0:00:13 i okay uh so he's is like a for that that line a white per some like to introduce the problem and then explain at what the ancient base it that hard thresholding to them then it's use the medical experiments and this might don't uh one in compressive sensing basically what we want is to uh sample S sparse signals they if you the your measurements and we construct the signal from the measurements Y uh however compressive sensing system are not always immune to noise we always had noise in the measurement him and a a or that was you showed uh part uh most of compressive sensing says team systems uh well combine all the noise but you you change yeah i'd a noise in the signal domain or in the them of and the line you can combine it uh here just as a single time are so you can just model the system us being uh your clean measurements blocks on some noise term or um however uh what happens to the measurements are not yeah are corrupted by sparse of uh impulsive noise okay is oh okay so what happens if the and measurements are corrupted by impulsive or sparse sparse a corrupted noise are most additional compresses just in systems on only assume a gaussian noise or bound the noise contribution a few papers that i'd is this problem uh well really when you have impulsive noise just noise has infinite variance for very large audience at the really so it just breaks all the the uh they got the theoretical guarantees of the least squares based i agree that a here are some motivation here we have an example of a sparse signal and we take a if run on on measurements uh here is the same as sure a and i only up with one because a like all the measurements here are like the construction this is the to get sparse signal reconstructed from the clean measurements sure using on P but here using in a game that the same algorithm we can we are a it from the the measurements you can see that the effect of only one of liar is pressed well all the the components so it breaks all the some shows so really the mode duration of this work is to develop a simple and robust algorithm that are capable of of there a faithful reconstruction from these uh corrupted measurements a what is the solution what but we take a week um a to the rows estimation of the theory to the rescue so we find is uh log L norms is just basically a concave a a function is not really an on what is i we change the lease to make use of the L two make we change it like these of B and actually we focus on the particular case um when P E was to which is that low range and norm cases very well known in the image processing community a a this uh the range and has some cops sort some uh a good at that just the first one is that is and everywhere continuous function and it's everywhere differentiable uh the second is that it's convex near near region so four points that are uh that's a we dean gamma or within in this uh a more uh it's it looks like an L two more and the third and most important one is that a large deviation are no cable be penalized by this not much "'cause" you from this behavior a a large deviation or the lot concavity just makes that the these norm is not have we penalise so we use this norm in a previous work uh we modified the basis pursuit algorithm so instead of minimizing the L one norm subject to a L two constrain what we need is find uh the signal with a one L one norm we other range and constraint on the fidelity they are are we have this reconstruction T is however we have a problem with this as he was very robust to noise but it was really a slow and complex uh to solve and for large uh signals or for a large uh problems it was impossible to small with the memory what's out so uh we will now come up with a uh or at different them based algorithm so we start with these idea L optimization problem really the in our we fine a a at the sparse vector X or an is a sparse vector X that minimize is these objective function which is are and a lot H channel uh fidelity constraint instead of doing these which is i name this an B and M P problem or a combinatoric problem a a we use to adaptive algorithm our basic an iterative hard thresholding algorithm this algorithm just goes out what G is the gradient of the direction function new is thus step that is or a us so step that is changing it with the time and we it go changing here you can think of that as a gradient projection algorithm or as a latin whatever a a projection algorithm to go we tentatively in it and in the opposite direction of the gradient and then you project you your solution onto the subspace space of a sparse signal which is basically these operator H is is doing which is the hard thresholding operator that gives the S O largest components of your signal uh and set the other wants to see you he just like a uh that's a and to we should of what they wouldn't does a degree you know the and function can be expressed as these weight it um being are function on where why minus V times X the is the error L at iteration T and this matrix W is defined and it's a diagonal matrix where each component of the diagonal is just is gonna square over a gamma square a the error or a squared he is like a lot of these uh sort of weight so what does we is that a whatever is inside um gamma here is are and example we down might was one so what it since i got we trust all the issue ornaments that are or when there are is a within a distance of a gamma of the two of the two uh a signal or a parent that true signal and the other ones we don't trust than that much so we give a that's a a a little weight to those me short in as you can look at those may short sir what i be the ones that are both highly are corrupted so we end up with these uh the range and uh it hard thresholding algorithm which is but basically uh the same almost the same algorithm as at least squares based algorithm the only difference really is a we are adding a multiplication by a diagonal and matrix so in terms of computational load it's all the same as the at the part of the algorithm but now the algorithm used a a was against an impulsive noise a here here well each oh how some um um that's a a gone and use in terms of the restricted isometry property uh it's a set there was it that some property is just of the condition there was a right B is at exactly the same as the a these squares it had a T part of than algorithm and we get to these reconstruction bound the first and in the ever bound is a an error that depends on actually the the norm are in X and it goes to zero as T goes uh to infinity and the second one is the noise model this and so you don't is now we have a constraint or a duration constraint on the noise instead of having an L two constraint we just but uh come under ancient all constraint on the on the noise and we get these uh exponential uh bar yeah here are some crucial a site mentioning uh to lights are go uh the selection of got my to go here why because if got is too large then a you are not going to reject too much of liar if got my is to a small you are going to reject on most everything in your uh in your measurements uh we don't have like our theoretical right and D for all the proper solution of proposed image for gamma however uh it because we have seen that this estimator based on one tile a has work uh fine is just the at points spite a one time mine as uh the twelve point five can white what fun time and basically with sending this gamma we are considered that twenty five percent of them assure men's are corrupted or we don't trust that twenty five percent well we draws the reminding uh seventy five percent of the measurements now i know there a like is that is the selection of the step size um you at each iteration uh a the optimal want is to select them use that that uh that a a T the maximal that buttons in the opposite direction however that's the doesn't have a close solution but we select that like solving this problem is our reweighted these a square problem for mean the matrix W and then having the this is problem it has now a close form solution is easily it can T be easily calculated and uh do with this is that and with this new we got and T that uh the lower tension oh fidelity to turn is is is more or or at least equal to a previous iterations so we are going in and i know this end and direction here as experimental setup for what we have uh go here the first uh experiment this experiment is performed using contaminated a gaussian noise where we have a gaussian noise plus on liars and the noise here uh use the contamination factor it which just like the percentage of uh a liar in the noise that comes from ten to the minus three oh to two a fifty percent uh the this i and shows the performance of the these these of based iht algorithm and of course when we have a well a all night the performance just draw are the right uh red one is the performance of the weighted median regression uh a goody than use uh a very was is based on the L one on but as the noise gets more to the perform the case also he the performance of a a it at a different racial algorithm with two choices of gamma uh the that one is a using they got a that it just playing based and the uh order statistics and the blue one ease a a knowing a priori the range of the clean a short and so it you uh know a priori the range of the clean a short as you just set yeah a was that i mean has of course the one the the better uh performance however the performance of our uh are estimated gamma the still is good and it's close to the other one without knowing anything or any having any prior information of the clean um a here are some sample with up a stable noise again the curve of the same few the performs of i is this a square based uh i i used D a a is more i'm i'll for uh one think when i'll five gets close to see that we have a more impulsive environment when a flight was one we have the count chi distribution and with all white was to we have the gaussian distribution which is the class go like a uh distribution so um of course and you but the performance as a a very good and for the more than is not only rub boss when we have uh mm impulsive noise but is also wrote most when we have a gaussian noise so the but think about this nist is for both bold in light and heavy tail an environment a here is an example of a corrupted up to a measurement without the stable noise but now of and the number of me short a or the number of samples we see here this plot L the green one is with a five people what's your point five which is a very high impulsive environment and we see that of course uh we need more samples what to compensate for the noisy um a sure elements uh when i'll like was to we have the gaussian case and was you know the performance of the low range base algorithm it's all the same as the performance of the leases quite a based algorithm which is optimal for the gaussian noise so we don't wear not losing out too much and an option vitamin how have a final example here are we down any mention shall they not this is a two fifty six like to P to six image uh we take some random how to mark me a actually a thirty two thousand how a run or how to manage instrument and we it some a couch in noise to them a so here is the the the the of bottom one is the corrupted men uh we perform a construction of core what that with the lease court base uh i it at a different temperature should go to that of course that are structure is not really good what about a here we use a cleaning algorithm just leading the measurements being all the liars before reconstructing a a the still the performance is something that that "'cause" we're losing information and here is the with the same a sure and the record be that the recovery of the tension in to par thresholding algorithm he just to for comparison i plot of the recovery of the with the least least squares are you used at algorithm but in the noise this case oh okay so that's let me conclude now a a we have presented a a simple and brought most uh it that if a to than but rock was again i'm impulsive noise um the performance some properties are studied and we see that it's uh rebels against heavy-tailed noise but also in like pale uh environments uh one of the future work is to a well leverage in the prior information on to this are in prior information like prior or information or bought of based compressive sensing like a to people in rice are working "'cause" these algorithm is not suitable for all those more is not only the hard it or deep hard the that hard thresholding algorithm uh and thank you very much so question uh when when you assume the noise is imposed to be people are to this previously uh because that means that the the noise itself self sparse source and you can simply so so i is so i a fast to put two at least squares i H T you will meant to five with the identity density and so put the the the estimate you noise as well as you a sparse coefficients i would've thought that this would be uh a much fairer comparison of you have be looked that so yes i've of of a look done uh those a those than i haven't the don no comprise but using images what have done comparison like our for this impulsive noise in this characteristics for the contaminated gaussian on measurements the performance is uh actually on the same is a is just the breakdown point is actually in how sparse you're your ms short of the the the corrupted the are of course it yeah you you have uh less of liars in mature immense your recall already is gonna be better but as you go a a a a a for their i mean more than a fifty percent of the measurements are corrupt again you well your performance gonna drop could you to have like in of measurements to recover that is sparse signal which is like the same behavior a we have uh the breakdown point here the percent comes not just for that breakdown point for the algorithm go the what in the in your chance it's uh the performance are are also by go what very similar to to to this one only problem with that that it this could be like a a tentative a gordon's people so first like find the sparse signal at then the the core of the short men's on it or rate or some people just sold it just one L one problem and fine a a lot of them but yeah don't compare some put the object you to me you the you assume you could do the same thing with a the actually is not it's not the uh no one uh such as strict model is a well yeah yeah it's uh so you could do any it's again is plus and he's but it's also that i and and and just put the uh all the men's the uh sensing again the the coefficients with impossible is much yes the this sys basically right of of the the and almost particularly very in chill of is it's community and the use it very extensively at least still and to use so did you make do can persons also the student but it's very well known that it's good for me impulsive noise yes sir of physics and for sparse signals so in that put it in the framework of K press an of course remotes oh i i well i have a look at other literature from the your P six oh part really a we have only like preview wars in our room with the lower inch and norm in the compressive sensing variable like you in the well not only for the no is part but also for the that's uh that's so you sparsity encouraging um one but that's so the own thing "'cause" what i to be only that have only have a look at the image processing literature not but you physics literature i Q uh uh i have two questions uh is the first questions for these for in this experiment a uh you um you shoes the the the power of the the uh the noise but the we i don't know the the power of them as a men's is uh as is the um is being or was be a is a is a big as the the noise more or or or smaller than the noise paul right i mean this as a of the measurements even the scale of the measurements are are yeah yeah it's just this is more done done done than the north okay yeah for this case and second one is the four the um uh and the principle of the intrusion the um your your or on algorithm um there is yeah it's here uh uh for for W and the big W a is uh as in it's um a little bit similar to the algorithm you three to you really use do have reason where to version of an a or yeah you to read you uh reading rustling working a reason uh i don't know see what is the difference between this one the the and the are proposed proposed and them um to run with your writer ounce in the the of a grid to but then look at it can concert that you to reach you to reading windy a us wrestle to know mean sure yeah O okay