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Article  |   September 2016
Discrete annular regions of texture contribute independently to the analysis of shape from texture
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Journal of Vision September 2016, Vol.16, 10. doi:10.1167/16.11.10
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      Ken W. S. Tan, J. Edwin Dickinson, David R. Badcock; Discrete annular regions of texture contribute independently to the analysis of shape from texture. Journal of Vision 2016;16(11):10. doi: 10.1167/16.11.10.

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

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Abstract

Radial frequency (RF) textures (created by applying a sinusoidal modulation of orientation to an otherwise circular texture) have been shown to be globally processed. RF textures differ from RF patterns (paths deformed from circular by a sinusoidal modulation in radius) in that the elements need not be constrained to a specific path. In the natural environment, objects differ from their background in texture, and a bounding contour can mark this textural change. This study examines the extent to which modulation of texture sums across space and whether the inclusion of a boundary between two areas provides a segmentation cue that limits the area over which summation occurs. RF textures were split into two annular regions and signal introduced to inner, outer, or both annuli Thresholds for the detection of RF modulation of orientation were not affected by the presence of a boundary. Further, it was found that the thresholds matched predictions for the independent contribution of the inner and outer areas to performance and that changing the relative phase of the modulation in the inner and outer annuli had no impact on performance, implying independent integration within the two annuli. Finally, integration of modulation information within the annuli was confirmed to ensure these results do apply to textures that are globally processed.

Introduction
Textures are commonplace in daily life, and they often provide information regarding the object being viewed. Unlike contours, shape information in textures is not constrained to a specific path; instead, the information is spread over area. It has previously been shown (Tan, Bowden, Dickinson, & Badcock, 2015) that modulated textures can be globally processed. In such textures, the shape information is summed as a function of polar angle (i.e., summed around the full 360°). However, as mentioned, textures are spread over area, and in this study, we were interested to know how information in globally processed textures was summed across radii. 
The textures used in the study by Tan et al. (2015) were based on radial frequency (RF) patterns—patterns created by sinusoidally modulating the radius of a circle as a function of polar angle (Wilkinson, Wilson, & Habak, 1998). RF patterns are a stimulus type that has been introduced to examine the visual processing of shape, and they produce deformation thresholds under optimal circumstances that qualify as hyperacuities (Bell, Badcock, Wilson, & Wilkinson, 2007; Loffler, Wilson, & Wilkinson, 2003; Wilkinson et al., 1998). RF patterns have proven to be an ideal tool to quantitatively demonstrate the phenomenon of global integration of shape information on a contour. Modulation amplitude thresholds for discriminating RF patterns from circles improve as more cycles of modulation (CoM) of a fixed frequency are progressively added to an otherwise circular contour. Thresholds are usually observed to fall at a rate that far exceeds that predicted by the probability of independently detecting local cues as they are added (probability summation [PS]; Bell & Badcock, 2008; Cribb, Badcock, Maybery, & Badcock, in press; Loffler et al., 2003; Tan, Dickinson, & Badcock, 2016; Wilson, 1980). This rapid rate of decrease of threshold has been used as an indicator for the integration of modulation information around the pattern and is commonly observed in low-frequency RF patterns (<8–10 CoM per 360°), where global integration of local information has, therefore, been shown to occur (Bell & Badcock, 2008; Dickinson, Han, Bell, & Badcock, 2010; Hess & Field, 1999; Jeffrey, Wang, & Birch, 2002; Loffler et al., 2003; Tan et al., 2015; Tan, Dickinson, & Badcock, 2013). 
RF patterns of different frequency but with the same number of CoM are discriminable at their thresholds for modulation detection, providing further evidence for their global processing (Dickinson, Bell, & Badcock, 2013). However, RF patterns with the same frequency but differing numbers of CoM are not discriminable at threshold, eliminating the number of CoM as a discriminating cue at threshold for detection. Further, patterns of differing frequency (three or six cycles/360°) with a single CoM are also not discriminable at their thresholds for detection, suggesting that it is the periodicity of repetition, across cycles, of features on the patterns that allow them to be discriminated. 
Modulation amplitude thresholds for RF patterns with the same number of cycles but with differing frequencies have been observed to be proportional to the reciprocal of the frequency of modulation (Dickinson, McGinty, Webster, & Badcock, 2012). One property of the patterns that varies with frequency in this way is the maximum local orientation deviation from a circle, leading to the suggestion that it is this property that is integrated across CoM and that determines the threshold for detection. Because the patterns are discriminable at their thresholds for detection, it has been suggested that analysis of the path at the points of maximum deviation from circular is used to determine the location of the points of maximum curvature (Dickinson, Cribb, Riddell, & Badcock, 2015; Dickinson et al., 2012). 
The maximum orientation difference from circularity (for patterns at threshold modulation) is near the zero crossings of the modulating sinusoid. This implies that modulation of the position of the path is not necessarily critical to the detection of modulation. This insight raises the possibility that it might not be necessary for the orientation information to be constrained to a contiguous path, which could mean that orientation modulation in a texture of elements could potentially also be globally processed. Support for this arises from a recent study demonstrating that integration can occur in a modulated texture that does not contain a contiguous path (Tan et al., 2015). In that study, a texture was created that was analogous to conventional Glass (1969) patterns in the sense that structure was implied by orientation but where the orientation of the texture conformed to those of an RF pattern; that is, elements conformed to the RF pattern's orientation for a particular polar angle but where the element's radial position had positional jitter such that it did not produce a contiguous path. The dot pairs preferred in conventional Glass patterns were replaced with Gabor patches, with the grating representing the local orientation in place of the alignment of the dot pairs. Achtman, Hess, and Wang (2003) have previously employed similar Glass patterns composed of Gabor patches and observed that the circular structure in these patterns was globally processed, but the work by Tan et al. (2015) measured integration of modulation of orientation causing the appearance to differ from circular. It was observed that, like conventional RF patterns, as more CoM were added to the texture, discrimination thresholds fell at a rate steeper than that predicted by PS, indicating the presence of global integration of orientation information implying shape around the pattern. 
Some research involving RF shape processing (e.g., Badcock, Almeida, & Dickinson, 2013; Poirier & Wilson, 2006; Schmidtmann, Gordon, Bennett, & Loffler, 2013; Wilkinson et al., 2000) proposes that the mechanisms involved in RF shape processing are different from those underlying the analysis of structure in Glass patterns, in that whereas RF shape detectors constrain both element orientation and position, Glass patterns only require elements to conform to a global orientation rule, and the mechanisms proposed to provide sensitivity to structure in Glass patterns are therefore insensitive to local positional information. For polar Glass patterns, the global orientation rule requires the local elements to have a common orientation with respect to the local radius. The RF textures introduced by Tan et al. (2015) were therefore different from conventional RF patterns in that the local elements were unconstrained in radial position. They differed from Glass patterns in that the orientations of the local elements were modulated with respect to the local radius as a function of polar angle, and the task was to detect that modulation—not to detect coherent orientation. The fact that the RF texture contains no contiguous paths would suggest that the RF shape detectors proposed by previous researchers (e.g., Schmidtmann et al., 2013) would have to be modified to allow for the processing of RF textures or that perhaps RF patterns and RF textures were processed by discrete systems. Glass patterns can be thought of as patterns implying flow (Kourtzi, 2004; Kovács & Julesz, 1992), that is, transformation of solid bodies (e.g., rotational motion for concentric Glass patterns, looming or receding motion for radial Glass patterns), with the flow arising from the trajectories of critical points in the object when it is transformed. RF textures, however, cannot easily be attributed to the flow fields that result from transformation of a solid body, and therefore sensitivity to such flow fields cannot be used to account for sensitivity to modulation applied to the radius. It therefore stands to reason that RF texture detectors have to inherently be about the shape information contained within the pattern and hence in their utility differ from Glass pattern detectors. 
Other researchers have employed sinusoidally modulated linear textures to investigate sensitivity to orientation modulation. For example, Kingdom and colleagues sinusoidally modulated orientations of Gabor element fields to investigate the spatial tuning of sensitivity to the orientation modulation (Kingdom, Keeble, & Moulden, 1995) and found such textures to be scale invariant; they further investigated the mechanism for this scale invariance (Kingdom & Keeble, 1999) and found that the size of the micro-pattern was vital, with the most critical factor being the micro-pattern's carrier spatial frequency. Kingdom and Prins (2009) also found that amplitude modulation (which they referred to as contour shape) aftereffects were reduced when the adaptor was a series of similar open contours instead of a lone open contour; this despite there being additional potential adaptations (from the surround contours). When the surround contours had orthogonal orientations, adaptation was reduced much less. This leads to the suggestion that linear textures and contours are processed differentially and, in addition, that multiple contours potentially constitute a single texture rather than separate contours. This was termed texture-surround suppression of contour shape (TSSCS). A later study (Gheorghiu & Kingdom, 2012) extended upon this to investigate how TSSCS operates and over what spatial extent. The findings of that study suggest that there are two components to the TSSCS phenomenon, one that is sensitive to opposite orientation texture surrounds and another that is sensitive to similar orientation texture surrounds. The former operates locally and disrupts contour processing, whereas the latter is spatially extended and prevents the shape of the contour from being processed as a contour. 
Although the aforementioned exemplar studies also employed sinusoidally modulated textures, they differ because the textures employed in these studies conformed to a continuous flow modulated along a straight line, rather than the one employed in the current study that occurred around a circle (which resulted in a closed flow). Closed contours have been shown to exhibit stronger pooling in comparison with open contours (Loffler, 2015; Schmidtmann et al., 2013; Wilkinson et al., 1998), and modulations on lines and closed shapes are processed differentially (Wilkinson et al., 1998); there is no current evidence supporting an assumption that this would be any different for modulated textures. While thresholds for determining polar Glass patterns have been proposed to be from global pooling of the orientation information throughout the pattern, thresholds for translational Glass patterns have not been shown to be determined by global processes (Wilson, Wilkinson, & Asaad, 1997). In the same vein, the textures used by Kingdom et al. (1995) could be thought of as having sinusoidal modulation applied to translational Glass patterns, whereas the textures we employ have that modulation applied to polar Glass patterns. They differ in that the polar examples give the impression of closure (Tan et al., 2015), which more closely resembles the bounding contours of objects and hence relate more to shape. 
There is also the additional benefit in using RF textures in that we can be certain that these textures are globally processed (Tan et al., 2015). It was also shown that retaining the same set of orientations but permuting the positions of the elements to disrupt the global structure, while retaining all of the orientation variation, eliminated the global integration. This revealed that the coherent modulation of orientation information as polar angle increased was critical to the integration of the orientation information around the pattern, supporting the conjecture that the modulation of orientation with polar angle represented shape information and not probabilistic summation of local orientation changes. Knowing that RF textures were globally processed, we were interested in characterizing the manner in which shape information from such a texture would sum across space and also whether the existence of a closed contour would constrain integration of information to within what might be considered an object boundary. 
Since the texture inside an uncamouflaged object's boundary is often different to the texture outside, it would seem undesirable to integrate texture modulation across the border. In order to see if the visual system is able to be selective, we used RF textures divided into inner and outer regions, and used a readily detectable contour to segment the regions in some instances, so that we could characterize the effect of texture differences and also that of segmenting contours. An area of texture was split into two regions; the regions were matched such that they were of equal area. Modulation was manipulated such that it either appeared only in the Inner, Outer, or in Both regions. For modulation that appeared in the Inner and Outer regions, their respective outer and inner regions were filled with nonmodulated circular texture. Potentially when one texture is surrounded by another area of differing texture, the surrounded area can appear to perceptually “pop out.” This is known as texture segmentation and is the apparent effortless partition of a visual stimulus into distinct segments based on spatial gradients in local feature properties (Heinrich, Andrés, & Bach, 2007). Several properties have been observed to be crucial for texture segmentation to occur, for example, element orientation, size, and contrast (Landy & Bergen, 1991; Malik & Perona, 1990; Nothdurft, 1993; Nothdurft, Gallant, & Van Essen, 2000; Thielscher & Neumann, 2003). Nothdurft (1985) has previously shown that as the orientation contrast between two abutting textures is increased, performance at a shape discrimination task gets better, but when orientation contrast is small, shape discrimination is poorer. Further texture segmentation was observed to arise not as a result of grouping caused by homogeneity of features but rather by detection of boundaries defined by adequate contrast along an attribute dimension (Nothdurft, 1991). In that study, Nothdurft showed that if two flow textures had a similar background orientation contrast (20°), the pattern with the lower border contrast (30°) was not apparent but the one with the higher border contrast (90°) was. Landy and Bergen (1991) used filtered noise textures (instead of the linear elements employed by Nothdurft) but again showed that shape discrimination performance for a segmented region declined with smaller orientation contrast between abutting textures. They also made the point that only when a region of texture is perceived as having a well-defined shape has texture segregation succeeded (Landy & Bergen, 1991). In the current study, modulation in the Inner and Outer conditions was presented at threshold, making them very similar to circular (which is what the nonmodulated segments would contain), and consequently, because of inadequate orientation contrast, segregation would be poor between the two texture regions, making it unlikely that the texture border would be perceived as having a well-defined shape without detailed scrutiny, in which case texture segregation would not have succeeded. Furthermore, the border is always circular in our stimuli and thus not informative with regard to shape, although its presence does depend on the texture modulation on the two sides of the border. A Filter-Rectify-Filter (FRF) model can also be used (see also Bergen & Landy, 1991) to explain how a border might occur between two regions of textures. First, the first-order filters (e.g., sensitivity profiles similar to oriented Gabors) detect luminance variation of particular orientations and their response is then rectified (or squared to provide an energy estimate), and the second-order filters then detect the variation in oriented energy estimates across the border by comparing activity for different orientation combinations. In the context of our stimuli, this model would only compute where there was a difference in the two regions of textures, indicating the location of a border, but would not inform as to how the information about orientation modulation would sum across the border, our primary focus in the current study. 
Here, thresholds in the Both condition were compared with predictions calculated, to estimate whether the signal in the two regions summed linearly or if, instead, the two regions contributed independently. To examine whether the presence of a boundary would hinder summation across the two regions, a set of stimuli was created in which a path was inserted at the boundary between Inner and Outer (Path condition). Should thresholds (in each condition) decrease on the addition of the boundary, this would imply that regions that contained signal were being treated discretely, reducing the noise contributed by the other region. 
The presence of a boundary generally signifies the limits of an object, and therefore it would be reasonable for the visual system to integrate information about the texture “belonging” to the object, only within the object. However, if one region of texture did not “claim ownership” of the boundary, it is also possible that the boundary might be viewed as something partially obscuring a homogenous texture in the background. In this circumstance, might the visual system integrate the texture across the boundary with minimal change in sensitivity? 
A previous study (Dickinson, Broderick, & Badcock, 2009) using Glass patterns showed that if an observer was aware of the location of the signal (that appeared in annular areas), their performance in the task was enhanced. This was argued to be due to the observer's being able to ignore regions of the pattern that did not contain signal; thus, the noise associated with ignored regions was also disregarded. However, the authors noted that this enhanced performance could be eliminated by interleaving conditions in which signal locations varied. Interleaving conditions resulted in an uncertainty of signal location, and observers had to monitor the entire pattern to ensure that the signal would be detected wherever it appeared. In the current experiment, to prevent the enhanced performance described above from occurring, the Inner, Outer, and Both conditions were interleaved. By doing so, it was ensured that the observer would have to attend to the entire stimulus to complete the task. RF texture was observed to be globally processed around a texture (Tan et al., 2015), but the manipulation of confining signal to specific annular regions allows us to test if summation of the modulation signal would also occur across distinct annular areas. Based on the assumption that the amount of noise present in the stimulus was constant (because of the need to monitor the entire stimulus), predictions were made for whether the Inner and Outer regions were contributing independently or in tandem, when the two regions were not separated by a boundary. If the boundary did cause the two regions to segment, then the noise in each section might be considered independently in the signal to noise analysis (i.e., only the noise in one half would be considered rather than all of the noise over the whole pattern), and observed thresholds might fall in comparison to the condition without the boundary. In such an instance, Inner and Outer in the Path stimulus set might then be expected to sum probabilistically (i.e., the signal would not be added across the boundary). 
Previously it has been argued that integration in patterns composed of concentric sampled RF pattern is constrained to annuli (Schmidtmann, Gordon, Bennett, & Loffler, 2013). In this study examining RF textures, therefore, we adopted the hypothesis that signal would be integrated around but not across the annuli. This hypothesis makes the following predictions: 
  •  
    An explicit circular path demarcating the boundary between the two annuli will have no influence on performance. That is, performance for the Both condition will be predicted by a probabilistic combination of performance for the Inner and Outer conditions.
  •  
    Adding a phase difference between the modulation in the Inner and Outer annuli will cause no deterioration in performance.
  •  
    Integration will be demonstrated around the Inner, Outer, and Both conditions.
These predictions are critically examined in three experiments. 
General methods
Participants
Each experiment had four participants, with a total of five participants across the three experiments. Two of the five participants were authors; the others were naïve to the experimental aims. Three of the participants (two of whom were authors) participated in all three experiments. All observers possessed normal or corrected-to-normal visual acuity and were experienced psychophysical observers. With the exception of one author who has a divergent squint (with normal acuity in both eyes) who completed all experiments monocularly by covering one eye with an opaque eye patch, all other participants completed experiments binocularly. Informed consent was obtained prior to commencement. The treatment of participants in this study complied with the guidelines set by the Human Research Ethics Committee of University of Western Australia and was in accordance with the tenets of the Declaration of Helsinki. 
Apparatus
Matlab 7.0.4 (Mathworks, Natick, MA) was used to generate the custom stimuli used in this experiment. The program was run on an Intel Pentium 4 CPU 3.0 GHz (1024MB RAM) and drawn on a 256MB frame buffer of a Cambridge Research Systems (CRS) ViSaGe graphics system (Cambridge Research Systems, Kent, UK). Stimuli were presented on a Sony Trinitron Multiscan G520 Monitor (resolution: 1024 × 768 [34.13° × 25.6°], refresh rate: 100 Hz; Sony, Tokyo, Japan) at a viewing distance of 65.5 cm. A chin rest was used to maintain viewing distance; at this distance, each square pixel subtended 2′ of visual angle. A darkened room (ambient luminance <1 cd/m2) was used as the testing venue. An Optical OP 200-E photometer (head model No. 265) and associated software (CRS) was used to calibrate screen luminance. The background luminance to all stimuli was set at 45 cd/m2. A CRS CB6 button box was used to record observer responses. 
Stimuli
Stimuli were composed of Gabor patches (sinusoidal luminance gratings weighted by a Gaussian envelope) with gratings that had a carrier spatial frequency of 6 c/°. The Gaussian envelope had σ = 0.067°, giving a diameter of the envelope of each Gabor patch at half maximum contrast of 0.157°. A textured pattern was generated by placing a field of Gabor patches within a circular aperture that had a radius of 8.53° of visual angle. The structure of the texture conformed to that of an RF pattern, which can be described by the following equation:  where R(θ) is the radius of the pattern at an angle of θ relative to the positive x-axis; R0 is the base radius; A is the amplitude of sinusoidal modulation of a circle expressed as a proportion of R0; ω is the frequency of modulation (RF number), which was always five in these experiments; and φ the phase of the modulation, which was always randomized.  
For the RF texture, an angle of α was formed between the axis of the patches and the perpendicular to the radius of the pattern. This orientation, α, was specified by:    
Gabor patches were arranged systematically in an implicit square grid. The distance between grid lines was 0.27°. To minimize the appearance of a regular arrangement of patches, they were jittered at random within ±0.1° of their original position. Patterns had an average of 3172.4 (SD = 5.46) patches per stimulus; this uncertainty arose from the random jittering of elements, resulting in a variable number of patches falling within the specified annular stimulus region that was to be drawn. Patch density within the stimulus was therefore 14.1 patches per square degree. The phase of the grating in each patch was determined randomly to be either positive or negative sine phase (with the center of each patch therefore at background luminance). In instances in which patches overlapped, gratings were added. Although, due to patches being constrained to the cells of a Cartesian grid, the overlap was minimal and did not result in the creation of any additional shape cues informative to the task. 
The centers of the stimuli had a circular area from which patches were excluded, because there were fewer elements at the center, making it impossible to represent enough different orientations to keep the orientation distribution uniform. This also had the benefit of producing two annular regions for comparison. Patches were excluded only if their centers were within 1° of visual angle of the center of the pattern, and because the position of elements was jittered in the vertical and horizontal, excluding this area did not produce any informative modulated contour at the inner edge of the textures. In Experiments 1 and 2, the number of CoM was always five cycles of an RF5 texture. In Experiment 3, the number of CoM varied between 1–5 CoM. 
The stimulus was always divided into two regions: an inner and outer region. In all experiments, the two annuli were matched for area. This choice was based on the premise that if noise was area dependent or number of elements dependent, similar-sized regions would contribute equally. The widths of the annuli containing signal for the Inner and Outer annular regions were 5.08° and 2.45°, respectively (see Figure 1 for graphical representation of this). 
Figure 1
 
Graphical representation of the annular regions used in the experiments. The colored overlays are to indicate the regions of Inner (red) and Outer (yellow) and were not present in the actual stimuli.
Figure 1
 
Graphical representation of the annular regions used in the experiments. The colored overlays are to indicate the regions of Inner (red) and Outer (yellow) and were not present in the actual stimuli.
Experiment 1
In Experiment 1, there were three conditions (Inner, Outer, Both) and two stimulus sets (Path, No Path). For the test pattern, RF modulation was introduced to either (a) only the Inner annulus, (b) only the Outer annulus, or (c) in Both annuli. The reference pattern always had concentric texture in both regions. The phase of the modulation was randomized across trials to prevent observers knowing where local deviations were most likely to be large. When modulation was introduced in either the inner or outer regions only, the adjacent unmodulated region contained concentric texture. When modulation occurred in both regions together, the modulation in the two regions was always in the same phase. In the Path set, a circle was drawn at the boundary between the Inner and Outer areas. A circular boundary was chosen as it did not contain any modulated shape information that might interfere with the judgment of modulation of the texture. The luminance profile of a section through the boundary between the two regions of texture was a D4 function (fourth derivative of a Gaussian; see Figure 2) with a peak spatial frequency of 6 c/° (same as the Gabor patches to maximize potential for co-processing). In the No Path stimulus set, a path did not exist on the boundary between the two regions, and only the threshold-level orientation modulation of the elements could provide an indication of the location of the boundary. 
Figure 2
 
Examples of stimuli used in Experiment 1. Both describes the condition in which modulation information was inserted into the two regions. Inner and Outer describe conditions in which modulation was inserted into the equally sized inner or outer annulus, respectively. Examples in the top row were in the No Path set and did not contain a D4 path at the boundary between the inner and outer area. Examples in the bottom row were in the Path set and contained a D4 circle at the boundary between the two regions. Examples are presented at a modulation level above threshold for ease of viewing.
Figure 2
 
Examples of stimuli used in Experiment 1. Both describes the condition in which modulation information was inserted into the two regions. Inner and Outer describe conditions in which modulation was inserted into the equally sized inner or outer annulus, respectively. Examples in the top row were in the No Path set and did not contain a D4 path at the boundary between the inner and outer area. Examples in the bottom row were in the Path set and contained a D4 circle at the boundary between the two regions. Examples are presented at a modulation level above threshold for ease of viewing.
Procedure
The Path and No Path sets were run in separate blocks, and the Inner, Outer, and Both conditions within each block were interleaved. In a two-interval forced choice task, participants were asked to indicate which interval contained the pattern they perceived to be most deformed from circular using a CB6 button box. As mentioned, performance can be enhanced when the location of signal elements is known to the observer prior to the experiment (for Glass patterns), but this effect can be removed by interleaving (Dickinson et al., 2009). Presentation order for the test and reference patterns was randomized. Patterns appeared sequentially and were on screen for 160 ms with an interstimulus interval of 500 ms. 
The method of constant stimuli was used to control stimulus presentation. There were nine amplitudes of test modulation (A in Equations 1 and 2), and 60 trials were completed for each of the nine amplitudes (for each of the conditions). This resulted in 540 trials per condition and 1,620 trials (for all three conditions) per block. Runs were self-paced with breaks as necessary. 
Results
The proportion of correct responses for each of the nine test amplitudes for each condition was calculated, and detection thresholds were obtained by fitting a Quick (1974) function:  where p(A) is the probability of correct responses for an amplitude A, Δ is the threshold in A for the discrimination of the test from the reference pattern at a 75% correct performance level, and Q is a measure of the slope of the psychometric function (Quick, 1974; Wilson, 1980). Two predictions were generated for determining whether information was pooled across area radially or whether information in distinct annular areas was treated independently. A prediction based on the possibility of linear summation of the signal in the inner and outer areas (Addition prediction) was derived from  where Δinner and Δouter are the Δ obtained from Equation 3 for inner and outer. A second prediction indicating expected performance when both regions were contributing independently to the threshold of the combined area (Independent prediction) was derived from    
The basis for this is that discrete variables that contribute independently do so in the manner of an orthogonal vector sum (Massof & Starr, 1980). As conditions were interleaved, the observer would have had to monitor the entire display (to adequately complete the task); as such, the viewing area that was to be considered for the Inner and Outer conditions was the same, and consequently, the associated noise was assumed to be constant. Addition and Independent predictions were made on this premise. 
The thresholds for the three conditions in both sets are similar (see Figure 3): Inner, M = 0.0136, 95% confidence interval (CI) [0.0094, 0.0178]; Outer, M = 0.0185, 95% CI [0.0143, 0.0226]; Both, M = 0.0108, 95% CI [0.0082, 0.0134], with substantial overlap in their confidence intervals. A 2 × 3 (Set × Condition) repeated-measure analysis of variance (ANOVA) showed that there was no significant main effect of Set (i.e., path presence), F(1, 3) = 2.26, p = 0.23, but there was a significant main effect of Condition, F(2, 6) = 8.12, p = 0.02. No interaction effects were observed, F(2, 6) = 0.76, p = 0.51. A post hoc Bonferroni multiple comparisons test showed a difference of means between Outer and Both, t(6) = 3.98, p = 0.02, but not for Inner versus Outer, t(6) = 2.54, p = 0.13, or Inner versus Both, t(6) = 1.45, p = 0.60. The lack of main effect for Set might have been due to one of two things: (a) the signal in the entire area was being treated as a whole, but the visual system might be ignoring the path that was present, or (b) the signal in each region was being treated independently, and as such there would be no interference from the path. The two Both conditions were combined (because no difference between Set was observed) and tested against their respective combination-type predictions (Equations 4 and 5). The two predictions were compared with a paired-samples t test and observed to be different from each other, t(7) = 8.76, p < 0.0001, d = 1.11 (see Figure 4). The combined Both condition was observed to be significantly higher than the Addition prediction (Equation 4), t(7) = 5.29, p = 0.001, d = 1.16, but not different from the Independent prediction (Equation 5), t(7) = 0.23, p = 0.828, d = 0.042. 
Figure 3
 
The threshold amplitude of modulation is plotted for the four participants for Path (filled symbols) and No Path (open symbols) sets for all three conditions. The black horizontal lines represent the mean thresholds (with 95% confidence intervals). Corresponding thresholds for Inner, Outer, and Both in each set were not statistically different from each other.
Figure 3
 
The threshold amplitude of modulation is plotted for the four participants for Path (filled symbols) and No Path (open symbols) sets for all three conditions. The black horizontal lines represent the mean thresholds (with 95% confidence intervals). Corresponding thresholds for Inner, Outer, and Both in each set were not statistically different from each other.
Figure 4
 
The threshold amplitude of modulation is plotted for the four participants for the combined Both condition and calculated predictions. Means are shown by the horizontal black bar (with 95% confidence intervals). The Both condition was statistically different from the Addition prediction but not the Independent prediction.
Figure 4
 
The threshold amplitude of modulation is plotted for the four participants for the combined Both condition and calculated predictions. Means are shown by the horizontal black bar (with 95% confidence intervals). The Both condition was statistically different from the Addition prediction but not the Independent prediction.
Discussion
The aim of the experiment was to investigate how the signal in an orientation modulated texture would sum across annular areas and if the introduction of a boundary between these areas would alter the summation across area. If the boundary between Inner and Outer caused an obligatory segregation of the areas, then thresholds for Inner and Outer would be expected to be lower for Path than for the No Path. This would be due to the noise of each region being analyzed independently (in the signal to noise analysis). Consequently, Path: Both would then be expected to adhere to the Independent prediction. However, this result of lower thresholds for Path versus No Path was not observed because no difference between Sets was found. Thresholds did not depend on the presence or absence of the path, which might be because the path did not segregate the textures, and we must hence assume that the noise is constant and is derived from the whole field or at least the same area of visual field in all conditions. It was possible that the similar thresholds for the Path and No Path conditions arose because the boundary might have been functioning as an overlying object partially occluding a continuous field of texture instead of a bounding limit for an object. When a target object is partially obscured by other objects in the foreground, the visual system is often able to infer that one part of the target object is connected to the other part even though it was partially obscured from view. In the case of our experiment, although the boundary does not obscure very much, it might have been viewed in the same manner as an occluder; as such, the noise across Inner, Outer and Both would be constant (as is also the case when no boundary is present), and our finding of the same result as that obtained in the No Path stimulus might be expected. However, even without a boundary present (No Path: Both), there was no evidence for linear summation of signal in the Both condition. This perhaps suggests that detectors for RF texture are annular in shape (see also Schmidtmann et al., 2013). 
As mentioned, an FRF model could be used to make predictions regarding how two regions of texture were to segment. Comparing modulated texture on one side of the border with unmodulated texture on the other could produce a modulated second-order signal because the orientation difference on the two sides of the border would vary. If the textural modulation at the edge of the Inner and Outer regions was sufficient to segment (even weakly), then independent contributions to the Both threshold might be expected, and the addition of a line might have no impact because the regions would already have been segmented. However, the evidence against this is that if the same border is being used in Experiment 1, the Inner and Outer conditions are essentially the same target (i.e., modulation on one side of the boundary or the other but essentially the same place), and yet the thresholds were quite different—so we conclude that performance is not explained by this mechanism. 
Experiment 2
The lack of linear summation of modulation signal across the Inner and Outer regions observed in Experiment 1 seemed to suggest that modulation information was not summed across radii. This implies that signal is integrated around the inner and outer annuli but not across the two regions. The improvement in threshold in Both is then assumed to be due to the statistical improvement in the probability of detection of signal in one of the two annuli when signal is present in both annuli. If this were the case, we would expect the thresholds for detection of signal across the whole stimulus to be insensitive to a displacement of the phase of modulation across the two annuli. We proceeded to investigate this possibility in Experiment 2
In this experiment, we manipulate the phase difference between the modulation in the Inner and Outer annuli. Modulation signal in the two annuli was presented such that it was (a) in the same phase, (b) 90° out of phase, or (c) 180° out of phase. If information was summed across radius, it might be expected that the in-phase stimulus condition would have lower threshold than the out-of-phase conditions. If the thresholds prove to be the same across these three conditions, this would provide corroborating evidence for the conclusions of Experiment 1
Stimulus
Detailed stimulus creation is outlined in the General methods section. Three conditions were run: Inphase, Outphase 90, and Outphase 180 (see Figure 5). In the Outphase conditions, modulation in the outer annulus was offset in phase by either 90° or 180°. The reference pattern was a circular texture for all conditions. All other stimulus details are the same as in the No Path set of Experiment 1
Figure 5
 
Example of stimuli used in Experiment 2. (A) Inphase: Modulation in the inner and outer regions is in phase. (B) Outphase 90: Modulation in the inner and outer regions is out of phase by 90°. (C) Outphase 180: Modulation in the inner and outer regions is out of phase by 180°. The number of CoM is always five. Examples are presented at a modulation level above threshold for ease of viewing.
Figure 5
 
Example of stimuli used in Experiment 2. (A) Inphase: Modulation in the inner and outer regions is in phase. (B) Outphase 90: Modulation in the inner and outer regions is out of phase by 90°. (C) Outphase 180: Modulation in the inner and outer regions is out of phase by 180°. The number of CoM is always five. Examples are presented at a modulation level above threshold for ease of viewing.
Procedure
This experiment employed the same procedure as Experiment 1. The three conditions were interleaved. As with Experiment 1, this resulted in 540 trials per condition and 1,620 trials (for all three conditions). 
Results
As before, a Quick (1974) function was fitted to the data to obtain thresholds. Thresholds from the three conditions were then compared with a one-way repeated-measure ANOVA. The analysis reflects the findings that there was no effect of condition, F(2, 3) = 0.925, p = 0.41 (see Figure 6). 
Figure 6
 
The amplitude modulation thresholds required for discrimination of all four participants are plotted for Inphase, Outphase 90, and Outphase 180 conditions. The black horizontal bars represent mean thresholds (with 95% confidence intervals). Inphase, Outphase 90, and Outphase 180 thresholds are not statistically different from each other.
Figure 6
 
The amplitude modulation thresholds required for discrimination of all four participants are plotted for Inphase, Outphase 90, and Outphase 180 conditions. The black horizontal bars represent mean thresholds (with 95% confidence intervals). Inphase, Outphase 90, and Outphase 180 thresholds are not statistically different from each other.
Discussion
In the previous experiment, it was demonstrated that annular regions of texture made independent contributions to the thresholds for detection of a stimulus that had modulation in more than one annulus. This result implies that such thresholds would be robust to the introduction of a phase difference between the modulation in two annuli. The current experiment manipulated the phase difference between the modulation in the inner and outer annuli and found that the detection thresholds for modulation in stimuli with 90° and 180° phase differences were not different from those found for stimuli with no phase difference between modulation in the two annuli. This result supports the conclusions of Experiment 1, that the modulation within the two annuli is analyzed independently. 
Experiment 3
Introduction
The findings of Experiment 1 and 2 suggested that when annular regions of modulated texture were present, each region contributed independently to the thresholds for detection of modulation within the whole texture. We started off the experiments by using textures we knew to be globally processed from a previous study (Tan et al., 2015), but in that study, the modulation was distributed across the whole stimulus. To be certain that integration was still occurring around the annuli, Experiment 3 measured integration around the annuli when the modulation was constrained to one or the other of the two annuli and when it was present across the whole stimulus. As an additional control, a parallel set of conditions was examined in which the annuli that would otherwise have had circular textures were left blank. That is, the circular texture was not presented. 
As has been shown previously (Tan et al., 2015), as larger sectors of modulation contribute to the RF texture, discrimination thresholds decrease. Some decrease can be expected when an increased number of local deformed sectors are presented due to PS of near threshold signals. In this instance, PS is defined as the increased probability of detecting local deformation from circular as more CoM are progressively introduced. Thresholds obtained from observer data tend to fall at a rate steeper than that predicted by PS (as CoM are added), and global pooling is evoked as a reason for this superior performance (Dickinson et al., 2012; Loffler et al., 2003; Tan et al., 2013; Tan et al., 2015). 
In this experiment, we ran two sets of stimuli, All and Only. The All set was similar to the stimuli used in Experiment 1 in that all of the Gabor patches were present. RF signal modulation was then presented in either the Inner, Outer, or Both regions; regions that did not contain RF signal modulation remained circular. In the Only set, the area of the regions that contained RF modulation were the same as the All set, except that the Gabors in the regions that did not contain RF signal modulation were removed. By comparing the two, we can use this as a test as to whether global integration is disrupted when some of the pattern carries no integratable signal. 
Stimulus
Detailed stimulus creation is laid out in the General methods, and the same method was used to smooth partially modulated patterns as in Tan et al. (2015). Two sets of stimuli were used in this experiment, an all set and an only set. The all set was in essence the same as the stimuli used in Experiment 1, whereas the only set had segments in which nonsignal Gabor elements were removed. In the inner only condition, the same area containing signal (as inner all) was employed; however, Gabor patches in the outer region were removed. The converse is also true for the outer only condition, in which Gabor patches in the inner region were removed. Both only was exactly the same as both all and was named only to differentiate that it was paired with the other two only conditions. Unlike in Experiment 1, in which only five CoM were employed, one, two, three, and five CoM were systematically introduced for each region. 
Procedure
The procedure was similar to that employed in the previous two experiments. There were two blocks of interleaved trials (All set, Only set). A total of 12 conditions were interleaved per set, four different CoM (1, 2, 3, 5 CoM) by three regions (Inner, Outer, Both). As in the previous two experiments, this resulted in 540 trials per CoM and 2,160 trials per region, which totaled 6,480 trials per block. 
Results
As before, a Quick (1974) function was fitted to the psychometric data to obtain thresholds. These thresholds were observed to conform to a power function when plotted against CoM (see Figure 7). The PS estimated here is derived from the slope parameter Q of the Quick function (see Equation 3). The rate of decrease of threshold is predicted to fall according to a power function with an index of −1/Q. This is the conventionally used method in examining the integration of shape information in RF patterns and also one that was used recently to examine the integration in RF textures (Bell & Badcock, 2008; Dickinson, Almeida, Bell, & Badcock, 2010; Dickinson et al., 2012; Loffler et al., 2003; Schmidtmann, Kennedy, Orbach, & Loffler, 2012; Tan et al., 2013; Tan et al., 2015; Tan et al., 2016). 
Figure 7
 
Threshold versus CoM with 95% confidence intervals plotted on logarithmic axes for four participants for all conditions with their respective probability summation predictions. Participants are sorted horizontally. The darker colors represent the All set, whereas the lighter colors represent the Only set. Probability summation predictions for each are represented by the dashed lines in the appropriate colors. All slopes were observed to be steeper than their respective probability summation estimates.
Figure 7
 
Threshold versus CoM with 95% confidence intervals plotted on logarithmic axes for four participants for all conditions with their respective probability summation predictions. Participants are sorted horizontally. The darker colors represent the All set, whereas the lighter colors represent the Only set. Probability summation predictions for each are represented by the dashed lines in the appropriate colors. All slopes were observed to be steeper than their respective probability summation estimates.
Six separate paired-samples t tests were run, and it was observed that slopes for all the conditions were steeper than their respective PS estimates, t(3) = 7.49, p = 0.0049 (Inner All), t(3) = 7.64, p = 0.0047 (Inner Only), t(3) = 8.25, p = 0.0037 (Outer All), t(3) = 3.36, p = 0.044 (Outer Only), t(3) = 7.06, p = 0.0058 (Both All), t(3) = 3.21, p = 0.049 (Both Only). Slope values and their respective PS predictions are shown in Table 1. Further paired-samples t tests were done comparing the slopes between the All and Only sets for the Inner, Outer, and Both conditions, and it was found that there were no significant differences between them: t(3) = 1.07, p = 0.36 (Inner), t(3) = 0.52, p = 0.64 (Outer), t(3) = 1.21, p = 0.31 (Both). 
Table 1
 
Slope values (and their 95% confidence intervals) for each participant for the different conditions and their respective PS estimates.
Table 1
 
Slope values (and their 95% confidence intervals) for each participant for the different conditions and their respective PS estimates.
Recently, there has been suggestions by Kingdom, Baldwin, and Schmidtmann (2015) and Baldwin, Schmidtmann, Kingdom, and Hess (2016) that signal detection theory (SDT) should be used to derive PS estimates. They argue that the conventional method for the calculation of PS (which we use in the current study) calculated by setting the index of the power function to the reciprocal of Q from Equation 3 should be rejected as it is based on considerations arising from high threshold theory (HTT). Following their work, we have also estimated thresholds using their methods, which are detailed in Kingdom and Prins (2016), by employing the Palamedes toolbox for Matlab (Prins & Kingdom, 2009). Using this method, the psychophysical data representing percentage correct performances were converted to d′ (using the Palamedes function PAL_SDT_2AFC_PCtoDP). These values were then fitted to a psychometric function: d′(gA)τ to determine the parameters g and τ. The fitted function allowed for the estimation of a 75% threshold and the ability to ascertain a slope describing the decrease in thresholds as CoM increased (based on this form of the psychometric function). 
A repeated-measures ANOVA was conducted to compare the SDT calculated 75% thresholds and their conventionally calculated counterparts, F(11, 3) = 30.47, p = 0.006. Although the ANOVA showed that an effect was present, further Tukey's multiple comparisons (alpha level = 0.05) showed that each corresponding 75% threshold calculated using the Quick function was not significantly different from its SDT-calculated 75% threshold counterpart for all relevant conditions (e.g., Inner All vs. Inner Only). 
The PAL_SDT_PS_PCtoSL function was used to estimate the amplitude (A) thresholds that would arise from PS among channels with the calculated properties. Since we employed random phase stimuli, and Dickinson et al. (2012) suggest that the critical cue for deformation thresholds are very local, it is reasonable to assume a large number of local channels may need to be simultaneously monitored. One possibility is that each local Gabor could represent a channel with the system needing to detect when its local orientation deviates from that expected in a circular pattern. That would require the monitoring of 3,172 channels (the number of Gabors in the stimulus). Another possibility is that signal may be restricted to 1 CoM, but with random phase stimuli, that would still require monitoring many locations. If we assume a channel for every degree of pattern rotation, then allowing for repetition on each CoM, the minimum number of channels monitored would be 72. Because the slope of the estimated PS function increases with reducing channel number, it is important to include this minimum value. Figure 8 shows the integration slopes derived from the observers' data and those predicted by PS for the two different proposed numbers of channels (3,175 channels was used instead of 3,172 because of rounding 1 CoM to the nearest whole number, i.e., 635 channels). Table 2 presents the slope values for the data and the PS estimates with 95% confidence intervals. 
Figure 8
 
Slope values for each condition and their respective PS predictions for 72 and 3,175 channels (and their 95% confidence intervals). The horizontal black line represents the mean. With the exception of Outer Only, all other conditions are statistically different from their respective two PS predictions.
Figure 8
 
Slope values for each condition and their respective PS predictions for 72 and 3,175 channels (and their 95% confidence intervals). The horizontal black line represents the mean. With the exception of Outer Only, all other conditions are statistically different from their respective two PS predictions.
Table 2
 
Mean slope values for all conditions and their respective probability summation estimates for 72 and 3,175 channels (with 95% confidence intervals).
Table 2
 
Mean slope values for all conditions and their respective probability summation estimates for 72 and 3,175 channels (with 95% confidence intervals).
Visual inspection of the data from Figure 8 shows that the mean slopes appear to be steeper than the two PS predictions. Repeated-measures ANOVA comparing these slopes to their 72 and 3,175 channel PS estimates showed these differences were significant, F(2, 3) = 119.5, p = 0.002 (Inner All); F(2, 3) = 42.3, p = 0.007 (Outer All); F(2, 3) = 71.9, p = 0.003 (Both All); F(2, 3) = 69.5, p = 0.003 (Inner Only); F(2, 3) = 11.36, p = 0.04 (Outer Only); F(2, 3) = 42.7, p = 0.007 (Both Only). Further multiple comparisons showed the data were significantly steeper than the two PS estimates. The only exception was the Outer Only condition, in which the differences in the slopes and the PS predictions did not reach statistical significance. The difference between means is similar in this condition, but the variability is greater. Therefore, it does not suggest that a different conclusion is strongly indicated for this condition. 
Because the slopes given by the data exceeded those predicted by PS on three accounts, it was concluded, once again, that the modulation in the textures used was being globally integrated. Addition and Independent predictions were then calculated for comparison (see Equations 4 and 5). To recapitulate, the two predictions refer to the manner in which signals from the Inner and Outer regions are combined. Predictions for Both All were calculated from thresholds for Inner All and Outer All, whereas predictions for Both Only were calculated from thresholds for Inner Only and Outer Only. 
The thresholds for the two Both sets were collapsed (since they were essentially the same stimulus) and compared with the calculated predictions (as described in Equations 4 and 5). A one-way repeated-measures ANOVA showed the presence of a significant effect, F(2, 7) = 29.55, p = 0.0003. Tukey's multiple comparison (alpha level = 0.05) showed that the combined Both was significantly different from the Addition prediction but not from the Independent prediction. The two predictions were also seen to be significantly different from each other. 
Discussion
This experiment was run as a check of our assertions that the textures we used in this study were globally processed despite the texture being constrained to particular annuli. Using the conventional Quick method for calculating PS, we found that the integration slopes for all conditions were steeper than their corresponding PS estimate, indicating global processing. Given that recent works (Baldwin et al., 2016; Kingdom et al., 2015) have suggested that an alternative method for calculating PS estimates derived from SDT might be more appropriate, this methodology was also used to calculate thresholds and PS estimates. The estimated improvement from PS in this method changes with the number of monitored channels. Therefore, two estimates for what might be considered local processing were generated using this method, one for 72 channels and one for 3,175 channels (justified earlier). Once again, we observed that the slopes from the data were significantly steeper than the PS estimates calculated, again indicating global processing. One exception was the outer only condition, but this appears to be a statistical aberration as the difference between condition means is similar whereas the variability between observers is greater. However, given that all the other conditions showed a similar trend (i.e., steeper than PS) when comparing their Quick-derived PS estimates, it seems unlikely that Outer Only was the only condition that was locally processed, and it was more likely that this was down to statistical noise. 
Addition and Independent combination rule predictions were generated for the Both data sets, based on the Inner and Outer condition, compared with the data, and as in Experiment 1, the data matched predictions for an Independent prediction, suggesting that each region's contribution did not sum linearly for threshold of the whole texture.1 
General discussion
Texture might be considered a surface property. In the context of visual processing, texture is exploited for a number of purposes in the analysis of a scene. A priority for the visual system is turning low-level luminance properties of the visual field into information that can be acted upon in the parsing of the visual field into objects. That this is often achieved through the recognition of boundaries that define the extent of those objects can be inferred from the observation that line drawings of a scene are often not judged as impoverished in meaning (Kourtzi & Kanwisher, 2000). A contrast in texture, however, can serve to delineate the boundary of an object (Bergen, 1991; Heinrich et al., 2007; Julesz, 1981; Tan et al., 2013), and the shape of the boundary can subsequently be analyzed to identify that object (Biederman, 1987; Kourtzi & Kanwisher, 2000). This process does not require a complex analysis of the textures either side of the boundary but requires only that there is sufficient contrast in texture at the boundary around the object (Nothdurft, 1985, 1991). Tan et al. (2015) showed that the visual system performs a global analysis of shape on boundaries defined by a contrast in texture that is as efficient as that performed on boundaries defined by explicit luminance defined paths. Moreover, if a texture contrast exists across an explicit luminance-defined boundary, then the texture contrast and explicit boundaries contribute independently to the shape analysis. 
Specific textures can also be generated on the visual field due to the finite integration times of the orientation selective neurons of the primary visual cortex. The motion of salient points in the visual field leaves streaks as their position is integrated over time. Transformations such as a scaling of the visual field that would occur as an observer moved through their environment result in a whole field textures that have a coherent flow. Textures that imply motion in this way have been referred to as static flow (Kovács & Julesz, 1993). The orientation fields examined in this study cannot easily be attributed to transformation of all or parts of the visual field due to motion through the environment or to motion of objects within the environment. They can, however, be regarded as texture counterparts of RF patterns. RF patterns have been used to demonstrate the integration of shape information around closed paths. The textures employed in this study show the same integration of information around the texture (Tan et al., 2015), but they cannot constitute a path from which shape can be derived. The signal that is integrated around the pattern must be derived in some way from the modulation of orientation. Tan et al. (2015) also demonstrated that modulation of orientation from radial is not integrated around the pattern, but modulation from concentric is. This observation perhaps provides a clue to the utility of the approximately concentric orientation texture. The boundaries of any object that has some depth in the visual field will be perpendicular to the frontal plane. The orientation of any texture on the surface of that object close to its boundary will then be biased toward concentric. Perhaps, then, analysis of the texture of an object might serve to approximate shape analysis. If this were the case, it might be assumed that the integration of orientation information might be somewhat constrained in its radial extent and also that, if it does provide an alternative measure of shape to that provided by an explicit boundary, it should be robust to the competing influence of such boundaries. 
This study therefore sought to determine how the signal in a globally processed texture would sum over area and whether the presence of a segmenting boundary would interfere with any potential summation. In Experiment 1, it was observed that thresholds for the equal-area annuli were unaffected when a boundary was added between the two regions and that the two regions made independent contributions to detection of modulation when the whole stimulus was modulated. In Experiment 2, we observed that altering the phase of modulation for one of the annuli with respect to the other did not affect thresholds, again suggesting that the visual system is using information from each annulus independently. In Experiment 3, we reassessed whether the textures we used were globally processed. Using two different methods for calculating PS estimates, we observed that thresholds fell at a rate that exceeded PS as CoM was increased on both accounts, suggesting global processing. When the data were again compared to predictions for how information would sum across a radius, the results were replicated, and they matched predictions for independent contributions. 
Schmidtmann et al. (2013) recently employed a stimulus composed of arrays of Gabor patches consisting of multiple concentric rings. They attempted to investigate how form structure was detected in noise. They proposed that this might either be the result of a global shape mechanism or a global texture (such as that of a Glass pattern detector) mechanism and sought to differentiate between the two by manipulating the location of signal elements. They presented two stimuli in succession (one with signal, one without), and observers were tasked to select the pattern they thought contained signal. Signal elements were either constrained to a specific path or spread across multiple concentric paths. The authors argued that if the visual system used a texture detector (such as a Glass pattern detector) to process such stimuli, then thresholds should not depend on how the signal was distributed across the pattern because a Glass pattern detector would be insensitive to signal position. If, however, there was a difference between having signal constrained to specific annuli versus spread across annuli or paths, it would indicate that the visual system was using specific form detectors, as these necessarily had constraints on element orientation and position. They found that fewer signal elements were required for detection if signal was constrained on a single path than if signal was distributed across multiple paths. Schmidtmann et al. argued that information in their stimuli was most efficiently integrated when signal fell on individual contours. They tested this against two predictions, one in which there were multiple shape detectors simultaneously processing the stimulus where the output from the detectors summed linearly prior to their individual thresholds (that is, they argued similar to a texture detector) and another in which all of the independent detectors were subject to individual noise and the resultant performance was given by probabilistic summation of their outputs. They observed that the latter prediction was closer to but slightly underestimated the performance observed when signal was spread across annuli. They argued that PS was a subideal strategy and that other strategies would predict slightly increased performance, but nonetheless, irrespective of strategy used, a single process that summed information across the display was inconsistent with their data. 
They noted that this did not argue against the existence of texture mechanisms but rather suggested that, in some contexts, texture mechanisms might be less sensitive than shape detectors and that depending on the arrangement of elements in the texture, the detectability may be limited by multiple shape processes. 
Like Schmidtmann et al. (2013), we do not observe linear summation of signal across annular regions. Although we have suggested that RF textures are processed by detectors that are different from those used for both RF contour patterns and Glass patterns, by virtue of the difference in what needs to be detected, we have found in this study and a previous study (Tan et al., 2015) that shape information is pooled around an RF texture despite the lack of precise radial positional placement (as in a conventional RF pattern). However, as shown in this study, we fail to observe linear summation of information across radially placed annular areas (as would be expected in Glass patterns, e.g., Dickinson et al., 2009; Wilson et al., 1997) and instead show that the visual system sums two regions of texture probabilistically. One plausible explanation for this might be that an RF texture detector that detects shape information is annular in shape, unlike what might be expected from other texture detectors such as a Glass pattern detector. In support of this notion, Dickinson et al. (2009) have found that the analysis of Glass patterns can be constrained to annular regions by selective attention to these regions, while here we observe that RF textures seem to operate only within annular regions, which leads to the suggestion of a difference between RF textures and Glass patterns. It is possible that the RF detector and RF texture detector would be used by the visual system to deal with information regarding shape and texture relevant to shape, whereas the Glass pattern detector might be more for visual information relating to translational movements (Badcock & Dickinson, 2009; Barlow & Olshausen, 2004). 
In summary, based on our findings, we conclude that signal in globally processed modulated textures does not sum across annuli, and this is consistent with the lack of signal summation across annuli of concentric paths of RF patterns as observed by Schmidtmann et al. (2013). 
Acknowledgments
This research was supported by an Australian Research Council Grant DP1097003, DP110104553, and DP130102580 to D. R. B. and an SIRF scholarship funded by the University of Western Australia to K. W. S. T. 
Commercial relationships: none. 
Corresponding authors: Ken W. S. Tan and David R. Badcock. 
Email: ken.tan@research.uwa.edu.au and david.badcock@uwa.edu.au. 
Address: The University of Western Australia,School of Psychology, Crawley, Australia. 
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Footnotes
1  While our focus was on the combination of signal from the two regions of the stimulus, and the explicit tests lead to the conclusion that the signal regions make independent contributions, it is important to note that adding noise outside the target region did degrade performance for the Inner condition. No such change is seen for the Outer conditions (see Figure 7). This may mean that the annular regions processed were less well matched to the stimuli for the Inner targets. This needs to be the subject of future research.
Figure 1
 
Graphical representation of the annular regions used in the experiments. The colored overlays are to indicate the regions of Inner (red) and Outer (yellow) and were not present in the actual stimuli.
Figure 1
 
Graphical representation of the annular regions used in the experiments. The colored overlays are to indicate the regions of Inner (red) and Outer (yellow) and were not present in the actual stimuli.
Figure 2
 
Examples of stimuli used in Experiment 1. Both describes the condition in which modulation information was inserted into the two regions. Inner and Outer describe conditions in which modulation was inserted into the equally sized inner or outer annulus, respectively. Examples in the top row were in the No Path set and did not contain a D4 path at the boundary between the inner and outer area. Examples in the bottom row were in the Path set and contained a D4 circle at the boundary between the two regions. Examples are presented at a modulation level above threshold for ease of viewing.
Figure 2
 
Examples of stimuli used in Experiment 1. Both describes the condition in which modulation information was inserted into the two regions. Inner and Outer describe conditions in which modulation was inserted into the equally sized inner or outer annulus, respectively. Examples in the top row were in the No Path set and did not contain a D4 path at the boundary between the inner and outer area. Examples in the bottom row were in the Path set and contained a D4 circle at the boundary between the two regions. Examples are presented at a modulation level above threshold for ease of viewing.
Figure 3
 
The threshold amplitude of modulation is plotted for the four participants for Path (filled symbols) and No Path (open symbols) sets for all three conditions. The black horizontal lines represent the mean thresholds (with 95% confidence intervals). Corresponding thresholds for Inner, Outer, and Both in each set were not statistically different from each other.
Figure 3
 
The threshold amplitude of modulation is plotted for the four participants for Path (filled symbols) and No Path (open symbols) sets for all three conditions. The black horizontal lines represent the mean thresholds (with 95% confidence intervals). Corresponding thresholds for Inner, Outer, and Both in each set were not statistically different from each other.
Figure 4
 
The threshold amplitude of modulation is plotted for the four participants for the combined Both condition and calculated predictions. Means are shown by the horizontal black bar (with 95% confidence intervals). The Both condition was statistically different from the Addition prediction but not the Independent prediction.
Figure 4
 
The threshold amplitude of modulation is plotted for the four participants for the combined Both condition and calculated predictions. Means are shown by the horizontal black bar (with 95% confidence intervals). The Both condition was statistically different from the Addition prediction but not the Independent prediction.
Figure 5
 
Example of stimuli used in Experiment 2. (A) Inphase: Modulation in the inner and outer regions is in phase. (B) Outphase 90: Modulation in the inner and outer regions is out of phase by 90°. (C) Outphase 180: Modulation in the inner and outer regions is out of phase by 180°. The number of CoM is always five. Examples are presented at a modulation level above threshold for ease of viewing.
Figure 5
 
Example of stimuli used in Experiment 2. (A) Inphase: Modulation in the inner and outer regions is in phase. (B) Outphase 90: Modulation in the inner and outer regions is out of phase by 90°. (C) Outphase 180: Modulation in the inner and outer regions is out of phase by 180°. The number of CoM is always five. Examples are presented at a modulation level above threshold for ease of viewing.
Figure 6
 
The amplitude modulation thresholds required for discrimination of all four participants are plotted for Inphase, Outphase 90, and Outphase 180 conditions. The black horizontal bars represent mean thresholds (with 95% confidence intervals). Inphase, Outphase 90, and Outphase 180 thresholds are not statistically different from each other.
Figure 6
 
The amplitude modulation thresholds required for discrimination of all four participants are plotted for Inphase, Outphase 90, and Outphase 180 conditions. The black horizontal bars represent mean thresholds (with 95% confidence intervals). Inphase, Outphase 90, and Outphase 180 thresholds are not statistically different from each other.
Figure 7
 
Threshold versus CoM with 95% confidence intervals plotted on logarithmic axes for four participants for all conditions with their respective probability summation predictions. Participants are sorted horizontally. The darker colors represent the All set, whereas the lighter colors represent the Only set. Probability summation predictions for each are represented by the dashed lines in the appropriate colors. All slopes were observed to be steeper than their respective probability summation estimates.
Figure 7
 
Threshold versus CoM with 95% confidence intervals plotted on logarithmic axes for four participants for all conditions with their respective probability summation predictions. Participants are sorted horizontally. The darker colors represent the All set, whereas the lighter colors represent the Only set. Probability summation predictions for each are represented by the dashed lines in the appropriate colors. All slopes were observed to be steeper than their respective probability summation estimates.
Figure 8
 
Slope values for each condition and their respective PS predictions for 72 and 3,175 channels (and their 95% confidence intervals). The horizontal black line represents the mean. With the exception of Outer Only, all other conditions are statistically different from their respective two PS predictions.
Figure 8
 
Slope values for each condition and their respective PS predictions for 72 and 3,175 channels (and their 95% confidence intervals). The horizontal black line represents the mean. With the exception of Outer Only, all other conditions are statistically different from their respective two PS predictions.
Table 1
 
Slope values (and their 95% confidence intervals) for each participant for the different conditions and their respective PS estimates.
Table 1
 
Slope values (and their 95% confidence intervals) for each participant for the different conditions and their respective PS estimates.
Table 2
 
Mean slope values for all conditions and their respective probability summation estimates for 72 and 3,175 channels (with 95% confidence intervals).
Table 2
 
Mean slope values for all conditions and their respective probability summation estimates for 72 and 3,175 channels (with 95% confidence intervals).
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