The Reduce operation is pretty clear.
So the idea is that, this level, we have a full resolution, or,
or whatever level, and then we want to take the next level down.
So here's a way of thinking about it, we'll just number these.
One, two, three, four, five, six, seven, eight, nine.
Nine pixels across.
I actually don't have to produce the pixels that would go under two, four,
six, and eight.
Because I'm just going to,
I only want half the number of pixels that I had to start with.
So in fact what this does is, it simply says I'm going to take some location,
here I use the middle, and I'm going to take what's called a 5-tap Filter.
And here are the filters, see it's one-sixteenth, four-sixteenth,
six-sixteenth, four-sixteenth, one-sixteenth.
And that's, that's what these numbers are, one-sixteenth, one-fourth,
three-eighth one-fourth, one-sixteenth.
Okay? That's the filter that I use to get
the odd pixel here in the middle.
Now that's just in one scan line.
I'm working with images.
How do I do this?
Okay. Go way back in your notes,
because I know you all are taking co, copious notes.
I almost said y'all.
When I say y'all, Megan's going to throw that monitor at me.
Okay?
Nothing against my friends from the south, but I come from New York,
went to school in Boston.
If I actually said y'all, I think I'd get shot by my relatives.
Anyway, You all have seen this before.
We talked about Separable Filters.
Right? So a separable filter was, instead of
applying a square filter, to the filter, I apply one, one dimensional filter
horizontally and then often it's the same one dimensional filter vertically.
So go back up and remember about separable filters.
The reduce operator uses a separable filter,
a 5-tap separable filter first in the rows, then in the columns.
All right.
That one was straightforward.
The Expand operator is a little bit trickier.
And you can tell that by looking at this awful picture.
In fact I'm going to pull that awful picture off by itself.
Here's the deal.
The idea is that on this picture only pixels that are, were real.
The question is, how do you make finer pixels from a coarser levels?
So here's the coarser level, here's the finer level.
And it turns out you do it two different ways.
Okay? So we're going to start with
the odd pixels.
All right?
So the odd pixels occur underneath pixels that are actually there.
So there are a couple of ways of thinking about this.
One is we're just going to assume that we're going to,
I'm just going to tell you what the filter is right there.
It's one-eighth, three-fourth, one-eighth.
Okay. That seems a little bit arbitrary until
you realize that's just kind of like if these were zeroes.
Okay.
And I took in all the values here, and
I made it the same 5-tap a four, but now doubled it.
So one-eighth, one-half, three-quarters,
one, these will be in the same ratio of one, four, six, four, one, as before.
Okay?
But I'm only pulling it off the odd pixels.
All right.
And so these points go away and
I'm only left with one-eighth, three-quarters, one-eighth.
By the way, we're going to provide the original
Burton Edelson paper that talks about reducing expanse.
You'll be able to, to look at that a, a little bit more carefully.
All right.
That's to reproduced a pixel, it's actually underneath one that's there.
Slightly trickier are the even pixels.
We have a pixel here that's underneath something that didn't actually exist.
So if I still do the six, four, one.
All right?
One, four, six, four, one above me.
Okay?
Well, when I'm on one of these even pixels, there's nothing directly above
me, and there's nothing to my ends, there's only these two here.
That's this one, and this one.
So looking five above me, only two of them are really there,
they're equally distant from me, what coefficients am I going to use?
I'm going to use one-half.
So to produce this finer value, I take one-half of that pixel and
one-half of the other and that actually sort of makes sense.
If you think about it, if I'm blowing up a picture, so
I'm going to go from say, seven pixels to, to 14.
I have to get space between two pixels that were there.
What value should I put in that space?
Halfway between the ones that were there before.
The one that's a little trickier to understand is if I'm blowing it up,
how come I just don't leave this original pixel alone.
And you can read in the paper, this is for the odd pixels,
you can read in the paper the motivation in terms of frequency theory.
In terms of how they do that.
All you need to know is that you now have a reduce operator and
an expand operator.