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A warm welcome to the lecture on the subject
of wavelets and multirate digital signal processing
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Let us spend a minute on what we talked about
in the lecture We have introduced the idea
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of a wavelet in the 2nd lecture and we have
done so by using the Haar wavelet Essentially
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where piecewise constant approximation are
refined in steps by factors of 2 at a time
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In today’s lecture we intend to build further
on the idea of the Haar wavelet by introducing
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what is called a multiresolution analysis
or an MRA as it is often referred to in brief
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So let me title today’s lecture We shall
title today’s lecture as the Haar multiresolution
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analysis And in fact let me also put down
here the abbreviation for multiresolution
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analysis MRA You see the whole idea of multiresolution
analysis has been briefly introduced in the
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context of piecewise constant approximation
So recall that we said that the whole idea
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of the wavelet is to capture incremental information
Piecewise constant approximation inherently
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brings in the idea of representation at a
certain resolution
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We took the idea of representing an image
at different resolutions in fact we use the
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term resolution when we represent images on
a computer 512 cross 512 is a resolution lower
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than 1024 cross 1024 and one way to understand
the notion of wavelets or to understand the
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notion of incremental information is to ask
if I take the same picture the same two-dimensional
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scene or same two-dimensional object so to
speak and represent it 1st at a resolution
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of 512 cross 512 and then at a resolution
of 1024 cross 1024 what is it that I am additionally
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putting in to get that greater resolution
of 1024 cross 1024 which is not there in 512
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cross 512 the Haar wavelet captures this
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So in some sense you may want to think of
the Haar wavelet as being able to capture
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the additional information in the higher resolution
and therefore if you think of an object with
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many shells so this is a very common analogy
you know if you think of the maximum information
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maybe as a cabbage or an onion informally
And if you visualize the shells of this cabbage
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or this onion like this then the job of the
wavelet is to take out a particular shell
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So the wavelet at the highest resolution wavelet
translates at highest resolution at Max resolution
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would essentially take out this at next resolution
it would take out this shell and so on So
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when you reduce the resolution what you are
doing is to peel off shell by shell in fact
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I think this idea is so important that we
should write it down We are essentially peeling
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off shell by shell using different dilates
and translates of the Haar wavelet And there
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again a little more detail different dilates
correspond to different resolutions and different
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translates essentially take you along a given
resolution
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So that is the relation between peeling off
shells and dilates and translates Now all
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this is an informal way of expressing this
we need to formalise it and that is exactly
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what we intend to do in the lecture today
Again we would now like to talk in terms of
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linear spaces So without any loss of generality
let us begin with the unit length for piecewise
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constant approximation I say without loss
of generality because after all what you consider
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as unit length is entirely your choice
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You can call 1 metre unit length you can call
1 centimetre unit length or if you are talking
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about time you could talk about 1 second as
unit length or unit piece and so on So unit
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on the independent variable is our choice
and in that sense without any loss of generality
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let us start make the focal point piecewise
constant approximation at a resolution with
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unit intervals So let us write that down formally
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Piecewise constant so you know the so-called
fulcrum or focal point is piecewise constant
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approximation on unit intervals And let us
sketch this to explain it better So what we
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are saying is you have this independent variable
again without the loss of generality let that
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independent variable be t you have unit intervals
on this And on each of these unit intervals
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you write down a piecewise constant function
essentially corresponding to the average of
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the original function on
that interval
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So this is the average of the function on
this interval this one on this interval and
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this one on this interval Now how can we express
this function mathematically with a single
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function and its translates So essentially
we want a function let us call it Phi of t
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now So what function Phi of t is such that
its integer translates can span this space
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what space 1st The space of piecewise constant
functions on the standard unit intervals What
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are the standard unit intervals
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The standard unit intervals are the open intervals
N N Plus1 for all N over the set of integers
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Now I wish to slowly start using notations
which is convenient So these notations script
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Z would in general in future refer to the
set of integers So I think we should make
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a note of this
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Script Z is a set of integers and this refers
to “for all” So what are we saying here
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let us go back We are saying we have this
space now again I must recapitulate the meaning
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of space linear space a linear space of functions
is a collection of functions any linear combination
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of which comes back into the same space So
if I add 2 functions it goes back into the
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same space if I multiply a function by a constant
it goes back into the same space if I multiply
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2 different functions in that space by different
constants and add up these resultants it would
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still be
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in the
same space
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In general we would say any linear combination
so we say a set a set of functions forms a
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space a linear space if it is closed under
linear combinations So we say linear space
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of functions
is a set of functions such that their linear
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combinations fall in the same set Now I am
making it
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a point to write down certain definitions
and derivations in this course and there is
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an objective behind that I believe that a
course like this is best learned by working
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with the instructor So although one could
just listen and try and remember that does
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not give the best flavour in a course like
this
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It does require in-depth reflection in thinking
and therefore I do believe
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that the student of this course would do well
to actually note down certain things and work
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with the instructor for it is then that the
full feel of the derivations or
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the full feel of the concepts would dawn upon
the students Anyway with that little observation
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and instruction let us go
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back to what we are doing here So you see
a linear space of functions is one in which
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any linear combination of functions in that
set fall back into the same set
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Now here there is a little bit of clarification
required You see in general if you consider
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the space of functions that we talked about
a minute ago namely the space of piecewise
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constant functions on the standard unit intervals
which are the standard unit intervals The
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intervals of the form open intervals of the
form N N Plus1 for all N over the set of integers
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then there is an infinity of such functions
And naturally when you talk about linear combinations
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you could have finite linear combinations
and you could have infinite linear combinations
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Now for this point in time when we talk about
linear combinations we are essentially referring
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to finite linear combinations That is just
a little clarification for the moment Well
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the idea could be extended to infinite linear
combinations too but I do not want to go into
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those niceties at this point in time they
would carry us away from our primary objective