So in this lesson, again, we're going to consider an empirical explanation. And it's a little bit complicated. But let me take you through it. And the idea is generally much the same as the idea that we've used throughout the course to explain these other phenomena in empirical terms. In terms of our cumulative experience. Our cumulative human experience with what's going on, on the retina so that we can use that information to behave correctly in the world even though we don't have any physical information because of the inverse problem about what's actually going on in the world in this case. What's actually going on in terms of the direction of motion. So the steps, and there are three of them here that I'm going to describe to you to explain this phenomenon of the aperture effects, the whole range of aperture effects in empirical terms. Involves first of all getting the directions that people actually see when a given aperture is applied. You want to do this quantitatively, not just say, well I see a shift in direction and it's downwards into the right in a circular aperture. And completely downwards in a vertical slit. You want to actually see what the real psychophysical result is. You do what's illustrated in this series of pictures. Time is going in this direction. You ask a subject to fixate on a spot initially in a circular aperture. And then you move a line across the aperture. Again, the line is going from left to right and you ask the person to judge what the direction is that they see. And they can repeat this as many times as they want. By adjusting this green arrow to the direction that they actually see, you can get them to report the subjective direction that they see in response to the line moving left to right in a circular aperture. So once you've done that, you could determine the directions of motion that people report in circular apertures to lines that are moving through the aperture in different orientations. So this orientation is 30 degrees, 35 degrees, 40 degrees and so on up to 60 degrees. So you want to have the line that's not always oriented. At 45 degrees, you want to have the line oriented at different angles so that you can test a range of possibilities. And the green arrow here in each case is the direction that a large number of observers reported for lines in these different orientations. So what you can see in these result, is that whatever the orientation of the line, the observers are always reporting their perception of the direction of motion that they see as being orthogonal, as being normal, To the orientation of the line. They're always reporting it as a direction that's indicated here on these little graphs as 0. That just means that it's exactly orthogonal. It's exactly normal to the orientation of the line that's moving through the aperture in each of these cases. So now with step 1, we know the direction that subjects are seeing in response to lines going through the apertures at different orientations. And the question we face next is, well, okay, can you explain the orthogonal direction that they see in empirical terms? So to do that we need to take the next step. Which is to collect the relevant empirical data about how lines do project onto an aperture, their frequency recurrence of lines projecting onto an image plain through an aperture. So the setup that we use to do this is the same as the setup we used to get the information for the flashlight effect. You make a simulated 3D space. A simulated visual space, and you populate that space now not with dots which is what we use to analyze the flash lag effect empirically, but you populate the space with rods, or lines if you like. Moving through the space at different speeds and in different directions and you ask, when they're passing through a circular aperture, what's the frequency of occurrence of their projection onto the retina? What's the empirical data that we're getting for this kind of simulation and does the frequency of recurrence of lines are projecting along to the retina or on to the image plain here is that explain it? And again, just as with the dots, you can do this with as much precision as you like. You can populate the space with millions of lines or rods moving in all different directions and speeds over the relevant range of human perception. And get the cumulative information about the frequency of occurrence that we've actually experienced over our history of existence on Earth or over as a species or over our lifetimes as individuals. And of course you can, this is a circular aperture. But you can make this aperture vertical or any shape that you like to get the needed information for other kinds of apertures. So what happens when you do this. What is the human experience with projected directions through the aperture? This is the same diagram that I showed you before, but now added onto it, are these little squiggles which represent in each case the frequency of occurrence of lines projected through a circular aperture from the artificial environment moving in different directions and at different speeds as well, and these little jagged squiggles show you the probability distribution for lines projected through the aperture when the orientation is this, this, this, this and so on. So you're testing from orientations that are 30 degrees to orientations that are 60 degrees. What is the frequency of recurrence of projected lines through the aperture in those situations? And these little squiggles are showing you that in each case, the lines moving through the aperture indicated by these little squiggles that the mode of that distribution is very similar to the green arrow that indicates the perceived direction that people report. Let's super impose the modes of the probability distributions onto the psycho physical results, and see if they correspond with the frequency of directions that are determined empirically. And the coincidence in each of these examples of the red line and the green line. Sometimes they're almost superimposed, but you can see that they're always very similar. The point of that is that the mode of the probability distribution of the projected directions accords very well with the perceived direction that people report. So once again, it's pretty clear that if you want to explain complex phenomenon like the aperture effect or the series of aperture effects because you can use the same strategy to get data to explain the downward direction of the vertical slit, aperture versus the 45-degree shift in direction for the circular aperture. That in all these cases, you could use this information to get a pretty good body of evidence that yeah this is a reasonable way to explain these aperture effects and if you have a better idea. Let's hear about it but there is a great deal of phenomenology to be explained here. And the empirical analysis done in this way makes a pretty good start on getting there. So the thing I haven't explained yet is why it is that the modes of these probability distributions correspond to the reports that people make in psychophysical testing. And it's just projected geometry again. It's not hard to see why the maximum frequency of the currents of lines moving through a circular aperture, are in fact going to be orthogonal to the orientation of the line. That's in fact, presented in the aperture. So let's look at that. It's not, I think, too hard to understand. These biases arise because lines moving through the aperture orthogonally will satisfy the aperture more often than lines moving in any other direction. You remember what I said, the meaning of satisfying the aperture was. Satisfying the aperture just means that the line is moving through it without revealing either one of its ends. And I think it should be pretty clear just thinking in terms of the geometry of the situation that there will be a shortest line indicated by the red line here that can move through the aperture and not reveal one of the ends as it does so. And that line is going to have to move orthagonally, because if it moves in any direction other than orthagonally, one end or the other is going to come into view. Now, that's not the only line of course that can satisfy an aperture. Think of a longer line, like this blue line. It can also satisfy the aperture but it has to be longer and it has to be moving in a different direction. It has to move from this position to this position across the aperture and it has to be longer to do so without revealing either end. Now since shorter lines are always parts of longer lines, it's always going to be the case that the shorter line is going to be, that's moving orthogonally, is always going to be the most prevalent, projected directions of lines moving in a circular aperture. And that direction is going to have to be orthogonal always to its orientation in the aperture. This is what we humans will always have experienced, and this is the reason geometrically for the prevalence in our human experience of lines. It's moving through the circular aperture, being seen as orthogonal to their orientation. It's just projected geometry. And you can use the same argument to apply to any other aperture. So we take the vertical aperture, again, there's going to be a shortest line length to satisfy the aperture. Other lines are going to have to be longer. The short lines are always components of longer lines and that simply means that always the direction that is most frequent is going to have to be determined by the geometry, the projected geometry. And this line, the shortest line, is going to be moving in this direction. So the direction that you see is not orthogonal, but now it's straight down because that's the bias that is present in human experience of lines moving through vertical slit. When they're moving left to right as we discussed. So all that is maybe a little bit much, but I think you get the general idea that just as speeds could be determined in the case of the flash-lag effect by the empirical experience that we human beings have always had of the difference between bots in the real world projecting onto the retina. So it's the case in direction that that can be explained empirically in the same way.
