Okay, great. Now, let's move on to the last type of
models. These are called interpretive models.
So we're going to look at an interpretive model of receptive fields, and the
question that we're asking is, why are receptive fields in V1 shaped in this
particular way? In other words, why do they look like
this? Why do they have this orientation and why
are they selective for bright or dark bars?
Another way of asking the same question is, what are the computational advantages
of such receptive fields? Now, this is the kind of question that
perhaps an engineer or a computer scientist would ask.
So why do we need to use these types of receptive fields?
Do they confer any advantages? So, let's look at one particular
interpretive model of receptive fields. The interpretive model that we look at is
based on the efficient coding hypothesis, so what does this hypothesis state.
Well, it states that the goal of the brain, through evolution for example, is
to represent images as faithfully and as efficiently as possible using the neurons
that it has. Now, these neurons have receptive fields
RF1, RF2, and so on. So here are some receptive fields of
neurons. And the question we're asking is, are
these the best way, or the most efficient way, of representing images.
And so, how do we represent images using these types of receptive fields?
Well, as an example, I can take these receptive feels are of three and four so
as take these two, and I can just add them.
So imagine that they are images, so here is a bright region in the image and a
dark region in the image bright region in the image and dark in the image so if i
think of these as just two image patches so these two receptive feels I can add
the two receptive fields. And what kind of an image can I
reconstruct? Well, if you add these two together,
you're going to get some linear combination of these regions.
So, you have that particular shape. So, it's going to be a really bright
region up here. Maybe a slightly bright region here and
here and here and here. Because you're adding the plus is here
with a little bit of the minuses over there, and you're going to get some dark
regions over here. And, so, what you have is an image that
looks something like that. So, given that you are adding these two
receptive fields, you have a image that you reconstructed that looks something
like, let's say, a plus sign. Now, if you're given a whole bunch of
these receptive fields, you can literally combine them in this particular way.
So this is just a summation sign over all the receptive fields, RF1, RF2 and so on
and each of these are weighted by some number.
So, these are the neural response, so a linear combination of them is going to
give you some particular image. Now, what is the goal here?
The goal is to find out what are the receptive fields, RF1, 2, and 3, and so
on that minimize the total squared pixel-wise errors between a given set of
images. So one of these images, perhaps the brain
is trying to optimize its representation for natural images.
And so we can look at the squared pixelwise errors between natural images
and the reconstruction of those natural images I had.
And we've also add an additional constraint, so you want them to be
efficient and so we want these responses for example to be as independent as
possible we don't want all the neurons to be firing at the same time for example
and so we can add the constraint that. We want these coefficients or these
responses r sub i to be as independent as possible.
So given that we have now this optimization criteria and this particular
idea of minimizing the total square pixel wise errors.
And also keeping these responses as independent as possible.
What are the receptive fields that achieve this objective?