no to that i'm not going to talk about that a problem and that inverse problem uh i'm actually look at the much simpler problem but i have a perfect set i know exactly but am looking for i put the device in the water okay and then i'm looking for nothing basically i one to reconstruct the what okay um so if i you if i do it before sold december paul inverse problem but they're we'll get its this pretty shocking right it's like finding can walter um in the if you look more careful that if picture of what you see is that with a trace from green to blue we show a basically says that this a in the time of flight probably if we remove this are you know these delay of time of flight we can get to the correct picture do so and what to get is not quite that effect is a lot of uh fluctuations run sense and that this file suggest that the positions um or not correct we have to um estimate this position so or two is to go row uh about it we can send was back to the manufacturer ask them to put them correctly on the circle and um you know it a distance and that they would probably say well uh you know uh on the piece of paper ever these simple but guess what these a physical devices is the best it can so we are start but these do watch okay and then the goal is to find the positions all of these sense um uh so how can we do that um if you are you know homogeneous medium do the simple religion should be didn't time of points and the pay was the senses basically um through this a constant C zero so if you know the time of flight you know the there was sense and what do so than we talk about their what distances um because there is uh very nice there i mean out to a nine john you that says if you put and a a points on the surface and if you for this particular metrics distance squared metrics which is basically the pairwise distances raised to the square that's all only the rank of this matrix is going to be for independent of N it is very easy to actually prove this is always like to three long but um so we're not result how can we use it to is another celebrate celebrated algorithm called multi dimensional scaling which basically through this uh matrix L if you put it on the left and right hand side of this and scored matrix uh applied no single value decomposition can find the exact positions of the sense so uh when i say that you can find exact positions what i really mean is that a can find them all to rigid transformation basically if you only have a pairwise was distances there is no difference between your topology and the one that is reflected okay well the one that is translate there or the one that is rotated so um it seems that the problem is solved because um i know the time of flight i can't D do use the um the pair was is a can use M the S i find a position actually continue stock well apparently um i should and or a couple of challenges head of first which prevents also from using go this met okay first thing all you know the or in you know and oh joint to do signal processing and in signal processing not think he's less so well time of flight are actually not that the first and source of certainty that's not actually the big deal because um well and the F is actually a robust against a a not but the more interesting ones are come the second source of on L certain certainty actually structured missing in trees and what happens here is that when the transmitter here a signal the ones which are in its vicinity can not hear the signal this is one of the limitations of this what so all of these red lines are going to be you right we don't have this information if i put it in a mid trick for well you know what have this use metrics two hundred back to hundred so i can up put actually uh you know numbers here it to take all my so instead i put colours here is top you know these numbers so on the left hand side basically what you know what we should on the right hand thought however the um the central band and these two corners are going to be race and we put zero because we don't know the value another source of uncertainty is actually what we call called right and the missing singing tree what happens is that if you plot the time of flight with the respect to the sensor in texas what you should get um is this news mandy four but what you will get uh we'll have you know a couple of sparks which do not make any sense and these are basically down years you have to discard them so you put zero again here no um if i you list the game means that some almost um pair was these sense are going to be a randomly this on well we seen that are going to be a riesz random of like the model let B C okay now um so we all left be such a picture you no longer have all the pair was this and if that's not enough uh well basically you know well in a okay oh you have to wait um so in a matrix form what it means is that you know you will have a couple of dots you know randomly as yet or you're and if that's not enough well you have these on known like that in the beginning of the talk i uh you know oh i i mention so well the reason here is that um but these are electronic get right and uh oh when you fired the transmitter it's not going to transmit the signal immediately each weights uh for a couple of seconds mark second sir and that but we don't know this V so we have to estimate these as well okay um you know just to uh less first um sort with these um a missing entries forget about this that um shift and time of flight um um so we had these amazing result that this stance squared matrix was ranked for we could use M T S we can no longer use it because may of these that there was these sense are missing now uh the question is there were not we can estimate these missing in tricks well uh this is actually a topic of matrix completion that had recently you know there are a lot of risk uh that's been lot of activities in the past to years and the question is um you know pretty five this state here we have a rank K matrix of dimension and by and some of the injuries are random me and then it turns out that on the the road conditions you can actually find the missing it um the one uh so right now they are you know a lot of uh and is out when we started this work the a couple of them so um we actually use um of the space um um the develop point want to now already and his students at stand for and the way that uh um this algorithm works is basically by projecting the uh at the metrics on the space of rank Q mattresses and then uh doing some kind of great great in these now um do is a catch and them in all these out algorithms that i know um you have to use you that the in trees or you raise randomly okay um so probably do true before i guess um i you know probably you know from of for this problem but uh well to the best of my knowledge this was the case and now um so but as i mentioned we have this structured missing trees these are in trees that we know we will never get a any observations about right so um i to space is not going to work for as a T um so we have to redevelop develop again these all the space to make sure that a a when if we have a structure missing trees this is going to work so before like a you know all these error bounds for uh for the classical a to space is no use for a um um and um well the theory a a is actually quite simple and you can find it in you know paper um i'm not going to bore you with that you know details of the proof sets order sartre but uh let me just mention you know the model that use so we no longer assume that the sensors are are actually say sensor circle you seem that the R uh a on these and we we uh be a and the way that we are going to capture the structure missing trees are great are are but a bit through the use um uh a a and with that if there's the transmitter here all the sensor um in flight three kill or not going to your anything okay so if the sensors uh or you know distributed uniformly at random you sound was and if you see assume that um hmmm the time of lights are going to be a random with probability P fix number and for the structure uh missing trees if we assume that the are fine is going to scale like school root of log over and then our our or and reads as follows that the distance between the um this squared matrix and in its estimate is going to be bound but boy these two true um uh what we should mention is that um we we did a assume anything about the noise so the noise can be deterministic random um you know you name so these bound whole you need full generality um the all that thing that they should mention is that this term goes to zero as an goes to infinity all everyone we control controlled easter in many many cases it goes to zero but i'm pretty sure you can come up with example take doesn't for instance for go in noise uh uh that are now a prior we were not you know interested in a find a distance score metrics what you wanted to do side you know finding to positions but as i said we can find the positions of to transformation and we have to make sure that you know what and uh we we we basically one it uh uh one to define the distance between the estimate and the right one in a way that doesn't depend on the rigid transformation it should be in it turns out the right way to do it is basically these four um which is in barrie on the different formation and it's going to be zero these diff the ins distance even all if E X you equals X i now um if we apply and the S after oh the space uh we can actually bounded if then basically the same rate that B D before is going to be it's same expression okay no uh for the uh so we had another other source of uncertainty which was these to like these constant that to have to measure here we assume that is going to be uh for every want for every transmitter is going to be say okay now a there is going to be a need to about it and that's fine these um this T zero um but the for the sake of them are not be true the details of these out with M uh what they should mention is that is probably is nonconvex so um i it's very difficult to find uh you know to you actually out prove the convergence uh we have if you're if the with and it converges numerically or we don't have any pro uh and the idea is again to use this property of the these sense square metrics metric is rank for wanna make sure that you know what we're fine is actually going to be as close as possible to the right form oh okay so um unofficially we had access to real data what um oh i cannot report these these you know these data as here so what we did is just some simulations that maybe the characteristic of uh the real data and then uh um a diffuse basically you know well what you still before it uh you know the the a or is going to be in the twenties and D meter is the number of them or two hundred and then uh the deviation is going to be half from that are is does that was the D is going to be D the metrics to real matrix if this is going to what to to be able to have and the if the you um you know you actually do well our them uh a fee that there are going to be a lot of deviations uh um from the from the circle so if you got yeah that this is the prince function that it have that all of these sense of are going to be on the thing bill but if we want that are with them see that they are not going to be a you know they're got not be to be place exactly the same so this the last um if you the the picture that we started that of if we actually remove the uh do you lace you know these constant you'll lay set we have to find out we'll get if speech or are you it before if we complete the distance a a three with a a a space you'll get is picture it's not very different from the previous picture a but if you find the positions and then you know a sold the inverse problem you get back into one the it's really important to calibrate the system is really important to find position and the even if all you know beforehand the the um the range was from uh one thousand uh four hundred to one thousand six hundred before the close from you know these value this one then you don't see any deviation a so it eight towards um you yeah thank you very much and i'll be have to answer questions if true or german thank you for this okay and we have time for one question please i um we just some on the might like best aging and and was like given a set up and you make get that yeah so you don and i one writing differentiation between you time at like measurement and and just an eight station at the sense that the "'cause" what we found like if a code is getting good trying to fight management it "'cause" i things like multiple and more fundamental fashion um so i actually the you're not have a seen the time of flights measurements so these there is this yeah these guys they have these estimators for the time of flight right and we seen that what were they you know what where we got from the is actually correct as a how likely as i think you know uh uh what is the and that kind of flight measurements i i get to get good ones because you got a dispersive medium um no i actually don't know the yours K can i think you it seems out okay so let's move to the the second work no the to not be in me