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Article  |   July 2011
Discrimination and identification of periodic motion trajectories
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Journal of Vision July 2011, Vol.11, 7. doi:10.1167/11.8.7
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      Charles C.-F. Or, Michel Thabet, Frances Wilkinson, Hugh R. Wilson; Discrimination and identification of periodic motion trajectories. Journal of Vision 2011;11(8):7. doi: 10.1167/11.8.7.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Humans are extremely sensitive to radial deformations of static circular contours (F. Wilkinson, H. R. Wilson, & C. Habak, 1998). Here, we investigate detection and identification of periodic motion trajectories defined by these radial frequency (RF) patterns over a range of radial frequencies of 2–5 cycles. We showed that the average detection thresholds for RF trajectories range from 1 to 4 min of arc and performance improves as a power-law function of radial frequency. RF trajectories are also detected for a range of speeds. We also showed that spatiotemporal global processing is involved in trajectory detection, as improvement in detection performance with increasing radial deformation displayed cannot be accounted for by local probability summation. Finally, identification of RF trajectories is possible over this RF range. Overall thresholds are about 6 times higher than previously reported for static stimuli. These novel stimuli should be a useful tool to investigate motion trajectory learning and discrimination in humans and other primates.

Introduction
Perception of periodic motion trajectories underlies our ability to extract repetitive patterns of movement. Many aspects of sports involve sensitivity to such complex motion. For instance, a tennis player predicts the ball's movement by analyzing the motion of the opponent's serve. Indeed, it was found that we are extremely sensitive to human motion, even when the motion is only represented by a series of moving point light sources attached to the major joints of a human body, known as “biological motion” (Johansson, 1973). When the points on a point light walker are individually analyzed, it can be shown that each of these motion trajectories is periodic relative to the translation of the walker (Tsai, Shah, Keiter, & Kasparis, 1994). In other words, the points complete a full rotation and end their trajectories at the same points at which they began. 
Although much of the motion literature has focused on various forms of translational motion (Adelson & Movshon, 1982; Williams & Sekuler, 1984) and optic flow (Gibson, 1950), almost none has focused on periodic motion. One of the few related studies is by Todd (1982), who used a series of line segments of which the endpoints moved along randomly generated elliptical or hypertrochoidal trajectories (planar projection of the trajectory of a circle that rolls on a fixed curve) so that they appeared to move as an object in three-dimensional space. The study showed that the difference between rigid and non-rigid motion could be accurately detected when trained observers performed the task by detecting slight differences in the trajectories of the endpoints of these line segments. Even under impoverished conditions where only parts of the rotation were shown, observers were still able to accurately detect the difference between rigid and non-rigid motion. In addition, using different speeds did not have an effect on performance. Importantly, the results imply that sensitivity to deformation of an otherwise elliptical trajectory forms the basis for discriminating non-rigid from rigid motion. It is therefore crucial to understand such sensitivity in the perception of periodic motion. 
Another potential factor in the perception of periodic motion trajectories involves accurate detection and identification of curvatures and closed contours. In their studies of curvature tracing, Jolicoeur and Ingleton (1991) and Jolicoeur, Ullman, and Mackay (1986, 1991) found extreme accuracy and efficiency in detecting whether two points belong to the same curved contour among a series of curved contours. The speed of curvature tracing (without eye movement) can be as high as 40 deg/s over a broad range of the visual field. 
Hypersensitivity has been shown to a class of static closed contours known as radial frequency (RF) patterns, as defined by sinusoidal deformations from circular contours (Wilkinson, Wilson, & Habak, 1998; Figure 1b). Sensitivity to these patterns was as low as 2–4 s of arc of radial deformation from circles for midrange frequencies between 3 and 5 cycles. Additionally, these RF shapes can be defined by motion (Rainville & Wilson, 2004). The extreme sensitivity to RF patterns makes them perfect candidates to investigate the sensitivity in perceiving periodic motion trajectories. 
Figure 1
 
(a) Radial frequency (RF) motion trajectories for 2–5 cycles. The difference-of-Gaussians (DOG) dot starts directly to the right of the fixation cross (which is the centre of coordinates) and moves around the trajectory (dashed line) for one complete revolution, in a clockwise manner (see arrows). The dashed lines are for illustrative purposes and were not present in the experiments. (b) The corresponding static RF patterns.
Figure 1
 
(a) Radial frequency (RF) motion trajectories for 2–5 cycles. The difference-of-Gaussians (DOG) dot starts directly to the right of the fixation cross (which is the centre of coordinates) and moves around the trajectory (dashed line) for one complete revolution, in a clockwise manner (see arrows). The dashed lines are for illustrative purposes and were not present in the experiments. (b) The corresponding static RF patterns.
In this study, we investigated the value of RF patterns as a tool to understand the perception of periodic motion. In particular, we examined whether motion trajectories defined by these RF shapes (Figure 1a and Supplementary Figure 1) could be identified and discriminated from circular trajectories. We established that: (1) RF motion trajectories were detected at small amplitudes of deformation (in the order of minutes of arc), (2) optimal detection required global processing, and (3) different RF motion trajectories could be identified. Our results indicate that these novel stimuli will be useful in the investigation of periodic motion perception. 
General methods
Apparatus
The stimuli were displayed on an eMac computer, at a frame rate of 112 Hz, with a spatial resolution of 800 × 600 pixels and a greyscale of 8 bits/pixel. Observers viewed the screen binocularly at a distance of 114 cm in a dimly lit observation room, such that the screen subtended 15.4 deg × 11.7 deg, and each square pixel had a width of 1.18 min of arc. Prior to testing, the monitor was gamma corrected using a Minolta LS-100 photometer, and a custom-written MATLAB script was used to generate lookup tables containing interpolated inverse gamma values. The mean luminance after gamma correction was 84 cd/m2
Stimuli
The stimulus (Figure 1a and Supplementary Figure 1) was a dot moving around an RF trajectory centred at fixation at angular speed v, such that the dot's polar location r at time t is 
r ( v t ) = r 0 ( 1 + A sin ( ω v t + ϕ ) ) ,
(1)
where r 0 is the mean radius, A is the radial modulation amplitude, ω is the radial frequency, and ϕ is the phase angle of the trajectory. Equation 1 is the motion generalization of a static RF pattern (Figure 1b; Wilkinson et al., 1998), produced by applying a radial sinusoidal temporal modulation to the radius r 0 of an otherwise circular trajectory. The mean radius r 0 was always set to 1.0 deg throughout the experiments. The range of radial frequencies ω tested was 2–5 cycles, as illustrated in Figure 1. This range was chosen as detection thresholds for static RF patterns asymptote at approximately 5 cycles (Wilkinson et al., 1998). The phase angle ϕ specifies the orientation of the trajectory. Changing ϕ has the effect of rotating the shape described by the trajectory. Throughout the experiments, the dot always moved clockwise around the trajectory for one complete revolution, with motion starting from (and ending at) a point directly to the right of the fixation point (which was the centre of coordinates of the trajectory). It is important to note that the RF trajectory shape itself was never displayed but was only defined by the dot motion over time. 
The dot itself is described by a radially symmetric difference of Gaussians (DOG): 
D O G ( R ) = 1.8 exp ( R 2 / σ 2 ) 0.8 exp ( R 2 / ( 1.5 σ ) 2 ) ,
(2)
where R is the radius of the DOG and σ was set to 7.1 min of arc such that the peak spatial frequency would be 2.74 cpd and the bandwidth would be 1.79 octaves at half-amplitude. These parameters were specified such that the dot's luminance profile sums to the mean luminance of the screen. To achieve precision of the dot's instantaneous spatial position, subpixel resolution was used to compute the dot such that a pixel's luminance was represented by the luminance of the continuous DOG function at the midpoint of that pixel. The dot subtended a visual angle of 43.8 min of arc across its width (±3.1σ), with a contrast of 100%. 
Experiment 1: Detection of RF trajectories
Observers
Seven experienced observers participated in this experiment. All observers except two authors (CFO, MT) were naive to the purpose of the study. All observers had normal, or corrected-to-normal, visual acuity and ranged in age from 24 to 43 years. 
Procedure
The method of constant stimuli combined with a temporal two-interval forced-choice (2IFC) procedure was used to estimate detection thresholds for RF trajectories of 2–5 cycles. To investigate the effect of rotational speed on detection, we conducted three separate conditions using polar coordinate angular speeds v of 180°/s, 360°/s, and 720°/s, equivalent to tangential speeds of 3.14 deg/s, 6.28 deg/s, and 12.57 deg/s, respectively. Each stimulus was composed of 57 frames displaying the dot at instantaneous points sampled at identical angular intervals of 6.43° around an RF trajectory. The last of these frames displayed the dot at the starting point, which was always directly to the right of the fixation point. The last frame was thus identical to the first frame, ensuring that the trajectory motion appeared to complete one revolution. The 57 frames were presented in rapid succession without an inter-frame interval such that the dot appeared to move around the RF trajectory. Each frame lasted for 35.7, 17.9, and 8.9 ms, respectively, for the three speed conditions, corresponding to stimulus durations of 2.0, 1.0, and 0.5 s, respectively. Note that the dot speeds in the three conditions differed only by changing the duration of a frame, while the number of instantaneous points sampled around the RF trajectory (i.e., the number of frames) was always the same (which was 57). Importantly, these settings ensured that all peaks and troughs of an RF trajectory were accurately represented, and observers reported that they experienced smooth trajectory motion. 
While the angular speeds are constant in the three conditions, the Cartesian (i.e., instantaneous linear) speed varies as the dot's polar location r changes when it moves around the RF trajectory. To examine such speed variation as a potential cue for detection, one additional experimental condition was introduced in which the Cartesian speed v′ was held constant at 6.28 deg/s by continuously varying the angular speed v over the course of revolution, using the following equation: 
v = r 0 v 1 + 2 A cos ( ω v t ) + A 2 cos 2 ( ω v t ) + A 2 ω 2 sin 2 ( ω v t ) .
(3)
Using Equation 3, the dot's instantaneous locations were sampled at unequal angular intervals around an RF trajectory to achieve a constant Cartesian speed (but a varying angular speed). Each frame still remained for 17.9 ms, identical to that in the corresponding (6.28 deg/s) constant angular speed condition. Given a constant Cartesian speed, the total number of frames required to complete one revolution inevitably increased as a function of the radial modulation amplitude A, as the length of the trajectory became longer. Fortunately, such increase was minor: the total number of frames ranged from 57 to 59 at the range of A we used, and a maximum 2-frame difference resulted in a change in stimulus duration by 36 ms only (cf. the average stimulus duration was 1.0 s). Importantly, pilot experiments show that observers could not reliably discriminate between the durations of two circular trajectories of such a small time difference. 
At the start of each experimental run, observers were informed of the speed and radial frequency to be used. The phase angle was set to 90°. The only parameter varied within a run was the radial amplitude. A centrally presented fixation cross remained on the screen over the course of a trial, and observers were required to maintain fixation throughout. Following central fixation for 500 ms at the beginning of each trial, observers were presented with two intervals of trajectory motion, temporally separated by a 1.5-s period of mean luminance (Supplementary Figure 1). One interval contained the target trajectory with radial deformations, and the other interval always contained a circular trajectory matched in mean radius and speed. The order of presentation of the two intervals was randomized from trial to trial. Observers were instructed to report by button press which of the two intervals contained deformations from a circular trajectory. No feedback was provided. 
Sixteen conditions (4 radial frequencies (2–5 cycles) × 4 speeds) were tested in separate blocks. For each condition, at least 5 levels of radial amplitude (40 trials/level) were tested to estimate a threshold. All seven observers participated in the 12 conditions using constant angular speeds of 3.14 and 6.28 deg/s and constant Cartesian speed of 6.28 deg/s. The order of these 12 conditions was randomized for each observer. Data collection was split into several sessions spanning a few days. Five of the same seven observers participated in the four conditions using constant angular speed of 12.57 deg/s. 
For each observer and condition, data collected in different sessions were combined for analysis. The proportion of correct responses for each radial amplitude was then calculated, and a psychometric function was derived by fitting a Quick (1974) or Weibull (1951) function to the data using maximum likelihood estimation. The point of 75% correct response on the curve was chosen as the threshold amplitude. An estimate of the standard deviation (SD) of the threshold amplitude was computed using a bootstrapping procedure. 
Results
The first experiment measured detection thresholds for periodic RF motion. Threshold amplitude decreases as a function of radial frequency in all four speed conditions tested. The average thresholds range from 1 to 4 min of arc for RF2–5. Figure 2 plots individual data, and Figure 3 summarizes the data by averaging over observers. These plots clearly show that the data fall along a straight line in log–log coordinates, indicating a power-law relationship between threshold and radial frequency. The exponents of the least-squares fits (the slopes of the lines in the plots) averaged −1.53 ± 0.18, −0.93 ± 0.10, −0.72 ± 0.17, and −1.03 ± 0.18, respectively, for constant angular speeds of 3.14, 6.28, and 12.57 deg/s and constant Cartesian speed of 6.28 deg/s. The corresponding estimates of y-intercepts averaged 11.0 ± 2.9, 5.36 ± 1.01, 3.58 ± 0.99, and 5.49 ± 1.50 min of arc. 
Figure 2
 
Individual results for Experiment 1, showing threshold amplitudes to discriminate RF trajectories from circular trajectories. In each of the four conditions, a power-law function was well fitted to the data for all observers. The error bars denote ±1 bootstrapped SD.
Figure 2
 
Individual results for Experiment 1, showing threshold amplitudes to discriminate RF trajectories from circular trajectories. In each of the four conditions, a power-law function was well fitted to the data for all observers. The error bars denote ±1 bootstrapped SD.
Figure 3
 
Results for Experiment 1, grouped over conditions. Here, geometric means across observers are plotted for each condition. The power-law fits were the same as in Figure 2. The error bars denote ±1 SEM.
Figure 3
 
Results for Experiment 1, grouped over conditions. Here, geometric means across observers are plotted for each condition. The power-law fits were the same as in Figure 2. The error bars denote ±1 SEM.
To evaluate the effect of using three different constant angular speeds on trajectory detection, one-way repeated measures ANOVAs were performed separately on two estimated parameters (slope and y-intercept) derived from fits to data of each of the five observers who performed all three speed conditions. No significant effect of speed was found for either slope, F(1,4) = 3.01, p = 0.16, or y-intercept, F(1,4) = 2.74, p = 0.17. These results are consistent with Todd's (1982) finding that speed does not affect detection of elliptical trajectories. There is, however, a trend toward shallower slopes and lower y-intercepts as the speed increases (Figure 3). Note that the apparent trend could be due to variations among observers' performance, especially for detecting RF2. The trend could be tested with more observers over a wider range of speeds. 
We also compared the effect of using constant angular speed and constant Cartesian speed (both at 6.28 deg/s) with paired samples t-tests on slope and y-intercept from fits for all seven observers. Again, no significant effects were found for both slope, t(6) = 0.34, p = 0.75, and y-intercept, t(6) = 0.61, p = 0.56. The result indicates that local speed variation in the constant angular speed condition does not affect detection performance for RF2–5. 
In the experiment, observers were required to maintain central fixation when viewing the stimulus. Nevertheless, it could be possible that eye movements had been made especially induced by the motion of the DOG dot. We thus conducted two control experiments to examine the effects of eye movements on RF trajectory detection, with five of the same seven observers using only RF3 moving at a constant angular speed of 6.28 deg/s. The experimental settings were the same as in the original experiment except that observers were instructed to track the motion of the dot by moving their eyes. The only difference between the two control experiments was that the central fixation cross was present throughout in one experiment (Figure 1 and Supplementary Figure 1) but absent in another. Results are shown in Figure 4. A one-way repeated measures ANOVA yielded no significant differences on detection thresholds among the original experiments and the two control conditions, F(1,4) = 0.03, p = 0.87. These results indicate that eye movements do not affect RF trajectory detection in the current experimental setup. 
Figure 4
 
Detection thresholds for three fixation conditions in Experiment 1, tested only on RF3 moving at a constant angular speed of 6.28 deg/s. The results show that RF trajectory detection is not affected by eye movements. The error bars denote ±1 bootstrapped SD.
Figure 4
 
Detection thresholds for three fixation conditions in Experiment 1, tested only on RF3 moving at a constant angular speed of 6.28 deg/s. The results show that RF trajectory detection is not affected by eye movements. The error bars denote ±1 bootstrapped SD.
To compare our results with those from the static RF patterns, we fitted a power-law function to detection thresholds for the comparable static RF patterns of 1.0-deg radius estimated by Wilkinson et al. (1998). The fitted slope and y-intercept are −1.26 ± 0.15 and 1.12 ± 0.19 min of arc, respectively. The slopes for the static and motion results fall within the same range, indicating that detection performance for both improves in a similar fashion with increasing number of cycles of deformation in a power-law relationship. However, sensitivity to RF trajectories is considerably worse than to static patterns, as shown by an approximately 6 times higher y-intercept for the motion data averaged from the three speeds (3.14, 6.28, and 12.57 deg/s). The lower sensitivity could potentially be explained by an increased processing demand arising from integrating motion signals over time as well as space in order to perceive the entire trajectory. In subsequent experiments, we examined whether global integration over these motion signals is necessary for optimal detection of the RF trajectories (Experiment 2) and whether these trajectories could be identified or discriminated from one another (Experiment 3). 
Experiment 2: Global integration
Background
Perception of complex motion has been shown to involve global processing over space–time (Warren & Rushton, 2009). In addition, Loffler, Wilson, and Wilkinson (2003) found that global processing is apparent in detecting static RF patterns up to 5 cycles of deformation. Furthermore, Hess, Wang, and Dakin (1999) showed that, although both local and global processing are used to analyze RF shapes as high as 8 cycles of deformation, detection performance is governed by global changes in the RF shape. Thus, it is reasonable to speculate that global contributions are required to process RF trajectories. We examined this possibility using a method described by Loffler et al. Specifically, the extent to which detection performance improves was tested as radial deformation was applied to an increasing portion (number of radial cycles) of an otherwise circular trajectory (Figure 5). If performance improvement is better than what would be predicted by a simple probability summation model (Graham & Robson, 1987) over local spatiotemporal units, global processing must be involved in RF trajectory detection. 
Figure 5
 
Stimuli for Experiment 2. The radial deformation of RF3 is restricted to a fraction of the trajectory leaving the remainder circular (the complete trajectory in white). Deformation was applied to N = 0.5, 1, or 3 cycles of the RF3 trajectory. Trajectories with 0.5 cycle of deformation were either a convex or a concave bump. The circular trajectory (in red) is superimposed on each RF trajectory for comparisons.
Figure 5
 
Stimuli for Experiment 2. The radial deformation of RF3 is restricted to a fraction of the trajectory leaving the remainder circular (the complete trajectory in white). Deformation was applied to N = 0.5, 1, or 3 cycles of the RF3 trajectory. Trajectories with 0.5 cycle of deformation were either a convex or a concave bump. The circular trajectory (in red) is superimposed on each RF trajectory for comparisons.
Observers
Five of the observers from Experiment 1 participated in this experiment. 
Methods
The method was essentially the same as in Experiment 1, except that thresholds were measured when the radial deformation was restricted to a fraction of the trajectory leaving the remainder circular. RF3 (ω = 3) was chosen for measurements, as this frequency shows the largest amount of global pooling in the detection of static RF patterns (Loffler et al., 2003). Deformation was applied to N = 0.5 or 1 cycle of the RF3 trajectory (Figure 5), using the method described by Loffler et al. (2003). Trajectories with half a cycle of deformation appear as a bump on an otherwise circular trajectory, defined as a cosine function: 
r ( v t ) = r 0 ( 1 + K A 2 ( 1 + cos ( ω v t + ϕ ) ) ) f o r θ c π ω v t θ c + π ω r ( v t ) = r 0 e l s e w h e r e ,
(4)
where θ c = (4πϕ) / ω represents the centre of the deformed region, and K specifies a convex (+1) or a concave (−1) bump. K was chosen randomly on individual trials of an experiment. Trajectories with 1 cycle of deformation were defined as the first derivative of a Gaussian (D1): 
r ( v t ) = r 0 ( 1 + B v t θ c σ g exp ( ( v t θ c ) 2 σ g 2 ) ) f o r θ c 2 π ω v t θ c + 2 π ω r ( v t ) = r 0 e l s e w h e r e ,
(5)
where θ c = (4πϕ) / ω represents the centre of the deformed region, and B and σ g are the two free parameters of the D1 set to match, respectively, the amplitude and the maximum slope of the original sinusoid described in Equation 1
Two parameters differed from those used in Experiment 1. First, a constant angular speed of 6.28 deg/s was the only speed used. Second, the phase angle ϕ was randomly chosen from 0°, 90°, 180°, 270°, or 360°, from trial to trial. This has the effect of rotating the shape described by the trajectory about its centre. Note that a phase angle of 360° results in the deformation region rotated by 1 cycle of the sinusoid from 0°, as defined in Equations 4 and 5. The dot's starting point was always directly to the right of the fixation point. We also tested RF3 with 3 full cycles of radial deformation (Figure 5), as in Experiment 1, using this speed condition and phase angles randomly chosen from 0°, 90°, 180°, or 270°, from trial to trial (note that 360° was not included here as it is equivalent to 0° for 3 full cycles of deformation). 
Thresholds predicted by the probability summation model were estimated based on the assumption that deformation was computed locally by independent spatiotemporal detectors (Graham & Robson, 1987). Probability summation over these local detectors leads to increase in performance (or decrease in threshold) as the number of cycles increases, in a power-law relationship: 
A t h = c N 1 / k ,
(6)
where A th is the threshold amplitude, c is a constant, N is the number of cycles, and k is the slope of the psychometric function. k was estimated by averaging the slopes of the psychometric functions across observers. 
Results
Figure 6 shows that threshold amplitude decreases as the radial deformation is applied to a larger part of the trajectory. The data fall along a straight line in log–log coordinates, indicating a power-law relationship between threshold and cycles of radial deformation. The exponent of the least-squares fit (the slope of the solid line) averaged −0.64 ± 0.03. In comparison, local probability summation predicts a slope (−1/k in Equation 6) of only −0.33 (dashed line), approximately half the observed slope. Interestingly, our results are consistent with those of static RF patterns (Jeffrey, Wang, & Birch, 2002; Loffler et al., 2003), though a steeper slope of −0.86 was found for static RF3. As discussed in these studies, deviation from probability summation is a clear indication of global pooling; in the present case, RF trajectories are pooled across space–time. 
Figure 6
 
Results for Experiment 2 examining potential global contributions in detecting RF3 trajectories against circular trajectories. The detection thresholds were well fitted by a power-law function (exponent = −0.64, solid line) against the number of cycles of deformation, implying improvements in average performance with increasing number of cycles. The dashed line gives the prediction of probability summation over locally independent detectors anchored at 0.5 cycle (Equation 6). Probability summation evidently underestimates the increase in performance. The error bars denote ±1 bootstrapped SD.
Figure 6
 
Results for Experiment 2 examining potential global contributions in detecting RF3 trajectories against circular trajectories. The detection thresholds were well fitted by a power-law function (exponent = −0.64, solid line) against the number of cycles of deformation, implying improvements in average performance with increasing number of cycles. The dashed line gives the prediction of probability summation over locally independent detectors anchored at 0.5 cycle (Equation 6). Probability summation evidently underestimates the increase in performance. The error bars denote ±1 bootstrapped SD.
We also examined whether randomization of the phase angle affected detection performance for RF3 with 3 full cycles of deformation. Thresholds from 4 random phase angles (0°, 90°, 180°, 270°) were separately estimated for each observer. A one-way repeated measures ANOVA showed no significant threshold differences among the 4 phase angles, F(1,3) = 0.31, p = 0.62. We subsequently averaged the thresholds by phase angles and compared them with RF3 thresholds estimated in Experiment 1, where phase angle was fixed at 90°. Figure 7 illustrates the comparisons of RF3 detection thresholds from the two experiments. A paired samples t-test on the results of five observers (CFO, MT, OK, DG, MD) showed significantly higher detection thresholds from Experiment 1, t(4) = 3.19, p < 0.034. One potential explanation is a practice effect, as these observers performed Experiment 2 after Experiment 1. We therefore introduced two new observers (LP and HH), who did not have prior experience on any of our experiments, to test for both fixed and random phase angle conditions only on RF3 with 3 full cycles of deformation. Each of the two phase angle conditions was split into two experimental runs, and the order of these four runs was randomized to prevent a practice effect. No changes to the ANOVA or t-test results were found by adding these two new observers, suggesting other unknown factors besides practice effect. 
Figure 7
 
Threshold amplitudes for detecting RF3 trajectories. The use of a fixed phase angle (Experiment 1) results in higher thresholds than randomizing phase angles (Experiment 2). The error bars denote ±1 bootstrapped SD.
Figure 7
 
Threshold amplitudes for detecting RF3 trajectories. The use of a fixed phase angle (Experiment 1) results in higher thresholds than randomizing phase angles (Experiment 2). The error bars denote ±1 bootstrapped SD.
One possibility is adaptation to the RF trajectories of a fixed phase angle over time throughout the trials, thus raising detection thresholds in Experiment 1. In contrast, adaptation could not occur in Experiment 2 as RF trajectories of different phase angles were interleaved. Nevertheless, further experiments are required to evaluate this possibility. 
Experiment 3: Identification of RF trajectories
Observers
All observers but one from Experiment 1 participated in this experiment. A new observer, not involved in previous experiments, also participated, which brought the number of observers to seven. 
Methods
We examined discrimination among trajectories of different radial frequencies (RF2–5) using a four-alternative forced-choice procedure. Observers were asked to identify trajectories ranging from 2 to 5 cycles in four different conditions. We first examined the effect of randomizing phase angles by contrasting two separate conditions, one with a fixed phase angle of 0° throughout the trials and the other with a phase angle of either 0° or 180° chosen randomly from trial to trial. In these two conditions, the stimuli all moved around at a constant angular speed of 3.14 deg/s, and radial amplitudes were set to three times the mean detection thresholds for stimuli of identical speed, as measured in Experiment 1. In the other two conditions, radial amplitudes were raised to seven times the mean thresholds, with the phase angle fixed at 0°. These two conditions differed only by the dot speed, where stimuli moved at constant angular speeds of 3.14 deg/s and 6.28 deg/s, respectively. All seven observers performed the first three experiments (all using a speed of 3.14 deg/s) in a randomized order. The fourth condition (using a speed of 6.28 deg/s) was conducted by five of the same seven observers. 
To familiarize the observers with the shapes of the trajectories, they were presented a screen showing sample static shapes (Figure 1b) of the four trajectories (RF2–5) at the start of each experiment. Each trajectory shape contained a digit indicating the number of cycles. Observers were told that the phase angles of the trajectories might change, such that they should generalize across the variants. On each trial, a single RF trajectory (chosen randomly from RF2–5) was displayed following a 500-ms fixation period on a centrally presented fixation cross. Observers were required to maintain central fixation during each motion presentation. They were instructed to indicate the trajectory's shape (number of cycles) by clicking a mouse on one of the digits (2, 3, 4, or 5) displayed on the screen at the offset of the stimulus. Feedback was not provided. 
For each of the four conditions, each radial frequency was presented at least 40 times in order to measure the proportion of correct identifications. The order of RFs displayed was randomized. 
Results
Experiment 3 measured proportions of correct identifications of RF trajectories. The results averaged over observers are illustrated in Figure 8. We first examined the effect of randomizing phase angles by comparing results from the two conditions using three times the thresholds (i.e., red vs. green line in Figure 8). A two-way repeated measures ANOVA was performed on the 2 phase angle conditions (fixed and random) by 4 RFs. The main effect of RF was significant, F(1,6) = 8.02, p < 0.03, but the main effect of phase angle was not, F(1,6) = 2.78, p = 0.15. The interaction effect was not significant, F(1,6) = 4.75, p = 0.07. The first observation is that performance declined significantly as higher RFs were displayed. While identification of RF2 averaged over 95% correct, performance dropped significantly to a range of 59–68% correct in RF4 or RF5 identification, although this range is still above chance, which is 25%. Further analysis shows that RF4 was often confused with RF5. Another major observation is that a general performance decline was not related to randomization of phase angle. 
Figure 8
 
Proportion correct identification responses averaged over observers, for Experiment 3. Average performance declines with increasing number of radial frequency cycles for three times threshold amplitudes, regardless of whether phase is randomized. In contrast, average performance remains above 72% correct when seven times thresholds were displayed for the two speed conditions. The error bars denote ±1 SEM.
Figure 8
 
Proportion correct identification responses averaged over observers, for Experiment 3. Average performance declines with increasing number of radial frequency cycles for three times threshold amplitudes, regardless of whether phase is randomized. In contrast, average performance remains above 72% correct when seven times thresholds were displayed for the two speed conditions. The error bars denote ±1 SEM.
With other parameters held constant, marked overall improvement in performance was found when the radial amplitudes were increased from three times to seven times the thresholds with fixed-phase RF trajectories of a speed of 3.14 deg/s (i.e., black vs. green line in Figure 8). This observation is confirmed by a two-way repeated measures ANOVA on 2 amplitude conditions (three and seven times the thresholds) by 4 RFs, where significant differences were found for both the main effects of RF, F(1,6) = 14.3, p < 0.01, and amplitude, F(1,6) = 8.85, p < 0.03, though not significant for the interaction effect, F(1,6) = 3.55, p = 0.11. Indeed, at seven times the thresholds (at a speed of 3.14 deg/s), performance stayed over 85% correct identification for the whole range of radial frequencies used, although a small but significant decline in performance was observed for higher RFs. 
We additionally examined whether using a higher speed (6.28 deg/s) would affect performance at seven times the thresholds (i.e., blue vs. black line in Figure 8). A two-way repeated measures ANOVA was performed on the 2 speed conditions by 4 RFs on the five observers who performed both speed conditions at seven times the thresholds. The main effect of RF was significant, F(1,4) = 9.14, p < 0.04, but the main effect of speed was not, F(1,4) = 2.57, p = 0.18. The interaction effect was not significant, F(1,4) = 1.55, p = 0.28. Thus, using a higher speed of 6.28 deg/s did not significantly affect trajectory identification, in spite of a tendency that RF5 identification became less accurate when a higher speed was used. 
Taking all results into account, while using three times threshold amplitude is sufficient to identify static RF patterns at over 90% accuracy (Wilkinson et al., 1998), our results indicate that a larger increase in radial deformation is necessary for accurate motion trajectory identification. 
Discussion
The purpose of our study was to investigate whether RF trajectories as a novel class of stimulus can be discriminated from circular trajectories (detected) and from one another (identified) and whether discrimination involves global processing. Experiment 1 shows that RF trajectories are well discriminated from circular trajectories with thresholds in the order of minutes of arc and that performance improves in a power-law relationship with increased radial frequency. Experiment 2 shows that global processing is evident, at least for RF3, based on our results showing that the increase in sensitivity of RF trajectories with more cycles of deformation exceeds that predicted by probability summation over local spatiotemporal detectors. Experiment 3 shows that RF trajectories can be identified with greater than 72% accuracy for the range of 2–5 cycles with radial deformations of seven times detection thresholds; however, only 2 cycles can be reliably identified at three times threshold (and 3 cycles when the phase angle was fixed). Importantly, the three experiments together indicate that we are very sensitive in discriminating these RF trajectories, and their processing is global. 
The speed variation of the dot's revolution could potentially have been a major factor in RF trajectory detection (Experiment 1). The way in which the speed varies might have affected how the RF trajectory was detected. Two different forms of speed variation were investigated: (1) keeping the polar angular speed constant and (2) keeping the Cartesian (i.e., instantaneous linear) speed constant. When the angular speed is constant, the Cartesian speed is inevitably varied over the course of revolution. Such variable speed introduces acceleration and deceleration of the dot's motion, which might have provided an additional cue to detect deformation from circular trajectories. Accordingly, the observer's response might depend on detection of speed changes alone rather than spatial changes of the dot's deviation from circularity. To ensure that speed variation did not influence detection performance, we employed a second condition in which the Cartesian speed was kept constant. Note that the angular speed was varied in this condition. Our results showed that there was no difference between the detection thresholds of the two conditions. This implies that performance was the result of shape integration over time rather than speed variation detection. 
We found that detection thresholds for RF motion trajectories were 6 times higher than those for static RF patterns. In addition, identification of RF motion trajectories requires amplitudes higher than three times the detection thresholds, but three times thresholds are sufficient for accurate identification of static RF patterns of the comparable radial frequencies. The differences indicate that extraction of deformation in motion is more difficult than simply processing static spatial contours. Memory is required to keep track of the different locations of the dot along the trajectory over an extended period of time (up to 2.0 s in our experiments), which requires extra processing as opposed to simple form processing of static RF patterns. Detecting RF motion trajectories is simply a more demanding task than the transient detection of static RF patterns, thus leading to higher detection thresholds and higher amplitudes needed for identification. 
In the current study, a DOG dot that moved around an RF trajectory was displayed. The DOG dot was chosen to ensure that no added orientation cues were present, so the dot motion was the only cue for trajectory detection. Nevertheless, it would be interesting to investigate whether orientation cues affect trajectory detection by moving an oriented patch along the RF trajectory. It would also be beneficial to investigate whether the size or spatial frequency of the moving dot has an effect on performance, as they were both fixed in the current study. 
It would be interesting to understand whether detection thresholds for RF trajectories are proportional to their radii, as is the case with static RFs (Wilkinson et al., 1998). Such constant proportion to the radius would suggest trajectory shape constancy, where detection is independent of the trajectory's size. Thresholds for our stimuli of 1.0-deg radius range from 1 to 4 min of arc, which expressed in proportions to radius (or Weber fractions) range from 0.017 to 0.067. However, the use of only a 1.0-deg radius in our experiments cannot address the issue of shape constancy. Our laboratory is conducting further experiments using a range of radii to investigate this question. 
Another future experiment is to investigate the detection of RF trajectories of higher numbers of cycles. While we found a power-law falloff of detection thresholds from 2 to 5 cycles, threshold as a function of number of cycles could asymptote (as in static RFs; see Wilkinson et al., 1998) or even rise for higher RFs. This may shed light on how the local and global cues interact in detecting the motion deformation. 
A major motivation of this study was to assess the utility of RF trajectories as a tool to study periodic motion perception. As the stimuli contain motion, it is likely that human MT+ (including middle temporal (MT) and medial superior temporal (MST) cortices) will be activated, as MT+ is thought to process coherent motion (Tootell et al., 1995). As we found global processing in RF trajectory detection (Experiment 2), later stages of the visual system such as the superior temporal sulcus (STS) may also be involved. Indeed, STS has been shown to be involved in the global processing of biological motion (Grossman et al., 2000). However, it is not clear whether STS responds to the repetitive periodicity of the individual points of the motion trajectories involved in biological motion or whether all the points that form the percept of biological motion are required. Perhaps a subpopulation of cells would respond to only one light point moving in a periodic trajectory. Our novel RF trajectories may help resolve this issue. 
As the entire trajectory is used to achieve optimal detection and identification, it is possible that the global shape of the trajectory is encoded to perform the task, in addition to extraction of the dot's transient motion. The trajectory's structure is possibly reconstructed by sampling different points along the RF trajectory. These points would then be stored in memory creating a mental representation of the entire structure. In addition, the dot's motion may leave motion smear, known as “motion streak” (Geisler, 1999), on cortical neurons to provide instantaneous local orientation signals along the RF trajectory. Indeed, it has been shown that orientation information is pivotal in discriminating the direction of motion, especially when the motion signals are ambiguous (Krekelberg, Dannenberg, Hoffmann, Bremmer, & Ross, 2003; Or, Khuu, & Hayes, 2010; Ross, Badcock, & Hayes, 2000). It is therefore reasonable to conjecture that the global spatial shape of an RF trajectory might be used for motion trajectory detection, as local orientation signals derived from motion streaks are integrated to give the global shape. The area V4 is thought to be involved in this “midlevel” form processing, as the area has been shown to be sensitive to simple shapes (Gallant, Braun, & van Essen, 1993; Gallant, Connor, Rakshit, Lewis, & van Essen, 1996; Pasupathy & Connor, 2002; Wilkinson et al., 2000). It is therefore possible that V4 will be activated when RF trajectories are perceived, if shape processing is involved. We are currently conducting fMRI experiments to examine this possibility. 
Supplementary Materials
Supplementary Movie - Supplementary Movie 
Supplementary Movie - Supplementary Movie 
Supplementary Figure 1. Movie demos for two individual trials of Experiment 1 (constant angular speed condition). On each trial, two intervals of trajectory motion were presented, one with radial deformations and the other a circular trajectory, in a random temporal order. (a) The first interval contains RF4. (b) The second interval contains RF5. 
Acknowledgments
The work was supported in part by NSERC Grant 7551 to F.W. and NSERC Grant OP227224 to H.R.W. M.T. was partially supported by the CIHR Training Grant in Vision Health Research. 
Commercial relationships: none. 
Corresponding author: Charles C.-F. Or. 
Email: cfor@yorku.ca. 
Address: Centre for Vision Research, York University, 4700 Keele Street, Toronto, ON, Canada. 
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Figure 1
 
(a) Radial frequency (RF) motion trajectories for 2–5 cycles. The difference-of-Gaussians (DOG) dot starts directly to the right of the fixation cross (which is the centre of coordinates) and moves around the trajectory (dashed line) for one complete revolution, in a clockwise manner (see arrows). The dashed lines are for illustrative purposes and were not present in the experiments. (b) The corresponding static RF patterns.
Figure 1
 
(a) Radial frequency (RF) motion trajectories for 2–5 cycles. The difference-of-Gaussians (DOG) dot starts directly to the right of the fixation cross (which is the centre of coordinates) and moves around the trajectory (dashed line) for one complete revolution, in a clockwise manner (see arrows). The dashed lines are for illustrative purposes and were not present in the experiments. (b) The corresponding static RF patterns.
Figure 2
 
Individual results for Experiment 1, showing threshold amplitudes to discriminate RF trajectories from circular trajectories. In each of the four conditions, a power-law function was well fitted to the data for all observers. The error bars denote ±1 bootstrapped SD.
Figure 2
 
Individual results for Experiment 1, showing threshold amplitudes to discriminate RF trajectories from circular trajectories. In each of the four conditions, a power-law function was well fitted to the data for all observers. The error bars denote ±1 bootstrapped SD.
Figure 3
 
Results for Experiment 1, grouped over conditions. Here, geometric means across observers are plotted for each condition. The power-law fits were the same as in Figure 2. The error bars denote ±1 SEM.
Figure 3
 
Results for Experiment 1, grouped over conditions. Here, geometric means across observers are plotted for each condition. The power-law fits were the same as in Figure 2. The error bars denote ±1 SEM.
Figure 4
 
Detection thresholds for three fixation conditions in Experiment 1, tested only on RF3 moving at a constant angular speed of 6.28 deg/s. The results show that RF trajectory detection is not affected by eye movements. The error bars denote ±1 bootstrapped SD.
Figure 4
 
Detection thresholds for three fixation conditions in Experiment 1, tested only on RF3 moving at a constant angular speed of 6.28 deg/s. The results show that RF trajectory detection is not affected by eye movements. The error bars denote ±1 bootstrapped SD.
Figure 5
 
Stimuli for Experiment 2. The radial deformation of RF3 is restricted to a fraction of the trajectory leaving the remainder circular (the complete trajectory in white). Deformation was applied to N = 0.5, 1, or 3 cycles of the RF3 trajectory. Trajectories with 0.5 cycle of deformation were either a convex or a concave bump. The circular trajectory (in red) is superimposed on each RF trajectory for comparisons.
Figure 5
 
Stimuli for Experiment 2. The radial deformation of RF3 is restricted to a fraction of the trajectory leaving the remainder circular (the complete trajectory in white). Deformation was applied to N = 0.5, 1, or 3 cycles of the RF3 trajectory. Trajectories with 0.5 cycle of deformation were either a convex or a concave bump. The circular trajectory (in red) is superimposed on each RF trajectory for comparisons.
Figure 6
 
Results for Experiment 2 examining potential global contributions in detecting RF3 trajectories against circular trajectories. The detection thresholds were well fitted by a power-law function (exponent = −0.64, solid line) against the number of cycles of deformation, implying improvements in average performance with increasing number of cycles. The dashed line gives the prediction of probability summation over locally independent detectors anchored at 0.5 cycle (Equation 6). Probability summation evidently underestimates the increase in performance. The error bars denote ±1 bootstrapped SD.
Figure 6
 
Results for Experiment 2 examining potential global contributions in detecting RF3 trajectories against circular trajectories. The detection thresholds were well fitted by a power-law function (exponent = −0.64, solid line) against the number of cycles of deformation, implying improvements in average performance with increasing number of cycles. The dashed line gives the prediction of probability summation over locally independent detectors anchored at 0.5 cycle (Equation 6). Probability summation evidently underestimates the increase in performance. The error bars denote ±1 bootstrapped SD.
Figure 7
 
Threshold amplitudes for detecting RF3 trajectories. The use of a fixed phase angle (Experiment 1) results in higher thresholds than randomizing phase angles (Experiment 2). The error bars denote ±1 bootstrapped SD.
Figure 7
 
Threshold amplitudes for detecting RF3 trajectories. The use of a fixed phase angle (Experiment 1) results in higher thresholds than randomizing phase angles (Experiment 2). The error bars denote ±1 bootstrapped SD.
Figure 8
 
Proportion correct identification responses averaged over observers, for Experiment 3. Average performance declines with increasing number of radial frequency cycles for three times threshold amplitudes, regardless of whether phase is randomized. In contrast, average performance remains above 72% correct when seven times thresholds were displayed for the two speed conditions. The error bars denote ±1 SEM.
Figure 8
 
Proportion correct identification responses averaged over observers, for Experiment 3. Average performance declines with increasing number of radial frequency cycles for three times threshold amplitudes, regardless of whether phase is randomized. In contrast, average performance remains above 72% correct when seven times thresholds were displayed for the two speed conditions. The error bars denote ±1 SEM.
Supplementary Movie
Supplementary Movie
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