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[Music]
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all right welcome back to the second
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lecture on eigenvalues and eigenvectors
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and in particular
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what i want to
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talk about is this a demonstration of
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the power of eigenvalues and
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eigenvectors through an application i'm
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going to call eigenfaces it's one of the
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very interesting early applications in
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some sense of data science and
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understanding the power
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of using
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representations of data or features of
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data to be able to make some actionable
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decisions and machine intelligence
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decisions and so this really started
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very early on and this eigenface
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application i want to talk about
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it does two things first it starts to
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generalize our concept of what we're
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going to be thinking about in terms of
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uh
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matrices themselves so far we've been
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thinking about it for this course in
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terms of the matrices represent systems
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of equations or something that comes out
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of a physics-based model but here we're
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going to start thinking about matrices
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just being collections of data and what
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does an eigenvalue an eigenvector for
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that data mean
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so uh more notes and a link to the class
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website is here so just below in the
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description you can hit that
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link and you'll go to the class website
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where you'll see a broader range of
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lectures as well as notes data code
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everything you need for this class
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okay
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so let's get to it so what i want to
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talk about with eigenfaces and
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eigenvalues is first i want to do is i
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want to bring in an image i want to
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bring in an image
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that is typically in pixel space and
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then i want to convert it into
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a vector is what i really would like to
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do and so here's some code both in
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python and matlab and encode matlab
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python this is going to find it on the
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website
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these are some important features that
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we might
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import from python in order for us to
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bring in images and to start thinking
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about as images as vectors
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so for instance here once i've got all
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of this
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these packages loaded i would say
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friends here f1c this is going to be a
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picture of
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roger federer for instance
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uh is the first picture of roger federer
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in color so that's what the f1 c stands
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for first federal picture in color
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i use the i am read command image read
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command so image dot image read
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one fed1.jpeg so it goes and in the file
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this folder pulls in this jpeg image
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and once you've pulled in this jpeg
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image
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normally what happens it's in pixel
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space so it has that representation
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and since it's a color figure it also
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has an rgb representation in other words
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each color picture has a red green green
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and blue pixel designation and so really
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what this is is the number of pixels in
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the x and the y direction plus three
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layers of of colors so it's it's
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basically
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it's a generalization of a matrix to
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what's called a tensor it's a collection
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of data which is a data cube in some
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sense right so it's got thickness 3
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which is the rgb along with pixels and x
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and pixels and y
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but what i'm going to do with this is i
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just want to use it as a matrix
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and so what i do is use the command rgb
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to gray which turns it from color to
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grayscale and so once i've done this now
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i've flattened this down and i can even
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resize it so if i've got a bunch of
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figures and i want to make them a
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standard number of pixels i've resized
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this to 120 by 80 pixels so that allows
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you to do this resize which allows us
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now to work in a common framework across
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all of our images because they're all be
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the same size
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i can also do this in matlab
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but
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again i am read image read command here
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it is here's a picture of let's say
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cloney
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jpeg so i have a picture of george
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clooney here i'll just show you in a
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minute and i pull this in
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i can do the same command rgb to gray
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here it is so it's going to convert it
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from a color map
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which is has the red green blue
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designation down to just a grayscale
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and then
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normally when you're in picture frames
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or picture uh pictures have a very
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different
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representation in terms of what the
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numbers mean and so
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the double command turns it now to
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double precision numbers so now i can do
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math with it so for instance if i find
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eigenvectors and eigenvalues whereas if
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i don't do this it's in a
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format called uint8 which means it's a
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bunch of integers that take values
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between 1 to 255 and you can't do
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standard matrix manipulations with these
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numbers
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finally i resize this just like i did
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before i am resize
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and i resize it to 120 by 80. so the
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point of these two codes is very simple
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how do i read in a piece of data which
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in this case is an image
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that i first turn it into a grayscale
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and then i make it so that all the sizes
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are the same 120 by 80. and so these two
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pieces of code are sort of like starters
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for you to be able to import images
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because part of what we're going to do
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is now start to manipulate images and
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start understanding what eigenvalues and
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eigenvectors mean in the context
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of
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face images
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so
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i have a code that does this i'm not
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going to go through the code in detail
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because you can download it off of the
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website but here for instance are a set
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of pictures and i have very few pictures
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actually normally when you want to do
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something like this you'd want to do
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a large number of pictures with a lot of
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data but for the purposes of showing
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some of the important features of what
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we're going to be looking at this will
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do just fine okay
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so what you're seeing here is in the top
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row i have five pictures and this is
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george clooney here you go so five
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pictures notice that i've cropped the
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face roughly the same for all of these
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images
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the second row i have barack obama there
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we go five pictures notice that there's
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quite a diverse set of backgrounds
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behind these faces right some are some
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are dark some are light
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in either case so i've got the the face
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pretty well centered in the frame
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uh margaret thatcher is here okay and
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again mostly these faces are looking
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head-on into the camera however there
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are some slight turns of the head um
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which are good and then matt damon here
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we go matt damon is right down here five
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pictures of matt damon so what i'm going
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to do with this is i'm going to take
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each one of these images
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which is i've down scaled them to 120
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pixels
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in this direction 80 in this direction
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so it's 120 by 80 for each one of these
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pixel pictures and what we're going to
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do is start to use eigenvectors and
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eigenvalues to start to identify for us
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common features
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in images
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uh and right away a couple things i want
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to highlight to you
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so we're here looking at these images
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and again i've already said it's 120 by
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80. what we're going to do with each one
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of these images is we're going to turn
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them into a vector so we're going to
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think about 120 by 80 being there are 80
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columns here
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and so we can take those 80 columns and
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stack them up on top of each other into
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a very long skinny vector
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or we can lay it across as a large row
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vector this is called data flattening so
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this is done commonly for many data
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sources in which you have
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various
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things that you measure for instance
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here this is the pixel space in x and y
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i could also take the rgb cube but data
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flattening takes all of these things and
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just takes them reshapes whatever tensor
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you have
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into
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a large vector
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test test okay
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so
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that's what we're going to do to collect
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this data and here i have 20 images
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which means i'm going to create 20
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vectors
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and because the 20 vectors are 120 by 80
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each vector is going to be length 120
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times 80.
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okay so this is what we have as our data
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this is going to com comprise our our
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matrix a
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by the way i could also look at this is
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sort of an interesting just observation
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which is i could take
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all of these george clooney pictures
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and i could average them so just add
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them together divide by five because i
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got five pictures same thing with obama
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thatcher
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and jason bourne and what you would find
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is the following right so here are these
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faces which are sort of their average
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faces so i start here with clooney
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there's obama
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and already you can see that in fact you
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can still see a lot of features here
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and you can still recognize that person
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even though the quality is pretty poor
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so one of the questions that we always
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ask is
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so clearly there's something here that
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your eye is using to make a recognition
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decision about who these people are
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even if i don't give you a high quality
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picture you know who they are you could
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label them because you know i'm working
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with four different people and it's not
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like you look here at george clooney and
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think that's margaret thatcher in fact
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it's pretty clear it's not or that you
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look here and see uh matt damon or that
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you think it's margaret thatcher or
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clooney or obama you you look at it and
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it's pretty clear that that's the matt
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damon picture so despite the low quality
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it's very easy for us to make these
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recognition decisions and part of what
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we want to get after here with
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eigenvectors and eigenvalues is how do
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we actually approach this to think about
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ways
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what are the features that we're
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actually maybe going after here that
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allow us to see
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and make these identification decisions
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so part of what we're going to do is
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we're going to go ahead and start to do
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an eigenvalue eigenvector analysis
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of this data and the way we're going to
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do it is we're going to take and arrange
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these images into let's say a matrix b
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and the matrix b is just going to be
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each one of those images remember i have
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20 images here and in general if you do
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uh you know production level industrial
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level
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data analysis this is going to be
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thousands millions tens of millions of
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images
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where you want to start understanding
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the features between faces
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but here we have 20 images which means
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there's me 20 rows to this matrix first
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i'm going to take the first image i'm
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going to vectorize it as a row vector so
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i'm going to take this thing and it's
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going to be a row whose length is 120
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times 80 long
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that's how long this first row is and
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let's say that's the first picture of
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george clooney and then the second
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picture of george clooney then i put
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barack obama's margaret thatcher's matt
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damon's all together
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20 rows
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which have 120 times 80 columns so
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that's my data matrix
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and i'm going to use that data matrix to
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start doing analysis of faces and this
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is again one of the earliest versions of
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really doing some very interesting data
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science and feature extraction from
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matrices that we have
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around around face recognition
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so the big thing i'm going to do with
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this matrix is create what's called a
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correlation matrix i'm basically going
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to take
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that matrix b
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transpose it multiply by itself and what
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this is allowing me to do is start to
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look at correlations among these vectors
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what does it mean to have correlations
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among the vectors
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well if i take
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one of those vectors which just say is
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the case image
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and i
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take the inner product with the jth
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image
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if i take the inner product of those two
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vectors
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what i'm really looking at is their
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correlation right so if the inner
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product is zero
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that means they have nothing to do with
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each other and if they're normalized
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vectors and if i take that inner product
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and it's 1 that means i have a high
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degree of correlation among them
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so what this is actually allowing me to
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do so notice how we've changed the
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context of what's going on in that
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matrix and that data when we take inner
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products now and these projections are
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telling us something about correlation
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how are two different
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columns
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or rows in this case how are two
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different rows of this matrix here
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correlated with each other and i can get
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that information by taking a dot product
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and if that dot product is zero that
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means they are orthogonally have nothing
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to do with each other but if the dot
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product is one and these let's say these
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were normalized then they would mean
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it's exactly the same image
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so what you're looking at to see is how
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does that dot product change across all
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those faces not only how do george
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clooney faces looks relative to each
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other but how did george clooney faces
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project or dot product with margaret
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thatcher for barack obama or matt damon
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so that's the kind of thing that we're
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actually calculating here in the c
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matrix as correlation
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so now we've
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turned our interpretation of this matrix
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the matrix is data it's not solving some
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3x3 system or solving for currents and
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with resistors and so forth this is
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actually just straight data and our
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inner product tells us something about
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correlations
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so how do you do that in python or
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matlab here are some basic code
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structures to do that
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so i
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am going to import from numpy
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from illinois called la because what
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we're going to do is eigenvalue
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decomposition i'm going to call it la is
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going to be this linear algebra package
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so i can just simply do v comma d so
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first of all i compute i create the
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matrix b and i already showed you how to
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read in images and so now you take these
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images you stack them on top of each
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other to make the b and then a matrix
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multiply b times b transpose
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so that's matrix multiply and that
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creates my correlation matrix c
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and then i can actually find its
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eigenvalues and eigenvectors just by
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simply here's the eigenvectors here's
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the eigenvacuumers by
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la dot i so this is calling the linear
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algebra package i could have just said
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linalch dot i c or las ixc whichever one
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but there's a little code structure for
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doing it
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and matlab
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here you go c is
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b times b and actually i think they have
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this backwards this just should have
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been just b times b transpose over here
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apologies for that
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okay and then you just use the x command
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and you get back v and d now notice here
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with i i'm actually doing something a
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little more sophisticated than up here
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which is i'm
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taking it taking that matrix c putting
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it in there
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and what i'm doing is
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i can ask i only want the top 20
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eigenvalues
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sorted by the largest magnitude so this
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is a very simple statement which allows
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me to
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get eigenvalues and eigenvectors to this
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in a very efficient way where i can
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specify what i want which is the largest
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magnitude and i only want 20
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instead of all possible eigenvalues
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eigenvectors you can do something
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similar here in python as well but this
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is the code structure for executing the
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eigenvalues and eigenvectors
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on this correlation matrix here okay and
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like i said apologies that that prime
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should have been over here
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okay
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so what do we get from this once we've
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executed these i want to talk about the
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interpretation
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and what i'm showing you here
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on this plot is i'm showing you five
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these are five faces they look like
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faces they look like very warped faces
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in some sense or
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kind of weird but what they are these
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are eigenfaces so i got the eigenvector
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which is of length 120 times 80. i
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reshaped it back to 100
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so it was length 120 times 80. now i've
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reshaped it back to 120 by 80.
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and this is the dominant eigenface
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this is the second eigenface the third
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eigenface so these are just eigenvectors
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but these eigenvectors
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remember their coordinate system
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and in this coordinate system you start
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to see the features
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that we can use to do face
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identification so let's talk about these
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features the first feature is
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interesting because what does it really
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have for us if you look at this right
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you see that what this first feature is
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is sort of in some sense the most common
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thing that all faces have
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so first of all
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it's round
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or oblong
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you got some eyes it has a little bit of
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a feature of a mouth but then you really
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you have this hair
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face separation
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so every single one of the faces i
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showed you
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there was clearly a feature where the
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forehead ends to the hair
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it has eyes and nose and mouth and this
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here really picks up
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a lot of that basic feature so
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it's very hard to tell the difference
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between any face because this is just a
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common feature across all the faces
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and in fact
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what i'm showing you here on this
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picture
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is the eigenvalues and they're what the
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actual values of these largest magnitude
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eigenvalues are notice the first one
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which is that one is much bigger than
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the others so it's quite a bit bigger
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look at this this goes to 10 to the 10.
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the next one's at 10 to the 8. so this
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is like two orders of magnitude bigger
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than the others
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okay so that tells you something
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important about what this eigenvector is
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picking up it's picking up these
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dominant features that all faces have
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the second eigenvector here it is
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you can start to see a little bit more
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of the eye structure ears mouth
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same thing with the third the fourth and
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the fifth and in fact one thing i want
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to point out it's around the third
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and the fourth that you start actually
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starting to see some teeth remember that
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out of those pictures originally there
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was only about
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four or five of the images where teeth
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was shown so eventually teeth shows up
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as one of the features in this thing so
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the idea here is all those faces could
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be represented as a linear combination
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of these
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features or these eigenvectors okay so
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another way to think about this from the
00:18:52
data science perspective is this is now
00:18:55
a feature space
00:18:57
in this feature space we can start now
00:19:00
to represent any generic face because we
00:19:04
can take a new face
00:19:05
and project it onto these eigenvectors
00:19:08
to get some idea of how
00:19:11
some new face or new person's face would
00:19:14
project relative to the other faces that
00:19:17
we have here
00:19:18
okay
00:19:19
so it's kind of an interesting concept
00:19:21
we've turned this idea of an eigenvector
00:19:23
into a coordinate system and this is an
00:19:26
eigenface coordinate system it allows us
00:19:29
to start thinking about how could i use
00:19:31
this to do face identification or
00:19:35
you know person identification in fact
00:19:36
this is one of the earliest things that
00:19:38
we started thinking about for this and
00:19:40
this is all the way back in the late 80s
00:19:43
so in fact just to make this more
00:19:44
concrete
00:19:46
i'm going to take the average
00:19:49
clooney face obama face thatcher or matt
00:19:51
damon face
00:19:52
and what i'm going to do with this
00:19:54
is i'm going to take those average
00:19:56
their average phase which is the average
00:19:57
over the five and i'm going to project
00:19:59
it on to those first 20 eigenvectors so
00:20:03
here's what george clooney looks like
00:20:04
projected onto those
00:20:06
here is what obama looks like projected
00:20:09
onto those
00:20:10
here's what thatcher looks like on those
00:20:13
here's what matt damon looks like on
00:20:14
those and notice
00:20:16
these are quite different and this is
00:20:19
exactly what we can use
00:20:21
to start to do
00:20:23
a classification task which allows us to
00:20:26
identify clooney versus matt damon what
00:20:29
is the difference well look at matt
00:20:31
damon look at these very strong
00:20:33
signatures
00:20:35
positive and then negative on the first
00:20:38
five eigenvectors whereas clooneys have
00:20:40
a very different structure there
00:20:43
same thing with obama has very different
00:20:45
structure here very low representation
00:20:47
on the higher modes
00:20:49
thatcher has something different so in
00:20:50
other words each one of these are like
00:20:52
keys
00:20:53
they're like
00:20:54
a key for representing one of these
00:20:57
faces so if i gave you a new face
00:21:02
uh and i didn't tell you if it was
00:21:04
george clooney or margaret thatcher
00:21:06
or barack obama or matt damon i just
00:21:08
gave you a face and i said okay i want
00:21:10
your computer to tell me
00:21:12
whose face this is
00:21:14
what you would do is you would take that
00:21:16
face
00:21:17
take it as a 120 by 80 image you would
00:21:19
take
00:21:20
you would reshape it take its inner
00:21:23
product on the first 20 eigenvectors for
00:21:25
instance and you would look to see does
00:21:27
it look more like this like this like
00:21:29
this or like this
00:21:31
and whichever one it looked most like
00:21:33
you would then make a classification
00:21:35
decision and say i think that's matt
00:21:37
damon because
00:21:38
when i project the face it looks most
00:21:40
like this set of weights onto the
00:21:44
eigenvectors remember we're all
00:21:46
projecting onto these eigenvectors which
00:21:49
is the feature space which are given
00:21:50
right by here i've only shown you the
00:21:52
first five but i can take the first 20.
00:21:55
okay and so this becomes a really
00:21:58
interesting concept
00:22:00
for
00:22:01
doing face recognition this is like i
00:22:03
said one of the earliest versions of
00:22:05
this was just saying let's just use
00:22:07
eigenvectors and eigenvalues as a
00:22:10
coordinate system
00:22:11
and it does a remarkable job in getting
00:22:14
you information about this and that
00:22:16
really started off this trend of
00:22:18
thinking about like wait a minute we can
00:22:19
use
00:22:20
a lot of these linear algebra methods
00:22:23
to build pretty sophisticated tools for
00:22:26
face recognition and of course in the
00:22:28
last decade or so that has all moved
00:22:30
towards deep learning but the concept of
00:22:32
a feature space
00:22:34
still exists there and this is really
00:22:37
what set down
00:22:38
the basis of understanding how to do
00:22:41
face recognition with this idea of a
00:22:44
feature space
00:22:47
so
00:22:47
let's continue on this little fun little
00:22:49
exercise and i'm going to go ahead and
00:22:52
take a picture for instance here's one
00:22:54
of the pictures in the training data set
00:22:56
this is margaret thatcher
00:22:58
and uh
00:23:00
actually i think i believe this was a
00:23:01
new image of margaret thatcher not in my
00:23:04
five that i contained or so i built this
00:23:07
right i took 20 pictures
00:23:09
i built this thing out and what i did is
00:23:11
i said let me take a new picture of
00:23:13
margaret thatcher let's project this new
00:23:16
picture on to the eigenvectors and
00:23:18
here's what i get
00:23:19
and what i could do is saying hey let's
00:23:21
try to reconstruct margaret thatcher
00:23:24
this picture in terms of this
00:23:26
eigenvector basis and notice what it
00:23:28
does it actually
00:23:29
gives you back a face that kind of looks
00:23:31
like margaret thatcher in fact i can
00:23:33
look at the error of this new face
00:23:37
against her five pictures and here's
00:23:39
where they are and so for instance one
00:23:41
of them is very close to one of the
00:23:43
pictures there so this is kind of an
00:23:45
interesting concept new new picture i
00:23:48
projected onto eigenvectors and actually
00:23:50
what i get out is something that looks
00:23:53
pretty similar to
00:23:55
uh
00:23:56
you know roughly it looks like margaret
00:23:57
thatcher
00:23:59
i can also take meryl streep here and
00:24:02
meryl streep played margaret thatcher in
00:24:04
the iron lady this was a movie i think
00:24:06
she was nominated for academy award of
00:24:08
course as usual and we could take her
00:24:11
projected onto margaret thatcher and
00:24:14
here's what we get it's actually quite
00:24:16
different than here and so part of what
00:24:17
you're doing is saying how does her
00:24:19
likeness
00:24:20
playing market thatcher actually stack
00:24:22
up to images of margaret thatcher and so
00:24:24
if i reconstruct with these onto the
00:24:28
eigenvectors i have
00:24:30
i get something that doesn't look like
00:24:31
margaret thatcher very much and in fact
00:24:33
the error between
00:24:35
this her representation and the other
00:24:37
pictures are quite large compared to
00:24:40
they can be pretty small
00:24:41
in the uh
00:24:43
with this other picture of margaret
00:24:45
thatcher finally i take hillary rodham
00:24:47
clinton there we go project onto here
00:24:50
and again she does not look
00:24:52
a whole lot like margaret thatcher
00:24:54
but she does look more like margaret
00:24:56
thatcher than
00:24:59
than we do
00:25:00
with uh with our representation of
00:25:03
cinema uh
00:25:05
uh
00:25:06
here as as someone playing margaret
00:25:09
thatcher so anyway these are kind of fun
00:25:11
games you can play
00:25:12
uh with these representations and again
00:25:17
it's all like coming down to
00:25:19
project onto this feature space which
00:25:21
are the eigenfaces and starting to
00:25:24
understand how the eigenfaces
00:25:27
really help us understand to do face
00:25:29
recognition whether i'm looking at meryl
00:25:31
streep or hillary rod and clinton how
00:25:33
they project how much do they look like
00:25:36
margaret thatcher versus a picture of
00:25:38
margaret factor i never had
00:25:40
and i project onto this
00:25:43
my little final example of this is i is
00:25:46
me
00:25:47
i take a picture of me there's my face
00:25:49
and i say i'm going to project myself
00:25:51
onto these 20 eigenvectors so there's my
00:25:54
projection onto the 20i vectors and if i
00:25:57
reconstruct myself in that 20
00:25:59
eigenvector space
00:26:00
here is my representation okay
00:26:03
and who do i look most like
00:26:06
i was kind of going to try to figure out
00:26:07
i was hoping i would look most like
00:26:09
super handsome george clooney but it
00:26:11
turns out
00:26:12
i don't what i look mostly like and you
00:26:16
could probably tell from the suit and
00:26:17
tie
00:26:18
i look very presidential so here's obama
00:26:21
and my best projection are on to obama
00:26:23
pictures
00:26:24
and then here's clooney i leak look
00:26:27
least like
00:26:28
thatcher and then i don't look too much
00:26:30
like matt damon except for one picture
00:26:32
of me of one picture of matt damon
00:26:34
that's probably when he was super
00:26:35
awesome jason bourne because probably i
00:26:38
probably look like something like that
00:26:39
you probably think i'm jason bourne
00:26:40
right so there we go presidential jason
00:26:43
bourne with a little bit of superheroes
00:26:46
from clooney that's
00:26:47
that'd be a nice way to think of oneself
00:26:50
but this is all using these concepts of
00:26:53
feature spaces to build out
00:26:56
really interesting representations of of
00:27:00
of ourselves of images and really
00:27:03
i think highlighting for us the role of
00:27:06
eigenvectors and eigenvalues in creating
00:27:08
a coordinate system in this case
00:27:11
the eigenphase coordinate system so
00:27:13
hopefully this has been a little fun
00:27:15
little uh
00:27:16
exercise and seeing the kind of things
00:27:18
you can do and again it comes down to
00:27:20
this concept of the eigenvalue and
00:27:22
eigenvector which are these very broad
00:27:24
general representation tools which are a
00:27:27
coordinate transformation into which
00:27:30
things become diagonalizable
00:27:32
and the eigenvectors themselves are
00:27:35
features here the features were these
00:27:38
eigenfaces but more generally in physics
00:27:41
based problems these eigenvectors become
00:27:44
these
00:27:45
in some sense the best coordinate system
00:27:47
for you to use for that system so that's
00:27:50
something we're going to keep building
00:27:51
on you'll keep seeing eigenvalues
00:27:53
eigenvectors all the time throughout
00:27:54
physics and engineering here i took it a
00:27:57
little bit different direction with
00:27:58
eigenfaces but really mostly for the
00:28:01
point of view to just highlight how
00:28:03
powerful a concept this is
00:28:05
if you want to download the data and
00:28:07
code for this it's all on the website
00:28:10
here you can click on the link on the
00:28:12
description of this page you get the
00:28:15
notes data everything you need and you
00:28:18
can play around with this when it's kind
00:28:19
of a fun little thing to play around
00:28:21
with
00:28:23
[Music]
00:28:30
you
