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.

^{2}.

*v,*such that the dot's polar location

*r*at time

*t*is

*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.

*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 (±3.1

*σ*), with a contrast of 100%.

*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.

*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:

*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.

*SD*) of the threshold amplitude was computed using a bootstrapping procedure.

*y*-intercepts averaged 11.0 ± 2.9, 5.36 ± 1.01, 3.58 ± 0.99, and 5.49 ± 1.50 min of arc.

*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.

*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.

*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.

*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).

*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:

*θ*

_{c}= (4

*π*−

*ϕ*) /

*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):

*θ*

_{c}= (4

*π*−

*ϕ*) /

*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.

*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).

*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.

*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.

*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.

*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.

*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.

*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.

**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.