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All that we ask for and that is not too unreasonable
is that the function has finite energy So
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let us at least put that down mathematically
What we are saying is we shall focus on functions
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with finite energy And what does energy mean
Energy is essentially the integral of the
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modulus square So if I have a function x of
t the energy in xt is the integral mod xt
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square over all T and this needs to be finite
all we are saying is this Incidentally this
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quantity has a name in the mathematical literature
or for that matter even in the literature
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of wavelets
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The energy as we call it in signal processing
is called the L2 norm by mathematicians And
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you know it helps to introduce the terminology
little by little from the beginning because
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if one happens to pick up literature on wavelets
these terms could be used So let us introduce
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that notation slowly So we say the L2 norm
of x is essentially mod xt square dt integrated
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over all t and to be very precise this needs
to be raised to the power half Similarly one
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can talk about LP norm you could talk about
of x
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And that would correspondingly be mod xt to
the power P dt integrated on all time and
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raised to the power 1 by P And of course P
here is a real number So for any real in fact
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real and positive You could talk about in
L1 norm you could talk about L2 norm you could
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talk about an L infinity norm what would L
infinity norm be let us take some examples
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What would an L1 norm be It would essentially
be integral mod xt the L2 norm we already
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know What would the L infinity norm be
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That is interesting so you see in principle
it would be something like this what now is
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this What do we mean by this You see as P
becomes larger and larger what are we doing
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we are emphasising those values of xt which
are larger So for a larger value of P we are
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emphasising those values of mod xt which are
larger And as P tends to a larger and larger
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and larger value as P tends to infinity we
are in some sense highlighting that part of
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xt which is the largest
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So in other words the L infinity norm of x
essentially would correspond to the maximum
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or the supremum you know the very largest
value that xt can attain all over the real
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axis So it has a meaning even as P tends to
infinity Anyway this was just to introduce
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some notations which we are going to find
useful And what we are saying in this language
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is that we are going to focus on functions
which belong now here you know we are going
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to start talking about functions that belong
to a space We say you know we say the space
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L2 what is the space L2
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L2 is over the real axis it is a space of
functions and it is a space of functions whose
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L2 norm is finite simple Similarly you could
have the space LP The space LP is a set of
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all functions whose LP norm is finite Now
the word space is used with an intent You
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see space really mean that if I take a linear
combination of functions that set it gets
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back to a function in that set So if I take
any finite linear combination of functions
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in a space LP the resultant is also in that
space in that set LP and that is why we call
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it a space
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So LP only LPs for any particular P are spaces
linear spaces they are closed under the operation
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of linear combination So
in other words we are saying let us focus
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our attention on the space L2 Now what we
have said in the Haar analysis that we talked
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about a few minutes ago is that if you take
any function in the space L2 I am in if you
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are adversary picks up any function in the
space L2 and puts before you a value e0 saying
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please give me an m so that when I make the
piecewise constant approximation on intervals
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of size T by 2 raised to the power m my error
square error is less than e0
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The proponent is able to come up with an m
which gives the answer And this could be done
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no matter how small the e0 is the proponent
will always come out with a suitable m That
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is the idea of what is called closure You
know so what we are seeing is when we do an
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analysis using the Haar wavelet in other words
when we start from a certain piecewise constant
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approximation on intervals of size say 1 for
example and then bring it to intervals of
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size half 1/4th 1/8th 1/16th as small as you
desire you can in principle go as close in
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the sense of L2 norm that means if I look
at the L2 norm of the error between the function
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and its approximation that L2 norm of the
error can be brought down as much as you desire
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And in that sense whatever the Fourier series
was doing after all what does the Fourier
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series do it allows you to bring the L2 norm
of the error between the functions and Fourier
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series as small as you desire for the reasonable
class of functions For a wide class of functions
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give me the Epsilon give me the e0 and I will
give you certain number of terms that you
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must including the Fourier series So the adversary
says well here is an e0 for you the proponent
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says okay include so many times in the Fourier
series and you can bring your error down as
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low as you desire
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The same kind of thing is happening here the
proponent-adversary principle Now this is
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a deep tissue that one function psi T is able
to take you as close as you desire to the
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function that you want to approximate And
by the way this is only one psi T which can
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00:10:05,650 --> 00:10:14,670
do it The whole subject of wavelets allows
you to buildup many such psi Ts Here we had
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a good physical a very simple physical explanation
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00:10:28,630 --> 00:10:42,460
We started from piecewise constant approximation
we said well when you want to refine your
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piecewise constant approximation you could
do it by using the Haar wavelet And this you
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could do to go from any resolution to the
next resolution Please remember here we are
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increasing the resolution or improving the
amount of information contained by factors
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of 2 each time And that is why we use the
term dyadic let me write down that term dyadic
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So what we have introduced in this lecture
is the notion of
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a dyadic wavelet and dyadic refers to powers
of 2 steps of 2 every time The Haar wavelet
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is then example of a dyadic wavelet and in
fact for quite some time in this course we
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00:12:55,720 --> 00:13:05,501
are going to focus on dyadic wavelets Dyadic
wavelets are the best studied they are the
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00:13:05,501 --> 00:13:16,040
best and most easily designed they are the
best and most easily implemented and I daresay
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the best understood So for quite some time
in this course we shall be focusing on the
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dyadic wavelet the Haar is the beginning
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I mentioned in the previous lecture that if
one understands the Haar wavelet and if one
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understands the way in which the Haar multiresolution
analysis is constructed many concepts of multiresolution
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00:13:41,260 --> 00:13:47,620
analysis would become clear What we intend
to do now after this in subsequent lectures
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00:13:47,620 --> 00:14:00,360
is to bring this out explicitly So let me
give you a brief exposition of what we intend
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00:14:00,360 --> 00:14:15,720
to do in subsequent lectures and then we shall
go down to doing it mathematically step-by-step
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00:14:15,720 --> 00:14:32,810
You see we brought out the idea of the Haar
wavelet explicitly here what is the Haar wavelet
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00:14:32,810 --> 00:14:52,430
we know we know what function it is and we
know that dilates and translates this function
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00:14:52,430 --> 00:14:59,800
can capture information in going from one
resolution to the next level of resolution
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00:14:59,800 --> 00:15:12,500
in steps of 2 each time Now how is this expressed
in the language of spaces after all we talked
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00:15:12,500 --> 00:15:20,650
about the space L2R L2R is the space of square
integrable functions So how can we express
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00:15:20,650 --> 00:15:32,780
this in terms of approximation of that whole
space so can we express this in terms of going
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00:15:32,780 --> 00:15:34,919
from one subspace of L2R to the next subspace
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00:15:34,919 --> 00:15:41,919
And in that case can be expressed this Haar
wavelet or the functions constructed by the
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00:15:41,919 --> 00:16:04,630
Haar wavelet and its translates and perhaps
also dilates in terms of adding more and more
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00:16:04,630 --> 00:16:20,689
to the subspaces to go from a course of surface
all the way up to L2R on one side and all
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00:16:20,689 --> 00:16:38,660
the way down to a trivial subspace on the
other So we are going to introduce this idea
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00:16:38,660 --> 00:16:44,520
of formalising the notion of multiresolution
analysis
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00:16:44,520 --> 00:16:52,679
We need to think of what is called a ladder
of the spaces in going from coarse subspace
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to finer and finer subspaces until you reach
L2R at one end and coarser and coarser and
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00:17:10,350 --> 00:17:26,510
coarser space until you reach the trivial
subspace at the other end Further we are going
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to see that the Haar wavelet and its translates
at a particular resolution at a particular
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power of 2 so to speak actually relates to
the basis of these subspaces
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So we are going to bring out the idea of basis
of these subspaces and how is the Haar wavelet
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captures what is called the difference subspace
in fact the orthogonal complement to be more
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formal and precise simple but beautiful and
what we do for the Haar will also apply to
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many other such kinds of wavelets Let us then
carry out this discussion in more detail in
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the next lecture where we shall formalise
whatever we have studied today for the Haar
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wavelet by putting down the subspaces that
lead us towards L2R at one end and towards
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the trivial sub space at the other thank you