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You know yesterday I told you that there is
this beautiful idea about just one function
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psi t its dilates and translates going all
over to capture incremental information now
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we need to state that formulae too but in
order to move in that direction we 1st need
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to bring in as I said another function which
will span V0 So we need to bring in this idea
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of spanning We say a set of functions we say
a set of functions let us say f1 to let us
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say fk and so on span a linear space if any
function in that linear space can be generated
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by the linear combinations of this set
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Now again there is this subtle distinction
between finite linear combinations and infinite
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mere combinations I do not wish to dwell on
those distinctions at the moment But what
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we are saying what we mean by span so when
we talk about the span of a set of functions
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we are talking about all linear combinations
of those functions and therefore the set or
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the space in fact the space of functions generated
by all linear combinations of that set So
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now we ask a question which should make our
life easy
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What function we asked this question before
but now we answer it what function suppose
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we call it Phi t and its integer translates
span V0 And the answer is very easy in fact
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if you were to visualize a function which
is 1 in the interval from 0 to 1 and 0 else
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lo and behold you have the answer So what
we are seeing is any function in V0 can be
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written like this summation n over all the
integers CN Phi t minus n so essentially integer
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translates of Phi
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And these are the piecewise constants here
Just to fix our ideas let us take an example
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here So what we are saying is for example
suppose I have this example of a function
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in V0 alright so I have 0 1 2 and 3 and so
on The values here is let us say 0.7 the value
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here is 1.5 the value here between 2 and 3
is 1.3 the value between minus1 and 0 let
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us say is 0.2 and so on then and this could
continue this function can be written as well
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dot dot dot plus0.2 times T plus1plus0.7 times
Phi t plus1.5 times Phi t minus1plus1.3 times
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Phi t minus2 and plus dot dot dot and so on
Simple enough not at all difficult to understand
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So we have this single function Phi t whose
integer translates span V0 now the subtle
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point is that if you were to go to any space
Vm so the same thing would carry forth So
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it is very easy to see that any space V can
be similarly constructed In fact they can
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be more precise we can write down Vm is essentially
the span overall n belonging to Z of Phi 2
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raised to the power m t minus n So just as
we looked at the wavelet yesterday and said
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it is a single function which can allow details
to be captured we now have this function Phi
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t which captures representations at a resolution
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It is a very powerful idea if you think about
it If you want to capture information at a
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resolution at a certain level of detail that
is all the information upto that resolution
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you have the function Phi If you wish to capture
the additional information in going from one
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subspace to the next you have the function
psi the wavelet Now we need to give this function
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Phi T a name we shall call it a scaling function
Now of course here the Phi T that I had drawn
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in this context is the scaling function of
the Haar wavelet for the Haar multiresolution
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analysis
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Now what is this multiresolution analysis
I have suddenly brought in this word So
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what is this Well this ladder of subspaces
that we
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are talking about here with these properties
is called a multiresolution analysis or an
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MRA for brief of course in this case the Haar
multiresolution analysis Now what properties
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we need to put them down formally once again
We
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have introduced 2 of them and one more subtly
now we need to put down axioms very clearly
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So let us put down what are called the axioms
of a multiresolution analysis So the 1st axiom
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is of course that is a ladder of subspaces
of L2R and we know what the ladder looks like
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such that axiom number-one union over all
integers closed Vm is equal to L2R Intersection
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overall integers Vm is the trivial subspace
with only the 0 element These are not all
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furtherů There exists a Phi T such that V0
is the span over all integer n of Phi T minus
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n Point 4 in fact you know it is not just
span there is something more
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This Phi T minus n over all n is an orthogonal
set this is a deeper issue here Now we will
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explain in more detail the notion of orthogonality
in
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the next lecture but for the moment let us
be content to put this down as an axiom Next
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if ft belongs to Vm then f 2 raised to the
power MT 2 raised to the power minus MT belongs
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to V0 So for example if ft belongs to V1 then
ft Y2 or f 2 raised to the power minus 1 T
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belong to V0 for all M belonging to Z
And if ft belongs to V0 then ft minus n also
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belongs to V0 for all integer n So these are
the axioms of multiresolution analysis that
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means this is
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what constitutes the multiresolution analysis
And here we have taken the Haar multiresolution
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analysis to build the idea up but the whole
abstraction is that
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we can have several different phis and then
to end this lecture the corresponding psis
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So where does the psi come in it comes in
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what is called the theorem of multiresolution
analysis
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Given these axioms there exists a psi belonging
to L2R so that psi 2 raised to the power mt
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minus n for all integer m and all integer
n span L2R This is a very very significant
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idea in other words this is exactly what we
said yesterday Take dyadic dilates and translates
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of a function psi and
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you can cover all functions or go arbitrarily
close to any function in L2R as
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you desire We have built this idea up from
the Haar example but in the next lecture we
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shall try and build a little more abstraction
into what we
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have done and proceed further from there Thank
you