Přepis řeči - BLIND EXTRACTION OF IMPROPER QUATERNION SOURCES

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

BLIND EXTRACTION OF IMPROPER QUATERNION SOURCES

Source Separation and Applications

Přednášející: Danilo P. Mandic, Autoři: Soroush Javidi, Clive Cheong Took, Cyrus Jahanchahi, Imperial College London, United Kingdom; Nicolas Le Bihan, GIPSA-Lab, France; Danilo P. Mandic, Imperial College London, United Kingdom