from a or mixtures a why is important uh so that the all source a solution of E D who was my uh but students clive who is uh was not a research assistant at imperial cyrus with is some that uh let be on from from france and and myself uh hmmm not a million vectors so click deadens a a uh that a very much used in computer graphics animation but does stand how however really in signal processing does till consider that a little bit exotic and the there's many you reasons for that so that has uh a a division algebra a which is a us skew you feel all there are four skew meaning that the uh a don't product is not computed and the use been used basically matrix manipulations a computer graphics because they provide that affect rotation and orientation so in many uh movies now you have a uh start doctors will perform some scenes and then based on those measurement this was the actor effectively N and and we produce the the scene so usually using with that was for that how this is only ticks manipulation so in adaptive signal processing we need to look at many other things like gradients uh the so called cool lot T N such like which all touch upon and we can face many many problems uh in this designing those things and is the there's been the recent these surgeons so good to don't while processing read several groups working very actively uh at imperial at the sent and there early on and and such like so several cost of is that uh uh are now we getting extended to quote that times the concept of set a lot or or sort so called of clarity but D distribution of signal is not rotation invariant the second thing is uh a leading directly from dies so called widely linear model which is the only model still able to deal with second order so but non circular data so called improper and just said problem we faced was the good at is the in fashion so if you are familiar with the modeling in the complex domain you would know that the gradient of a real number does not exist so typically our cost functions are are powers they real however to design say uh i i mess that the log or deals with really you need to perform from the derivative or can't of that deal function the cost meant does not a lot to do that and you need to do this or the so called looking at and and the does of the many ways but uh you which you can use to a to between functions in a two and fashion and C to introduce a called C are calculus well you can you can uh uh a a the and is for a special class or or the old function and those have functions of a complex number that and Z conjugate so we are going to do a very similar things and the with that and the men uh so that this talk is really about blind source extraction not separation and the glass as extraction i think of the paper i uh i think was the first was by build force and i'm not good with french names so not a little force was the first author and i have a i have a lower that's only quite a lot of about this method from my only and i'm what's skip from reagan and to get that we can do develop several algorithms most of them in the complex domain using so called augmented complex statistics to deal with the noncircularity of distributions and this is a a time now to extend it extended it to the quaternion them and okay how help contents that ends uh when we think about a new technology and so more and more we have facing so called vector sensors so instead of having i i don't and that a or a single univariate to single single-channel census we which we then combine some howling to matrices how about those sensor being multi dimensional and this is happening very much many new technologies a based on vector sensor so the says and all mention is the and then meter so this guy did him measures wind speed and that actions symbol and this particular one is the by your instrument score collaborating with S and it have some out sonic roles and this one is two D so you measure simultaneously in speed and that so it's a vector sensor if you would like to measure in three D you have those problems on top as well second second two sensors for the work to did now with the elderly or unsure a virtual reality the uh what these sensors so that's P D national what motion uh sense so you measure simultaneously exploits said position that was and so on and if you stick when you of those two are person then then you can look at their gate the but it of into to full or you can do one animation for movie uh so because uh i i but it much in processing so uh here's what we do so you this is a bit far that a typical we far and you can read their minds which all over the place and this bowls with those kind of crown this some top uh three D i'll the sonic and of meters so what to are doing you go trying to model a a boat optimal all remote control oh do in farm and also to provide prediction i gaze then the guests or any well vibrations patients any problem that can break their buying by modeling green using a vector sense so are are either had in the been modeling with have many many uh results uh a going back again to to this uh model or and going to what's movies so if you have a style doctor or anybody skilled performance some second that two seconds and such like and you can synthesise a hey i like or or you can see so but it wasn't actor here so for those of you who are familiar with computer games this comes from be assumed right or which is from nineteen ninety to six so that night S a six was the first time that could that employed in commercial uh uh adventures and uh if you remember the difference between this game in anything before these fidelity on movement was and not mostly better that's someone applications and now i'll be we get everything short and go go to towards my a separate to my talk uh what and that's the to a division algebra so the only division algebra is a reels complex what that neon and an could onions if you have a read the uh new scientist from two weeks ago terms two pages dedicated to look at all neurons where the thing that the string is your in anything a can serve that can be explained but it coming in to but then do was a very but in wonderful fields the problem is very simple you have to sacrifice some doing so if you going from the young two complex numbers you lose order so complex field is not order two but two plus two J smaller not bigger than one was J it doesn't exist go from the complex will look what that means you lose uh common activity of the product so X times Y is not cool white time sex and that in poses many other problems for instance you have left and the right eigen values and the the uh right hand i i can analysis is now established how other but these still no got to deal with left and i can as and so on so i'll just give the sketch of the proof later but are not prove it a would by could to know it is safe from from C and i guess say that we need to rush uh the i don't think that these interesting think is the so called in lucien so what ten in this though right for four minutes that that and the because of the real part and the imaginary part of V consisting of these i J K which are was uh imagine in numbers and unit vectors so the difference between a are inc that neurons he's in the other for i J K and so on just unit vectors uh the the not in the axes but as in the code that in them and so imagine number so we have to be a careful about the or because i J equals K but J eyes minus uh again so if it's in why uh to do not a are so let's think about the problem of rotation so if you want to use the the perform rotation using all around that is in a three then you would have to go to get a lot X why and and is a which is someone that giving this matrix so we have to it three times in the quoted and domain because it's a compact for a for it's or three this is a a a a that known in in it all the four and this is the rotation okay so the benefits of then if you are performing and any image related problem or any track and you not the on and such like then this would be typical trace a a fight coming from those all are matrices X Y Z as in the with that in the menu just move act like that there's not problem yeah the thing is that division algebra allows to to do not to get stuck in so called given by law so for those of you are familiar with gyroscopes the usually have three axes so one second and so and they wrote it so it is possible that to of the axis coincide if you model the problems in the are three so this is a physical a scope but this is a but today interpretation however so you losing one degree of freedom so if you member the up paul eleven at some point once spinning in one lane because the charter school but fault it was in the model local that time this cannot not happening that and of course than the story but to need is much smaller because in the real but been not for the need for covariance matrices for a uh for for the dimensions and then six so the covariance it's made in the with that in the menu need one covariance matrix and streets of the covariance method so setting but got very quickly to two to something to my talk uh this is a key slide and from then i'll just go to application uh is a whitening at modelling so i'll introduce use it by thinking about why building at modeling in the complex of man think about the standard mmse estimator uh you want to estimate these signal Y in terms of the like a sir and so uh uh are mutually in the band then then the optimal estimate is a a linear model H is a coefficient vector and X is that a good as some vector now in the complex the people usually assume that is still holds but you can look information here you don't have to have a mission you can have a transpose it of coefficients are just to as much how are think this way so well D complex number of has the real and imaginary part and both of them are functions so the really much and a part of this and uh but X are is X X to get or two and then X imagine is X minus that's going to get or that's some minus missing some way so that effectively why are are is a function of X an X to get and line might in it is a function of X and X conjugate so if that mutual then then D a linear model becomes that a be linear model expressed that and i miss this is G this some other of G not and this way you can capture the complete second order statistics in the complex domain so anything you develop so far in C is optimal only force local satellite data right D distribution is rotation invariant but all the real data are noncircular and it's very simple then then you can extend it to the quaternion domain this way so can for channels and for different a we that there's and this is the co or it vector to which comprises you V G so i'm conscious the time of just got so clarity so this is quite a got fact affected by by stuck a lot i was considering we and and this is a polar target of in speech so clearly speech a much stronger for some that ashes she's that from the others and i was thinking of case we is a complex a back to because you have intensity an and so it's to the complex and i was trying to bottle simultaneously both but is also we're not so would for some or G for some of them the perfect for some that were not cool that i discovered that this is because they start that modeling in C assume assume could a lot of two of these distributions so as if you have a circle here for data for non circular data and all of them one non circular because the been at the side was either there from the C to the C and so on if have to use of white linear model and can some some benefit oh so we can develop no one you good D and for for uh for a quick that and since a conjugate gradients so please look at the i-th i-th was composing letters uh john two thousand then eleven and then a few elements coded in uh lms was developed in two thousand nine because this for because many false this is one of the form uh then widely linear quote an anonymous as this for and to show you that you widely linear model works better than the linear model for sale noncircular read so there's laura jean high and medium regime but the good airline every time out performs the the black line meaning the widely linear model is ideally suited for that so this is my top i'll skip all the mass so we caff this blind source extraction diagram which most of you are familiar with that there's these separation matrix W doesn't mean why but a low you like run a predictor which in this case you want to be near if a prediction that are which and tries both these set that and this of the optimization problem so we need to more a a you and be here sort of the type of and i'll show how this course for E G you could that work out how uh this simplifies actually because that at all uh you can work out the how the sorts uh uh these in the paper so have to result one is for a synthetic sources but have to non set of close the source sources which of both synthetic and then looking at performance index but that the the extraction weight or is that no "'cause" one element or not and the second is a work E G so this is the performance index clearly if you lose a we use white living at what didn't in predictor or and the sources is that in sources of us then the performance is really good but as for non circular with that don't date that the standard what a more of the i don't model was not sufficient and then the think it was that in but computer interface and also and then E G research you reject blend your data because that contaminated by active artifacts and does that if that's not only come from the power line and from the eye blinking if you think about it because two lines and we of horizontal and vertical move so in the complex domain say is on than of a vertical movement can can be model but it would be uh course on than but for you O G is a complex number you to wise and an order L complex numbers as a with that so why not develop an algorithm in the group and the main that extracts a a moments both from the right and your left i and we did that for various scenario but people building all they are as a a a a nice i've row such like and the obviously you have some group so all the from tell channels but one put that neon all do medium channels second quaternion and all the uh the or channels that the that the and you can see it on be on the blinking uh and the you chi use channels that obviously because the sources is close to the front to that then the front of channel so much more affected by active then say the channels as the back and such like we use the for you G channels as a reference so that are not part to the mixing model or and else it just use them uh to combat performance of our model beat the out it and here the results so this is how we did it uh so this is the power spectrum of the original signal and the artifacts to folks i blinks which are low frequency and fifty hz mains and we perform extraction wipe one one so why would extraction because in by a data often you have many sensors so in and me G U the one hundred twenty sensors so of mixing matrix is one twenty by one twenty which is more than ten thousand quid fish so need about one million course which to convert but as you interested in only one or two sources so why not have a much smaller model and extract only source is based on sun property that we can control like sparsity or a predictability and such like so in of uh less and diagram uh we first extract the the a line noise in the one living at the and standard the main was also able to extract it what well however when it comes to the i blink so used the power i links uh i i don't know how fast a complete so maybe three four five hz so that here so clearly the the the the widely linear extraction now to was able to both uh extract eye blinks from the mixture and so press the uh fifty hz interference whereas as the standard al gore was not able to do that so it's a also the blinks and interference in mine but very high and at this point i just the want to show that there's somebody many is going on special about the diagonalisation of those made it's is in two make any convergence proof and in conclusion a a quaternion so it's that a natural to work in the good then in the man and please of interest it web set or talk to me and my students well that think are to match yes E E S okay i'll a very quickly uh if you look here uh i'm i think a much time to talk about simulations so this is that for in few sir to map my three D all on something kind of it clearly but the air temperature or or or density correlated with speak but it's much slower moving so can be that into the model but the S this yes you can have three D in speed and for the mentioned and temperature take it is a real uh and try to say so that using we model what happens is that the matrix is very ill conditions and it the output symbols up because the been been dynamics and uh use practically smaller than to a mix of the mean in that but the main you to the same but do group being or or uh elements in to covariance variance and so the covariance might this is different and not only the make this is a main stable but also the uh incorporation of beans the indoor air density uh inc enhances prediction so if you think about the matrix everybody as a child you you had this image possible you that those styles that you move to sort out some image i don't know the snow white or whatever that so all the information is that in those styles but went the a scrambled you can maybe guess what it is but you know you don't know what it is that's the real domain main i you've those tiles that's also can see the image it's kind of quaternion and model thing so it's much hand here are used much more robust