Perceived visual speed has been reported to be reduced during walking. This reduction has been attributed to a partial subtraction of walking speed from visual speed (F. H. Durgin & K. Gigone, 2007; F. H. Durgin, K. Gigone, & R. Scott, 2005). We tested whether observers still have access to the retinal flow before subtraction takes place. Observers performed a 2IFC visual speed discrimination task while walking on a treadmill. In one condition, walking speed was identical in the two intervals, while in a second condition walking speed differed between intervals. If observers have access to the retinal flow before subtraction, any changes in walking speed across intervals should not affect their ability to discriminate retinal flow speed. Contrary to this “direct access hypothesis,” we found that observers were worse at discrimination when walking speed differed between intervals. The results therefore suggest that observers do not have access to retinal flow before subtraction. We also found that the amount of subtraction depended on the visual speed presented, suggesting that the interaction between the processing of visual input and of self-motion is more complex than previously proposed.

*v*minus a proportion

*k*of the walking speed

*w*:

*v*and a linear effect of walking speed that is independent of the visual speed. If

*k*= 1, the walking speed is completely subtracted from the retinal speed, constituting a coordinate transform from a retinocentric to a world-centric frame of reference. Durgin et al. found values of

*k*between 0 and 1, indicating less than complete subtraction (Durgin & Gigone, 2007; Durgin et al., 2005).

*v*

_{t}to that of a standard stimulus with speed

*v*

_{s}. According to the direct access hypothesis, walking speed will not affect the ability to discriminate visual speeds in such a task, so the perceived speed difference will only depend on the visual speed difference Δ

*v*=

*v*

_{t}−

*v*

_{s}. However, if observers only have access to retinal flow after subtraction, any differences in walking speed Δ

*w*=

*w*

_{t}−

*w*

_{s}will also determine the perceived visual speed difference:

*w*and thus is a function of both the difference in visual speed (i.e., retinal flow) and the difference in walking speed between the two intervals. Following Freeman et al. (2009), we therefore presented visual speeds under two different conditions. In

*homogeneous*trials, the walking speeds in the intervals with test speed and standard speed were equal (Δ

*w*= 0). In

*heterogeneous*trials, walking speeds differed between intervals (Δ

*w*≠ 0). If observers have direct access to the retinal flow, they base their judgments on Δ

*v*only. Hence, we would not expect any difference in discrimination performance between these two conditions. However, if observers only have indirect access to the retinal motion, varying Δ

*w*will effectively introduce noise into the visual speed difference Δ

*homogeneous*trials, the walking speed was the same in both intervals. In

*heterogeneous*trials, all six possible pairs of different walking speeds were presented in the two intervals. The three different walking speeds were all tested twice in the homogeneous condition, to make the total number of trials equal to that in the heterogeneous trials. Homogeneous and heterogeneous trials were randomly intermixed in the same session. The experiment was run in five separate sessions of 288 trials. Each session was performed on a different day and consisted of three blocks of 96 trials, with short breaks in between.

*w*= −0.8, −0.4, 0.4, or 0.8 m/s), for all the trials in the heterogeneous condition collapsed together and for the homogeneous condition. The maximum-likelihood procedure described by Wichman and Hill (2001) was used for fitting, with lapse rate included as a free parameter.

*w*shifted away from the PSE in the homogeneous condition. As a consequence, the discrimination threshold across all Δ

*w*was slightly higher in the heterogeneous condition, as predicted by the indirect access hypothesis.

*F*(1.470, 10.293) = 44.129,

*p*< 0.001, after Greenhouse–Geisser correction for asphericity). Thus, in trials where participants walked faster in the test interval than in the standard interval (Δ

*w*> 0), the PSE shifted upward, whereas the reverse happened when the walking speed was lower in the test interval. There was a small but significant interaction between visual standard speed and walking speed difference (

*F*(1.779, 12.456) = 4.062,

*p*= 0.048, after Greenhouse–Geisser correction). This interaction was caused by an increase in the proportion of walking speed that was subtracted for higher standard speeds (0.20, 0.54, and 0.64, for standard speeds 1, 2, and 3 m/s, respectively). Visual standard speed did not have a significant effect on the PSEs (

*F*(2,14) = 2.564,

*p*= 0.113).

*F*(1,7) = 4.515,

*p*= 0.036, one-tailed, repeated measures ANOVA). This corresponds to the change in PSEs for individual walking speed differences in the heterogeneous trials, both showing that visual speed discrimination was affected by the changes in walking speed. Neither the main effect of visual standard speed (

*F*(2,14) = 2.661,

*p*= 0.105) nor the interaction effect (

*F*(2,14) = 0.005) was significant. The threshold differences between the homogeneous and heterogeneous conditions were solely caused by PSE shifts and not by changes in the sensitivity to visual speed. Analysis of the discrimination thresholds for the individual walking speed differences Δ

*w*between the two intervals did not show a significant difference (

*F*(1.395, 9.766) = 2.139,

*p*= 0.175 after Greenhouse–Geisser correction).

*k*of the walking speed increased with the visual speed. This is not accounted for by the simple linear model of Equation 1, proposed by Durgin et al. (Durgin & Gigone, 2007; Durgin et al., 2005). Our results therefore suggest a more complex non-linear compensation process. This parallels recent findings in studies on the compensation for the retinal effects of smooth pursuit eye movements (Freeman, 2001; Goltz, DeSouza, Menon, Tweed, & Vilis, 2003; Souman & Freeman, 2008; Souman et al., 2006; Turano & Massof, 2001; Wertheim, 1994). One possibility is that the subtracted walking speed

*kw*depends on visual speed

*v,*introducing an interaction between the two terms in Equation 1 (see Figure 4a). Interactions of this sort are supported by studies that have found that visual speed influences perceived walking speed (Mohler et al., 2007; Pelah & Barlow, 1996; Prokop et al., 1997; Rieser et al., 1995). Another indication for such an interaction is the effect of the characteristics of the visual scene. Subtraction effects are stronger for a simulated empty hallway or ground plane than for a cluttered scene with nearby objects (Durgin, Reed, & Tigue, 2007). Moreover, the amount of subtraction is reduced when observers look sideways during locomotion in a simulated hallway, seeing only laminar flow, instead of forward (Durgin et al., 2005). An alternative interpretation of these findings is that changing the visual scene does not change the amount of subtraction

*k,*but the visual speed signal itself (

*v*in Equation 1). This is similar to some models of compensation during smooth pursuit eye movements (Freeman, 2007; Freeman & Banks, 1998). In particular, a compressive transduction function of visual speed

*v*to estimated visual speed

*k*would produce larger PSE shifts for higher visual speeds (see Figure 4b).

*w*. However, this is unlikely for several reasons. First, studies on perceived walking distance suggest that the perceived walking speed is a non-linear function of actual walking speed (Bredin, Kerlirzin, & Israël, 2005; Mittelstaedt & Mittelstaedt, 2001). Second, Durgin et al. (2005) report different amounts of subtraction for walking in place on a treadmill (

*k*≈ 0.20) and normal overground walking at approximately the same speed (

*k*≈ 0.36). Treadmill walking differs in several respects from normal walking. When walking in place on a treadmill, vestibular cues to walking speed are likely to be less salient than when walking overground. Hence, the estimated walking speed

*v*may be lower for treadmill walking than overground walking, explaining the different amounts of subtraction found by Durgin et al. (2005). On the other hand, it is not clear to which degree vestibular cues contribute to perceived walking speed when walking at an approximately constant speed (Glasauer, Amorim, Vitte, & Berthoz, 1994; Mittelstaedt & Mittelstaedt, 2001). Third, perceived walking speed is influenced by stepping frequency (Durgin et al., 2007). Since stepping frequency when walking in place on a treadmill tends to be higher than with normal walking (Alton, Baldey, Caplan, & Morrissey, 1998; Murray, Spurr, Sepic, Gardner, & Mollinger, 1985; Stolze et al., 1997), this means that the same walking speed can lead to different estimates of walking speed by the perceptual system and, consequently, to different amounts of subtraction.

*k*of the walking speed (see Equation 1). The standard deviation of the distribution was chosen to be 0.35 m/s (estimated from pilot data on the homogeneous condition). As in our experiment, five trials per visual test speed were run for all combinations of three walking speeds (0.6, 1.0, and 1.4 m/s), with equal walking speeds run twice. For each test speed, the proportion of trials in which the estimated visual speed in the test interval was higher than that in the standard interval was computed. A cumulative Gaussian was fitted to these proportions as a function of test speed.

*k*= 0, walking speed differences do not matter and all psychometric curves lie on top of each other (barring sampling noise). However, as

*k*increases, the curves for different walking speed differences start to drift away from each other.

*w*in the heterogeneous condition will differ from each other, discrimination thresholds will be higher in the heterogeneous condition than in the homogeneous one, and the goodness of fit of the psychometric function will be lower in the heterogeneous condition. To assess the statistical power of these three effects, we simulated 10,000 experiments with 8 observers for a range of subtraction proportions (from 0 to 1). In each simulation run, the

*F*-value for a repeated measures ANOVA on the PSEs for different walking speed differences and the

*T*-value for the threshold differences were computed. The associated

*p*-values were then determined from the corresponding null hypothesis distributions. If changes in walking speed do not affect discrimination performance, the

*F*-values for the PSEs in the heterogeneous condition conform to a

*F*(3, 21) distribution (with the degrees of freedom determined by the 4 walking speed differences and the 8 observers). Under the null hypothesis, the

*T*-values for threshold differences between the homogeneous and heterogeneous conditions would follow a

*T*(7) distribution (with df = 8 observers − 1).

*p*-values of the simulated

*F*- and

*T*-values were determined from the null distributions. These

*p*-values indicate how unlikely the obtained value is under the null hypothesis that observers have direct access to the retinal flow. Figure A2 shows the distributions of these probability values as a function of

*k*in boxplots. As can be seen from this figure, the

*F*-value for the PSEs becomes quickly significant from

*k*= 0.1 (more than 75% of the simulations produced a

*p*-value < 0.05). For the

*T*-value associated with the threshold difference, this only happens once more than 50% of the walking speed is subtracted (

*k*≥ 0.5). Thus, an ANOVA on the PSEs in the heterogeneous trials constitutes a more powerful test of subtraction effects than a

*T*-test on the threshold difference between heterogeneous and homogeneous trials. From the figure, it seems that the

*F*-value becomes slightly less significant again for higher proportions of subtraction (

*k*> 0.7). However, this is an artifact of the restricted range of test speeds we used in our experiment (and simulation). For higher proportions of subtraction, the central part of the psychometric curve shifts out of the test speed range, making the estimated PSEs less reliable.

*k*in heterogeneous trials. However, this decrease was only small and hence not useful as a test of the effect of subtraction on visual speed discrimination. Further simulations showed that whether the goodness of fit in the heterogeneous trials improves or decreases with

*k*depends on the amount of noise in the sensory estimates. For larger standard deviations of the underlying Gaussian distribution (1.0 m/s instead of 0.35), deviance hardly changes as a function of

*k*anymore. For smaller standard deviations (0.1 m/s), deviance increases with

*k*.