Introduction

The sensory basis of knowledge of one’s own positition during locomotion consists of proprioceptive, vestibular, and visual signals. Vision may support the knowledge of one’s own positition by relationship to known landmarks in the environment (Judd and Collet 1998; Gillner and Mallot 1998), or by providing visual motion cues about direction, speed, and duration of movement, which must be integrated to achieve a measure of the distance traveled (Peruch et al. 1997; Bremmer and Lappe 1999; Riecke et al. 2002). Such integration may be similar to the integration of vestibular or proprioceptive inputs in the process of path integration (Mittelstaedt and Mittelstaedt 1980).

Path integration estimates the course of an extended movement by integrating short pieces of the movement to yield the total path (Mittelstaedt and Mittelstaedt 1973; Maurer and Seguinot 1995). If the path involves turns, the direction and amount of turn are also taken into account. Path integration has been studied extensively with blindfolded subjects in order to determine the ability to use proprioceptive and vestibular cues for the estimation of path length and turns (Loomis et al. 1993; Etienne et al. 1996; Loomis et al. 1999).

Visual motion cues also support the estimation of travel distance. Human observers are quite accurate in discriminating the path length of two visually simulated forward motions even when the velocities of the two motions differ (Bremmer and Lappe 1999). The ability to visually discriminate two path lengths is not trivial because optic flow, the visual velocity pattern generated by self motion, is ambiguous without information about the scale of the visual environment. When the distances to all visual elements are increased by a particular factor, and the forward speed is multiplied by the same factor, then the visual speeds in the optic flow in terms of degrees per second remain the same. Thus the ability to discriminate the distances of two simulated self-motions must rely on knowledge of the relationship between the environments.

Clearly discernable changes of the view of the environment between two succesive self-motions (such as a visible increase in eye height above the ground) can be used to re-scale the use of the optic flow information to allow accurate discrimination also between environments (Frenz et al. 2003). This suggests that path length estimation from visual motion is based on an estimate of one’s own velocity relative to the environment rather than directly on the pattern of optic flow. This estimate of self-velocity, however, has to be derived from the optic flow information.

Knowing one’s own position in the environment after a period of movement requires more than being able to discriminate movement distances. One needs to convert the absolute estimated movement distance to an updated position. This requires explicit use of the scale of the environment. Experimental investigation of an observer’s ability to convert absolute movement distances to an updated position involves relating a particular movement to a particular target position.

Redlick et al. (2001) presented participants with an initial target at a given distance within a visually simulated hallway, then extingushed the target and subsequently used optic flow to simulate a movement towards the now invisible target. Participants indicated the point when they felt that they had arrived at the target position. From this, the simulated travel distance was calculated that perceptually matched the visual distance to the initially shown target. For constant velocity movement, participants usually responded too early. For instance, for an initial target distance of 16 m they responded when they had travelled only 12 m. This suggests that the travel distance was overestimated: with a true movement of 12 m subjects felt they had covered the distance to the target that was initially 16 m away.

Frenz and Lappe (2005) conducted similar experiments but with a reversed order procedure. First, participants experienced a simulated movement of a given distance over a flat terrain. Then they adjusted a target to match the distance of the simulated movement. Participants on average placed the target too close. For instance, for a true movement of 5.5 m they placed the target at 4 m. This suggests that the simulated travel distance was underestimated. Such underestimation was found independently of the depth cues provided in the visual environment and of the reporting measure used (target adjustment, walking the same distance blindfolded, verbal distance in multiples of height above ground) (Frenz and Lappe 2005; Lappe et al. 2005).

Frenz and Lappe (2005) reproduced the overestimation found by Redlick et al. (2001) when they used the experimental procedure of showing the target first and requiring subjects to report the arrival at the target, as was done in the Redlick et al. study. However, this was only true for large target distances. When the procedure of Redlick et al. (2001) was used with small target distances, an underestimation of travel distance was observed.

In order to resolve this apparent incompatibility between the two data sets, we compared travel distance estimates in both experimental procedures over a large range of distances using the same equipment and subjects. We show that travel distance estimates are similar in both paradigms for short distances but become very dissimilar for large distances. We propose a path integration model based on leaky integration over the covered space that can explain all these observations.

Methods

Participants

10 subjects (3 females, 7 males, between 20 and 53 years old) participated in the study. All had normal or corrected-to-normal vision. All experiments were approved by the York University Human Experiment Subject Committee.

Setup and stimuli

The experiments were conducted in the Immersive Visual Environment at York University (IVY, Robinson et al. 2002). The subject was seated in the center of a 2.4  ×  2.4  ×  2.4 m cube, the walls of which were rear projection screens (Fig. 1a). Visual stimuli were presented on the walls by four CRT video projectors (Barco) running at 120 Hz frame rate and 1,024  ×  768 pixel resolution each. Floor and ceiling images were generated by four identical video projectors. Visual images for stimuli were generated by a nine node Linux PC cluster with one dedicated machine per projector and one additional machine for controling the stimuli and synching the displays. Stimuli were presented in stereoscopic vision using LCD shutter glasses (CrystalEyes) with 60 Hz frame rate per eye. The stereoscopic rendering was yoked to the head position and orientation to always provide the subject with the correct view of the scene from the current view point. Head position and orientation was tracked with a 6DOF head tracker (Intersense 900).

Fig. 1
figure 1

Setup and stimuli. a The Immersive Visual Environment at York University (IVY) is cubic room with 2.4 m side length. Each wall of the room is a rear-projection surface. The projection is driven by a nine node linux-based graphics cluster. Projection is stereoscopic and yoked to the head position of the observer. The subject was seated in the center of the cube. b The stimuli used in the experiment simulated forward movement inside a stereoscopic virtual corridor inside IVY. The walls of the corridor consisted of randomly colored panels 0.5 m wide and 3 m high. The floor and the ceiling were unstructured gray

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Visual stimuli simulated the inside of a hallway, 2.4 m wide and 3 m high, that extended from the initial viewing position 200 m into the distance (Fig. 1b). Because of the distance from the observer and the resolution of the screens the visible end of the hallway did not appear to expand during the simulated movement. The walls of the hallway were formed by colored panels 0.5 m wide and 3 m high. Each panel was painted a randomly chosen color. During the simulation, the colors of panels were refreshed at individual, randomly chosen intervals in order to discourage observers from tracking any particular panel over time. The random color changes also discouraged observers from memorizing the panel at which the target was located in the move-to-target condition (see below). The floor and the ceiling of the hallway displayed uniform gray surfaces.

The simulated movement consisted of forward motion of the viewpoint within the hallway with a velocity of 1, 2, or 4 m/s. Subject responses were registered using a game pad (Sony) connected to the computer. The game pad contained a response button and two adjust buttons.

Procedure

Two different main procedures were used in this study (Fig. 2). In the first, the adjust-target condition, the seated subjects pressed the response button to start the trial. Immediately thereafter a simulated forward movement commenced with speed and distance chosen randomly from a predetermined set of values. The distances were 2, 2.83, 4, 5.66, 8, 11.31, 16, 22.63, 32, 45.26, and 64 m. Each distance was run with at least two different speeds: 0.5 m/s for distances between 2 and 16 m, 1 m/s for distances between 2 and 32 m, 2 m/s for distances between 4 and 32 m, and 4 m/s for distances between 8 and 64 m. The simulated movement stopped when the predetermined distance was reached. Next a visual target appeared at a random position between 0.25 and 1.75 times the distance of the movement from the observer. The target was a frame filling the inside of the hallway with a cross in the center. The subjects moved the target back and forth with the two adjust buttons on the game pad. Each button press scaled the distance of the target by 0.99 or 1.01, respectively. The button repeated automatically until released. The subjects adjusted the target such that the distance of the target from the origin (indicated by a green line displayed on the floor below the feet of the subject) matched the distance of the movement simulation. When the subjects felt that the target was at the correct distance they pressed the response button and advanced to the next trial.

Fig. 2
figure 2

Procedures for Experiment 1. a In the adjust-target condition, first a visually simulated forward movement with a randomly chosen distance and speed was presented. Then a visual target appeared at a random position between 0.25 and 1.75 times the distance of the movement from the observer. The subject adjusted the target such that the distance to the target matched the distance of the movement simulation. b In the move-to-target condition, each trial began with the presentation of a static target in the hallway at a random distance. When the subject pressed a button, the target was extinguished and visually simulated forward movement with a randomly chosen speed was presented. The subject had to monitor the movement along the hallway and stop it by pressing the button again when the position of the target was reached

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In the second condition, the move-to-target condition, each trial began with the presentation of a static target in the hallway at a distance that was randomly selected from the set of predetermined distances. Distances, and speeds for the subsequent movement, were the same as in the adjust-target condition. When the subjects pressed the response button, the target was extinguished and simulated forward movement began with the randomly chosen speed. The subjects had to monitor the movement along the hallway and press the button again as soon as they felt that the position of the target was reached.

The two conditions were run in blocks with order of blocks randomized between subjects. Before data collection began, the subjects were given a few practice trials to familiarize them with the task. No feedback about performance was given in the practice trials or in the experimental trials. The order in which the conditions were run was randomized between subjects. To further reduce the influence of practice effects subjects usually completed two blocks of each condition. The second block was run at a later time when other conditions, including further conditions presented below, were run in-between. In each block, each of the distance and speed combinations described above was presented once. The number of trials per block was 32.

For the adjust-target condition, the distace to the adjusted target was the dependent variable. For the move-to-target condition, the distance covered during the movement from start to button press was the dependent variable.

Results

Adjust-target task

We first tested whether distance and speed of the simulation influenced the target placement. Because not all speeds were used with every distance we grouped conditions into three speed groups for the analysis. Each speed group covered the range of distances used with the respective speeds. The three speed groups were low (0.5 and 1 m/s, distances over the range of 2–16 m), medium (1 and 2 m/s, distances over the range of 4–32 m), and high (2 and 4 m/s, distance over the range of 8–64 m). For the low speed group, an ANOVA with factors speed and distance gave a significant influence of distance (F(1,6) = 45.6, P < 0.001) and a small but significant difference between speeds (F(1,6) = 20.7, P < 0.001) as well as a significant interaction (F(1,6) = 2.3, P < 0.05). For the medium speed group, an ANOVA with factors speed and distance gave a significant influence of distance (F(1,6) = 63.4, P < 0.001), no significant difference between speeds, and no significant interaction. For the high speed group, an ANOVA with factors speed and distance gave a significant influence of distance (F(1,6) = 90.4, P < 0.001), no significant difference between speeds, and a significant interaction (F(1,6) = 2.3, P < 0.05). Since speed influenced the target placement only slightly and only for small distances we collapsed the data over speeds and analyzed the dependence of target placement on distance.

Figure 3a plots the median target placement as a function of travel distance over subjects. The figure shows that subjects generally underestimated the distance of the movement, i.e., placed the target too near, in the adjust-target condition. For example, when the distance of the movement was 16 m subjects on average placed the target at a distance of 12.3 m, i.e., 3.7 m , or 28%, repectively, too close. The underestimation was especially pronounced for large distances, reaching 14 m (22%) for 64 m of movement. For short distances, target placement was closer to correct. For example, the avearge target placement for the 4 m movements was 3.83 m, i.e., only 17 cm (3%) too close.

Fig. 3
figure 3

Results of Experiment 1. a Adjusted target distance versus actual distance of the movement simulation. Data points are the medians over ten subjects. Error bars give the inter-quartile range. The dashed line indicates veridical perfomance. The continuous line is the fit by the leaky integration model (see section “Leaky-integration model”). The adjusted target distance is overall smaller than the actual travel distance. Thus, the perceived travel distance underestimates the actual travel distance. b Travel distance in the move-to-target condition vs. distance to the initially seen static target. The actual travel distance is overall smaller than the distance required to reach the initial target. Therefore, the perceived travel distance overestimates the actual travel distance. c Data from a and b in a plot of perceived versus actual travel distance. In the adjust-target condition, the perceived distance corresponds to the distance to the adjusted target. In the move-to-target condition, the perceived distance is given by the initial target, since this is the static distance that perceptually corresponded to the distance of the movement. Note that the error bars are vertical in this condition because the dependent variable, the travel distance to the target, is on the x-axis

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Experiment 1: move-to-target task

In the move-to-target condition, an ANOVA for the low speed group with factors speed (0.5 and 1 m/s) and distance (2–16 m) gave a significant influence of distance (F(1,6) = 91.4, P < 0.001), no significant difference between speeds, and no significant interaction. For the medium speed group, an ANOVA with factors speed (1 and 2 m/s) and distance (4–32 m) gave a significant influence of distance (F(1,6) = 87.0, P < 0.001), no significant difference between speeds, and no significant interaction. For the high speed group, an ANOVA with factors speed (2 and 4 m/s) and distance (8–64 m) gave a significant influence of distance (F(1,6) = 90.4, P < 0.001), no significant difference between speeds, and no significant interaction. We therefore collapsed the data over speeds again.

Figure 3b shows the median travel distance until button press as a function of initial target distance in the move-to-target condition. For large distances, subjects overestimated the distance of movement, i.e. they pushed the response button too early. For an initial target in 64 m distance the subjects on average pushed the button after a movement of 54 m. Thus, a movement of 54 m corresponded to a static target distance of 64 m, which is an overestimation of the movement distance by 10 m or 18.5%. For short distances, the responses were more accurate with a slight tendency to underestimate the distance of the movement, i.e., to press the button a bit too late. For example, for an initial target distance of 8 m subjects on average pushed the button after a movement of 8.7 m, which gives an underestimation of 9%.

Discussion of experiment 1

Although the data in Fig. 3a, b look similar the conditions actually show opposite misestimations. For distances over about 12 m, there was underestimation of travel distance in the adjust-target condition and overestimation of travel distance in the move-to-target condition. In the adjust-target condition, estimated travel distance corresponds to the placement of the target, which was on average too close. For example, for a travel distance of 11.3 m the median adjusted target distance was 9.9 m, which corresponds to an underestimation of about 12%. In the move-to-target condition, the distance traveled until button press corresponds to the distance at which subjects felt that they had travelled as far as the initial target distance. For example, for an initial distance of 45.26 m the median travel distance was 34.6 m. Thus, a travel distance of 34.6 m corresponded to a target distance of 45.26 m: an overestimation by 24%.

To better illustrate this difference, Fig. 3c plots the perceived distance of the movement as a function of the actual distance of the motion simulation for both conditions. In the adjust-target condition, the actual distance simulated was the distance of the motion simulation, and the perceived distance was the distance to the adjusted target. In the move-to-target condition, the actual distance of the simulated movement was the distance traveled until button press, and the corresponding perceived distance was the initial target distance, since this is the static distance that perceptually corresponded to the distance of the movement. Figure 3c replots the data to illustrate that perceived distances are smaller than the actual distance of the movement in the adjust-target condition but larger in the move-to-target condition, especially for distances larger than 12 m.

These results are consistent with earlier work (over a more restricted range of distances) that showed underestimation in the adjust-target condition (Frenz and Lappe 2005) and overestimation for large distances along with underestimation for small distances in the move-to-target condition (Redlick et al. 2001; Frenz and Lappe 2005). The results further show that these estimation errors occur over large distances and with identical subjects and identical stimuli. It is therefore likely that the difference is due to inherent differences in the tasks. We sought for a coherent explanation for both effects.

Distance over- or under-estimation in a particular experimental setting can be explained in many ways. The combination of underestimation in one task and overestimation in another, however, restricts the possibilities.

First, let us look at a candidate explanation that does not work. The perceived distance to a static visual target is usually underestimated (Luneburg 1950; Foley 1980; Loomis et al. 1992). This underestimation is even more pronounced in virtual environments (Knapp and Loomis 2004; Thompson et al. 2004; Plumert et al. 2005). When we asked subjects to verbally indicate the distance to the initially presented target in meters we also found a severe underestimation of the distance (Fig. 4). Note that interpreting subjects reported distance in meters is complicated because it involves an explicit scale. However, as explained below, any coherent underestimation of static distances could not explain the misestimation of travel distances we report which has both over- and underestimation.

Fig. 4
figure 4

Underestimation of static distances. The plot shows the median reported distance to a static target versus the true distance to the target. The distance to the target is perceptually foreshortened

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Let’s assume that, in the move-to-target trials, the subjects initially saw the target at a particular distance and formed a mental representation of the target distance which, according to the underestimation of static distances, is less than the true target distance. If the subjects then used this distance representation during move-to-target trials, they should have pressed the button early, because the movement reached the represented distance earlier than the true distance (note that this assumes that the estimation of travel distance is essentially veridical). Thus, an understimation of static distance may explain the data of the move-to-target trials. It would predict, however, different results in the adjust target trials. If we again assume that travel distance estimation is veridical, then, at the end of the movement in the adjust-target trials the subjects should have a correct representation of the travel distance. If the subjects then adjusted the static target to match that distance representation the target would have been set further than the value of the distance representation, because the perceived distance to the target was less than its actual distance to the target. This would predict that data in Fig. 3a should be larger than veridical (above the dashed line) by about the same magnitude as the data in Fig. 3b is lower than veridical. Clearly, this is not the case.

The argument works the same way if one begins with the adjust-target condition and then tries to predict the data for the move-to-target condition. An assumed misrepresentation of static distance may explain the data in one condition but will then fail to predict the data in the other condition. The combination of underestimation in one task and overestimation in the other task constrains the possible explanations.

It is noteworthy that the data in in Fig. 3a, b are similar, however, in that the amount of over-/under-estimation increases with increasing travel/target distance. Moreover, the variance, plotted in Fig. 3 by the inter-quartile range, also increases with increasing distance. These observations suggest that the cause of the misestimation involves some form of decay over distance.

Leaky-integration model

Path integration is an important concept in distance estimation from vestibular and proprioceptive signals (Mittelstaedt and Mittelstaedt 1973, 2001; Maurer and Seguinot 1995; Loomis et al. 1999). In path integration, the individual is assumed to track the amout of space covered by an extended movement—and the direction, which we do not use here because our movement is one-directional—by accumulating segments of individual movements over the course of the entire movement. Mathematically this amounts to integrating position throughout the movement. Misrepresentation of the length of the entire movement may arise if the integration of the new position uses a misrepresentation of the momentary position change (essentially a gain issue) or if the integration is leaky. We wanted to explore whether such leaky integration could fit our data set, including both the adjust-target and move-to-target conditions.

Leaky integration assumes that a state variable, such as the current distance from the starting point, is incremented with each step by the distance of the step with a gain factor k, but that it is subsequently slightly reduced in proportion to a leak factor α. Thus, the state variable is continuously incremented according to the movement but has a tendency to decay by itself.

In path integration, the integration over the path segments can be performed in time, i.e., the state variable is updated in successive time intervals, or over space, i.e., the state variable is updated with every step taken (Mittelstaedt and Mittelstaedt 1973). For perfect integration these are equivalent and most models for path integration integrate the positions over time. If the integration leaks, however, there is a difference between integration over time and integration over space. The former predicts that the state variable decays with time, even when the individual does not move. The latter predicts that the state variable decays only when there is movement. Furthermore, integration over time predicts that the distance estimate, i.e., the value of the state variable at the end of the movement, depends on the movement duration. If the movement takes longer, there is more decay through leakage. Hence, movements that cover the same distance but do so with different speeds are predicted by such a leaky integrator model to result in different estimates of path length. This was not the case in our experiments where speed was found not to be a significant factor. We therefore explored a leaky integrator model integrating over space.

The model is explained in detail in the Appendix. For the adjust-target condition, we assume that the state variable is the distance from the origin. The state variable is incremented by an amount proportional to the length of the step size with a proportionality constant, or gain factor, k. In the mathematical formulation the step size is considered infinitesimal, but conceptually it can be thought of as having a small but finite length. Leakage occurs with every spatial step and is proportional to the current value of the state variable with a proportionality constant, or leak rate, α. The model predicts that longer distances lead to more decay such that the percentage of underestimation increases with the distance of the movement. This is qualitatively in accordance with the data.

In the move-to-target condition, the task begins with a given distance that has to be covered with the subsequent movement. We assume that the state variable is the distance to the target, which has to be nulled by the movement. The state variable is decremented in every step proportional to the length of the step size. The proportionality constant is again k, the same value as used in modeling the adjust-target data. The leakage occurs with every spatial step and is proportional to the current value of the state variable according to the leak rate, α, also the same value as used above. This model predicts that two processes lead to a reduction of the perceived distance to the target: the decrement according to the forward movement and the leakage of the integrator. Thus, with ongoing movement the distance to the target becomes overproportionally smaller because of the leakage, and the point of perceived distance zero is reached early. Again, this is qualitatively consistent with the data.

The appendix details how the point at which the target is reached is calculated from the initial distance using the leaky integration model. In order to see how the model agrees with the data quantitatively we fitted the model parameters k and α to the data from the two conditions. The best fit model is generated with k = 0.98 and α = 0.0076 and is plotted as continuous lines through the data in Fig. 3.

The leaky integration model explains both underestimation in the adjust-target condition and overestimation in the move-to-target condition with the same set of parameters. This is because the leakage works towards reducing the current estimate of travel distance in the adjust-target condition, but works towards reducing the current estimate of distance to the target in the move-to-target condition. A similar proposal has been made by Mittelstaedt and Glasauer (1991), but for a leaky integrator over time.

The gain factor k describes whether the increment in each step of the integrator is larger (k > 1) or smaller (k < 1) than the actual step size. Because the best fitting k is slightly smaller than one (0.98) the input to the integrator is slightly smaller than the veridical distance covered by each step. Thus, the curves in Fig. 3a, b are both below the line of slope 1. However, for other values of the parameters, the leaky integration model may yield other combinations of over- and underestimation. Figure 5 presents examples of model fits to the data from three individual subjects to show the variability in responses and the ability of the model to capture this variability. The performance of subject JS (k = 0.99 and α = 0.011) is very similar to the average performance shown in Fig. 3. Subject LRH (k = 0.79 and α = 0.011) shows more pronounced underestimation in the adjust-target condition, and in the move-to-target condition underestimation for target distances below 30 m and overestimation for targets beyond 30 m. The model captures this behavior and thus demonstrates that it can yield underestmation and overestimation of the travelled distance depending on the target’s distance. Subject MBC (k = 1.25 and α = 0.015) shows strong overestimation in the move-to-target condition, and some overestimation for short distances and underestimation for large distances in the adjust-target condition. Here, the best fitting gain factor k was larger than one, indicating that the input to the integrator is larger than the veridical distance covered by each step.

Fig. 5
figure 5

Fits of the leaky integration model to the results of Experiment 1 for three individual subjects. Each data point is the average of between two and four trials, hence no error bars are shown. The individual results illustrate the variability in perfomance between the two task. The model captures this variability. The best-fit parameters of the model differ between subjects but are identical for both conditions for each subject

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Experiment 2: updating of target distance

The leaky integration model holds that in the move-to-target condition the distance to the target decreases over the course of the movement, because of the leakage. Thus, if the movement runs only part of the way the remaining distance to the target should be overproportionally decreased. To validate the model, we tested this prediction in an experiment that combined initial vision of the target with later adjustment of the remaining target distance.

Procedure

The experimental trials in Experiment 2 consisted of three phases, as shown in Fig. 6, which were all taken from the experiment described above. First, the seated subject pressed the response button to start the trial. Upon button press, a static target was presented in the static hallway at a distance that was randomly selected from the set of predetermined distances (4, 5.66, 8, 11.31, 16, 22.63, 32, 45.26, and 64 m). When the subjects pressed the response button again, the target was extinguished and simulated forward movement began with the speed and duration randomly chosen from a predetermined set of values. The speeds: were 1 m/s for distances between 4 and 16 m, 2 m/s for distances between 4 and 64 m, and 4 m/s for distances between 22 and 64 m. The distance of the simulated movement covered only either 70.7 or 50% of the true distance to the target such that the movement ended at the next-smaller or the second-to-next-smaller distance from the original set of distances. For example, if the initial distance was 16 m, the movement ended after 11.31 or 8 m. The total number of trials was 36.

Fig. 6
figure 6

Procedure for Experiment 2. Each trial began with the presentation of a static target in the hallway at a random distance. The subjects estimated and memorized the distance to the target. When the subject pressed the button, the target was extinguished and visually simulated forward movement with a randomly chosen speed was presented. The simulated movement stopped after part of the way to the target, either when 50 or 70.7% of the distance to the target was reached. Then the visual target re-appeared at a random position between 0.25 and 1.75 times the remaining distance from the observer. The subject adjusted the target such that the distance to the target matched the remaining distance, i.e., such that the target appeared to be in the same spot in the hallway as before

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The simulated movement stopped when the predetermined distance was reached. The subsequent adjustment procedure now concentrated on the final target distance, i.e., the remaining part of the distance from the initial target to the observer (e.g., if the initial distance were 16 m and the movement covered 11.31 m the final target distance was 4.69 m). The visual target re-appeared at a random position between 0.25 and 1.75 times the final target distance. The subject adjusted the position of the target such that the distance of the visible target from the subject (indicated by a green line on the floor below the feet of the subject) matched that final target position after the movement simulation, or, in other words, such that the visible target was at the same place in the hallway as it was initially.

Results and discussion

From the best fit parameters of the leaky integration model to the previous adjust-target and move-to-target conditions one can calculate the prediction for the perceived final target distances, i.e., the distance that the observer adjusts. This is given by Eq. 5 in the Appendix. The prediction and the data from the experiment are shown in Fig. 7.

Fig. 7
figure 7

Results of Experiment 2. Adjusted target distance versus actual remaining target distance after the part-way movement simulation. Data points are the medians over ten subjects. Error bars give the inter-quartile range. Squares present data from trials in which the movement covered 70.7% of the initial distance to the target. Diamonds present data from trials in which the movement covered 50% of the initial distance to the target. The dashed line indicates veridical perfomance. The continuous lines are the predictions of the leaky integrator model with its parameters taken from Experiment 1. The adjusted target distance is overall smaller than the actual remaining distance to the target. This underestimation of remaining target distance is stronger when a longer path had been traversed. These effects are predicted by the model

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The predictions of the leaky integrator model can be summarized as follows. First, the adjusted target distance is smaller than the true target distance, because of the leakage and because the gain is smaller than one. Second, for longer initial distances and therefore longer movement distances the undershoot in the adjusted target distance becomes progressively larger as more and more leakage occurs. Because the final target distance is a constant proportion of the initial distance (50 or 70.7%) this predicts that the undershoot should also increase with increasing final target distance. Third, for the same final target distance, the adjusted target distance is smaller in the condition that went 70.7% of the way than in the condition that went 50% of the way. This is because the former covered a longer distance and therefore involved more leakage.

All of these predictions are borne out by the data (Fig. 7). First, the adjusted target distances lie below the line of slope 1, i.e., they undershoot the true final target distance. Second, the undershoot becomes progressively larger with increasing final target distance. This dependence of the undershoot on final target distance was significant (F(1,16) = 16.2, P < 0.001, ANOVA with factors travel percentage (50% or 70.7%) and final target distance). Third, the adjusted target distances were significantly smaller in the 70.7% condition than in the 50% condition (F(1,16) = 25.2, P < 0.001, ANOVA with factors travel percentage and final target distance). Figure 7 shows that the data matches the predictions very well in the condition that went 70.7% of the way but somewhat overreaches the prediction in the condition that went 50% of the way to very far targets.

General discussion and conclusions

This study started out from apparent differences in the ability to gauge distance from optic flow in two different experimental paradigms. Motion based distance estimates undershot the true distance of a simulated self motion when the distance from the starting point had to be indicated (Frenz and Lappe 2005; Lappe et al. 2005) but overshot the true distance of a simulated self motion when the arrival at a previously specified location had to be indicated (Redlick et al. 2001; Frenz and Lappe 2005). Comparing both methods in the same setup and with the same set of subjects yielded several new findings. First, distances are underestimated in the adjust-target condition and overestimated in the move-to-target condition especially for large movement distances. Second, the dependence of under-/overestimation on movement distance is not linear, as earlier results had suggested (Redlick et al. 2001; Frenz and Lappe 2005), but resembles a logarithmic function. Third, the misestimation occurs even in a fully immersive virtual environment that contains ample depth cues from stereo vision and motion parallax from tracked head movements. Fourth, the misestimation cannot result solely from a general underestimation of static distance because this would predict opposite results: if the static target in the move-to-target condition is perceived too near to the observer than, the adjusted target in the adjust-target position should also be perceived too near to the observer. To indicate the veridical movement distance the target would then have to be placed further from the observer so that its perceived distance matches the true travel distance. The data show, however, that the target is commonly placed too near.

Perceived static distance is often underestimated in virtual reality when compared to distance estimates in real environments, depending on factors of the display (Knapp and Loomis 2004; Thompson et al. 2004; Plumert et al. 2005). Our own measurement of perceived distance to the static target also suggests underestimation (Fig. 4) that is stronger than that typically found in real world measurements (Luneburg 1950; Foley 1980; Loomis et al. 1992). It may be that our distance misestimates from motion also exaggerate the errors that occur in equivalent real life situations. However, the use of virtual technology is the only way to study the usage of pure visual signals during self-motion without vestibular or proprioceptive inputs. Sun et al., (2004) studied distance perception to static targets in real world experiments with methods very similar to ours. They also compared distance indication in a condition that involved walking to a remembered stationary target with distance indication in a condition in which a target had to be adjusted to reflect the distance of a prior walk. Error analysis suggested a common representation of walking distance in the full cue situation (vision, proprioceptive, and vestibular information present) but a dependence on condition when only proprioceptive and vestibular information were available. It is difficult to compare this directly to our results, however, because their conditions always included proprioception and vestibular cues.

Based on the differences in estimation of travel distance in the two conditions we propose a leaky integration model. In this model the parameter necessary for the completion of the experimental task is continuously updated with new movement information. The task-relevant parameter is the distance from the origin in the adjust-target condition and the distance to the target in the move-to-target condition. The continuous updating follows the same rules and parameters in both conditions such that the distance from the origin is incremented with a particular gain for every instantaneous movement, and the distance to the target is decremented according to the same gain value. The integration is assumed to be leaky with the same leakage factor in both conditions. Because of the leak, the distance from the start point in the one condition and the distance to the target in the other condition are reduced over the course of the movement. This results in underestimation of the travel distance from the origin in the adjust-target task and at the same time in an overestimation of the distance covered in approaching the target in the move-to-target task.

Our model integrates over space. Integrating travel distance over time is not consistent with the data, since it would predict a dependence on duration of the movement, and hence speed, which was not observed. Integrating in spatial coordinates is interesting because it predicts that leakage should only occur during movement. If leaky integration occurred over time, on the other hand, leakage should occur as time progresses even if the observer remains stationary. Thus, because of integration over space the integration is automatically yoked to the observer movement. Integration over space predicts that the amount of leakage increases with the distance of the movement (confirmed in Experiment 2).

In order to allow integration to occur over space, the visual system must convert the visual motion signal from every—theoretically infinitesimal—spatial step to the momentary distance gain by that step. This involves an additional integration process, that of motion to position, before the signal can be used in the distance integrator. The effectiveness of such integration is captured in the model by the gain factor k. If k = 1, each infinitesimal step is converted accurately into the corresponding distance covered during the infinitesimal movement. Our observation that k is on average slightly smaller than one suggests that the transformation from visual motion to distance slightly underestimates the instantaneous travel distance.

Our mathematical model assumes an infinitesimal step size but it may be that the step size used by human subjects is finite such that the first integration is performed over a particular step distance and the result is added to the state variable in an iterative manner. Our data give no indication of this, but it may be interesting for further study. Moreover, the integration gain from motion to momentary distance may depend on the availablity of visual depth cues such as stereo or perspective foreshortening, and may in more natural situations also access proprioceptive or vestibular information about instantaneous velocity and movement distance. It would be interesting to investigate whether the availability of such cues modulates the value of the gain and whether or not this affects the integration constant α.

The underestimation of static distance that we observed cannot explain the difference between the distance estimates in the two conditions. However, static distance estimates are required to fulfill the task in both conditions. It may thus be that the foreshortening of static distances also modulates the task performance. We explored including the static distance foreshortening in the model by an additional transformation step in which perceived static distance was calculated from true static distances according to Fig. 4 and vice versa. In this variant of the model, initial target distance in the move-to-target condition was first converted into a perceived target distance and the subsequent integration of movement distance was performed until the perceived distance was reached. In the adjust target condition, the final estimate of movement distance was considered to be a perceived distance and this was then converted into a true distance. The best fit by this model was considerably worse than the fit by the original model. We therefore concluded that the underestimation of static distances did not contribute to the estimation of travel distance from visual motion.

To conclude, our results show that humans are capable of estimating travel distance from visual motion alone, by using an integration process, but one that leaks over the spatial extent of the movement. Our model suggest that the task to be performed (estimating the cumulative distance from start or estimating the remaining distance to goal) determines the variable to be integrated over the movement, but that the integration process itself is the same.