Yeah.
He says.
Thank you very much everybody
thank you Francesco for
inviting me here today to present well
I will present Today's just an overview
of all that is wavelet analysis and
I'm frequency and I was in general and
some of the application engineers some of
these up locations are classic operation
of wavelets and not only with
Letson some are more targets or
research and
some of these is some of the results.
In the past.
So I will present some of the results and
some of the technical.
This is a short i Whino my talk.
I will try not to worry too much with the
math and if I do please in the right me I
will skip a little bit that we we're
going to start historically from
the reason why wavelets are being you
know I've been developed in the past and
why they're used today and
why they represent a powerful.
Analyses and
out of position system for signals.
And I will explain what a wavelet is and
we will explain how he walks and you.
Construct the way of a transform and
use it and with some examples through
one of the it to the signals and we will
see some of the most important way but
types of Fortune we're going to be short
on that otherwise it will probably take
twenty minutes just talk about whether it
was a placation the noise in compression
and there were imaging
structuring money doing and
then some conclusions and for conclusion
I will present you some of that and
then see story with a theory so
let me start with tell you what.
Wavelets are women as are mathematical
functions their mother Mica functions they
behave like waves and they want to wave
it derives from French on the letter
it was coined by John I
think it was a physicist and
he was starving functions they could
do about that amicably to the signals
because he had the need to
have long seen only I for
consumers Aleutian and short signals with.
Low frequency resolution.
Wavelets ever varying frequency so
they change the frequency they analyze
the signal and they have
a limited duration and his or me.
What wavelengths do OK.
Wavelets mainly analyze the sing
a different level resolution we can see
what wave of Doesn't this way it
will provide or not in time for
Consider presentation and that's apart
from the classical theory a theory
that provides a constant fragments it was
Aleutian for for the signalling allies and
for your representation why
are they needed so why do we need.
For it always is very good and it's very
good represent the single they don't
change very much with what they have a
very fine structure they say but you see.
But for most of the real signal we have to
stand the signal to pray to the city and
most of the time we have time for
a quinsy complex time frequency behaviors.
Therefore we need a tool
that is able to adjust
to decode if they're sick of the signal
to give us the best representation.
Without being too much information.
Some of the fields of up occasion for
away OK a little engineering and
come from a different unit so
the first one comes to mind that
civil engineer mechanical computer
science communication is a geologist term
you can apply them a little bit
every way and that's what you're
seeing a prosimian all is medical imaging
structured on the nose in compressions by.
Representation system in the kitchen
clustering and we will see some of these
today but for a limited time we will
be able to actually touch hold.
So.
That you're not what we mean by station or
not when we talk about stationary we
have to imagine a signal a dozen changes
going to stick through time and
I have two examples that.
Those are these want to sound as
though it is its frequency year and
as we see you can extend this from
mining to Fiji to plus infinity and if
the signal is not going to change here we
have a composition of different things and
also here we see that because I'm fricken
sick of these signal is not going to
change so
we can define station audio more rigorous.
But in general the concept is that one
at eighty is represented by light.
So I would best represent a single doesn't
change with time well so well as for it
in you know seven playing that any better
the function can be represented as a C.
is a complex financial basically
I can take a basis sinusoid
sign it was one and I can combine
these bases the represent my sig.
These actually defines a transform and
well the transformed it takes from one
domain are present in another the ME and
usually we used transform because
the representation in the second the main
is better than the one in the first I mean
by that I'll bet it in the sense that they
enter the signal is concentrated in fewer
confusion and therefore I have more sparse
in the representation so I don't need to
keep everything by me only what's
important I keep only will simple and.
As you see here those intervals a very
famous another and talk about the but as
you see the signal multiply by the complex
as planets in any integrated through.
So these allows me to
define They say they.
Energy of the sea and I do that doing the.
Square of the.
X. of omega.
I for you transformed from a signal and
what I have is a representation of
the energy don't think I'm a signal this
case these is that you're the grandmother
I was factored into the signal that we had
before with three tones and we see very
well that these representation represent
the three tones very separate as it is so
we can recognize the three
phones are part of them I see.
But what happened were
real signals that we.
Are in our ward every day most of us
can we can see the new station and
so therefore they have kind of
these to change through time and
they have complex time frequency go
through six this is just an example I
generically it is a trip signal easily
no trip so frequency from ten to five
hundred hertz and as time stime
evolves the frequency increases so
what happens if I think the footage
transform of the signal and
I applaud it but this is what happens so
I have a flight line and
zoom in a little bit here so
you see the scale maybe you don't but
this scale doesn't change my eyes almost
flat you have a little bit of variation.
If I only have these representation on
my signal I don't understand that they
volution for a passage of time exactly so
but I would like to have actually is
this and this is like I'm jumping ahead
a little bit but this is a time for
a can sort of presentation the maps my
chair to the times I can see plain and
this is a pain by the short time for
instance for.
So some other examples reducing
of like for you to can see that.
So this is a response from 3. X.
assume either to a violent impact so
basically it is a sensor sitting on the
table literally and I am are down with so
that the actual response of this NG with
the full so you have a pic of me here.
I know it goes does it ever rebound.
This is a real signal is not stationary
it's almost impulsive in certain places
and he has a complex time for
consecutive days.
The second singer has a little particular
probably you never seen sing like that
before coming from this very moment
this is an explosion when we basically
record overpressure wave
coming from this board and
sort of this we were safe were far
away don't worry about it got hard but
as you see here well the happen is that
the pressure wave arrives in the signal
almost instantaneously we actually
measure the time the next when the signal
to go from zero to the peak is about fifty
microseconds in thirty microseconds this
in the rise to the peak and then decays
this point actually goes to zero and
what the F. there is that they came so
this is clearly no stationary This is
impulse of six and the last one is the
voxel time series from functional M.R.I.
scan on it's the thighs right so
this is a signal measured by when it
was an image in of of anymore there's
been an I set eyes on the table and
what we actually measuring here is
a blot responds to I am is a show
from the magnets and these are the direct
representation of out the brain
in the particle of all he is
using oxygen in the body.
So through the study of these
signals we can understand
the functioning of the brain
in this case of the rock but
also the research done on the rock to get
me through a slave to certain levels.
So this signals a complex.
Is not like the two one before but
for sure is no stationary and
he has certain characteristics for which
and I was the single to wavelets it leads
to to a nicer presentation right now so
let's see some shortcomings
of Free Transform why while we can use
a production while the free accounts for
most of this is going to
represent I know a lot of prison.
From the signals that we saw before
we had a local representation so
here these features are local to
the signal there are no global and
also some of the features
in this NG along.
The time information is lost so
I take the Free Transform and
the say I have information
about frequency but
I don't know where the information
falls in the time of illusion see.
And.
Most signals of a frequency
counter the changes with time so
that the signal evolves the time frequency
counting changes and we'll see that.
So we need time and
frequency of opposition so
our Can we achieve that one simple what we
can achieve that is their window signal so
OK we assume the signal station so
is changing but if I take a small enough
window maybe this NG of the full in that
window can be considered stationary and so
ninety four to six or
no vote Prize in Physics for
first time I think the composing of
additional But I meant a way for
they called time frequency and
those time frequency Uphams over time for
a solution and if I consider a solution so
we can we can think of a new formulation
of the Free Transform using the concept
of time frequency out windows and
these formulation leads to well that's
called a short time for atmosphere.
So in the short time for
transform we simply take the classical for
its own or I would multiply
the sing of these each function and
as each function is a window we know there
is a time and also a frequency code so
moving these windows through the time
domain we actually isolating different
portions I would signify reach up with
all the window we compute the freedoms.
We can then the finest battleground is
on them that the absolute value square
of them by of my short time that I.
Know that we had to find is
a representation there is another.
Question that comes to mind so are the why
choose my windows What's a bet size or
form of that when the for my signal well.
We know that if the window is too short
they're not going to be enough samples
the when I take the foot it runs for
at the results I'm going to get from that
transform is not going to be very high.
But I am localize in time.
On the other way if I enlarge the window
and take more samples and will The happens
that now I have enough samples to have
a more precise idea of what frequency
content is in that window but
in the same time a losing time look at it.
So the concept is displaying
over here when this is gumbo or.
And this is the time for can see plane so
a time frequency of them is just the box
and time for can see plane wave
spread in time and spread in or
can see in his lock eyes in a position
you in time see frequency and
once we actually choose.
The size of this box we can change it so
this box is feats for that analysis So
let's say you have a signal that as a.
Requires a constant time for Consider a
solution then this representation is very
good then they are put the does and I was
is leads to a good representation this
these days is represented here I have a
chair I can assume the same resolution for
all the duration of the signal and eyes in
this in this way it's to a tie a tie line
that goes from zero to Max and
want to describe my frequency evolution.
And other fingers were mentioning
is that we can not arbitrarily.
Choose a very high resolution
time a very outer solution for
this is because these are representation
by the as a Mercer principle so
it can be proved that.
The variation in time and
event multiplied by the.
Variation in frequency is always bigger or
equal to one of the two
yes.
That's where basically when you
start changing the size of this
box through your analysis you come out
where world is there were a percentage and
what will say that that counts at least in
the wavelength but you cannot do that with
these for Malaysia.
There is a evolution now that is called
The stock will transform the visit we
can be seen as a marriage
between short time for
a transfer and continues whether
that means more for which we have.
A better resolution and I'm from
a seaplane but you lose some of that.
You have more than a senior position so
you need to use much more information to
represent your signal that you will
do just with of course you have more
information so
it's a trade off is always a trail.
So as a summary here if we look at
well forget does is the vibe you know
your frequency X. is in beans of
the fine thickness of the guy and
if you look at gob or well Garber said
well I can do better than that I can
window it so now i have cost them
a box tiling on my time focusing
on.
My features are my signal require that
these block change inside and once my and
all this is progressing well I need to
do something else that is not enough so
when we actually think of time and
time frequency out them the changes result
Lucian to a time we have thinking
about the concept of the wave and.
The first wave let you know now this
is a it is a more formal definition
where the wavelength is but
the first wave label was invented by our.
Think Alfred art and it was a.
Piecewise function where.
You can take that function in
terms live in the way that and
what you end up with is of
the normal basis of of of or of for
your sick actually you have often
on the basis for it to a bar and.
To the proper use of
the bays you can actually
project a single door the bays and I have.
Those projections I visually
that wave liquefaction.
But over I wave it is a function
first of all is a functional
feel to it we call it the we
belong to the well to OF OUR and
there is integral from minus
infinity to plus if it is so
as is ZERO me as a uniform normal human or
Soviet Union or and
we can think about now this function in
Star Mother fine with the way that
if we translate it through time and
once we do that if you
if we do that properly.
We generate the dictionary
time frequency atoms and then
they were transformed is nothing less than
the in a problem between the function and
each time frequency out on their
I've by the translation and
and and
the dilation of this wave of function.
And it continues with a chance from
is the one that we see down the fine.
So we now we now have a formal definition
of what wave or transform is well.
Let's see the connection between that and
frequency most in general Alina if it so
we can think about their wave Latron's
form as an a product of the se but
we can also write as a convolution and
derive is a convoluted and
just need to change will be
the definition of the few of us
right in this way so
if I if I write these and then I take
that if nation of Fiesta bar and
I take a fit transform of that.
I end up with this one and now let's say I
want to vote away this quantity is zero so
this quantity is zero by definition
is zero so what does it mean.
And means that basically for it runs from
a function easy to four means that there
is no energy to see if there is no
energy D.C. I can think about that
function of the function present they are
shown in their wave in the free of the ME
as a band pass filter and and that's very
important because this is one of the first
that the recognized actually wavelength
can be implemented as a set of filters.
Very special few of those but nevertheless
fifty or so we can do a filter operation
to paint is a leaner permission
to attain at a presentation is
a millionaire.
So in the end we can think about the
official the way of a transform like a say
as a as a result of a filtering
operation between our our function and
about impossible to find by the particle
wave length scale and their position.
So more of these later we actually will
see out the evidence I implemented
physically but now let's think
about the time frequency playing so
our time frequency playing can be seen.
Henri can be seen by.
By it can be represented by this
bitchery or so a wavelength.
Larger the main.
As a.
Last Look at these issues in time
by as more applies in frequency.
On the other on the other side when I
take these way of life and I shrink it
OK Well they OK and is a moral
clause time look at his initial But
most spreading inference and
if you look at these and
we take all the delegation and all the
translation what we end up wit we call.
Time frequency play and as you imagine if
you do that continuously you have a lot of
overlap of these boxes so if you want
to implement is in the in a computer
with with the speed that
is consistent with.
Almost real time operation let's say
a prosperous and I was by you don't have
to wait hours for this and I was to finish
you want to you want to presentation
of this concept there's a little more
efficient that doesn't include all does or
than and to do that we need to actually
think about the bases as an often
don't know basis so we need to go from
a continuous representation to a discrete
a presentation and doing that we need
to actually build orthonormal basis for
that for this ng.
But before we do that I would
like to present sample here and
this is just me say child OK so
let's see if we actually played.
OK Well that was me and
say child now if I use the short time for
it once for this is what I of pain and.
And the read the read
portion is the time for
a can see I'm frequency portion
that contains i energy so
they enter the signal on my voice recorder
to my computer same child basically.
As energy to provide gaits through
the duration of the war and
from the idea I live in in blue here but
as you see these that is
the frequency axis there is I for
can see component that as
a line of frequencies that.
Progress through this so now if I actually
think the continuous wavelengths form and
I try to analyze this with the interior of
evidence for this is what I got that they
feature is quite different so we see
that the wave of transform in this case
is able to better as a way some features
on my signal in the time fitness of.
And that's going to be very usable for
some of their politicians were no ceiling.
So when we think about the script is a
show go from a continuous time to discrete
time OK and
it is possible to think about these.
This critize indeed exists the you and
that is and we do that with a dyadic three
So basically we say those in the can
only change is a power of two and
these shows does not run them but
is one of the choices they we can take
the lead this choice leads to a very
nice or the normal bays for for
my way of so now I don't have all the
overlapping that I hadn't in the time for
can see playing but those boxes lined up
very well and will sit out the line well.
So if I do that my
wavelength bases became so
now my indexes are not in our anymore but
are there on the Z.
so.
If I have to think about my signal of this
point and I'll do a project my signal in
the wave as I will I will take my signal
and I will divide it this way and
then a product again between my function
and the way of a basis are my way of
a good feature and so as you see my way
back when fissions av a J. index and
a cane the J. index indicates the scale
and the Canucks indicate the position all
functional.
So now I can do this efficiently
as Actually I would be go by so
and is the length of my sig and so
this is this is very good because I want
to commend is a venture into the S.P.
process or want to have it on my computer.
To understand how the different
level of resolution works
we need to the final this color moving
multiple solutions approximation and
multiple solutions
approximation basically.
The find our signal behave
between levels all resolution.
If we go back to the reviews equation over
here is actually to this in a product.
We can recognize actually we can the find
what we call the partial signs and
we call these the J D because it
comes from details at this point.
And we'll see that they
that well that means.
But I can write these and
I can type of these as the difference
between two levels or resolution so
that representation of a signal F.
at two scales and so when I do that.
Ultimately I can think about these
difference in reserve Lucian and
the signal in that difference
as the projection of my signal.
In a subspace subspace
the is in the five but
resolution a two digit and
if I do that I call the subspace V.J..
And I call my signal and I say my signal
belongs to be zero I'm defining a set
the suspect is there very pretty corporate
with is there and that's that and
they can define my signal and
of it all they spawn all at two o'clock so
we can see the V.J. they veejays our
method for every J. belong to Z.
the J. isn't going to be Jamie's one and
eventually does include two and
V.J. is always part of what you saw.
So now I can think of other thumb on
projection on the my function in this
one V.J..
And I called is F J and
I had it with Peavey Jr.
So P.B.J. F. is such that the difference
the norm between my function and
the projection is minimized so
it's ultimately the sense.
If I want to I'm a morning record
was the Phoenician of all the.
Resolution for submission is that's it
I've like not to say too much
on this part of the theory.
But one point I need to make is very
important is that due to the finish
of the mill to resolution process and
mission that was first the fine by my lot.
I can define another function fifty fifty
anything inherently relate to the way
of life and it's called scaling function
these function fealty to scaling function
is the function that were used
to span the subspaces V.J. So
he's an author of a basis for B.J..
It is a natural progression but
you don't have to they're up they say
now they're the future way
of Adak respond I spaces.
You know.
I'm not sure about.
I don't think I've ever
seen anything that.
Was possible now to define.
The difference between V.J. minus one
of B.J. it called these W J And so
is the difference us face the company
the mentor is space to be J. and
I can arrive in German is fine as a V.J.
plus.
W. J. in the same way I can
define a projection even Jim and
his friends in the project and B J plus
the projection would double take and
if I look at these I know that V.J.
minus one is a finer percentage because
I go one level Resolution five.
These is equal to the sum of
a coarse approximation I before.
Plus the tails so this is very
important because allows me to think
of my signal a different level resolution
and formally defined the concept
of scaling missing on
different levels of which.
So the projection of my signal to J.
I'm going to be.
Approximations and the projection of my
signals in Dublin Jamie call detail and
all see that is very important because at
this point I can actually build my way of
a tree so I can start from a point there
is my signal and I apply in this case
filtering the permissions to get
the signal of the next level resolution.
Out far the Y. go on the tree
it will depend on the signal
because that the latest the latest
leave on the street is the coarser
pursue mation on my signal
at that particular subspace.
Therefore.
There is a point where we stop and this is
up to us that the side where we need to
stop for that for that particular signal
there is there's not a rule the say
will stop by you the pain in the signal
if you have a long as you're seeing
what you need to obtain from then on.
OK I will glance through
it through a little bit but
through the properties of the scaling
function it's possible to define to field
the operators H N G H N G us
a CA that we the subspaces
V J N W J So whenever I want to
projection of a signal the same B.J.
I applied the field rage whenever I want
to projection of the signal in Dublin J.
I applied the fields are G..
Doing that I can actually
arrive there pursue missions.
As a combination of H. N.
G.'s and some of the properties of this
this field there is that there when you
get it pairs before you have an example
actually this is an example or
way of like hold up a sheaf or and
those of the field ISIS
say that with a wave.
So we see that the field for H. is a low
pass filter the field there G.'s and
i Pods field.
And some of the property.
Is of this filter are they
CAN YOU GET IT papers is very important
otherwise there will be complimentary
to each other and they need to
be because they represent two suspects
that are complimentary to each other.
So that I do I actually compute that with
a little small but now we see that we can
go from a concept of a space in
a projection in a continues the main to
a discrete domain and we can use filter to
do that well this is actually I would do
it we take the signal I fifty and
we pass it through doing a few of that H.
N.-G. that we just saw before a doubt with
a dispute as we need to down sample and
we need to do that to conserve
the total energy of hours.
Day output of the H. filter is them
passing and through a set on either H. and
you feel and he's done sample again and
you see I would go so that up with a G.
you will be my first detail
here that output of G.
of the second stage will be my second and
I need to hear out with of H.B.
of there is my approximation so if it was
really my way of representation at this
point is there an aide to the two and
the one and these represent a Also So
the information on this quick fissions
is an hour for me to actually go back.
And reconstruct the sick and the way of
a crisp with a single exactly the opposite
way you had to compose it I up sample this
case you know when you don't simply throw
away stuff well when you up
sample you have to have it was.
Because you don't know what you're
trolling for unless you keep it somewhere.
But you apply the complement filters right
at this point the complement of age in
the complement of G. you some of
the together your sample again and
eventually use the information
before you have your sing.
Their way of what to have a perfectly
structure of thought all wavelengths that
have come but support like the way of some
of the wavelets don't have comments or so.
They say they have infinite support and
time but they have a compass reporting for
us so
they have different got their states.
And questions.
I just want to briefly mention this
concept here it's concept of waiver pack
because I'm going to use it
later in one of the examples but
basically think about now
the Waverley three is the signal
the first level of a person may show
detail detail a person mission details
now what about if I think those details
and I started composing those that as
well what are you going to
do actually you're going to.
Top your time frequency plane
in a little different way that
you will do just use in a way.
But what they are allowed to for
you to do is take a.
Set of Ottomans or
leaves at one level of the tree and
the collection of those queer fish and see
here are two more presentation for six so.
A single computer presented by
the leaves at the end of this tree or
a optimal combinational leads to the tree
there are going to the site will go to
multiple nations these for
a particular presentation.
Of like the present you with
some examples now OK So
let's take the signal that I had before
this is actually well there's a little bit
of a variation there's a Doppler signal is
a Doppler so they frequency is changing
going from my focus is on Pluto to
a low frequency I am blue tooth and
I overlaps and so
there is some noise of a lot of it.
So this is the way of letting going
to use it looks like a wave and
is going to act with this signal using
that construct that we saw before so
the first level of the composition
is just going to give me this result
the first approximation still
looks a lot right this NG and
all you find out if you like later
on to continue reading about
this topic is that approximation look
a local like the average of your sick.
But the detail this Joe is
much more different and
it contains the variations in this case of
the I or frequency component of the signal
in this case is that it's and part of the
signal contained this portion here you can
see over here if I do that again I doubt
when I get that other person mission and
other tail I can do that again and I can
do that again and so they treat composed
by a four in all the details is my
way of a representation for my sick.
I can clearly see.
The changing of the frequency in
this portion of the signal and
be reflected in the details
here is a little
mosque by the nose in the first level but
it's very clearly their own.
And there is very little very little
energy left in the rest of the band and
the person mission is almost
all Opus it is a lot this
is a lot of of my original seeing so
it almost looks like
the original Dr Without it was.
So you can start thinking about is not
well you know if I think just to sing
a while I vote for Simitian is almost
like I'm the noise in this and
you can do that you can assume that's
true for some simple signals and you can
reconstruct the signal from one level for
summation and have a clean version of your
of your signature start with Usually
things aren't all that easy and
we'll see later on you have to do a little
more than just get at the noise signal.
But A to B. is ample now we have
a very big image processing so.
I can do the basic two approaches that
I can do any major producing are the No
using a compression.
Let's see let's see the noise it actually
actually is you first what happens when
I just to compose my image with with a to
do a bit so down here I have a to do with
it is actually the same wavelength I
used before but now I extend these into.
Duty space so
I have a sculling function here and
I have a waiver presentation for
each direction so I have.
Vertigo and I have oblique lines and
so the projection of these
to the space with a major will give me
a different representation of the image so
the low pass looks like a lot like the
original one by the skin and you see that.
This is the result of the down simply so
you don't sample an image you have a lower
resolution in the X. and Y. axis and
so you have a smaller image.
And then you have the details the details
are just I Pasco efficient and
in all the direction so this is.
This is what is on though
this is vertical invisibly.
And you have that iteration so
you see the tree in the styling.
So the first the first this is just
to level resolution the first level
resolution will give you the big
Fourth out when Lana will be actually.
Outsize and the second will
give you the second tiling.
In this portion of the figure over
here and then it becomes more.
Someway votes very very briefly are I
think about that our way of late is
the first wave that actually was
invented by Italy it was not a kind
eyes a wave like yet because there
was no formal theory at the time or
wavelets may wavelength and
frequency band limit
function as much as for.
And that they'll be she waved
a family that she very famous for
inventing a combat support with
a family where the size of the support.
Is minimal for any given order that means
as modern as other function that they use.
Guarantees the number of that if it is I
can take the function equal to zero and
once a said there.
At the beach she grunted that she gives
me a wave lay with the with a short
the support support but all that's
important because I can do I for
one second that there's issues with that
as as a change the older my way of what
they wave the Change One interesting thing
of the she is that if you take the BE
she one is equal to one
you have there are way so
really there are way of late it is is
the beginning of the machine with and
this is the way of late in this discussion
functional society I can go up to
I used to be she's twenty to twenty
five depending on the type one I was yet
to do and as you go I order those few
of theirs became sharper and sharper So
if you want to hide rejection outside
the band of the signal Well you trees and
I hold their wavelength when the support
is actually larger motherfucking
silicon ization these is better.
And vice versa.
So out of does I think I can
find something for me good but
doubly sure no doesn't have
a specific percentage there
I have there that I buy the properties
that you set for the wavelength and
the properties of the filter bank that you
divide so you up dainties wavelength by
their I Ving the filter
with Asians all the way.
Those are plot of the interactively the
one I did there in they don't have a close
for most wave I don't have a cause because
they're there I have to say after the B.
she that I've discovered this or there was
a problem with symmetry other wavelengths
as you see here there is there is no
symmetry and the next iteration was they
seem like there is expansion of the B.
She better get the symmetry and
so for Di I don't have all that but
you can find some of the references.
So I'd like to see very very very
quickly now some of the applications and
hearing wavelength.
We talk about the nosing a little bit so
when we have a signal they say my signal
is at fifty I can think
about the signal as.
Ody general signal Geo T.V.
this corrected by a noise W.T. and
whatever like to do I would like
to estimate F A T such that F.F.T.
mind you tease me much so
I'd like to eliminate the no right.
So if I think now of the way of
a representation I can represent
my wave like ids.
Let's now look at the indexes for
a second and
see this is the scaling profit
multiply going by that we have it so
that we have a just give me a set of
features different I've always pollutions
one of the good things are there wavelet
is that if I choose the right way.
And my signal as energy concentrated for
some particular feature of the wave
length or they come up with a spouse
representation so I have servant
with fish and they have energy and
I'll buy the fish and that don't
have to but they're very low now.
So I can think about keeping only
local fish and that actually I am and
that's the concept that's a concept
of threshold ing that way for
transport to just give way to the noise.
So the assumption is that your signal
will concentrate diluted with fish and
so on your noise will spread around all
the level of your little position so
you just keep those I
added with fission and
eventually you get rid of
the noise just the children that.
Are just decreasing the amplitude.
So let me take again their one
the signal that I had before and
when I actually apply wavered then always
in choosing a threshold Kleefisch and
the fine like these with Sigma is that
variance of my estimate noise I have
to have a way to estimate when I was right
and so once I do that usually a good.
Threshold is three sigma and
this is close to trees so
you can play a little bit with it to see
which one is better there are many ways
to choose the threshold but
basically the concept is.
The gorilla the noise and
that if I can start your second and
if you have this Doppler here with
the noise the original signal is in red so
one side the nose overlapped the blacks
it was my the nose version to
the original one and you see.
How good a job there with.
This is opposed to leaving a filtering for
for
getting rid of the noise I mean
a few to assume that the noise and
I frequency portion of a spectrum and the
low frequency portion is your signal so
you you you low pass your signal
get a read of that I frequency and
resulting in a low POS version of
your signal that's true if your no
is actually on that frequency but you're
always everywhere you also keeping some of
the know using your locals hard.
Because I build the signal like that.
Does the this is this is something I
actually give myself this was these days
was there was some kind of
there's that but then I was.
So sick and then I was in for to be so
this guy did I was fine and so
I took a line I hear and
I had some gushing noise through it and
now what I actually don't know is this so
I apply the same concept I I told
Michael fissions in that domain
I have an average version so
you see is not part of it right but if I
need to keep every African second pointed
out will end up with a virgin always
version days very much like so
the way we express quality for
image is very subjective and
depends very much on the application in
general you can use the Yes on but yes and
I wasn't thinking to find you.
Visual perspective.
So there are some indexes that.
Body quality based on the visual
property of the human system.
If you look at there is evil
What's life from the dinner was it
you see the day Gosh Mr So
there we have it is a very good job
to pull out the gusher Unfortunately I
can push this a little more and
remove some more noise Well you know
does that in some of the features in
the feathers you know at I'm
going to start the blind again.
And you're going to lose advantage.
Compression so all the way by
compression by compression we mean that
we want to present our signal and
its Causes possible to the radio but
we don't want to keep all day and
object of this so if if we represent
our signal in a trance from the main
that we do coding on this so
we keep only what's important
we can keep these coding
as a representation of my signal whenever
I need it I can transform it back and
attain an equivalent version so
what happens if I do that too
using wavelets is that I can I have
a presentation of my signal and that
retains ninety five ninety six percent of
the energy of the original one by only.
Keeping in this case six point five
percent of the Kleefisch right so
I really have a ratio of days like in
this case almost two hundred to one.
It's pretty.
Now of like displaying some of
the more advance apply kitchens away
is part of the research
I did in the past for
the collaboration with Professor Emory
University here in Georgia as well.
I was approached by the.
At a problem of isolating
portions of the brain response
of some of the rats that
she was working on.
Too and eyes which portion of the brain
actually out the of any given time and
she was doing that through
a technique called.
Seed based correlation so
you take a portion of a bra.
Look all the other portion of your brain
there actually walking like that and and
you either want out clothes or
go right to that then you put
the zero the one that I'm tickled.
While I say maybe we can do something
using with its display out these walks
in the scanner or in this case the rat
is in this kind of this is a lady there
alright but
what you get is a is a slice version.
A set those lies.
Of your brain are presented
their images of.
The blogs a response of the functioning
of a brain and given time.
And if you take just one little volume and
you plot it through time
whether you have is a signal that
looks more or less like that so
the dots of set for a for
a scan is a three D..
So you has two dimensions and then.
If you look at the contours think
of the signal this signal as
in frequency as the K. of these thigh and
so is along out a collision.
So if you look at the Tehreek long
out the correlations on the good for
some of the applications
needed to realize and
the code the signal so we need in Jordan
we say that this isn't why we need
in a position that widens the signal so we
can apply whatever transform actually and
they all took a relational function
does other coalition function of Oxo.
Will look like that up that
way little more so we look so
much better now than before.
Depositional or I though we came
up with it takes we have like we
fissions as a feature for
cost and so for any given time for
any given voxel there were Confucians
are computed and I don't want the cost or
use so we clustering regions in this case
with liquid fish and more similar to.
Other across the brain when we recompose
they emerge that we have from the scanner
and we call is the region with the I
live a correlation there with pain from
the class thing we have that
the results that we have pain
a very much resemble They not
only of the brain itself so
each one of these regions this color
differently our portion of the context of
the right brain and
this is probably the reason we are know so
me any special relation we can assume any
anatomical correlation between the regions
of the brain where just applying the
algorithm again in the results at this is
very powerful because implies that we can
divide different functions of the brain
without assuming that the function
is actually being implemented.
Why is this important because in
Britain is the functional going to T.V.
we would like to see out the really
different regions of the brain I interact
and when you know doing so
your brain is always doing something for
your wrist thing you're
sleeping in something that's.
So to study that you need to have and
understand the world regions of
the brain active any given time what.
Evolution through the time what frequency
are more prevalent through time.
Through these anomalies as we can build
basically our way of late three days away
of a packet tree and they were called with
a pack of functional clustering three of
different clustering and as we see each
one of these Leafs is cut out their eyes
by a center frequency in these regions so
that the first one from B.
from zero to one odds second that
first level decomposition source to
the second stage will be the first
year is zero to zero point five and
the zero point five to
one is out of the way for
packet they compose the frequency
spectrum of that of the signal.
As you see certainly
some of this cluster ing
results are not very clear
there is a lot though.
You know randomness in them but some other
is a very clear very well defined well
doesn't tell us is that the one the very
well defined are present in very specific
behavior that are at the core fragments
of range that one there are not very well
defined and more representing of
Brownlow behavior inside the brain.
Next that I want to show you what they
see based correlation looks like so
this is the result of them
by sea based correlation
different depending on the position of a
seed you can isolate different regions or
brain but you see that
the result of these regions is
is far from the result of divisions
do things and with a clustering.
And what can I do now with these
I can take two different regions
that got their eyes by two different
color so I say the blue region here and
the orange region and I can do.
What we call a coherent thought so the
queen is between the signals it is still
regions tell us when these two regions
are positively correlated negatively
correlated when their positively
correlated their working together when
they are negative or
correlated their working in that is so
that tells us something about
the function of the brain.
We know it is true I want to note is
a lot of problems that look like that so
we have a frank a C. band and
we have a recurring problem of AI a little
correlation I means that
there is almost like.
The father of correlation on to
correlation in the behavior of the brain
for the foundry.
And this is written states of the Mises
honest and if those eyes and or
doing anything they implies and also those
are calles found some of these results
there are similar to the result
obtained by correlation I think and
should use that.
There is a spawn time use of salacious in
the regions of the brain that seems like
to overcome the conscious
function of the brain itself and
he speculated it repeats itself.
This plot is attained taken.
The time signals from
two of these regions.
So as you imagine here each one of these
originally scan posed by a lot of Oxo So
OK so you want to paint a stime signal
that is representative of this region
you do that every job that I almost
literally that voxel in this region and
you do that for
two regions then you have to sit OK
once you have these two signals to do
away victory and then I'll just go into.
We did the wave liquid and I was it
with these plot is obtained by using
the stockholders there's a mix between you
can be seen as a as a trade off between
short time for
transform it continues with us yes.
We think about that in
this way yes it could
be interesting to actually explore
that little to see if we can find it.
Or presentation is most right.
And it was.
OK Next is ample I want to show
you in the last one as well
is equivalent applies to structure and
one.
There are several applications and many
applications I will say to waive that for
starch and monitoring from the diction
a crock that propaganda or
that machinery to normal
it isn't a pretty or
the signal from rather of of a cutter
to crux of the Filipinas structures
these was an application for a bridge
that is so we think about bridge and
we think about traffic passing to the
beach reach those bridges are built for
lasting forty to fifty years so they
are not there MONTAGNE through time but
there are no replace because
it's a Stanley expensive and
cumbersome to replace a bridge or
you have to close the traffic for
quite a long time I want to have
over here these two pictures is that
they the collapse of by fifty five I
think I was in a common two thousand and
seven when that happened and
we actually went past to study.
Some of the some of the data
come from this accident and
this is the comment I will be
that bounce of reject leave
many of you probably have seen the movie
of this bridge just bouncing like a.
Rubber rope Devils because when they build
the bridge and in think about the resonant
frequency of the structure and the wind
as it's excited those resonant frequency
until the bridge collapse we are very
likely to have the movie and the pictures
from this event because a photographer at
the studio just on the edge of the bridge
and when this was up and he was call and
he put his camera there and
he made the movie and it to be so
we now have the document.
But let's think about structure and
let's compute those are celebrations so
isolation from a uni X.L. So to me
the place on ought to be in this case and
that's me excited in the being with
the hammer so am I marrying on the beach
in the first case is the beam is pristine
the second case there is a damage
well do you see any difference between
these two things I don't I mean my high
doesn't I mean there is a difference
really maybe a little bit like there is
some energy if it is not but
you know there is nothing to tell me I pay
the second one is damaged the first one
is no but I know because a simile this.
Actually does that are from this data from
a simulation but we also the this part.
Now lets say I think that we have a packet
transform and I compute the energy for
each wave a packet can be
defined very simple like that.
So this is the normalize energy and
I plug anonymize energy for
Mark Weiss oration so
i also the distribution ion of G.
through the back of the composition my
signal doesn't tell me very much but
if I compute the difference all of
a sudden what I have is a spike and
this spike just into five
the location of them.
So the packet the very good
to the text father of the.
And well it is in general for these and
I was with a wavelet from the family
it was a way but I think twelve that she
thought it was a good compromise between
vanishing number of money moments and
support of the way.
And this in this case the X.
axis is the number of the element so for
each element of a fan
I'm missing Malaysian.
You have the output of.
A virtual Salumi you take the output
to compute the wave I could transform
from the damage and the image right and
then you compare that you so
that way the difference of the two
actually results in display.
Right.
When you don't really you in real case
you have to have a set of assorted So
let's say you covering a being right
I mean a solution is the bean.
In a real case there will be engineer
the Goes inspecting the structure to first
find out if it's them and then if you
found some damage is the voting then
is this structure an instrument if you
want to if you want to skip the step or
you want to streamline the structure
before it's damage then you need to think
about distribute sensor networks of
sensors isn't bad in the structure so
you have to assume that is either present
of the structures being deployed them and
if I deploy that after I deploy
kind of a dance dance and
they tour around the location I'm
interesting to order right and so
that output for each mission for each node
isn't it or it can be process using with
the concept that we are they were
representing most impulses here.
And that.
Is the is a reaction of this
difference of overlap and.
Because if you think about the propagation
of the waves inside the structure
it's kind of a god the wave right is
a being so there are other walls and
there's kind of a uniformity
in the center so.
Vibration from one side by get this
through the damage is just a crack right
and only doubt it in the interim would
use it so that no man added to the crack
creates a reflection that reflection
increased and I do you see that but.
I.
Guess we did I did that yeah that was that
was my dissertation there's a.
Gosh I have to go back and see that I and
I don't remember to my mind was that but
it was Leno's as with roping
three other men I think.
Yes Then I get five thousand
omens that damage was
luck lies through a gushing
function in a certain position and
it was spread uniformly in that X.
Y. and Z. order and it's not.
Yes Yes I wasn't Yeah I think I think
you're right but I wasn't using any
I was using and only.
You modeled it in that case you
modeling the reflection at the boundary
between two of them it's right that you
can not doing what I did You cannot
mold the reflection inside one out and
one on me supposed to be uniform so
you have this continue to be you have a
mist and the discounting we spread through
the borders of several islands let their
idea into each other that's a very simple
way to do it the most more complex way to
do is to actually some weight variation
confuse variation in the properties
of that it's so there was
a variation to broke within the stiffness
of those islands but it was finite for
each other it was and
you give our inside that in itself.
Doesn't answer OK.
The next step will be to think about
a bigger became so well that means
when you have a sensor night where there's
a really been deployed inside the beam and
you have somebody that drives like a truck
so we do this sort of a spare human.
And I myself was actually installing
assume he was on the needed back
of a bridge and
when I ate in we were possible by
the displacement of the back it's
thirty to forty centimeters and
you the first time you thought but
you don't expect that much and
it's a lot of energy that goes for
the structure in that one and
there's a lot of the space in the energy
through to the structure itself so
we thought the more complex supplication
So we know that these these.
These.
Signal that is generated by us and
by the tracks is and was the surest signal
and it's very hard to maul because you
have to know the source of them all that
but you come August across that way so
you can assume that the signal is
constant and certain bands and
it changes Zaldy if you take short
the way so what we did with the compose
the signal we use and where the packets
and we do being for me looking for
more does is focus is there up with
from several sensors to one coherent and
so each op with of of each band that goes
into five there with the packets will
then condense into a beam
format the informant was about.
Ultimately to do that but this is ample of
the being bothered by them farmers that
we use and if we do that and we repeat the
difference between them much of that much
what we have is an energy amount
the basically defy the damaging
not only location by out so
frequency and angle of the form.
So we arrive to the simulation of
these were never the ploy because
the final implementation require about
three A.M. there are three hundred fifty
sensors that we can get authorization
by day a flow to the bottom of this but
they should to go in stealth so many
sensors underneath operational bridge and
I don't know where there is or
stand at this point because.
That was when I got away.
So those.
Luol the box up with its own
way of life and with analysis.
For a conclusion of like the present
a short story a way of life and
I'll be very brief year for
you if you know seven are nineteen or
nine in forty six Gabo or
seventy's mode and
eight is gross man for money is a sure way
to transform is very important maybe you
are going to have a basis I think it was
the first eight is the wanted to be she
did really expand this field and now a lot
more data is ocean pursue mation describe
the evidence from casket I'll grant
the terribly we'll finish the banks and
color from a every cow's or wave packets
and this is there's no the complete
history there's so much more that for
reasons of time we can show and
those are some of the faces of the people
that I mention through this talk.
And those are some of the research
is that you can actually go and
dig out if you want to know
something more about wavelets and
those are also the resources they
use to prepare these presentations
those are some of the books
that headline writers.
Thank you very much for.
Thank you.
And I believe on time.
I think about that I guess you
can if yours are they know what
the damage is going to be so
I guess you can you know that I point to
stress from this article in
the particular join approach.
I know exactly sure I was dying.
Though to have I have a resolution on that
mash and to simulate a lot of other things
mostly when you have a complex German for
me it was easy it was just being so
like a rectangle but
I go on and divide it but
if you have to give a complex
structure they have to.
You have to account for the curvature
of the functions around the curve
usually demagogic diction on.
Our space structure is done through.
He was sensor so that sensors on the
surface because you have metal when I was
a composite it's in these implication I
was actually working with a concrete so
that the propagation inside the beam
was uniform if you have a composite
finally you have a repetition of
different frequency different levels so
you have to account for
that there's actual level.
Thank you.