect these may well be the same bank of channels and the depth of adaptation is proportional to the prior activation. For the Fraser illusion we postulate that the banks of channels are different, perhaps with discrete spatial frequency sensitivity, and that the sensitivity of the second bank of channels is increased in proportion to the activation of the first by the context orientation. The activation due to the test line is again indicated by red bars. The perceived orientation of the test line (dashed red line) is given by the orientation of the resultant of the vectors describing the activity across the bank of channels.
Figure 10B represents the sensitivity of the bank of channels represented in Figure 10A postadaptation to a line with an orientation of 30°. The reduction in sensitivity, or gain, of the channels is assumed to be linearly related to their activation in their initial state (Figure 10A) by the adapting line. A test line is then presented to the bank of channels and their individual activity calculated in the same manner (the relative activation is given by the height of the point of intersection of the line illustrating the test orientation with each of the channels). Figure 10C represents the response of the channels to the test line as vectors. The polar space in which they are represented is double the angle of orientation (Clifford, 2002). The orientation of the resultant vector in the double angle space is represented by the dashed blue line. 
For the TAE, the model assumed a reduction in sensitivity (due to adaptation) of one-third of the activity in the channel whose preferred orientation matched that of the adaptor. The reduction in sensitivity of other channels was smaller and in proportion to their relative activation by the adaptor. In the modeling of the Fraser illusion sensitivity was increased by one-half by the presence of the context. This resulted in a ratio between the most and least sensitive channels being the same in each case. In this case, though, the bank of channels represented in Figures 10D and 10E would not be the same. The bank of channels in Figure 10D would be sensitive to the orientation of the Gabor patches and the bank in Figure 10E would have larger, perhaps second-order, receptive fields. The increase in sensitivity in the bank of channels in Figure 10E is assumed to result from input from the bank of channels represented in Figure 10D in an orientation-specific manner. In this modeling the sensitivity of the channels of Figure 10E was increased by an amount proportional to the activation in the channels of Figure 10D with the same preferred orientation. Perceived displacements of the test line were derived over the full range of adaptor/context orientations between −90° and 90°. The predictions are displayed in Figure 11
Figure 11
 
The TAE and Fraser illusion predicted by the models illustrated in Figure 10 as a function of orientation difference between the adaptor/context and test. The data points represent the orientations indicated by the resultant vectors in the double angle space at ten orientations of the adaptor/context. The solid lines are fits of first derivative of Gaussian (D1) functions to the data. The conventions used are the same as those used for the psychophysical TAE and Fraser illusion data.
Figure 11
 
The TAE and Fraser illusion predicted by the models illustrated in Figure 10 as a function of orientation difference between the adaptor/context and test. The data points represent the orientations indicated by the resultant vectors in the double angle space at ten orientations of the adaptor/context. The solid lines are fits of first derivative of Gaussian (D1) functions to the data. The conventions used are the same as those used for the psychophysical TAE and Fraser illusion data.
The predictions of the model are well fitted by the D1 function. Moreover, the standard deviations of the fits of the TAE and Fraser illusion predictions are similar (TAE: 25.6° ± 0.5° [95% CI] and Fraser illusion: 28.8° ± 0.8°) with the Fraser illusion being somewhat larger than the TAE. No attempt was made to optimize the model yet it predicts the psychophysical data well, including the slightly larger standard deviation in the D1 for the Fraser illusion. 
In conclusion, given that the shape aftereffects of the RF pattern can be accounted for by local contextual interactions, the most parsimonious explanation for our results is that global shape analyzing mechanisms integrate the perceived local positions of elements of the shape, which may be modified by local contextual interactions in order to accommodate misperception of orientation. In the case of the TAE the effects of adaptation serve to enhance the perceived difference between successively presented stimuli. The Fraser illusion might be a consequence of the subversion of a system for enhancing the salience of partially occluded paths. These local effects are sufficient to account for the misperception of shape in these experiments but that does not preclude the existence of higher level contextual effects. 
Supplementary Materials
Acknowledgments
This research was supported by Australian Research Council Grants DP0666206, DP1097603 and DP110104553 to DRB. 
Commercial relationships: none. 
Corresponding author: J. Edwin Dickinson. 
Email: edwin.dickinson@uwa.edu.au. 
Address: School of Psychology, University of Western Australia, Crawley, Perth, WA, Australia. 
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Figure 1
 
Adaptor stimuli used in Experiment 1. (A) A radius and orientation adaptor stimulus with radius and orientation modulation. (B) A radius only adaptor with radius modulation only. (C) A circular orientation only adaptor with orientation modulation only. (D) An adaptor with radius and orientation modulation but with the radii of each of the patches permuted (permuted). The amplitude of modulation, A of Equations 1 and 2, for the adaptors was 0.1 when applied and zero otherwise. Test stimuli were RF patterns that varied in amplitude around zero and had three cycles of modulation in 2π radians.
Figure 1
 
Adaptor stimuli used in Experiment 1. (A) A radius and orientation adaptor stimulus with radius and orientation modulation. (B) A radius only adaptor with radius modulation only. (C) A circular orientation only adaptor with orientation modulation only. (D) An adaptor with radius and orientation modulation but with the radii of each of the patches permuted (permuted). The amplitude of modulation, A of Equations 1 and 2, for the adaptors was 0.1 when applied and zero otherwise. Test stimuli were RF patterns that varied in amplitude around zero and had three cycles of modulation in 2π radians.
Figure 2
 
Aftereffects of adaptation to the four conditions of Experiment 1. The four conditions are grouped, with the labels for the conditions corresponding to the labels for the adaptors displayed in Figure 1; (A) radius and orientation modulation; (B) radius modulation only; (C) orientation modulation only; (D) radius and orientation modulation with permuted radii (permuted). Error bars are 95% confidence intervals. A substantial aftereffect is experienced after adaptation to all four adaptor types.
Figure 2
 
Aftereffects of adaptation to the four conditions of Experiment 1. The four conditions are grouped, with the labels for the conditions corresponding to the labels for the adaptors displayed in Figure 1; (A) radius and orientation modulation; (B) radius modulation only; (C) orientation modulation only; (D) radius and orientation modulation with permuted radii (permuted). Error bars are 95% confidence intervals. A substantial aftereffect is experienced after adaptation to all four adaptor types.
Figure 3
 
An illustration of the timecourse of a trial in Experiment 2. An adaptor is presented for 160 ms followed by a test interval with the same duration after an interstimulus interval of 640 ms comprising the grey background.
Figure 3
 
An illustration of the timecourse of a trial in Experiment 2. An adaptor is presented for 160 ms followed by a test interval with the same duration after an interstimulus interval of 640 ms comprising the grey background.
Figure 4
 
The tilt aftereffect (TAE) as a function of the difference in orientation (Δ orientation) between the adapting and test patches. A positive (negative) value for Δ orientation indicates that the orientation of the test patches is anticlockwise (clockwise) of the adapting patches. A positive (negative) TAE indicates repulsion of orientation of the test patches in an anticlockwise (clockwise) direction from the orientation of the adapting patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1). The colored bar is used to associate the size of the TAE to a color for use in displaying the local TAE in the modeling of the TAE across the stimulus as a scalar field.
Figure 4
 
The tilt aftereffect (TAE) as a function of the difference in orientation (Δ orientation) between the adapting and test patches. A positive (negative) value for Δ orientation indicates that the orientation of the test patches is anticlockwise (clockwise) of the adapting patches. A positive (negative) TAE indicates repulsion of orientation of the test patches in an anticlockwise (clockwise) direction from the orientation of the adapting patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1). The colored bar is used to associate the size of the TAE to a color for use in displaying the local TAE in the modeling of the TAE across the stimulus as a scalar field.
Figure 5
 
Decline of the magnitude of the TAE with distance in the visual field at an eccentricity of approximately 2° for the stimuli used in Experiment 2. The value of the TAE when the adaptor and test were spatially coincident was derived from the fit to the data in Figure 4 pertaining to an eccentricity of 2° of visual angle (dashed red line). The orientation of the adaptor was 20° clockwise of the test orientation. Increasing distance between the centers of the adaptor and test clusters of patches is represented along the x-axis and the TAE represented on the y-axis. A Gaussian function is fitted to the data. The function is constrained to have an amplitude equal to the TAE at 0° distance (at an eccentricity of 2°) and a mean of zero degrees separation. The function, therefore, only has one free parameter, the standard deviation. Error bars are 95% confidence intervals.
Figure 5
 
Decline of the magnitude of the TAE with distance in the visual field at an eccentricity of approximately 2° for the stimuli used in Experiment 2. The value of the TAE when the adaptor and test were spatially coincident was derived from the fit to the data in Figure 4 pertaining to an eccentricity of 2° of visual angle (dashed red line). The orientation of the adaptor was 20° clockwise of the test orientation. Increasing distance between the centers of the adaptor and test clusters of patches is represented along the x-axis and the TAE represented on the y-axis. A Gaussian function is fitted to the data. The function is constrained to have an amplitude equal to the TAE at 0° distance (at an eccentricity of 2°) and a mean of zero degrees separation. The function, therefore, only has one free parameter, the standard deviation. Error bars are 95% confidence intervals.
Figure 6
 
The perceived shape (black line) of a circle after adaptation to a RF pattern with an amplitude of sinusoidal modulation of 0.1 in zero phase, as predicted by a TAE field. Panel A shows the prediction for observer ED and B the prediction for observer VB.
Figure 6
 
The perceived shape (black line) of a circle after adaptation to a RF pattern with an amplitude of sinusoidal modulation of 0.1 in zero phase, as predicted by a TAE field. Panel A shows the prediction for observer ED and B the prediction for observer VB.
Figure 7
 
Thresholds for the detection of modulation within a path of Gabor patches. Thresholds are lowest for patterns with modulation in both radius and orientation, slightly higher for patterns solely with modulation of radius, and substantially higher for modulation of orientation alone. Error bars represent 95% confidence intervals.
Figure 7
 
Thresholds for the detection of modulation within a path of Gabor patches. Thresholds are lowest for patterns with modulation in both radius and orientation, slightly higher for patterns solely with modulation of radius, and substantially higher for modulation of orientation alone. Error bars represent 95% confidence intervals.
Figure 8
 
The Fraser illusion as a function of the orientation difference between the orientation of the patches and the axis of the path of patches. A positive orientation difference indicates that the axis of the path is anticlockwise of the orientation of the patches that comprise the path. The Fraser illusion for a positive (negative) orientation difference is negative (positive) indicating that the perceived orientation of the axis of the path has been drawn towards the orientation of the patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1).
Figure 8
 
The Fraser illusion as a function of the orientation difference between the orientation of the patches and the axis of the path of patches. A positive orientation difference indicates that the axis of the path is anticlockwise of the orientation of the patches that comprise the path. The Fraser illusion for a positive (negative) orientation difference is negative (positive) indicating that the perceived orientation of the axis of the path has been drawn towards the orientation of the patches. Error bars represent 95% confidence intervals. The function fitted to the data is the first derivative of a Gaussian (D1).
Figure 9
 
The predicted perceived distortion of a circular path due to the Fraser illusion in a path of orientation modulated Gabor patches. The amplitude of orientation modulation was 0.1. Panel A is the prediction for observer ED and Panel B the prediction for VB.
Figure 9
 
The predicted perceived distortion of a circular path due to the Fraser illusion in a path of orientation modulated Gabor patches. The amplitude of orientation modulation was 0.1. Panel A is the prediction for observer ED and Panel B the prediction for VB.
Figure 10
 
Proposed analogous mechanisms for the TAE (left column) and Fraser illusion (right column). A bank of orientation selective channels (blue Gaussian functions) forms a closed loop representing the full spectrum of possible orientations. In Panel A (D) an adaptor (context) orientation activates the channels sensitive to that orientation to a degree that is proportional to the height of the Gaussian sensitivity profile at that orientation. The activation of each channel is represented by the height of the red lines (this height is the height of the Gaussian at the point it is intersected by the adaptor or context orientation). The sensitivities of the bank of channels represented in Panel B (E) are modified by the activation shown in Panel A (D). For the tilt aftereffect these may well be the same bank of channels and the depth of adaptation is proportional to the prior activation. For the Fraser illusion we postulate that the banks of channels are different, perhaps with discrete spatial frequency sensitivity, and that the sensitivity of the second bank of channels is increased in proportion to the activation of the first by the context orientation. The activation due to the test line is again indicated by red bars. The perceived orientation of the test line (dashed red line) is given by the orientation of the resultant of the vectors describing the activity across the bank of channels.
Figure 10
 
Proposed analogous mechanisms for the TAE (left column) and Fraser illusion (right column). A bank of orientation selective channels (blue Gaussian functions) forms a closed loop representing the full spectrum of possible orientations. In Panel A (D) an adaptor (context) orientation activates the channels sensitive to that orientation to a degree that is proportional to the height of the Gaussian sensitivity profile at that orientation. The activation of each channel is represented by the height of the red lines (this height is the height of the Gaussian at the point it is intersected by the adaptor or context orientation). The sensitivities of the bank of channels represented in Panel B (E) are modified by the activation shown in Panel A (D). For the tilt aftereffect these may well be the same bank of channels and the depth of adaptation is proportional to the prior activation. For the Fraser illusion we postulate that the banks of channels are different, perhaps with discrete spatial frequency sensitivity, and that the sensitivity of the second bank of channels is increased in proportion to the activation of the first by the context orientation. The activation due to the test line is again indicated by red bars. The perceived orientation of the test line (dashed red line) is given by the orientation of the resultant of the vectors describing the activity across the bank of channels.
Figure 11
 
The TAE and Fraser illusion predicted by the models illustrated in Figure 10 as a function of orientation difference between the adaptor/context and test. The data points represent the orientations indicated by the resultant vectors in the double angle space at ten orientations of the adaptor/context. The solid lines are fits of first derivative of Gaussian (D1) functions to the data. The conventions used are the same as those used for the psychophysical TAE and Fraser illusion data.
Figure 11
 
The TAE and Fraser illusion predicted by the models illustrated in Figure 10 as a function of orientation difference between the adaptor/context and test. The data points represent the orientations indicated by the resultant vectors in the double angle space at ten orientations of the adaptor/context. The solid lines are fits of first derivative of Gaussian (D1) functions to the data. The conventions used are the same as those used for the psychophysical TAE and Fraser illusion data.
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