Exploring the Effects of Self-Overlapping Spaces on Distance Perception and Action Judgments

Boletsis’ VR locomotion typology [2] defined four VR locomotion types: Motion-based, such as arm-swinging [11]; Roomscale-based such as Natural/Real Walking [14]; Controller-based such as joysticks; and Teleportation-based such as point and teleport [3]. These locomotion types are defined based on their characteristics of Interaction (physical or artificial), Motion (continuous or non-continuous), and Interaction space (open or limited). There has been plenty of work studying the differences in spatial understanding between these characteristics.

Paris et al. compared four artificial motion types (two continuous and two non-continuous) in their study testing how well video game locomotion techniques perform in navigational tasks. They found that continuous locomotion performed better in a blind-pointing task than the non-continuous methods, but there was no significant difference within the continuous motion types [15]. The benefit of continuous locomotion on spatial understanding is further supported by Cliburn et al. [7]. In their study, they found that teleportation was able to navigate a scene faster, but that participants navigated to erroneous locations more often than the free-roam locomotion.

Ruddle and Lessels found that information gained through translational movement is more important for building spatial understanding than rotational-movement and visual-information [20]. This hierarchy of information-source importance is supported by a study from Sigurdarson et al. Their study found that adding a physical rotation to participants did not increase their performance in pointing tasks [23]. However, it appears that rotational information could be important in non-continuous locomotion; when comparing concordant teleportation (the user physically rotates) and discordant teleportation (the user artificially rotates), Cherep et al. found that concordant teleportation performed better in a navigational task; though both performed worse than real walking [6]. Zanbaka et al. compared real walking to joysticks (3DOF- and 6DOF-tracking) in navigation and cognitive tasks. They found that participants who navigated through real walking scored higher in various aspects of a cognitive questionnaire and produced better sketch maps [32]. In summary, continuous, physical locomotion (i.e., natural walking) tends to have better performance in navigational tasks.

Even though real walking provides these benefits to spatial cognition, it cannot be universally implemented. One of the largest barriers to this is the amount of physical space needed as virtual environments grow. There have been several classes of methods introduced to overcome this limitation; one of which is known as redirected walking. Redirected walking works by manipulating the mapping of physical movement to virtual movement so that they are not one-to-one. This is done either through rotational gains applied to the user’s view or turns so that a straight line in VR is a curve in the real world or through translational gains so that a distance traveled in VR is greater than the distance walked in the real world [13]. Several studies have observed that the benefits of real walking on spatial understanding are maintained through redirection. Peck et al. compared redirected walking vs walking-in-place vs joysticks in naive and primed search tasks. They found that participants locomoting with redirected walking were able to complete searches in less distance traveled, repeated routes less often, made wrong turns less often, and performed better at pointing tasks, suggesting that they were able form better cognitive maps [16]. Langbehn et al. conducted a similar study comparing redirected walking vs joystick vs teleportation; in their study, they found that participants in the redirected walking condition were better able to recall the position of objects in the scene, and they found no significant differences between most of their pointing tasks [10]. Though this could be due to the characteristics of the environment, as Cherep et al. found that landmarks and geometric boundaries reduce the cognitive costs of teleportation [6]. There is another method that redirects the user, though it manipulates the environment rather than the user. Suma et al. introduced Change Blindness Redirection; by leveraging the change blindness phenomenon [24], they swapped the position of doors and hallways to have users exit a room in a different direction than they came in from. In their study, they found that participants were able to generate sketch maps that resembled the intended virtual space, only one out of 77 participants explicitly noticed the change, and they found no difference in pointing task performance when measured before or after the change [25].

The method to overcome the physical space limitation that we are most interested in is Impossible Spaces. Impossible spaces utilize overlapping geometry to have multiple virtual locations exist in the same physical space. In their initial study, Suma et al. found that the overlap detection threshold varied between small and large rooms (55.57% and 31.06%, respectively) [26]. Plenty of other work has been conducted on studying detection thresholds. Vasylevska and Kaufmann experimented with the effect of hallway shape on overlap detection. They found that a long, C-shaped hallway had a much lower detection rate (\({\lt}\)40% at 40% overlap) than a short, straight hallway (\({\gt}\)70% at 40% overlap) [28]. They then expanded on this work to further study more complex hallways; path asymmetry and alternating the direction of turns had the greatest effect on lowering overlap detection with curved corridors exaggerating these effects further [29]. The aforementioned studies also measured distance estimation through a blind walking task. Suma et al. observed that participants tended to overestimate the physical difference between two points, but these estimations were close to what the distances would be in a non-overlapping space; this suggests that users were either using local visual information or considering rooms relative to each other [26]. Vasylevska and Kaufmann observed similar results, and the path complexity added to the overestimation [28]. A couple of studies have specifically investigated the effect impossible spaces have on spatial judgments. Robb and Barwulor measured pointing task performance in possible and impossible spaces. They found that while error increased in the impossible spaces, the direction participants were pointing was where the object would have been if the space was possible [18]. This further supports the possibility that users rely on relative information in spatial judgments. Robb and Barwulor further explored this in their study that had participants estimate the size of rooms in possible and impossible spaces, while either considering one room at or a time or both at once. They found that when measuring independently participants were more accurate, regardless of the overlap, and in the simultaneous condition, they underestimated room sizes while preserving the relative ratio between the rooms [19]. An important next step is to investigate how these underestimations and size judgments affect one’s actions within the environment. To achieve this we designed an action-judgment task that asked participants to choose which button in two overlapping rooms would be a shorter walk.

Another aspect of spatial perception and understanding that we are interested in is distance estimations. In their 2013 review, Renner et al. identified three classes of methods for measuring distance estimation: verbal estimates, perceptual matching, and visually directed actions (e.g., blind walking). They found that regardless of the method egocentric distances in VR are almost always underestimated (around 75%) and that blind walking is the most often used method [17]. In a more recent review, Dong et al. also summarized methods used for distance into three categories: perceptual, direct action, and indirect action. The verbal and perceptual matching categories from Renner et al. fall into the perceptual category. Perceptual methods involve reporting the perceived distance either verbally or through a comparison task. They are simple and easy to understand but they might not be entirely suitable by themselves. Direct action methods involve bodily movements and reporting the perceived distance through some motion directly along the distance (e.g., walking the perceived distance). They are accurate and better integrate the perception-action coupling, but they are more constrained to space and can be more fatiguing. Indirect action methods differ from direct action methods such that they involve movements not directly towards the target (e.g., walk at an angle away from the target then turn and walk towards it). They better expand distance estimations to the 3D space, but they are more complex and less accurate [8]. Locomotion and interaction can benefit from distance estimations; in a series of experiments, Waller and Richardson found that walking around and interacting with a virtual environment can improve distance estimation (from 50% to nearly veridical) and that the interaction does not need to be task-oriented for this improvement [31]. In a study to better understand the effect of visual and body-based information on distance estimation, Campos et al. found that body-based information has a higher “priority” than visual information when walking [5]. As stated in the previous section, Suma et al. and Vasylevska and Kaufmann found that when estimating distances between points in overlapping rooms that participants overestimated but that their estimations were close to what the distances would be in non-overlapping rooms [26, 28]. In contrast to most previous work that measured direct, through-the-wall distances between two points, and to be in line with our action-judgment task, we had participants estimate the length of the paths they would walk to the buttons from a specified point. Due to the magnitude of the distances that would be estimated, we could not safely use a direct action task such as blind walking, instead, we used a variation of the perceptual distance matching task from Dong et al. [8] as it is an accurate perceptual method.

This study was approved by our institution’s review board (IRB2021-0870).

As we were interested in how the amount of overlap in an impossible space influenced decision-making, we utilized a repeated-measures within-subject design in which we manipulated two variables: (1) the amount of physical space occupied by both rooms (overlap percentage), and (2) the difference in distance to the button in Room A and the button in Room B from a predetermined judgment location (path delta). We ended up with five overlap percentages (10%, 28.75%, 47.5%, 66.25%, and 85%) and six path deltas (0.1 m, 0.3 m, 0.5 m, 0.7 m, 0.9 m, 1.1 m) for a total of 30 trials. The order of the trials was fully randomized based on each participant’s ID; participants could see all combinations of overlap and path deltas in any order.

Participants wore an Oculus Quest One HMD with the Guardian Boundary disabled to explore a virtual scene built using the Unity game engine. The total size of the virtual scene was 7\(\times\)5.5m and consisted of two environments: (1) the experimental impossible space and (2) a simple environment used to report perceived distances in the impossible space. The environments were loaded and unloaded around the user so that they would not have to move physically to travel between them.

The first environment consisted of a hallway 7 \(\times\) 1.5 m in size. This hallway connected two rooms, each of which having a depth of 3 m and variable length width that ranged from 1.6 m to 6.9 m (a 0.1 m offset was used to avoid clipping textures).¹ The walls for a given room were loaded when participants approached the doorway and unloaded after leaving the room and moving slightly away from the doorway; this ensured that participants never saw any of the far room when at the entrance to the closer room. In the far back corner of each room was a button that would turn from red to green when pressed. As seen in Figure 1 the wall between the rooms and hallway is thick. While not ideal for maximizing usable space, we wanted to limit how much could be seen in the hallway and a room at the same time. The judgment location was represented by a platform with two panels in front of it.

The second environment was a room that was 2 \(\times\) 3.75 m. This room was used for the distance estimation task and involved the users moving the back wall. The room was empty apart from a button on the left side and a wooden bar that spanned the length of the room. This bar was placed to dissuade users from moving forward as during a pilot study, some participants would want to walk the estimated distance.

We systematically generated trials using the environment dimensions, the percent overlap, the desired path delta, and a fixed room size. A list of all 30 trial configurations can be found in Table 1.

In general, percent overlap is calculated with the following formula where \(p_{o}\) is percent overlap, \(L_{a}\) is the length of room A, \(L_{b}\) is the length of room B, and \(L_{t}\) is the total length of the environment:

This formula can be flipped to find the length of a given room:

In order to keep the dynamic room size within the bounds of the environment and large enough to be entered we had to shift the range of static room sizes for some overlap percentages.

To calculate the path delta, we first found the length of the path from the judgment location to a given room’s button. This was computed by finding the distance from the judgment location to the far edge of a room’s doorway and then the distance from that point to the room’s button. The path delta can then be computed by subtracting the distance to Room B’s button from the distance to Room A’s button (see Figure 1). The path delta is affected by the room sizes and the position of the judgment location. Since the room lengths are locked to control the percent overlap, in order to change the path delta, we change the offset of the judgment location from the center of the hallway. Given the total length of the environment (\(L_{t}\)), the width of the rooms (\(W_{r}\)), half the width of the hallway (\(W_{h}\)), and the length of the rooms (\(L_{a}\) and \(L_{b}\)), the following process can be used to calculate the offset for the judgment location graphically:

It is worth noting that the actual distance calculated is not an exact value for what would actually be traveled since it is a perfectly straight line and goes to the very edge of the doorway, but it provides a close enough estimation for our purposes.

All configurations were calculated with A as the static room, but when generating trials, the configuration of the two rooms had a 50% chance of being mirrored. This is done by switching which room has the static size and changing the sign of the offset; this maintains the overlap percentage and the absolute value of the path delta. This was done so that the closest button was not always in the same room.

Since most previous work has only investigated spatial judgments and not actions within these spaces, the primary task for this study was an action-judgment decision based on two distances, namely the distance required to walk to the location of each button in each room. Note that this is not distance “as the crow flies,” but the distance participants would have to physically walk from the current location in order to reach each button. Participants were tasked with determining which of two buttons was a shorter walk away from their current position (Section 3.3 for more details). They were asked not to use any tricks like counting steps; we wanted their decisions to be based on visual and body-based information.

After indicating which button they felt was a shorter walk away, participants were then asked to estimate the actual distance they would have to walk to reach each button. This involved moving a wall in an infinite hallway until it matched how far they believed the button was; this is similar to the perceptual distance matching explained in the review from Dong et al. [8], though they were not able to see the target at the time of the task. While they did not report absolute distances, Campos et al. [5] used a similar task to measure distance traveled.

For each trial, we recorded: (1) the button they thought was closest, (2) the estimated distance to button A, and (3) the estimated distance to button B.

We hypothesized the following:

Participants were recruited through the psychology department’s Sona system. Twenty-two people completed the study. Sixteen (72.73%) identified as female and six (27.27%) identified as male. Twenty-one (95.45%) were between the ages of 18 and 21, and one (4.55%) was between the ages of 22 and 25. Nineteen (86.36%) identified as White, two as Asian (9.09%), and one (4.55%) as Native Hawaiian or Pacific Islander. Three (13.64%) reported having 20\(+\) hours of experience in VR (20, 60, and 200), thirteen (59.09%) reported having between 1 and 5 hours of experience, and six (27.27%) reported having no experience.

Participants were first instructed to sign the informed consent form and complete a pre-survey. After completing this, they were given an explanation of what they would be doing in the virtual environment, supported with a tablet drawing of a sample configuration (participants were not made aware of the use of impossible spaces). After the explanation and any questions about the task they put the HMD on and completed three test trials. The test trials were to ensure they understood the task and all participants completed test trials using the same random seed (999). Participants were presented with self-overlapping geometry in each of the test trials.

During a pilot study, it was noticed that the height of the virtual camera would sometimes drift slowly due to the Guardian boundary being disabled. To counteract this, we had all participants go through a calibration process where we talked them through enabling the Guardian system, setting up the stationary boundary, and then disabling the Guardian system. This was effective in eliminating the camera drift.

Participants started the experiment in the center of the connecting hallway. Each participant completed 30 trials. During these trials, participants were instructed to enter both rooms and press a button located in the back corner, ensuring that participants actually moved through the full length of the room. Participants were allowed to choose which room to enter first. After both buttons had been pressed, participants returned to the center of the hallway where a new button was now visible. Pressing this button began the “choice” phase of the trial where participants were instructed to stand on a “judgment location” that was offset from the center of the hallway along the x-axis and marked with a circle; participants were told that the judgment location would never be in the center of the hallway. Two panels were placed in front of the judgment location along with the instructions to press the panel corresponding to which room’s button participants felt was a shorter walk from their current location. Participants were required to walk to the judgment location prior to selecting which button was closer.

After making this selection, the secondary environment was loaded around them. In this environment, they were asked to move the back wall by using the thumbsticks on the controllers and press the button to confirm the distance. The text above the button says “Confirm Distance to A/B,” respectively. They first estimated the distance to the button in Room A and then estimated the distance to the button in Room B. After confirming the distance to B, they were placed back in the primary environment with a new button in the center of the hallway that has text above it saying “Begin Next Trial.” They were asked to complete trials until they were placed in a room with large text saying “Thank you for participating.” Progress updates about the number of trials remaining were provided to the participant through text above the panels. After completing all trials, they completed a post-survey, consisting of the Igroup Presence Questionnaire [22] and Simulator Sickness Questionnaire [4], and a short semi-structured interview was conducted.

To determine how participants’ likelihood of correctly selecting the closest destination, we employed a GLMM with a binomial distribution and logit link function.

The buildmer library was used to identify the model that best fit the data; this final model incorporated fixed effects for the variables Unsigned Path Delta, Overlap Percentage, and Offset From Center, along with an interaction term between Overlap Percentage and Offset From Center, and the random effects of individual IDs. The model accounted for a considerable amount of the variability in the data, with a conditional \(r^{2}\) of 0.299 and a marginal \(r^{2}\) of 0.282.

Fixed effects and their respective 95% confidence intervals are summarized in Table 2. Significant main effects for Path Delta and Overlap Percentage were observed, as well as a significant interaction effect between Overlap Percentage and Offset From Center. These results suggest that the likelihood of making a correct choice is influenced both by the delta between the path distances and the global overlap and is further moderated by whether the correct choice is the closest door. The random effect of ID accounted for individual variability among the 22 participants, revealing a variance of 0.0783 and a standard deviation of 0.280.

Five-fold cross-validation confirmed the model generalizes well with a mean Area Under the Curve of 0.770. All model assumptions were verified and met the criteria for validity. Of the original 660 data points, 23 were identified as outliers based on standardized residuals and were removed from the analysis. A marginal effects plot further elucidating the impact of the significant predictors is shown in Figure 2.

To determine how participants’ distance estimations were impacted by the self-overlapping spaces, we fitted a linear mixed model to the data. The model of best fit included Overlap as a fixed effect and no interaction effects. Participant ID was used as the random effect. The model demonstrated a strong overall explanatory power with a conditional \(r^{2}\) of 0.772 and a marginal \(r^{2}\) of 0.004. The marginal r2 was small; however, it should be noted that the bulk of this explanatory power is present in the modeled random effects, not the fixed effects.

The random effect of ID accounted for individual variability among the 22 participants, revealing a variance of 0.0406 and a standard deviation of 0.201. Fixed effects and their respective 95% confidence intervals are summarized in Table 3. A significant main effect for Overlap was observed. These results suggest that participants estimates grew slightly less accurate as the proportion of the self-overlapping space increased.

Five-fold cross-validation confirmed the model generalizes well with a ratio of root mean square error to a standard deviation of 0.4904. All model assumptions were verified and met the criteria for validity. No outliers were identified based on standardized residuals. Figure 3 visualizes the effect of overlap percentage on participants’ estimations.

The interaction effect between overlap percentage and offset from center was unexpected, as it seems to indicate the participants’ position in the hallway introduced a selection bias as overlap increases. Even more surprising, the bias appears to favor the button present in the room that was farther from participants, not closer. To further explore this effect, we evaluated whether it could be explained as a consequence of depth compression (i.e., if rooms were actually smaller, as perceived by participants, would that result in a change in the correct outcome). Our primary question of interest was whether modeling depth compression in our analysis could eliminate the interaction effect between overlap amount and offset from center and thus explain this bias.

To test this, we fitted a linear model predicting how far each destination would be when accounting for the depth compression observed in participants estimated distances. The model of best fit included Actual Distance and Overlap as a fixed effect and the interaction effect between them, see Table 4. Participant ID and Actual Distance were used as random effects. The model demonstrated a strong overall explanatory power with a conditional \(r^{2}\) of 0.874 and a marginal \(r^{2}\) of 0.054. While the conditional \(r^{2}\) captured a very large amount of variability across participants, the marginal \(r^{2}\) suggests that the fixed effects of our model could only account for a small portion of the variance in our data.

We then used this model to predict the compressed distance to each destination as modeled for each individual participant. Finally, we evaluated whether the correct outcome of any trial would change given these compressed distances. However, distance compression did not alter the outcomes of any trials and thus cannot alone explain the interaction effect between Overlap and Offset from Center. Further research will be required to better understand this effect.

The results from the post-study interview were inconclusive. The interview contained questions asking if participants noticed anything “odd” or “unnatural” about the environment, but the interview questions were not robust enough to get a clear answer. The majority of participants responded in the negative and/or made no mention of impossible spaces. However, after they were debriefed on impossible spaces and prompted again, participants tended to say they did notice the impossible spaces and it did feel weird. Based on this sudden shift, we believe there was either some confusion about the questions or the participants were subject to some sort of response bias.