5 Results and Discussion

Figures 3 and 4 show theoretical curves that depict certainties for motion coherence, transparency and a unitary (phase) 1-d velocity for both additive and multiplicative transparency. From the graphs, one can see that the interpretation of transparency was most likely when the plaid components moved at high speeds. With respect to changes in relative contrast (fig. 3 b) or modulation depth (fig. 4 b) the certainty value for transparency increased relative to the certainty measure for coherence. This suggests that transparency is more likely as relative contrast decreases[ 12 ]. To the conditions shown in figure 3 b, the model would predict bi-stability because certainties for coherence and transparency are similar in magnitude. Notably, visual perception also alternates between coherent/transparent interpretations to these stimuli[ 11 ]. Also, as we manipulate orientation differences, one can see from the curves that transparency is most likely when component orientation differences are small and move in opposite directions. Finally, one should note that to the case of multiplicative transparency (fig 4 ), the phase velocity was assigned the highest certainty under almost all conditions. One should note, however, that the magnitude of each certainty measurement may be adjusted by altering the magnitude of the Lagrange multiplier for each model. Moreover, to transparencies composed of a carrier grating and contrast envelope, one of the perceived velocities is defined by the phase velocity[ 5 ], although it is rarely observed in isolation.

   
Figure 4: Motion certainties for coherence, multiplicative transparency and phase (1-d) velocity . The image was defined by , where m refers to the envelope's modulation depth and c is a constant. We varied: (A) : Image speed. The two 1-d structures were equal in contrast and orthogonal in spatial orientation. (B) : Modulation depth for fixed speed, where each 1-d structure moved at 1 pixel per frame. The two 1-d structures were orthogonal in spatial orientation. (C) : Relative orientation. The magnitude of spatial frequency and temporal frequency was fixed. Each 1-d structure moved at at 1.0 pixel per frame. The envelope spatial frequency was fixed at 1/8 of the spatial frequency of the carrier.

These results are plausible. Consider (as an example) two gratings that are equal in spatial orientation, differ in spatial frequency and move at different velocities. A coherent model cannot fit the image data because (in the absence of prior information and/or spatial integration) it is spatially degenerate. However, the image still has two degrees of freedom. These degrees of freedom are better explained by a 1-d transparent motion model. In 1-d, a transparent motion model also has two unknown parameters which take into account the variation in the image measurements. A model of coherent motion cannot.