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Research Article  |   December 2007
Extrafoveal viewing reveals the nature of second-order human vision
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Journal of Vision December 2007, Vol.7, 13. doi:10.1167/7.14.13
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      Chara Vakrou, David Whitaker, Paul V. McGraw; Extrafoveal viewing reveals the nature of second-order human vision. Journal of Vision 2007;7(14):13. doi: 10.1167/7.14.13.

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

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Abstract

For the majority of visual tasks, performance in extrafoveal vision can be equated with that at the fovea simply by a change in spatial scale of the stimuli (magnification). We sought to exploit this association to examine the nature of second-order vision. More specifically, we investigate the relationship between the scale of second-order vision and the scale of its first-order input. We find that sensitivity to second-order stimuli can be equated across visual space, but only for stimuli that are magnified in every respect (identical scaling of both first- and second-order characteristics). In other words, sensitivity to stimuli which posses a fixed ratio between the scale of first-order input and second-order spatial scale can be equated across the visual field using a single magnification factor. Moreover, stimuli which possess quite different ratios of first- and second-order scale can be equated across eccentricity using the same magnification factor. This argues for a strict relationship between second-order vision and the scale of its first-order input and reflects a parallel arrangement of dedicated second-order mechanisms having a common eccentricity dependence.

Introduction
Our ability to make judgments regarding visually presented stimuli varies across the visual field according to the required task. For example the detection of large, rapid-moving objects in the periphery of the visual field remains almost as good as if the objects were near central vision (Levi, Klein, & Aitsebaomo, 1984; Whitaker, Mäkelä, Rovamo, & Latham, 1992). On the other hand, our performance in tasks demanding precise spatial localization of objects declines rapidly on moving away from the central visual field (Klein & Levi, 1987; Whitaker et al., 1992). For many visual tasks, performance in the periphery can be made equivalent to that at the fovea simply by applying a scaling or magnification factor to the stimuli used (Johnston, 1987; Rovamo & Virsu, 1979; Virsu & Rovamo, 1979; Watson, 1987). The direct implication of this statement is that peripheral vision is qualitatively identical to central vision, simply differing in quantitative terms. The physical process of stimulus magnification reflects the physiological observation that, at a cortical level, neural sampling declines systematically as a function of eccentricity—a relationship commonly known as cortical magnification. However, the scaling (or magnification) which stimuli require for performance to be made equivalent to that at the fovea has been found to vary dramatically (over 100-fold) and be highly dependent on visual task (Levi et al., 1984; Levi & Waugh, 1996; Whitaker et al., 1992). In the case of the two aforementioned tasks, judgments of relative position require a dramatic increase in magnification on moving away from the fovea, while some judgments of motion require virtually no magnification at all. The neural mechanisms underlying this observation remain unclear. One possible explanation is based upon a view of visual processing whereby different stimulus characteristics (or different types of visual judgment) are processed at different levels of the visuocortical pathway (Drasdo, 1991). Therefore, the magnification that each task requires simply reflects the magnification factor of that particular neural stage. The ubiquitous cortical magnification factor identified physiologically could, in reality, reflect just a single level of processing (most likely primate area V1). 
In the present study, we compare scaling factors for two types of stimuli that are known to involve processing by different cortical streams. The first involves the ability of the visual system to detect, localize, and analyze stimuli that are defined by variations in luminance (first-order information). These type of stimuli are thought to be analyzed by neurons early in the primary visual cortex (area V1) which behave as simple linear filters, signaling the difference in average luminance between excitatory and inhibitory regions of their receptive fields (Hubel & Wiesel, 1962; Movshon, Thompson, & Tolhurst, 1978; Shapley & Lennie, 1985). The output of this stage is sufficient to identify the stimulus in Figure 1a as a luminance grating tilted clockwise of vertical. The second type of stimulus involves contrast or texture (known as the ‘carrier’) whose amplitude is modulated by a relatively low spatial frequency envelope. This modulation must be analyzed by a dedicated ‘second-order’ mechanism, since there is no net variation in luminance across the stimulus at this relatively coarse spatial scale, effectively making it invisible to a linear mechanism. The absence of significant crossover effects in psychophysical tasks such as adaptation supports the view that first- and second-order stimuli are processed via dedicated pathways (McGraw, Levi, & Whitaker, 1999; Nishida, Ledgeway, & Edwards, 1997; Schofield & Georgeson, 1999). Second-order stimuli are thought to be analyzed by specialized neurons located in cortical areas V1 and V2 (Merigan & Maunsell, 1993; von der Heydt & Peterhans, 1989; von der Heydt, Peterhans, & Dursteler, 1992). Figure 1b represents an example of a second-order stimulus, consisting of an oriented envelope tilted clockwise of vertical—with a relatively fine-scale vertical carrier grating. 
Figure 1
 
Examples of stimuli used in the experiment consisting of a Gabor patch tilted clockwise: (a) first-order stimulus and (b) second-order stimulus. In the experiment, the Gabor could be tilted either clockwise or anti-clockwise.
Figure 1
 
Examples of stimuli used in the experiment consisting of a Gabor patch tilted clockwise: (a) first-order stimulus and (b) second-order stimulus. In the experiment, the Gabor could be tilted either clockwise or anti-clockwise.
Models of second-order vision suggest that the detection of a non-linear structure, such as a second-order stimulus, involves two filtering stages with a non-linear step in between them (Chubb & Sperling, 1988; Lin & Wilson, 1996; Sutter, Sperling, & Chubb, 1995; Wilson, Ferrera, & Yo, 1992). The fine-scale elements (carrier) are first analyzed by linear neurons tuned to relatively high spatial frequencies, whose output is then subjected to a non-linearity such as rectification. The rectified output undergoes a second linear filtering by neurons tuned to a much coarser spatial scale that are able to analyze the low-frequency envelope. The second-order system therefore relies upon input from small first-order filters and, by definition, must be at a later stage. There is sufficient physiological (Zhou & Baker, 1993, 1994) and psychophysical evidence (Graham, Beck, & Sutter, 1992; Levi & Waugh, 1996; Victor & Conte, 1991) that support this filter–rectify–filter cascade as an adequate system for processing second-order information. However, the relationship between the second-order visual system and the spatial scale of its first-order input still remains a controversial issue (Dakin & Mareschal, 2000; Kingdom & Keeble, 1999; McGraw et al., 1999; Sutter et al., 1995). 
While a few behavioral studies have attempted to compare directly peripheral visual performance for first- and second-order stimuli (Smith, Hess, & Baker, 1994; Smith & Ledgeway, 1998; Solomon & Sperling, 1995), the methodology employed in this type of comparison is absolutely crucial. Specifically, stimulus size is the decisive factor (Rovamo, Virsu, & Näsänen, 1978; Virsu & Rovamo, 1979; Virsu, Rovamo, Laurinen, & Näsänen, 1982; Whitaker, Latham, Mäkelä, & Rovamo, 1993; Whitaker et al., 1992), with the consensus view that (at least for first-order stimuli), provided visual stimuli are enlarged sufficiently, there is no loss of observer performance in peripheral vision. This type of methodology is known as ‘spatial scaling’ (Johnston, 1987; Watson, 1987) and involves the measurement of visual performance across eccentricity for a range of stimuli differing only in magnification. If spatial scaling holds true, the resulting data can be simply shifted along the size axis to equate performance across eccentricity. 
This spatial scaling methodology immediately introduces a potential dilemma for studies of second-order vision, since there is not just a single spatial scale to consider (such as spatial frequency for first-order stimuli; Figure 2a). Rather, both the spatial frequency of the envelope and its carrier can be varied independently to create a two-dimensional parameter space for second-order stimuli ( Figure 2b). For example, along direction ‘B’ in the figure, the spatial frequency of the carrier remains fixed, but that of the envelope varies. Direction ‘A’ is of considerable interest, since stimuli along this oblique axis are the only ones to conform to a strict spatial scaling methodology in that they are magnified versions of each other in every respect—both carrier and envelope spatial frequency vary together such that the ratio of their spatial scales remains fixed. This is true for all stimuli within parameter space that lie along this oblique direction (such as A′ and A″). 
Figure 2
 
(a) Example of the scaled stimuli used in the first-order experiment. All stimuli are simply magnified versions of each other, with magnification increasing to the right as spatial frequency decreases. (b) Examples of the second-order stimuli used in the experiment. Stimuli along A conform to spatial scaling, according which stimuli are scaled in every respect (i.e., both the spatial frequency of the carrier and the contrast envelope varies in combination) such that they are simply magnified versions of each other. Along A, the magnification decreases in the direction of the arrow. The same principle holds along A′ and A″, but for different ratios of carrier/envelope spatial frequency. Stimuli along B do not conform to spatial scaling since only the spatial frequency of the contrast envelope changes relative to a fixed carrier frequency.
Figure 2
 
(a) Example of the scaled stimuli used in the first-order experiment. All stimuli are simply magnified versions of each other, with magnification increasing to the right as spatial frequency decreases. (b) Examples of the second-order stimuli used in the experiment. Stimuli along A conform to spatial scaling, according which stimuli are scaled in every respect (i.e., both the spatial frequency of the carrier and the contrast envelope varies in combination) such that they are simply magnified versions of each other. Along A, the magnification decreases in the direction of the arrow. The same principle holds along A′ and A″, but for different ratios of carrier/envelope spatial frequency. Stimuli along B do not conform to spatial scaling since only the spatial frequency of the contrast envelope changes relative to a fixed carrier frequency.
In measuring sensitivity to such second-order stimuli in peripheral vision, at least three potential outcomes could emerge. Firstly, the visual system might not care about carrier spatial frequency, and provided the spatial scale of the envelope is suitably enlarged in peripheral vision (by moving along direction B in Figure 2b), sensitivity to all stimuli might be equated. Conversely, the spatial scale of the carrier and its relationship to that of the envelope might be absolutely critical, and only when both are magnified appropriately (such as moving along direction A) might peripheral sensitivity be equated to foveal sensitivity. Third, it could turn out that the carrier and the envelope require such diverse and independent levels of magnification in peripheral vision that meaningful comparison of foveal and peripheral sensitivity to second-order stimuli becomes impossible. To address this issue, we measured sensitivity to second-order stimuli across the two-dimensional parameter space shown in Figure 2b in both foveal and peripheral vision. 
Methods
Apparatus
Stimuli were presented for 500 ms on a 20-in. Electron d2 monitor. The non-linear luminance response of the display was linearized by using the inverse function of the luminance response as measured using a Minolta CS-100 photometer. The host computer was a Motorola Starmax 4000/200. All stimuli were generated using the macro capabilities of public domain software NIH image™ 1.61. The contrast resolution of the display could be enhanced by independent control and combination of the three color guns (Pelli & Zhang, 1991). The resolution of the monitor was 1024 × 768 pixels with a refresh rate of 75 Hz. Pixel size was 0.377 mm. A wide range of viewing distances from 0.309 m to 13.59 m was used in order to vary the spatial characteristics of the stimuli. 
Observers
The observers who took part in the experiments were two of the authors (CV and DW) and an additional observer naïve to the purpose of the experiment (JVMH). All observers where necessary wore appropriate refractive correction for the relevant viewing distances. 
Stimuli and procedures
First-order stimuli consisted of a Gabor patch tilted obliquely either clockwise or anti-clockwise from vertical ( Figure 1a). The mathematical representation of the Gabor element was  
L u m i n a n c e = L + ( exp ( x 2 + y 2 ) 2 σ 2 × L C sin ( 2 π f ( x ± y ) + ϕ ) ) ,
(1)
where L is the mean luminance of the background (30 cd m −2), C is the stimulus contrast (Michelson contrast), and f is the spatial frequency of the sinusoidal grating. x and y are the respective horizontal and vertical distances from the center of the stimulus, and ϕ is the phase of the sinusoidal grating, relative to the peak (center) of the Gaussian envelope and was randomly set. Stimulus contrast C was varied to provide an estimate of contrast threshold. The standard deviation of the Gaussian envelope ( σ) was systematically varied in combination with the spatial frequency such that there were always 0.565 cycles per σ
Second-order stimuli were composed of a contrast modulation of a vertical sinusoidal grating of spatial frequency f carr. The contrast modulation was in the form of a Gabor patch, again tilted either clockwise or anti-clockwise of vertical. Its spatial frequency was defined by f env and, again, the size of its Gaussian envelope was varied in combination with its frequency such that there were always 0.565 cycles per σ. The spatial frequency of both the carrier and the envelope could be varied independently. The mathematical representation of the second-order element was  
L u m i n a n c e = L + ( ( 0.5 L sin ( 2 π f c a r r x ) ) · ( 1 + ( exp ( x 2 + y 2 ) 2 σ 2 × C sin ( 2 π f e n v ( x ± y ) + ϕ ) ) ) ) .
(2)
 
Examples of the second-order stimuli used are illustrated in Figure 1b. As the above equation shows, the contrast of the unmodulated carrier of the second-order stimuli was always 50%. For these stimuli, C refers to the contrast of the modulation of this carrier. 
A consequence of varying size and spatial frequency in combination ( Figure 2a) is that low spatial frequency targets occupy relatively large regions of visual space. The definition of stimulus eccentricity for all stimuli was in relation to the center of the Gabor patch. Justification for this is that, close to threshold contrast, the parts of the Gaussian-windowed stimuli away from the stimulus center fall below threshold and the perceptual spatial extent of the patch converges to the peak (Fredericksen, Bex, & Verstraten, 1997). Viewing was monocular using the dominant eye (right eye for observers CV and DW, left eye for observer JVMH) and, for foveal presentations, fixation was central and was aided by a small black fixation cross. Peripheral presentations were always in the nasal visual field so as to avoid the blind spot, and again a small black fixation mark was provided in order to maintain appropriate fixation. 
Contrast detection thresholds were determined by a two-alternative forced choice method, in which the observer had to decide whether the relevant grating was tilted clockwise or anti-clockwise from vertical. Stimuli were presented within a rectangular temporal window of 500 ms duration. Presentation of the grating was announced with a brief auditory tone, but no feedback was provided. 
Threshold was determined by a method of constant stimuli in which any one of 9 different contrast levels could be presented during a trial. The step size between levels was 0.05 log units. The contrast range was centered so as to provide an adequate coverage of the psychometric function for detection. A total of 40 trials were presented at each of the 9 contrast levels. The resulting psychometric function was analyzed by logistic regression (Maxwell, 1959) to provide an estimate of the contrast threshold. This is defined as the stimulus intensity resulting in a performance half-way between 50% and 100% correct responses; i.e., contrast threshold was estimated at 75% correct response level: 
y=50+501+e(xμ)θ,
(3)
where y is the percentage of correct responses, x is the contrast of the test stimulus, μ is the contrast level corresponding to the 75% of correct responses, and θ represents the slope of the psychometric function at the midpoint area. Contrast sensitivity was defined as the logarithm of the reciprocal of contrast threshold. 
Results
Sensitivity to first-order stimuli
Figure 3 shows the measured contrast sensitivity as a function of spatial frequency for first-order gratings in central and peripheral vision up to 20 deg. The foveal data reflect the descending limb of the traditional contrast sensitivity function for achromatic stimuli with sensitivity declining approximately linearly with increasing spatial frequency. In the periphery, lower spatial frequency stimuli (larger stimuli) are required to obtain sensitivity values equivalent to those at the fovea. This is reflected in the sequential leftwards shift of the functions with increasing eccentricity. The eccentricity dependence of the data can be accounted for by applying scaling factors to the peripheral curves in order to superimpose them upon the foveal function (Watson, 1987). If linear spatial scaling holds true, the scaling factors (SF) can be calculated by 
SF=1+E/E2,
(4)
where E is the eccentricity and E2 is a parameter representing the eccentricity at which the stimulus must double in size to maintain performance equal to that at the fovea. 
Figure 3
 
Sensitivity as a function of spatial frequency across four different eccentricities for first-order stimuli. Data for three observers—DW (upper), CV (middle), and JVMH (lower figure). Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 3
 
Sensitivity as a function of spatial frequency across four different eccentricities for first-order stimuli. Data for three observers—DW (upper), CV (middle), and JVMH (lower figure). Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
An estimated value of E 2 was used to shift the peripheral curves along the size (spatial frequency) axis towards the foveal function. A single function was then used to fit the entire data set. The function to describe the scaled contrast sensitivity was a version of a contrast sensitivity function used by Rovamo, Luntinen, and Näsänen (1993), namely 
S=Smax·((1+(fpeakf)k)1k)·((1+(ffpeak)k)1k),
(5)
where Smax represents peak contrast sensitivity (height on the y-axis), fpeak represents the scaled spatial frequency at which the function reaches its maximum, and k is a constant which determines the sharpness of the curve at its knee point. The product in the first bracket of Equation 5 governs the sharpness of the decrease in contrast sensitivity at low spatial frequencies. Since we examine only the high-frequency end of the CSF, we set the product of the first bracket to unity, transforming Equation 5 into 
S=Smax·((1+(ffpeak)k)1k).
(6)
 
Equation 6 was fitted to all first-order data. Multiple iterations of the application of Equation 4 to the data sets for each observer resulted in an E 2 value which minimized the sum-of-squares deviation around the curve fit to the combined data. The resulting scaled data are shown in Figure 4. The introduction of an appropriate E 2 value (equating simply to a horizontal shift of the original data sets) serves to collapse the data across all eccentricities. Similar E 2values were found for each observer: 2.13° for observer DW, 2.53° for observer CV, and 2.65° for observer JVMH. 
Figure 4
 
The data of Figure 3, scaled to account for eccentricity. For each observer, the data from Figure 3 were scaled by an estimated E 2 value, and the residual variance was calculated using Equation 6 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E 2 value was found which minimized the overall residual variance for each observer. This method accounted for 91.1% (DW), 90.7% (CV), and 84.4% (JVMH) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 4
 
The data of Figure 3, scaled to account for eccentricity. For each observer, the data from Figure 3 were scaled by an estimated E 2 value, and the residual variance was calculated using Equation 6 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E 2 value was found which minimized the overall residual variance for each observer. This method accounted for 91.1% (DW), 90.7% (CV), and 84.4% (JVMH) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Sensitivity to second-order stimuli
We measured contrast sensitivity functions (CSF) for contrast modulation of sinusoidal luminance gratings (the carrier) (see Figure 2b). At the fovea, we measured CSFs for various envelope spatial frequencies for four different carrier spatial frequencies: 2, 4, 8, and 16 c/deg. Extrafoveally (5, 10, and 20 deg), the carrier and envelope spatial frequencies were chosen in a way to provide adequate measurement of sensitivity within the constraints imposed by stimulus characteristics and physical viewing distance. Figures 5 represent the second-order contrast sensitivity functions for the three observers at four eccentricities. In terms of parameter space, this type of plot represents stimuli lying across the dimension B in Figure 2—the variation in sensitivity as a function of envelope spatial frequency (size), with carrier spatial frequency remaining constant. 
Figure 5, 5a, 5b
 
Sensitivity to envelope modulation as a function of envelope spatial frequency. Data for all observers (DW, CV, and JVMH) at 0, 5, 10, and 20 deg eccentricity. Different symbols denote different carrier frequencies as specified in each legend. Note that, for peripheral presentations, DW used a slightly different range of comfortable viewing distances than CV and JVMH, resulting in a small difference in stimulus spatial frequency. However, the fcarr/ fenv spatial frequency ratio for all observers was always maintained.
Figure 5, 5a, 5b
 
Sensitivity to envelope modulation as a function of envelope spatial frequency. Data for all observers (DW, CV, and JVMH) at 0, 5, 10, and 20 deg eccentricity. Different symbols denote different carrier frequencies as specified in each legend. Note that, for peripheral presentations, DW used a slightly different range of comfortable viewing distances than CV and JVMH, resulting in a small difference in stimulus spatial frequency. However, the fcarr/ fenv spatial frequency ratio for all observers was always maintained.
At the fovea the second-order CSFs display the same inverted U-shape as first-order stimuli but with a more gradual high spatial frequency decline than the first-order CSFs ( Figure 3). Note how the peak of these foveal functions is at much lower spatial frequencies that the peak of first-order CSFs. Beyond a carrier frequency of 4 c/deg, peak sensitivity of the functions is reduced—indicated by the vertical shift of the sensitivity values. This is not surprising given that first-order contrast sensitivity declines with increasing spatial frequency, as shown previously. Any given contrast modulation of a fixed-contrast carrier would be expected to become less and less visible as contrast sensitivity to the carrier itself declines. 
What is also noticeable is the rather restricted range of spatial frequencies examined for the second-order stimuli. Previous investigators have stated the importance of keeping the spatial frequency of the contrast envelope significantly below the spatial frequency of the carrier used in each stimulus, otherwise progressively greater first-order artifacts can be introduced (Smith & Ledgeway, 1997). This was the major concern for the sensitivity at the high-frequency end. The spatial frequencies of both the first- and second-order structures of the stimuli (the carrier and the envelope, respectively) were chosen in such a way that the spatial frequency of the carrier was always 2.84 times higher than the envelope spatial frequency. At the low spatial frequency end, the limiting factor was the actual size of the Gabor patch (contrast envelope) and the very short viewing distances required. 
The primary purpose of the study was to determine whether a process of spatial scaling (stimulus magnification) could equate second-order vision across the visual field. However, this appears far from obvious from the data in Figure 5 given that the shape of the peripheral sensitivity functions hardly resembles the foveal ones at all (a prerequisite for spatial scaling). Nevertheless, if we divide the complete data set of Figure 5 into separate sub-sets, each representing a common carrier frequency to envelope frequency ratio ( f carr/ f env), a much clearer picture emerges. This revised type of analysis is the one which strictly adheres to spatial scaling, in that we only consider data which arise from stimuli which are magnified versions of one another in every respect (along dimension A in Figure 2b). Figures 6, 7, and 8 show these data (re-plotted from Figure 5) for the three observers, DW (a–d), CV (a–d), and JVMH (a–d), respectively. Each graph shows sensitivity functions at the four different eccentricities for stimuli having the same ratio ( f carr/ f env). In other words, all data points within each graph have been obtained with the same stimulus geometry—each point differing only in stimulus magnification. Different graphs (a, b, c, d) represent different ( f carr/ f env) ratios (i.e., the same direction yet different locations within the two-dimensional parameter space shown in Figure 2b (A, A′, A″, etc.). 
Figure 6, 6a
 
(a–d) Modulation sensitivity plotted for each of the four fcarr/ fenv ratios for observer DW. Data taken from Figure 5 but plotted for fcarr/ fenv ratios individually: (a) fcarr/ fenv = 2.84, (b) fcarr/ fenv = 5.68; (c) fcarr/ fenv = 11.36, and (d) fcarr/ fenv = 22.7. (a′–d′) The panels a–d, scaled to account for eccentricity. For all fcarr/ fenv ratios, the graphs a–d were scaled by a common estimated E2 value, the residual variance was calculated with Equation 5, and then summed across the four fcarr/ fenvratios. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit, apart from panels b′ and c′ for observers DW and CV in which the parameter k was fixed ( k = 3) in order to capture the relatively sharp peak. Using this iterative procedure, an E2 value was found which minimized the overall residual variance. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 6, 6a
 
(a–d) Modulation sensitivity plotted for each of the four fcarr/ fenv ratios for observer DW. Data taken from Figure 5 but plotted for fcarr/ fenv ratios individually: (a) fcarr/ fenv = 2.84, (b) fcarr/ fenv = 5.68; (c) fcarr/ fenv = 11.36, and (d) fcarr/ fenv = 22.7. (a′–d′) The panels a–d, scaled to account for eccentricity. For all fcarr/ fenv ratios, the graphs a–d were scaled by a common estimated E2 value, the residual variance was calculated with Equation 5, and then summed across the four fcarr/ fenvratios. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit, apart from panels b′ and c′ for observers DW and CV in which the parameter k was fixed ( k = 3) in order to capture the relatively sharp peak. Using this iterative procedure, an E2 value was found which minimized the overall residual variance. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 7, 7a
 
As for Figure 6, but observer CV.
Figure 8, 8a
 
As for Figure 6, but observer JVMH.
Figure 8, 8a
 
As for Figure 6, but observer JVMH.
This manipulation produces data sets which appear eminently suited to a process of spatial scaling since they are of similar shape at different eccentricities, simply shifted leftwards (i.e., magnified) relative to the foveal data set (Watson, 1987). This is obviously less convincing at higher fcarr/fenv ratios due to the sparsity of data—an unavoidable consequence of poorer sensitivity and the spatial limitations of the stimuli which we described earlier. For each observer, all four graphs (at each of the four fcarr/fenv ratios) were simultaneously scaled by eccentricity-dependent scaling factors derived from a common E2 value (Equation 4). 
The scaled functions are shown to the right of each unscaled data set— Figure 6 (a′–d′) for DW, Figure 7 (a′–d′) for CV, and Figure 8 (a′–d′) for JVMH. An iterative procedure was used to find a common E 2 value that resulted in the minimum overall variance for each observer, using Equation 5 as a template. Optimum E 2 values were found to be 1.60° for DW, 2.43° for CV, and 1.62° for JVMH. Despite the constraint (which we purposefully set) that the same E 2 value was used to scale each stimulus condition, the overall variance explained by the scaling procedure was good (86.1% for DW; 70.3% for CV; and 86.5% for JVMH). Comparison of the scaled functions reveals that, while different f carr/ f env ratios possess distinct sensitivity functions (peak sensitivity declines and shifts to lower spatial frequencies as f carr/ f env increases), each share a common variation in scale (magnification) as a function of eccentricity. 
Discussion
Contrast sensitivity functions for luminance-defined stimuli were collected at the fovea and at three eccentric locations. As is well documented, increasing eccentricity requires progressively larger size (lower spatial frequency) stimuli in order to maintain foveal performance; hence, the sequential leftwards shift of the eccentric functions across the spatial frequency axis ( Figure 3). The effect of eccentricity can be accounted for by applying a simple eccentricity-dependent scaling process ( Equation 5), and the E 2 values that we found correspond well with previous studies which examined the eccentricity dependence of luminance-defined stimuli (e.g., Rovamo et al., 1978, found an E2 value of 2.5°). 
Sensitivity functions for second-order stimuli ( Figure 5) appear relatively flat in comparison to the steep first-order functions of Figure 3 but still tend to display a band-pass shape, in agreement with previous studies (Gray & Regan, 1998; Hutchinson & Ledgeway, 2004; Kingdom, Keeble, & Moulden, 1995; Landy & Oruç, 2002; Schofield & Georgeson, 1999; Sutter et al., 1995). Initially, we plotted sensitivity as a function of envelope spatial frequency for common carrier spatial frequencies. The emerging functions did not lend themselves to a process of scaling across eccentricity. Nevertheless, when the contrast sensitivity functions were plotted for fixed ratios of fcarr/fenv, the introduction of a single parameter E2 enabled us to collapse functions from all eccentricities on to the corresponding foveal sensitivity functions (Figures 6, 7, and 8). This observation indicates that second-order vision can be equated across the visual field and there is no qualitative difference in performance between central and peripheral vision. This in itself highlights the fact that second-order vision is a dedicated pathway that possesses an orderly arrangement across the visual field. Peripheral second-order visual filters are identical to their foveal counterparts aside from a change in spatial scale (magnification) provided the magnification is considered in every respect (i.e., both carrier and envelope spatial scale must be magnified together). 
We now address the possibility that our results reflect the detection of first-order artifacts within our stimuli. Our use of a narrow-band, sinusoidal carrier (a necessity in order to examine the issue of relative spatial scale) gives rise to first-order sidebands (or ‘beats’) whose orientation could be a cue to discrimination. Furthermore, the critical factor in determining the nature of these first-order artifacts is the f carr/ f env ratio of the stimuli. This may provide the reason why our second-order stimuli scale in a similar way to the first-order data. To preclude this possibility, we measured sensitivity to the contrast modulation of broad-band, random noise texture. The texture was drawn randomly from a uniform luminance distribution and was of 50% Michelson contrast when unmodulated. Importantly, the texture elements were small relative to the period of the second-order contrast modulation in order to minimize first-order luminance artifacts due to ‘clumping’ (Schofield & Georgeson, 1999). There were always 10 elements per modulation period. Two of the authors gathered data in exactly the same way as before, but for this new carrier. Figure 9 (top) shows unscaled sensitivity functions which closely resemble the functions for low fcarr/fenv ratios in 6Figures 6, 7, and 8. Furthermore, the data scale extremely well across eccentricity, revealing E2 values which agree well with those for the sinusoidal carrier data. This confirms that first-order artifacts do not determine the scaling of second-order sensitivity across the visual field and shows that our grating carrier data genuinely reflect the response of second-order visual mechanisms. 
Figure 9
 
(Top) Sensitivity for contrast modulated gratings with noise carrier as a function of envelope spatial frequency. Data for two observers (DW and CV) at 0, 5, 10, and 20 deg eccentricity. Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. (Bottom) The data of top panel scaled to account for eccentricity. For each observer, the data were scaled by an estimated E2 value, and the residual variance was calculated using Equation 5 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E2 value was found which minimized the overall residual variance for each observer. This method accounted for 97.79% (DW) and 95.6% (CV) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 9
 
(Top) Sensitivity for contrast modulated gratings with noise carrier as a function of envelope spatial frequency. Data for two observers (DW and CV) at 0, 5, 10, and 20 deg eccentricity. Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. (Bottom) The data of top panel scaled to account for eccentricity. For each observer, the data were scaled by an estimated E2 value, and the residual variance was calculated using Equation 5 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E2 value was found which minimized the overall residual variance for each observer. This method accounted for 97.79% (DW) and 95.6% (CV) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Previous psychophysical and neurophysiological studies have provided evidence that second-order detectors initially receive input from spatial frequency- and orientationally-tuned first-order detectors whose outputs are then subjected to some kind of non-linear transformation in order to reveal the second-order spatial structure in the image. This has left open the debate of whether there may be selectivity for spatial frequency in the contribution of first-order filters to second-order detectors. The generalized model of filter–rectify–filter allows for a wide range of inter-stage connectivities. The nature of these connections has suggested that the second-stage filters are tied to specific first-stage filters, such that high frequency filters at the second-stage are coupled to high frequency first-stage filters, whereas low frequency filters at the second-stage are coupled to relatively low frequency first-stage filters (Kingdom & Keeble, 1999). Similarly, Sutter et al. (1995) have argued that second-order channels have a preferred carrier frequency that is 3–4 octaves above their preferred envelope frequency. However, the very fact that we can perceive second-order stimuli at different fcarr/fenv ratios indicates that there must be a substantial range of connections across first- and second-order scale. Dakin and Mareschal (2000) have suggested that second-order channels have no preferred carrier frequency and are simply more sensitive to higher carrier frequencies than lower ones. 
Our own results indicate that second-order vision across the visual field is inherently organized on the basis of the ratio between carrier and envelope spatial frequencies. In simple terms, it cares fundamentally about the relationship between its own spatial scale and that of the first-order input which it receives, and subsequent analysis of the output of second-order filters is arranged on the basis of this relationship. Thus, while second-order vision consists of units with a wide range of preferred f carr/ f env ratios at each eccentricity (and sensitivity is determined by this ratio), all share a common eccentricity dependence. The observation of carrier frequency selectivity in second-order vision is compatible with both neurophysiological (Baker, 1999; Baker & Mareschal, 2001; Mareschal & Baker, 1998; Zhou & Baker, 1994) and psychophysical findings (McGraw et al., 1999). While the lack of significant crossover adaptation between first- and second-order vision is now well established (Nishida et al., 1997; Schofield & Georgeson, 1999), McGraw et al. (1999) investigated crossover adaptation within the second-order pathway for stimuli of different carrier spatial frequencies and orientations. They demonstrated that adaptation to second-order stimuli transferred across carrier orientation but not across carrier spatial frequency. In other words, second-order stimuli of sufficiently different fcarr/fenv ratios do not interact. Thus, while carrier orientation information is lost to subsequent analysis of second-order vision through a process of pooling (Arsenault, Wilkinson, & Kingdom, 1999; McGraw et al., 1999), information regarding relative spatial scale is strictly retained. As further evidence for the importance of relative spatial scale, it should be noted that illusory interactions between first- and second-order vision, such as the Fraser and Zöllner illusions of orientation, are critically dependent upon the fcarr/fenv ratio of the stimuli used (Skillen, Whitaker, Popple, & McGraw, 2002). If this ratio is held constant, the magnitude of the illusory effects is also constant, demonstrating the phenomenon of ‘scale invariance’ (Jamar & Koenderink, 1985; Kingdom & Keeble, 1999; Sutter et al., 1995). Scale invariant analysis has distinct real-world relevance, where objects are continually changing in size either through their own motion or self-motion of the observer. Either way, while the spatial scale of the object's first-order texture and its second-order variation in texture might change dramatically, the ratio between the two is maintained, thereby permitting continued analysis of the object within the same fcarr/fenv processing stream. 
In the motion domain, a number of studies have previously attempted to compare sensitivity to first- and second-order vision across eccentricity. Smith et al. (1994) and Smith and Ledgeway (1998) measured the loss in sensitivity to a stimulus of fixed spatial frequency with increasing eccentricity and found similar rates of sensitivity decline for first- and second-order stimuli. This is precisely the converse experimental strategy to that used in the present study, where stimulus size is varied in order to equate sensitivity across eccentricity. The dangers of using a target of fixed dimensions across the visual field have been well documented (Poirier & Gurnsey, 2005; Watson, 1987), and it is not a trivial matter to relate findings from this type of experiment to those which adopt a strict spatial scaling methodology. Solomon and Sperling (1995) varied the envelope spatial frequency of a fixed carrier in order to establish resolution limits for second-order motion in peripheral vision. They found that second-order resolution limits, although being much lower, varied with eccentricity at approximately the same rate as first-order resolution thresholds. Again, it is difficult to relate this to our findings, particularly since few of our sensitivity functions approached the resolution limit (Figure 5). Nevertheless, our observed E2 values for first-order vision (2.13, 2.53°, 2.65°) are not too dissimilar to those for second-order (1.60°, 2.43°, 1.62°), particularly when considered in the light of the 100-fold difference in E2 values between other tasks (Whitaker et al., 1993). What seems clear is that, in order to fully characterize the properties of peripheral second-order motion sensitivity, a strict method of spatial scaling should be adopted. 
Conclusions
The results presented here demonstrate that stimuli defined by both first-order and second-order characteristics can be detected equally well in the extrafoveal visual field, given appropriate magnification. Furthermore, second-order vision, for which there exist a plethora of possible combinations of second-order spatial scale and first-order input, exhibits a reassuringly straightforward organization with respect to eccentricity. Stimuli which possess a fixed ratio between first-order input and second-order spatial scale can be related to any other stimulus of identical geometry anywhere else in the visual field, by the application of a single scaling (magnification) factor. Stimuli that possess different geometry (a different ratio between first- and second-order spatial scale) possess their own individual sensitivity profiles, yet all share an identical relationship with respect to eccentricity. 
Acknowledgments
David Whitaker is supported by the Leverhulme Trust. Paul V. McGraw is supported by the Wellcome Trust, UK. 
Commercial relationships: none. 
Corresponding author: Chara Vakrou. 
Email: c.vakrou3@bradford.ac.uk. 
Address: Department of Optometry, University of Bradford, Richmond Road, Bradford, West Yorkshire, BD7 1DP, UK. 
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Figure 1
 
Examples of stimuli used in the experiment consisting of a Gabor patch tilted clockwise: (a) first-order stimulus and (b) second-order stimulus. In the experiment, the Gabor could be tilted either clockwise or anti-clockwise.
Figure 1
 
Examples of stimuli used in the experiment consisting of a Gabor patch tilted clockwise: (a) first-order stimulus and (b) second-order stimulus. In the experiment, the Gabor could be tilted either clockwise or anti-clockwise.
Figure 2
 
(a) Example of the scaled stimuli used in the first-order experiment. All stimuli are simply magnified versions of each other, with magnification increasing to the right as spatial frequency decreases. (b) Examples of the second-order stimuli used in the experiment. Stimuli along A conform to spatial scaling, according which stimuli are scaled in every respect (i.e., both the spatial frequency of the carrier and the contrast envelope varies in combination) such that they are simply magnified versions of each other. Along A, the magnification decreases in the direction of the arrow. The same principle holds along A′ and A″, but for different ratios of carrier/envelope spatial frequency. Stimuli along B do not conform to spatial scaling since only the spatial frequency of the contrast envelope changes relative to a fixed carrier frequency.
Figure 2
 
(a) Example of the scaled stimuli used in the first-order experiment. All stimuli are simply magnified versions of each other, with magnification increasing to the right as spatial frequency decreases. (b) Examples of the second-order stimuli used in the experiment. Stimuli along A conform to spatial scaling, according which stimuli are scaled in every respect (i.e., both the spatial frequency of the carrier and the contrast envelope varies in combination) such that they are simply magnified versions of each other. Along A, the magnification decreases in the direction of the arrow. The same principle holds along A′ and A″, but for different ratios of carrier/envelope spatial frequency. Stimuli along B do not conform to spatial scaling since only the spatial frequency of the contrast envelope changes relative to a fixed carrier frequency.
Figure 3
 
Sensitivity as a function of spatial frequency across four different eccentricities for first-order stimuli. Data for three observers—DW (upper), CV (middle), and JVMH (lower figure). Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 3
 
Sensitivity as a function of spatial frequency across four different eccentricities for first-order stimuli. Data for three observers—DW (upper), CV (middle), and JVMH (lower figure). Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 4
 
The data of Figure 3, scaled to account for eccentricity. For each observer, the data from Figure 3 were scaled by an estimated E 2 value, and the residual variance was calculated using Equation 6 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E 2 value was found which minimized the overall residual variance for each observer. This method accounted for 91.1% (DW), 90.7% (CV), and 84.4% (JVMH) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 4
 
The data of Figure 3, scaled to account for eccentricity. For each observer, the data from Figure 3 were scaled by an estimated E 2 value, and the residual variance was calculated using Equation 6 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E 2 value was found which minimized the overall residual variance for each observer. This method accounted for 91.1% (DW), 90.7% (CV), and 84.4% (JVMH) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 5, 5a, 5b
 
Sensitivity to envelope modulation as a function of envelope spatial frequency. Data for all observers (DW, CV, and JVMH) at 0, 5, 10, and 20 deg eccentricity. Different symbols denote different carrier frequencies as specified in each legend. Note that, for peripheral presentations, DW used a slightly different range of comfortable viewing distances than CV and JVMH, resulting in a small difference in stimulus spatial frequency. However, the fcarr/ fenv spatial frequency ratio for all observers was always maintained.
Figure 5, 5a, 5b
 
Sensitivity to envelope modulation as a function of envelope spatial frequency. Data for all observers (DW, CV, and JVMH) at 0, 5, 10, and 20 deg eccentricity. Different symbols denote different carrier frequencies as specified in each legend. Note that, for peripheral presentations, DW used a slightly different range of comfortable viewing distances than CV and JVMH, resulting in a small difference in stimulus spatial frequency. However, the fcarr/ fenv spatial frequency ratio for all observers was always maintained.
Figure 6, 6a
 
(a–d) Modulation sensitivity plotted for each of the four fcarr/ fenv ratios for observer DW. Data taken from Figure 5 but plotted for fcarr/ fenv ratios individually: (a) fcarr/ fenv = 2.84, (b) fcarr/ fenv = 5.68; (c) fcarr/ fenv = 11.36, and (d) fcarr/ fenv = 22.7. (a′–d′) The panels a–d, scaled to account for eccentricity. For all fcarr/ fenv ratios, the graphs a–d were scaled by a common estimated E2 value, the residual variance was calculated with Equation 5, and then summed across the four fcarr/ fenvratios. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit, apart from panels b′ and c′ for observers DW and CV in which the parameter k was fixed ( k = 3) in order to capture the relatively sharp peak. Using this iterative procedure, an E2 value was found which minimized the overall residual variance. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 6, 6a
 
(a–d) Modulation sensitivity plotted for each of the four fcarr/ fenv ratios for observer DW. Data taken from Figure 5 but plotted for fcarr/ fenv ratios individually: (a) fcarr/ fenv = 2.84, (b) fcarr/ fenv = 5.68; (c) fcarr/ fenv = 11.36, and (d) fcarr/ fenv = 22.7. (a′–d′) The panels a–d, scaled to account for eccentricity. For all fcarr/ fenv ratios, the graphs a–d were scaled by a common estimated E2 value, the residual variance was calculated with Equation 5, and then summed across the four fcarr/ fenvratios. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit, apart from panels b′ and c′ for observers DW and CV in which the parameter k was fixed ( k = 3) in order to capture the relatively sharp peak. Using this iterative procedure, an E2 value was found which minimized the overall residual variance. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 7, 7a
 
As for Figure 6, but observer CV.
Figure 8, 8a
 
As for Figure 6, but observer JVMH.
Figure 8, 8a
 
As for Figure 6, but observer JVMH.
Figure 9
 
(Top) Sensitivity for contrast modulated gratings with noise carrier as a function of envelope spatial frequency. Data for two observers (DW and CV) at 0, 5, 10, and 20 deg eccentricity. Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. (Bottom) The data of top panel scaled to account for eccentricity. For each observer, the data were scaled by an estimated E2 value, and the residual variance was calculated using Equation 5 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E2 value was found which minimized the overall residual variance for each observer. This method accounted for 97.79% (DW) and 95.6% (CV) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
Figure 9
 
(Top) Sensitivity for contrast modulated gratings with noise carrier as a function of envelope spatial frequency. Data for two observers (DW and CV) at 0, 5, 10, and 20 deg eccentricity. Note that progressively lower spatial frequency stimuli are required for the eccentric locations in order to reach the same level of performance as that at the fovea. (Bottom) The data of top panel scaled to account for eccentricity. For each observer, the data were scaled by an estimated E2 value, and the residual variance was calculated using Equation 5 as a template. The three parameters of the curve fit were allowed to float in order to minimize the residual sum-of-squares deviation of the combined eccentricity data around the curve fit. Using this iterative procedure, an E2 value was found which minimized the overall residual variance for each observer. This method accounted for 97.79% (DW) and 95.6% (CV) of the variance in the data. Circles: 0 deg, squares: 5 deg, diamonds: 10 deg, triangles: 20 deg.
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