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So here is the result of the analysis.

And it's quite simplified in this case because we're only considering

lines at four orientations.

The lengths of the projected lines just expressed in terms of, for convenience,

the number of pixels in the analysis.

But we're only considering lines that are 0,

that would be a horizontal line at 30 degrees oblique,

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at 60 degrees oblique, and at 90 degrees, that would be a vertical line.

So we're talking about four classes of lines here, each color coded.

The blue line here, for example, is the straight horizontal line,

the red line here is the vertical line for these other color coded lines.

So, that's the result and you see and it's the point of showing you this graph

that they all have different frequency of occurrence, that's not surprising.

I mean you really wouldn't expect if you think about it lines in the real world to

project under the retinas.

All with the same frequency, given that they have different lengths and

orientation.

They're all going to be different and these graphs, these lines,

these curves show you the way in which they're different.

Over a range of line lengths, that's a large one.

And this is the probability of their occurrence.

So some lines are occurring more frequently at different orientations and

lengths than other lines.

That's what you'd expect and that's in fact what the data show you.

Now let's look at that in an even more simplified way.

We're now just considering the horizontal line, that's blue.

The blue line, the blue curve here versus the red line, the vertical line.

And you see if you just consider these curves for any particular length,

lets take this length indicated by the dotted line.

There's going to be a different amount of space under the two curves.

Meaning that you have the wherewithal to actually tell you to make

a prediction based on the frequency of occurrence of vertical lines at

a certain length versus horizontal lines at a certain length.

You're going to be able to tell,

what's your prediction if the hypothesis is that it's the frequency of

recurrence in our human that's arranged the perception of line lengths to generate

the way in which we see these phenomenon that I've been telling you about?

That you have the wherewithal to make that comparison.

And if you do this, many, many times for the whole range of lengths and

for the whole variety of orientations, it's not just vertical and horizontal but

the whole range of different projections that we've been talking about from

horizontal through oblique to vertical and back again to horizontal.

You can compare what you see in psychophysical testing,

what we talked about before with the expectation, the prediction,

that derives from the frequency of occurrence

of lines in human experience over as I say the eons.

And let's look at those predictions.

So here's what we saw before, this is the perceptual function.

Remember what we saw was that indeed we see vertical lines

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as longer than horizontal lines, the horizontal line is 1,

the vertical line is up here at about 10% greater.

But remember that we see this strange McDonald's arches effect,

that the peak of line length deviation,

the peak difference between horizontal and vertical is not straight vertical but

a line that's a little bit off vertical, 20 degrees or so off vertical.

So that's a phenomenology,

psycho-physically determined phenomenology, that's very subtle and,

scratch your head and say, well, gah, whyever should that be?

But when you make the prediction based on frequency of our experience,

this is exactly what you get.

You get these McDonald arches derived from predicting

the expectation of line length that you should see for lines of

all of these different orientations and of course at different lengths, too.

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Based on the frequency of occurrence that we human beings have experienced,

since day 1 so to speak.

So, I think to end this lesson, that's to me at least,

a very impressive demonstration of how experience can explain

stuff that would be almost impossible just to come up with intuitively or

in any other way than documenting this workaround to the inverse problem.

Remember I said that the inverse problem is that you

don't know what's out there based on the image on your retina.

The same image on your retina can be a line in different

orientations of different lengths at different distances.

You are stymied unless you have a workaround.

And the workaround that seems to work very well in this case,

and I should say that it works well in all the other cases too.

And of course I don't have time to describe it.

You can read about them if you like in the resources for the course.

This is a very good demonstration that the workaround is an empirical one.

And that you can explain all this phenomenology in terms of our

human perceptual experience.