0:00:46from
0:00:49a or mixtures
0:00:51a
0:00:52why is important
0:00:54uh so that the all source a solution of E D who was my uh but students
0:00:59clive who is uh was not a research assistant at imperial
0:01:03cyrus with is some that
0:01:06uh let be on from from france and and myself
0:01:11uh
0:01:15hmmm
0:01:18not a million vectors
0:01:23so
0:01:24click deadens a a uh
0:01:25that a very much used in computer graphics animation
0:01:29but does stand
0:01:30how however really in signal processing does till consider that a little bit exotic
0:01:35and the there's many you reasons for that
0:01:38so that has uh a a division algebra a which is a us skew you feel all there are four
0:01:45skew meaning that
0:01:47the uh a don't product is not computed
0:01:51and the use been used basically matrix manipulations a computer graphics because they provide that affect rotation and orientation
0:01:59so in many uh movies now
0:02:02you have a uh start doctors will perform some scenes and then
0:02:06based on those measurement this was the actor effectively N
0:02:10and and we produce the the scene
0:02:12so
0:02:13usually using with that was for that
0:02:16how this is only ticks manipulation so in adaptive signal processing we need to
0:02:21look at many other things like
0:02:23gradients
0:02:24uh the so called cool lot T N such like which all touch upon
0:02:28and we can face many many problems uh in this designing those things
0:02:33and is the there's been the recent these surgeons so good to don't while processing read several groups working very
0:02:39actively
0:02:40uh at imperial at the sent and there early on
0:02:44and and such like
0:02:45so several cost of is that uh uh are now
0:02:48we getting extended to quote that times the concept of set a lot
0:02:52or or sort so called of clarity but D
0:02:55distribution of signal is not rotation invariant
0:02:58the second thing is uh a leading directly from dies so called widely linear model
0:03:03which is
0:03:04the only model still able to deal with second order so but non circular data so called improper
0:03:10and just said problem we faced was the
0:03:13good at is the in fashion
0:03:15so if you are familiar with the modeling in the complex domain
0:03:18you would know that the gradient of a real number
0:03:22does not exist
0:03:23so typically our cost functions are are powers they real
0:03:28however to design say uh
0:03:30i i mess that the log or deals with really
0:03:32you need to perform from the derivative or can't of that deal function
0:03:37the cost meant does not a lot to do that
0:03:39and you need to do this or the so called looking at and
0:03:43and the does of the many ways
0:03:46but uh you which you can use to a to
0:03:48between functions in a two and fashion and C
0:03:52to introduce a called C are calculus
0:03:55well you can
0:03:56you can uh uh a a the and is for a special class or or the old function
0:04:02and those have functions of a complex number that and Z conjugate
0:04:06so we are going to do a very similar things and the with that and the men
0:04:10uh
0:04:11so that this talk is really about blind source extraction not separation
0:04:16and the glass as extraction i think of the paper i uh i think was the first was by build
0:04:21force and
0:04:22i'm not good with french names so
0:04:25not a little force was the first author
0:04:27and i have a i have a lower that's only quite a lot of about this method from my only
0:04:32and i'm what's skip from reagan
0:04:34and to get that we can do develop several algorithms
0:04:37most of them in the complex domain
0:04:40using so called augmented complex statistics to deal with the noncircularity of distributions
0:04:45and this is a a time now to extend it
0:04:48extended it to the quaternion them and
0:04:51okay how help contents that ends
0:04:53uh when we think about a new technology and so more and more we have facing so called vector sensors
0:05:00so instead of having i i don't and that a
0:05:02or a single univariate to single single-channel census we
0:05:06which we then combine some howling to matrices
0:05:09how about those sensor
0:05:11being multi dimensional
0:05:12and this is happening very much many new technologies a
0:05:16based on vector sensor
0:05:17so the says and all mention is the and then meter
0:05:20so this guy did him measures wind speed and that actions symbol
0:05:25and this particular one is the by your instrument score collaborating with S
0:05:29and it have some out sonic roles
0:05:31and this one is two D so
0:05:33you measure simultaneously in speed and that so it's a vector sensor
0:05:38if you would like to measure in three D
0:05:40you have those problems on top as well
0:05:43second second two sensors
0:05:45for the work to did now with the elderly or unsure a virtual reality
0:05:49the uh what these sensors so that's P D national what motion
0:05:53uh sense
0:05:55so you measure simultaneously exploits said
0:05:57position that was and so on
0:05:59and if you stick when you of those two are person
0:06:02then then you can look at their
0:06:04gate the but it of into to full or you can do one animation for movie
0:06:09uh so
0:06:10because
0:06:11uh
0:06:12i i but it much in processing so
0:06:14uh here's what we do so
0:06:16you this is a bit far that a typical we far
0:06:19and you can read their minds which all over the place and this bowls with those kind of
0:06:25crown this some top
0:06:27uh
0:06:27three D i'll the sonic and of meters
0:06:30so what to are doing you go trying to model a a boat
0:06:34optimal all remote control
0:06:36oh do in farm and also
0:06:38to provide prediction i gaze then the guests or any well vibrations patients any problem that can break their buying
0:06:44by modeling green
0:06:45using a vector sense
0:06:47so are are either had in the been modeling with have many many
0:06:51uh results
0:06:52uh a going back again to to this uh model or and going to what's movies
0:06:57so if you have a style doctor or anybody skilled performance some second that two seconds and such like
0:07:04and
0:07:04you can synthesise a
0:07:06hey
0:07:07i like or or
0:07:08you can see so but it wasn't actor here
0:07:11so for those of you who are familiar with computer games
0:07:14this comes from be assumed right or which is from nineteen ninety to six
0:07:18so that night S a six was the first time that could that employed in commercial
0:07:23uh
0:07:24uh adventures
0:07:26and uh if you remember the difference between this game in anything before
0:07:30these fidelity on movement was and not mostly better
0:07:34that's someone applications and now i'll be we get everything short and go go to towards my
0:07:39a separate to my talk
0:07:41uh what and that's the to a division algebra
0:07:45so the only division algebra is a reels
0:07:47complex what that neon and an could onions
0:07:50if you have a read the
0:07:52uh new scientist from two weeks ago terms two pages dedicated to look at all neurons
0:07:57where the thing that the string is your in anything
0:08:00a can serve that can be explained but it coming in to but then
0:08:04do was a very but in wonderful fields the problem is very simple
0:08:08you have to sacrifice some doing so
0:08:10if you going from the young
0:08:11two complex numbers
0:08:13you lose order
0:08:14so complex field is not order
0:08:17two but
0:08:17two plus two J smaller not bigger than
0:08:21one was J
0:08:22it doesn't exist
0:08:24go from the complex will look what that means
0:08:26you lose uh
0:08:28common activity of the product so
0:08:30X times Y is not cool white time sex
0:08:33and that in poses many other problems for instance
0:08:37you have left and the right eigen values
0:08:40and the the uh right hand
0:08:42i i can analysis is now established
0:08:45how other but these still no got to deal with left and i can as and so on
0:08:49so i'll just give the sketch of the proof later but are not prove it
0:08:53a would by could to know it is safe from from C
0:08:56and i guess say that we need to rush
0:08:58uh
0:08:59the
0:09:00i don't think that these interesting think is the so called in lucien
0:09:04so what ten in this though right
0:09:07for four minutes
0:09:10that that
0:09:11and the because of the real part and the imaginary part of V
0:09:15consisting of these i J K
0:09:17which are was uh
0:09:19imagine in numbers and unit vectors so the difference between a are inc that neurons
0:09:24he's in the other for i J K and so on just unit vectors
0:09:28uh
0:09:29the the not in the axes
0:09:30but as in the code that in them and so imagine number so we have to be a careful about
0:09:35the or because i J equals K
0:09:37but J eyes
0:09:38minus
0:09:41uh again so
0:09:42if it's in why uh to do not a are so let's think about the problem of rotation
0:09:48so if you want to use the the perform rotation using all around that is in a three
0:09:52then you would have to go to get a lot X
0:09:55why and and is a which is someone that
0:09:58giving this matrix so we have to it three times
0:10:00in the quoted and domain because it's a compact
0:10:03for a for it's or three
0:10:06this is
0:10:06a a a a that known in in it
0:10:08all the four
0:10:09and this is the rotation
0:10:13okay so
0:10:13the benefits of then if you are performing and any image related problem or any track and you not the
0:10:18on and such like
0:10:20then
0:10:21this would be typical trace
0:10:22a a fight coming from those all are matrices X Y Z
0:10:26as in the with that in the menu just move act like that there's not problem
0:10:30yeah the thing is
0:10:31that division algebra allows to to do not to get stuck in so called given by law
0:10:36so for those of you are familiar with gyroscopes
0:10:39the usually have three axes
0:10:42so one second and so
0:10:44and they wrote it
0:10:46so it is possible that to of the axis coincide if you model the problems
0:10:51in the are three
0:10:52so this is a physical a scope but this is a but today interpretation
0:10:56however
0:10:57so you losing one degree of freedom
0:10:59so if you member the up paul eleven at some point once spinning in one lane because the charter school
0:11:04but fault it was in the model local that time
0:11:07this cannot not happening
0:11:08that
0:11:09and of course than the story
0:11:11but to need is much smaller because in the real but been not for the need for covariance matrices for
0:11:16a uh
0:11:17for for the dimensions and then six
0:11:19so the covariance it's made
0:11:21in the with that in the menu need one
0:11:23covariance matrix and streets of the covariance method
0:11:26so setting
0:11:28but got very quickly to
0:11:30two
0:11:32to something to my talk
0:11:34uh this is a key slide and from then i'll just go to application
0:11:38uh is a whitening at modelling so i'll introduce use it by thinking about why building at modeling in the
0:11:43complex of man
0:11:45think about the standard mmse estimator uh
0:11:48you want to estimate these signal Y in terms of the
0:11:51like a sir
0:11:52and
0:11:53so
0:11:54uh uh are mutually
0:11:56in the band then then the optimal estimate is a a linear model
0:12:00H is a coefficient vector and X is that a good as some vector
0:12:04now in the complex the people usually assume that is still holds but you can look information here
0:12:10you don't have to have a mission you can have a transpose it of coefficients are just to as much
0:12:15how are think this way
0:12:16so
0:12:17well D complex number of has the real and imaginary part and both of them are
0:12:21functions so the really much and a part of this and
0:12:25uh
0:12:25but X are is X X to get or two and then X imagine is X minus that's going to
0:12:31get or
0:12:32that's some minus missing some way
0:12:34so that effectively
0:12:35why are are is a function of X an X to get and line might in it is a function
0:12:40of X and X conjugate
0:12:42so if that mutual then
0:12:44then D a linear model becomes that a be linear model
0:12:47expressed that
0:12:49and i miss this is G
0:12:50this some other of G not
0:12:54and
0:12:55this way you can capture the complete second order statistics in the complex domain
0:12:59so anything you develop so far in C is optimal only force local satellite data
0:13:04right D distribution is rotation invariant
0:13:07but all the real data are noncircular
0:13:09and it's very simple then then you can extend it to the quaternion domain
0:13:13this way
0:13:14so can for
0:13:15channels and for different
0:13:17a we that there's
0:13:18and this is the co or it
0:13:20vector to which comprises you V G
0:13:23so i'm conscious the time of just
0:13:25got so clarity
0:13:27so this is quite a got
0:13:29fact affected by by stuck a lot i was considering we and
0:13:33and this is a polar target of in speech
0:13:35so clearly speech a much stronger for some that ashes she's that from the others
0:13:39and i was thinking of case we is a complex
0:13:43a back to because you have intensity an and so it's to the complex
0:13:46and i was trying to bottle simultaneously both
0:13:49but is also we're not so would for some or G
0:13:53for some of them the perfect for some that were not cool
0:13:55that i discovered that this is because they start that modeling in C
0:13:59assume assume could a lot of two of these distributions so
0:14:02as if you have a circle here
0:14:04for
0:14:05data
0:14:06for non circular data and all of them one non circular because the been
0:14:10at the side was either there from the C to the C and so on
0:14:13if have to use of white linear model
0:14:15and can
0:14:16some some benefit
0:14:17oh so we can develop no one you good D and for for uh
0:14:21for a quick that and since a conjugate gradients so please look at the i-th i-th was composing letters uh
0:14:27john two thousand then
0:14:29eleven
0:14:30and then a few elements coded in uh lms was developed in two thousand nine because this for because many
0:14:35false this is one of the form
0:14:38uh then widely linear quote an anonymous as
0:14:41this for
0:14:42and to show you that you widely linear model works better than the linear model for sale noncircular read
0:14:48so there's laura jean high and medium regime
0:14:51but the good airline
0:14:52every time out performs the
0:14:54the black line meaning the widely linear model is
0:14:57ideally suited for that
0:14:59so this is my top i'll skip all the mass
0:15:01so we caff this blind source extraction diagram which most of you are familiar with that
0:15:06there's these separation matrix W doesn't mean why
0:15:10but a low you like
0:15:11run a predictor which in this case you want to be near
0:15:15if a prediction that are which and tries
0:15:17both these set
0:15:18that
0:15:19and this of the optimization problem so we need to more
0:15:22a a you and be here sort of the type of
0:15:25and i'll show how this course for E G
0:15:29you could that work out how uh
0:15:30this simplifies actually because that at all
0:15:33uh you can work out the how the sorts uh uh these in the paper
0:15:37so have to result
0:15:39one is for a synthetic sources but have to non set of close the source sources
0:15:44which of both synthetic
0:15:46and then looking at performance index
0:15:48but that the
0:15:49the extraction weight or is that no "'cause" one element or not
0:15:52and the second is a work E G
0:15:55so this is the
0:15:56performance index
0:15:57clearly
0:15:58if you lose a we use white living at what didn't in predictor or and the sources is that in
0:16:02sources of us
0:16:03then the performance is really good
0:16:06but as for non circular with that don't date that the standard
0:16:09what a more of the i don't model was not sufficient
0:16:13and then
0:16:14the think it was that
0:16:15in but computer interface and also and then E G research
0:16:19you reject blend your data because that contaminated by active artifacts
0:16:23and does that if that's not only come from the power line and from the eye blinking
0:16:27if you think about it
0:16:29because two lines and we of horizontal and vertical move
0:16:33so
0:16:33in the complex domain say is on than of a vertical movement can can be model but it would be
0:16:38uh course on than but for you O G is a complex number
0:16:42you to wise and an order L complex numbers as a with that
0:16:46so why not develop an algorithm in the group and the main
0:16:49that extracts
0:16:50a a moments both from the right and your left i
0:16:54and we did that for various
0:16:56scenario but people building all they are as a a a a nice i've row such like
0:17:01and the
0:17:04obviously
0:17:04you have some group so all the from tell channels but one put that neon
0:17:09all do medium channels second quaternion
0:17:11and all the uh the or channels that
0:17:14the that the
0:17:15and you can see it on be on the blinking uh
0:17:18and the you chi use channels that
0:17:20obviously because the sources is close to the front to that
0:17:24then the front of channel so much more affected by active
0:17:27then say the
0:17:29channels as the back and such like
0:17:31we use the for you G channels as a reference so that are not part to the mixing model or
0:17:36and else
0:17:37it just use them uh to combat
0:17:40performance of our model beat
0:17:42the out it
0:17:43and here the results so
0:17:44this is how we did it
0:17:46uh
0:17:47so this is the power spectrum of the original signal
0:17:53and the artifacts to folks i blinks
0:17:55which are
0:17:56low frequency and fifty hz
0:17:58mains
0:17:59and we perform extraction wipe one one so why would extraction
0:18:03because in by a data often you have many sensors
0:18:06so in and me G U the one hundred twenty sensors
0:18:09so of mixing matrix is one twenty by one twenty which is more than ten thousand quid fish
0:18:14so need about one million course which to convert
0:18:17but as you interested in only one or two sources
0:18:20so why not have a much smaller model and extract only source is based on sun
0:18:24property that we can control like sparsity or
0:18:28a predictability and such like
0:18:30so in of uh less and diagram
0:18:32uh we first extract the
0:18:34the a line noise
0:18:36in the one living at the and standard the main was also able to extract it what well
0:18:41however when it comes to the i blink
0:18:44so
0:18:44used the power i links uh
0:18:46i i don't know how fast a complete
0:18:49so maybe
0:18:50three four five hz
0:18:51so that here
0:18:52so clearly the the the the widely linear extraction now to was able to both
0:18:58uh extract eye blinks from the mixture
0:19:01and so press
0:19:02the
0:19:02uh fifty hz interference
0:19:05whereas as the standard al gore was not able to do that so
0:19:08it's a also the blinks and interference in mine but very high
0:19:13and at this point i just the
0:19:15want to show that there's
0:19:16somebody many is going on special about the diagonalisation of those made it's is in two
0:19:22make any convergence proof
0:19:24and in conclusion a a quaternion so it's that a natural to work in the good then in the man
0:19:29and please
0:19:30of interest
0:19:31it web set or talk to me and my students well that think are to match
0:20:02yes
0:20:07E E S okay i'll a very quickly
0:20:10uh if you look here uh i'm
0:20:13i think a much time to talk about
0:20:14simulations so
0:20:16this is that for in few sir to map my three D all on something kind of it
0:20:21clearly but the air temperature or or or density
0:20:24correlated with speak
0:20:26but it's much slower moving
0:20:28so
0:20:29can be that into the model but the S this yes you can have three D in speed and for
0:20:34the mentioned and temperature
0:20:35take it is a real uh and try to say
0:20:38so that using we model
0:20:40what happens is that the matrix is very ill conditions and it
0:20:43the output symbols up
0:20:44because the been been dynamics
0:20:46and uh
0:20:47use practically smaller than to a mix of the mean
0:20:51in that but the main you to the same
0:20:54but do group being or or uh elements in to covariance variance and so the covariance might this is different
0:20:59and
0:21:00not only the make this is a main stable but also the
0:21:03uh incorporation of beans the indoor air density
0:21:07uh inc
0:21:07enhances prediction
0:21:09so if you think about the matrix everybody as a child you you had this image possible you that those
0:21:14styles that you move to sort out some image
0:21:17i don't know the snow white or whatever that
0:21:20so all the information is that in those styles but went
0:21:24the a scrambled
0:21:25you can maybe guess what it is but you know you don't know what it is
0:21:28that's the real
0:21:29domain main
0:21:31i you've those tiles that's also can see the image
0:21:34it's kind of quaternion and model thing so it's much hand here are used much more robust