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1. The demonstration programs for our super-threshold visual phenomena were created in Adobe Flash CS3 and were programmed in Actionscript 2, a scripting language that is built into the Flash programming environment.
2. We presented the displays in a classroom situation, to 26 American University students between the ages of 19 and 24. The size of the stimulus in terms of visual angle was dependent on the row in which students sat; students' chairs ranged from approximately 3.0 meters to 6.7 meters from the screen. The projected size of the image on the screen was 1.2×1.8 meters (i.e., observers in the front of the room saw a projection that was approximately 22×31 deg of visual angle, and observers in the back of the room saw a projection that was approximately 10×15 deg of visual angle). The display was controlled from a Macbook Pro connected to a Sanyo PLC XT25 theater projector. The demonstrations have been presented to both small and large public audiences (for instance, at Vision Science Society's Demonstration Night and the Best Illusion of the Year contest in 2008 and 2009). The phenomenal differences between peripheral and foveal viewing are robust over a wide range of viewing configurations, distances, and scales. For the purposes of most demonstration programs (with the exception of the Kanisza illusion, whose procedure will be discussed below), we were interested in documenting the existence of the effect.
The American University Institutional Review Board approved the experimental protocol for these experiments.
Student observers at American University were given a response form with questions concerning the displays. Students were informed orally and in writing (on the first page of the response form) about the conditions of the study, that participation in the study was anonymous and voluntary, that their responses would be part of a data set that may be published in scientific proceedings, that turning in a completed response form indicated their informed consent to be part of the study, that they could turn in a blank response form or hold on to the response form if they did not wish to participate, and that there was no penalty for not participating. The presentation of the trials corresponded to potential responses on the response form. The demonstrations were presented on the classroom projector system. After each demonstration was presented, the participants wrote down their responses; when all participants had finished recording their responses to a demonstration, the next demonstration was presented.
3. The motion energy analysis was detailed in [26]. In brief, the analyses were performed on a series of still images (movie frames) created with the aid of a Flash-Video converter (MacVide). The method is illustrated with a dropping solid ball (Fig. 1A). For each illusion, we first computed the Michelson contrast of each point (x, y) in each movie frame and then placed all the images at different time points in the three-dimensional (x, y, t) space. For a dropping solid ball, projections of the resulting three-dimensional volume in the x-t and y-t planes are shown in Figures 1B and C.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018719.g001 Motion energy analysis for a solid dropping disk.A) A series of frames depicting a solid dropping disk. B&C) First-order motion plots in the x-t and y-t planes. D&E) Projections of DC-removed and rectified second-order dropping disk movie in the x-t and y-t planes. F–I) Fourier analysis of the first-order and second-order motion energy of the solid dropping disk in the fx-ft and fy-ft planes.
To compute first-order motion energy in the horizontal and vertical directions, we first computed the Fourier power spectrum of the three-dimensional volume using Matlab 7.4. We then projected the three-dimensional Fourier power spectrum onto the fx-ft and fy-ft planes. In Figures 1F and G, polar plots of the Fourier power are summed over every 15 degs in the fx-ft and fy-ft planes, respectively. Note that the different directions in the fx-ft and fy-ft planes represent different speeds in the horizontal and vertical directions, respectively. For any point in the fx-ft and fy-ft planes, the larger the slope of the line connecting the point to the origin, the faster the motion. In both the fx-ft and fy-ft planes, we define motion energy, , in a particular quadrant, i, as the sum of Fourier power in that part of the Fourier space. The total motion energy, whose sign determines the direction of motion, is defined as(1)In the fx-ft plane, positive motion energy signifies motion from the left to the right; negative motion energy signifies motion from the right to the left. In the fy-ft plane, positive motion energy signifies motion from the top to the bottom; negative motion energy signifies motion from the bottom to the top. In both planes, zero motion energy signifies no motion. To take into account contrast-gain control in motion systems [40], a normalized measure of motion energy,(2)was computed and used to estimate the presence or absence of horizontal and vertical motion in a display. In Eq. 2, Etotal is the total Fourier energy in the fx-ft plane or the fy-ft plane. Etotal includes energy in the four quadrants and on the axes. For the motion stimuli in Figure 1, nMEx = 0.00, and nMEy = 0.75, reflecting no motion in the horizontal direction but significant top-to-bottom motion in the vertical direction. To compute second-order motion energy, we computed the Michelson contrast of each point in each movie frame, removed the DC component in each frame by subtracting from the contrast images of each movie frame the mean x–y image of all the movie frames, applied a full-wave rectification (square) on each point of all the resulting images [38][39] and placed all the resulting images at different time points in the three-dimensional (x, y, t) space. For the dropping solid ball in Figure 1, projections of the resulting three-dimensional volume in the x-t and y-t planes are shown in Figures 1D & E. The remaining steps of the analysis are identical to those in the first-order analysis. For the stimuli in Figure 1, the normalized second-order motion energy in the horizontal direction is nMEx = 0.00, signifying no motion in the horizontal direction. The normalized second-order motion energy in the vertical direction is nMEy = 0.74. Therefore, consistent top-to-bottom motion energy is present in the first- and second-order motion systems.
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