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.

*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*.

*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

*C*refers to the contrast of the modulation of this carrier.

*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.

*SF*) can be calculated by

*E*is the eccentricity and

*E*

_{2}is a parameter representing the eccentricity at which the stimulus must double in size to maintain performance equal to that at the fovea.

*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*

_{max}represents peak contrast sensitivity (height on the

*y*-axis),

*f*

_{peak}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

*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*

_{2}values were found for each observer: 2.13° for observer DW, 2.53° for observer CV, and 2.65° for observer JVMH.

*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.).

*f*

_{carr}/

*f*

_{env}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

*f*

_{carr}/

*f*

_{env}ratios) were simultaneously scaled by eccentricity-dependent scaling factors derived from a common

*E*

_{2}value (Equation 4).

*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.

*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

*E*

_{2}value of 2.5°).

*f*

_{carr}/

*f*

_{env}, the introduction of a single parameter

*E*

_{2}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).

*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

*f*

_{carr}/

*f*

_{env}ratios in 6Figures 6, 7, and 8. Furthermore, the data scale extremely well across eccentricity, revealing

*E*

_{2}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.

*f*

_{carr}/

*f*

_{env}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.

*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

*f*

_{carr}/

*f*

_{env}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

*f*

_{carr}/

*f*

_{env}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

*f*

_{carr}/

*f*

_{env}processing stream.

*E*

_{2}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

*E*

_{2}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.