**Estimated time contracts or dilates depending on many visual-stimulation attributes (size, speed, etc.). Here we show that when such attributes are jointly modulated so as to respect the rules of perspective, their effect on the perceived duration of moving objects depends on the presence of contextual information about viewing distance. We show that perceived duration contracts and dilates with changes in the retinal input associated with increasing distance from the observer only when the moving objects are presented in the absence of information about the viewing distance. When this information (in the form of linear perspective cues) is present, the time-contraction/dilation effect is eliminated and time constancy is preserved. This is the first demonstration of a perceptual time constancy, analogous to size constancy but in the time domain. It points to a normalization of time computation operated by the visual brain when stimulated within a quasi-ecological environment.**

*size constancy*: Our assessment of the objective (or

*distal*) size of objects remains largely unaffected by changes in the retinal (or

*proximal*) size entailed by the viewing distance (Boring, 1942). Size constancy depends on the presence of information about the viewing distance (Holway & Boring, 1941), to which linear perspective provides a strong cue (Aks & Enns, 1996; Fineman, 1981). Based on a conjecture of Gorea and Hau (2013), we designed a series of experiments to test whether our perceptual system also shows

*time constancy*

^{1}in a 3-D world as it is represented by its 2-D projection.

*foreshortening*of the rolling ball (i.e., different combinations of the ball's proximal attributes: size, speed, and length of the motion path). Our stimuli therefore can be described as different combinations of temporal duration and foreshortening of the rolling ball. The stimuli were presented in two different contextual conditions: either on a uniform gray background (hereafter referred to as the

*flat*condition; for details, see Figure 1a, c, and d and Methods) or on a linear perspective projection of a checkerboard “floor” below a blue “sky” (hereafter, the

*perspective*condition; Figure 1b and e). The flat condition allows testing whether the foreshortening of the moving ball, in the absence of other information about viewing distance, does systematically distort the perceived duration. The perspective condition, on the other hand, is meant to reveal whether contextual information about the viewing distance (in the form of simple linear perspective cues) can eliminate these distortions, provided that perspective rules are respected.

*constant-length*experiment) or by letting the distal trajectory length vary while keeping the distal speed constant (

*constant-speed*experiment). This allowed us to control for the potential use of the trajectory length (or speed) as a proxy for duration judgments. As anticipated, the rolling ball assumed one of four levels of foreshortening. In the perspective conditions (Figure 1e), each level was always associated with one of the four vertical locations on the screen (corresponding to four depth planes in the virtual dimension), so that speed, path length, and size scaled in agreement with the perspective rules. In two of three flat conditions, however, the different levels of foreshortening of the rolling balls were randomly associated with the balls' vertical locations on the screen (Figure 1c). As a consequence, the distal

*event constancy*(i.e., the ability to recognize the event as having the same properties in the virtual dimension despite changes in its 2-D representation on the screen) over locations was compromised. (Note that we will hereafter use the term

*foreshortening*to indicate proximal changes in the stimulus in both the flat and perspective conditions.) In a third flat condition the level of foreshortening of the rolling ball scaled systematically with the vertical location of the ball, just like in the perspective condition (Figure 1e). In this case, the four different durations were always obtained by varying the length of the trajectory of the rolling ball (hence keeping constant its distal speed). This flat condition was meant to isolate the pure contribution of the perspective-rendered background to duration perception.

^{2}) or displayed a linear perspective projection of a checkerboard “floor” (mean luminance = 2.3 cd/m

^{2}) below a blue “sky” (10.3 cd/m

^{2}). In the latter case the four

*y*-locations translated into four depth planes equally spaced in log units as seen by the observer (1.8, 3.2, 5.8, and 10.5 squares of the checkerboard floor). The width of the squares as rendered on the screen was 14.6 cm for the central square at the lower edge of the display, decreasing to 0.2 cm at the level of the horizon. When placed at the corresponding elevations, the four ball sizes translated into a unique size in the virtual 3-D world: The largest ball placed in the nearest depth plane had the same size (in virtual dimension) as the smallest ball placed in the most remote depth plane. Note that in order to distinguish the characteristics of the moving ball in the virtual 3-D world from their 2-D projection on the screen, we use the terms

*distal*and

*proximal*, respectively.

*y*-locations on the screen (Figure 1c). An additional control experiment with a flat (uniform) background was run, where the

*y*-location of the moving ball on the screen covaried with the foreshortening of the ball just like in the perspective condition (flat with fixed vertical positions; Figure 1d).

*SD*= 11.87) paticipated in the perspective and flat conditions of Experiment 1 (constant-speed experiment), eight (two women and six men; mean age = 38.13 years,

*SD*= 12.46) in the perspective and flat conditions of Experiment 2 (constant-length experiment), and eight (three women, five men; mean age = 32.37 years,

*SD*= 14.68) in the control experiment with flat background and fixed positions; two observers (the authors) were shared by the three groups. The number of participants was chosen so as to permit us to obtain sensible estimates of standard deviations for the random-effects terms of the models used in the analysis (Bates, 2010; for details on the models, see the Conjoint measurement models subsection). All had normal or corrected-to-normal vision and gave their informed consent to perform the experiments. The study was conducted in accordance with French regulations and the requirements of the Helsinki Convention.

*ψ*of a given stimulus was modeled as where the indices

*i*∈ {1, 2, 3, 4} and

*j*∈ {1, 2, 3, 4} indicate, respectively, the level of physical duration (

*D*) and foreshortening (

*F*) of the stimulus (see Stimuli and apparatus subsection). If perceived duration is not affected by variations in depth, then

*F*should be equal to zero for all

*j*. When two stimuli are compared, we assume that observers base their decisions on the noise-contaminated variable Δ: so that

*ψ*will be judged as longer than

_{ij}*ψ*when Δ > 0 (with

_{kl}*ε*representing a normally distributed judgment error). For model identifiability, we anchored the perceptual scales by setting the scale values for the first stimulus level of each dimension to zero (i.e.,

*D*

_{1}=

*F*

_{1}= 0). Following Gerardin, Devinck, Dojat, and Knoblauch (2014), we scaled the estimated values of

*D*and

*F*so that they would be on the same scale as the sensitivity index

*d*′ (Green & Swets, 1966).

*D*and

*F*are normally distributed across observers. Within the mixed-effects modeling framework, the perceived duration of a given stimulus is modeled as a linear combination of

*fixed*(

*D*and

*F*) and

*random*, or observer-specific, effects (

*u*); for example, for a given observer

_{s}*s*, the perceived duration of a stimulus with physical duration level

*i*and foreshortening level

*j*is where Σ indicates the variance-covariance matrix for the multivariate Gaussian distribution of the random (observer-specific) effects

*u*. This model can be formulated as a generalized linear mixed-effects model (Knoblauch & Maloney, 2012) with observer as the grouping factor. We estimated its parameters by maximum likelihood using R (R Core Team, 2015) and the lme4 library (Bates, Maechler, Bolker, & Walker, 2014).

*D*and

*F*) in our experiment showed a clear linear dependence on the stimulus levels, we introduced an additional simplification and fitted an

*additive-linear*model, where these scales are modeled as linear functions of the stimulus indices. In the additive-linear model the decision variable can be notated as where the

*β*values are the linear slopes;

*i*,

*j*,

*k*, and

*l*are the indices indicating the levels of the stimuli; and

*δ*indicates the differences between the indices. This model can also be formulated as a generalized linear mixed-effects model, where the slope for each observer

*s*is modeled as a sum of fixed (

*β*) and random (

*b*) effects:

_{s}*p*(Δ

*> 0|*

_{ijkl}*s*) corresponds to the probability that the observer

*s*judges the stimulus with physical duration and foreshortening levels

*i*and

*j*, respectively, to last longer than the stimulus with physical duration and foreshortening levels

*k*and

*l*.

*independence*model where the fixed effect of foreshortening is set to zero (that is, a model where

*β*= 0). Across the two models (additive-linear and independence), we kept constant the structure of the random components

_{F}*b*. Since the independence model is nested within the additive-linear model, we compared the two models using a likelihood-ratio test. Additionally, we tested the effect of the perspective background at the within-observer level by fitting both conditions (flat and perspective) with a single model that included parameters for the changes in the slopes between the two conditions, and by comparing these models via likelihood-ratio tests with reduced models where the changes in slopes were forced to be zero.

*saturated*model where we introduced an additional coefficient

*β*that was applied to a product of the indices to obtain an additional interaction term that could account for interactions between physical duration and foreshortening. As with the independence model, this saturated model was compared to the additive-linear models with likelihood-ratio tests (keeping the structure of the random effects constant).

_{FD}*d*′ (Gerardin et al., 2014; Green & Swets, 1966). The strength of this modeling approach is that it allows measurement of the effect of different, heterogeneous physical variables on a common perceptual scale, even when these variables are defined on different domains (here, temporal and spatial). Hereafter, the notation

*D*refers to the coefficients coding for the contribution of physical duration, and

*F*to the coefficients coding for the contribution of foreshortening. They denote the contribution to the perceived duration of their physical counterparts. We estimated the set of scale values that best capture observer's judgments of the perceived duration difference between the stimuli in each pair by maximizing the likelihood of observer's responses under the additive model (for details, see Methods). The estimated scale values (averaged over observers) are represented in Figure 3 (bottom panels) as a function of the stimulus level (1 to 4), where stimulus level refers to either physical duration (the four durations used, in increasing order; open symbols) or foreshortening (the four combinations of proximal attributes, ordered from near to far; shaded symbols). The values for

*F*show different trends depending on the experimental condition: They decrease in the flat conditions, indicating a contraction of perceived duration with increasing levels of foreshortening (from near to far), while they stay around 0 in the perspective conditions. This indicates that the net effect of perspective foreshortening, in the flat condition, is to make the perceived duration contract, the more so as the speed, size, and length of the moving ball decrease. This effect is abolished in the perspective condition.

*D*and

*F*is their linear dependence on the stimulus levels. This linearity is likely a consequence of the logarithmic spacing of the stimuli, which (in accordance with Weber's law) made them perceptually equidistant (Rogers, Knoblauch, & Franklin, 2016). We took advantage of this linearity to model the data at the group level with a generalized linear mixed-effects model, following the approach described by Rogers et al., where the estimated scale values are treated as linear functions of the stimulus levels. As a consequence, each contribution to perceived duration (physical duration or foreshortening) can be specified by a single parameter, the slope of the linear function, with the advantage of largely reducing the number of parameters (for details, see Methods). We called this model additive-linear because it is based on the assumption that all observers share the same underlying linear shape of the perceptual scales, although they might differ in sensitivity (the slope). We tested this assumption by comparing the more parsimonious additive-linear model to a hierarchical version of the additive model of Ho et al. (2008), which was fitted at the group level and does not make any assumptions about the underlying shape of the perceptual scales (for details, see the Conjoint measurement models subsection). We compared the models using the Akaike information criterion (Akaike, 1974), a measure of the relative quality of statistical models. We did the comparison separately for each experiment and condition: Differences in the Akaike information criterion ranged between 18 and 36, and in all cases favored the additive-linear model. Thus the assumption that all observers have linear perceptual scales allows for a model that uses fewer parameters while fitting the data equally well as the additive model.

*χ*

^{2}(1) = 11.81,

*p*< 0.001, and constant-length,

*χ*

^{2}(1) = 11.89,

*p*< 0.001, experiments, confirming that foreshortening induces in these conditions a significant contraction of the perceived duration. Additionally, the effect of foreshortening was significant in the control experiment with a flat background and fixed vertical positions,

*χ*

^{2}(1) = 7.84,

*p*= 0.005, where stimuli appeared at the same screen locations as in the perspective condition. Although this condition contains some information on the 3-D layout, it appears that it is not sufficient to cancel the effect of foreshortening. Moreover, this result rules out any potential confound due to the balls' location differences between the two flat conditions (with fixed and random trajectory locations). In contrast, the same test (additive vs. independence model) performed on the results of the perspective conditions (i.e., with the perspective background) did not reach significance for either the constant-speed,

*χ*

^{2}(1) = 0.59,

*p*> 0.25, or the constant-length,

*χ*

^{2}(1) = 2.28,

*p*= 0.13, experiment. The effect of foreshortening on the apparent duration, as summarized by the slopes of the component scales, is represented in Figure 4 for all experiments and conditions. Overall, for the stimuli tested, foreshortening made a contribution of about 23% to perceived duration in the flat condition (ratio of the slopes for foreshortening and physical duration) and only 2% in the perspective condition.

*χ*

^{2}(1) = 53.33,

*p*< 0.001, and constant-length,

*χ*

^{2}(1) = 5.91,

*p*= 0.01, experiments, revealing that the contribution of foreshortening to perceived duration depended on the presence or absence of perspective cues. Conversely, no significant change in slope between flat and perspective conditions was found for the contribution of physical duration, in either the constant-speed,

*χ*

^{2}(1) = 2.84,

*p*= 0.09, or constant-length,

*χ*

^{2}(1) = 0.36,

*p*> 0.25, experiments. To sum up, in agreement with our hypothesis of time constancy, we find that duration judgments are unaffected by changes in the proximal aspects of the stimulus (foreshortening) only when those changes are made in conditions where information about viewing distance is present (linear perspective cues).

*p*s > 0.5) for all experiments and conditions. We therefore conclude that modeling the interaction between temporal duration and foreshortening as a simple additive contamination of the former by the latter provides an adequate description of observers' responses.

*SD*= 0.07) and 0.22 (

*SD*= 0.08) in the flat and perspective conditions of the constant-speed experiment, respectively; 0.54 (

*SD*= 0.35) and 0.44 (

*SD*= 0.19) in the flat and perspective conditions of the constant-length experiment; and 0.55 (

*SD*= 0.51) in the experiment with a flat background and fixed positions. For the observers who performed both the flat and perspective conditions, we did not find any significant difference in the discrimination thresholds—constant-speed:

*t*(7) = 1.11,

*p*= 0.30; constant length:

*t*(7) = 1.35,

*p*= 0.22.

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^{1}We point out that because time (duration) does not exist by itself (i.e., it is always associated with an event), the notion of

*proximal stimulus*is undefined in the time domain. For the current purposes we call proximal the

*ensemble*of physical features that reach the senses and define the event. Accordingly, with respect to our experiments, the term

*constancy*indicates that two different proximal events (different sizes and speeds but identical durations) are judged to be perceptually identical in the presence of 3-D cues but different in their absence. Hence, constancy according to this definition refers to the fact that

*different proximal stimuli in a 2-D world*become

*perceptually equivalent in a 3-D world*.