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.