okay
I this week started will come with afternoon welcome to the
distinguished serious to of Electrical and Computer
engineering this afternoon we have the pleasure to
welcoming doctor richard meier new he is the
Victor Cameron professor over of Engineering at Rice University
he received a PhD from the University of Illinois at urbana-champaign
a master's degree in puritans Conson
I'll he the
has a very distinguished trajectory in the academic
world of signal processing I he has received many awards including the
a hole in our ya know Young Investigator award
unicef Young Investigator award as well
I'm he's on IEEE Fellow
he has received the SBA where with pioneer word
in many other words in in in this societies of IEEE
I'll he in 1989 he
found it a very interesting and successful a nonprofit
publishing project I'm I'm not sure if you will
say something that we are good it is a is a is a very
interesting project funded by the
Hewlett Foundation NSF in the name typed the
making available high-quality educational content on the internet
so so it's been adopted by many at them again Solutions is a loser really
interesting project that he he founded I'll
he controller Winnipeg he was born in Winnipeg can either
according to some statistics is the call the city in the world
with a population over 600
thousand I was I spent six months in Helsinki and I thought that was a call
this the Syrian though
well that the I guess Winnipeg incidences up there
I'll rich
into it research interests the are in the area of signal and information
processing
I am his earlier work was
in the area of time-frequency analysis and
in the image modeling me away with its and more recently he has been leading
in pioneering the working compressive sensing it's a it's a really
I revolutionary aspect of looking at the Howell signals
are acquired in sensed
who today his lecture is is going to focus and describe
this area what is really interesting this
in this area is that he can be applied in many areas can be applied in optics
can be applying
trust could be in signal processing and many many areas or
it's a it's a very fundamental a technique to
house has received considerable attention in many
many areas of of did IEEE
without for a further delay a welcome
join me in welcoming rich to the distinguish years
where here
for may wish to soundcheck
people hear me OK okay so I
thank you very much for that kind introduction thanks very much for that
I invitation to come here and speak today it's great to be here
I it's great to be here was not minus 45 degrees
below zero like it leisure and now Winnipeg
at Christmas I and what I'd like to do a little bit
I today is talk about this area above compressive sensing it's a little bit
uva
tricky thing to talk about here because there's actually a lot out
absolutely world-class work coming out to the University Delaware
about what I'm gonna try to do is to convince those have you who are not
already working
and doing world-class work in this area here at two joined
join the team and I move into the future defect a quick shot hit
how many people think they know a little bit about this area
Trahan so have hap hap split
okay good so there's really two aspects
that to talk the first will really be about that
why I and the how how it all works and why we would like to do it
and then out towards the second happen to talk all talk a little bit more
I specifically about what deferred come into different groups around the world
are doing
I what we're doing in my group and hopefully by the end you'll see that
this is an area that
that that has a broad research agenda with a lot of different possibilities
right that a lot of people get involved and can we make it toll interactive
I nor in the East Coast but can we make this really interactive
okay what may interact cell as you know there that there's an arms race
I in in digital sensing right up pushing the
the resolution that capture rate the quality of digital sensors
and making that every year better and better right used to be a big deal table
1 megapixel camera
now everybody carries around at told lot of people have a 12 megapixel camera
right
and and where we're very interested in to particular problems the first is
soap that what what how will we continue to drive
this ever-increasing quality of sensors how we ate drive this and second what
are we gonna do with all the data
really two separate but interrelated problems one
making higher res sensors and second what do you do with all the date out in
a sense
and the reason why worrying about the data up front is very very important
because we are in a world
absolutely exponent at super
almost super exponential growth up the monopod obtain a generated every year
case ought to take back take a look at 2010
in 2010 we generated about 12 Hundert at 1250 billion gigabytes of data
pay that more bets than stars in the universe
it's growing by roughly I 38 about
roughly 40 to 50 percent per year that's the compound interest rate
case I'll growing extremely quickly and
there are some very important moment that we passed through recently
first in 2007 we passed the point where the total amount of world data generated
exceeded the total amount of world storage
actually money data coming in from sensors was more than all I love
the memory on card s kicks at RIRA and everything in the entire world
which means in 2007 we moved into a different regime
up thinking about data that it is not always necessarily going to be something
you can archive and look at later
but increasingly date is going to be something at ever all that you use it
now or lose it
right so triage the idea tree I is going to become increasingly important
K another important aspect is that the increases in the amount of data
generated from all kinds of sensors
is growing much much faster then increases in transmission rate I've
communication systems K communication systems are getting better
but they're getting better at a much lower rate and
iight we're generating data or that were even able to
to store data nice to talk to one example its second it very illustrative
example it that
it's a camera system don't buy darpa called our guests
camera K this is a 1.8 gigapixel camera
k 1.8 gigapixel camera it actually was not that expensive to create
because all lies is that all it is is a tile have a whole bunch of cell phone
cameras
hundreds up they would like to run this camera at a video I'll rate
about I can frames per second that 1.8 gigapixel just to give you a sense of
how big this here's
if you think about 10 centimetre resolution magic flying this up in the
air
at 10 centimetre resolution this camera can look at the entire
greater Los Angeles area at one picture
k so madge in looking at most in the eastern seaboard
with just a few of these cameras K massive amounts of data
and now let's look at the datapipe that they have to connect the same to a
satellite
k the data rate their transmission access time which is
for state-of-the-art military spec is 274 megabits a sec
so that basically a three thousand times
ratio from the amount of data generated to the
the the amount of data you can actually get out the platform so the way that
this platform is actually use today
it's flown data stored on to some hard drives that are on the platform the
platform it landed
the drives are taken out and they're reloaded it flow of loaning it
K this is very similar to wanting to send files
from here to los angeles if your file is large enough
it is faster to put it on a tape and FedEx it
and to connect by the FTP channel right so this is increasing the world that
we're going to be
at moving into this is Ben reported in all kinds of different
I different publications there's a for people more action less is a fabulous
%uh special report in The Economist magazine a body %ah
or year or so ago and there was just recently a a very nice at articles in in
science on this
exact prop so today we're gonna talk about
is first what's wrong right what's going on with digital sensing systems
%ah why do we go to all the work to acquire all of this data
when we don't have the data pipe for example that can get the date of our
platform why why we do this
and then we're gonna look at just one way out in the particular when I talk
about is this
idea of a randomized dimensionality reduction or compressive sensing
as a way to both enable us to bill higher
quality high resolution sensors and also as a way
to squeeze down came on a day that we need to get out
those platforms where talk man in two parts about how it works first phiri
will do some math IOC is a map is actually not that complicated
and then well what look at some examples and depending on time will do more
or less up the example is everybody okay with that
agenda ok saw I let's just look at what goes on inside your standard digital
camera when you want to take a picture so I want to take a picture
Nicholas so I get out my mice my handy camera
I asked my quest how many samples do I need my quest ask me back
well what spatial frequency resolution are you interested in with your camera
that defines the Nyquist rate that tell me how many megapixels like
so I have analog a light writers light coming in
through the lens is a might my camera system and
shit just the standard paradigm that
in and not think that single image pricing is to just uniformly sample
in this case color bearer array uniformly sampled that light
and that gonna generate and samples and for the rest of the talk Annes
millions right megapixels so and it's gonna be a big number
so first thing we do a sample the second thing we do is realize we have too many
samples
right and no one here starship images on their camera
so we compress everybody here knows at least knows about
J Pak writer Jay pag 2009 idea here
is to take those and numbers and somehow
compress them down to fewer numbers just K numbers
that somehow capture all the intrinsic information and the picture nicolás
be using far fewer numbers right we want k to be much less than
and the way that this works it actually really quite simple
but these the so-called transform Coulter's all they do is they take that
that the sample of right that the megapixels
up samples and they take a wavelet transform
or %ah something like a DCT discrete cosine transform
and these are orthogonal 121 unitary are
101 transform so by in pixel values over there
become n wavelet coefficients and even if
I you've you've never really looked away but before that well the information is
in the wavelet domain is
equivalent to the information that is in the image domain
but you see a thing called it in a very different way so in this case
I with the color matte blue being small coefficients are zero coefficients
you see that what we've done is taken energy from the picture
and we compacted into just a few corporations
right just these coefficients that's not really a shot just that
the red and the yellow large corporations at all arrester small
so basically all Jay PEQ 2000 does are similarly which a peg is essentially
take away my Transformer DCT keep only 2k big corporations
there's only KFM and for all a everything up everybody got this
a very important and this property is called
sparsity right that we have a sparse representation of
the picture of Nicholas in the wavelet domain yessir
I I could use it but to thank you
mean to lazy worker ok
I think it's just cuz if the back projection at best a little if I missed
a second
so you might think thank you though you might think that this is the 3rd gen
I for specific property imagery but believe me
this property sparsity is really a general property have
nature that there's all kinds of different classes a
no I'm serve did no way like signals actually have this partially property
here's another example
it %ah at bat shirt data back yells out
in order to echo locate as they fly around looking for yummy mops
and it's a very difficult able to acquire because they have a large time
and with products at very low
long in time but it also had a very high Nyquist frequency
really weight loss fast day so you need to sample the signal at about
140 killer K very very high
frequency in order to acquire without aliasing
but if instead of looking at this in the time domain or the frequency-domain
we look in the time-frequency domain right that cup or a domain which is
sorta the 1d
an allergy of the wavelet transform up there you see again with the same color
map
nearly all beef pork or fish in 30 in just a few of them are large
and in fact at any given point in time they're just three frequencies
basically a harmonic set up frequencies that are active
to stand up somehow the intrinsic information being
I 70,000 uniform samples per second
which is the Nyquist view we can say really instantaneously
they're sorta 3 sideways going on per second three is a lot less than
hundreds about right week so let's just
look a little bit assists send some more notation talk about
for mathematical models for the sparse
signals so what is a sparse eg not what the back tre
axe remember to Blake and and it has only
k large coefficients at so we're gonna call a sparse vector
and let me to step back for second and say for the next 20 min 15 minutes
let's not think about signals that are sparse in the wavelet domain at the curb
or domain but let's think about single-gender canonically sparse
meaning sparse in the time domain so a factor in Matlab
all heroes with just a few K spikes
everybody got that ok Sol I
I if we're to sort the entries that this factor
sort them and then look at the magnitudes they would decay because
their sordid
and then after K all that rest to the coefficients should be zero
so this organic of sparse signal so now we're gonna do I i thought experiment
and think about concerns the first
sir important point to talk think about what is the fat a ball sparse signals
look like
it was a sad a false par single look like we're very used to do with
as engineers billing model right so I think I've
the setup spar singles as a model are also very used a linear models
right we love subspaces we love Kalman filters writer based on linear subspaces
all at Wiener filter tallest coaster
so what did ask yourself what is a Sat Nav sparse signals ok like
and left due to very specific case a signal that are
for real I'm and have three entries and one other entries is zero
first three entries and one of them is here what that not complicated right
what's so complicated about that
right so that there could be in one of the three spots right
sold as a set of all such signals look like what you think about it
it's actually a very sir bizarre set actually bizarre set if the Union
up three subspaces the union of three
two dimensional subspace so now your job for the rest of the stock is to sort of
take this picture
close your eyes and imagine now that you're not in three-dimensional space we
live in a million dimensional space
that's where r signals Lev and the signals might be
all zeros except 90 at let's say 10,000 locations
that means the fat a sparse signals
is the Poinciana million dimensional space that level on all about 10,000
dimensional subspaces
that are aligned with the coordinate axes does anybody know what they look
like
can if one hand it's like totally bizarre brenner handed
very comforting to look at look at this and if they come in as a sea urchin
right it's just that saying that pointy and sticking up in all directions
and it sticks out along the coordinate axes
everybody got that the reason why sticks out on court in taxes because
most any other corners or $0.01 so you can only text and along the axis
who's got okay so that sparse said
of I in practice we never have sparse signal of
we have approximately sparse signals so you might imagine taking this
%uh sparse signal and adenoids that would add from five down here
I but a more general model is something that's called a compressible signal
and a compressible signal is basically a segno
that have the power light decay
have a power-law decay that is fast enough such that if you chop it off at a
given K
and you integrate all this energy that's really not very much
rice you can occur just a small error by truncating to this K
terms everybody with me on this effect is a very
doesn't really what happens when you look at the wavelet transform but
nicholas
there aren't really that many exactly zero coefficients
they're all just really small so when you sort have a power
but I got case an hour to settle power-law signals look like where the
power-law
have extremely rapid decay Kayla turns out he is also a very weird space
K it's basically a pincushion right it's basically like
a sphere with the air sucked out at take the sphere
attach it at the coordinate axes and in fact clear out
these have a a name there LP ball right there LP balls
everybody here knows but a p/e norm what is the PNR
it's just take the peak power out here elements in your vector and add them up
if you do too people's to that the energy
right but the thing that is very different than the P norms you would
have studied
in usual classes is we always study the kind XP norms
right kind x1 where p is bigger than one
fact we're not interested in those at all where only interested in fact a
signal of
that have a abounded pain or writer Peter Blessed One for example
where p is actually less than 1 so but this is a highly non-convex at
just like up there why the non-convex take any two points
I miss that and thinking a line connecting them no signal on that line
will be in the sack
right so this is these are nonlinear said that when you add signal you leave
the set
everybody got me up Sol
I now we've got that that little better math now we can pose couple questions
and then we can
forget get on with what what compresses sent I sensitive
the first question that that are the first problem I have with the standard
with sensing
is the Saurabh if any fish in sea that we build this camera
think back to the Argus we build this camera that acquires n measurements
right in like with samples
all later then process to compute fairway but transformer them
and then throw away nearly all the entries that back
seems inefficient right why bill that camera
with all it's moving parts in order to just keep a few numbers
but inefficiency argument but there's actually a much better
theoretical argument that everything to do with sampling theory and Nyquist
theory that we teach and say DSP
is based on a linear model of the fair point processes linear
the recap recut riadi signals that we can sample without aliasing live in a
band limited subspace
linear the I reconstruction problem sink interpolation go back an ally is linear
everything to sampling is linear whereas other day and the
be transformed to the wavelet domain everything they
to do with compression is extremely nonlinear
this process I've keeping only take a big corporations and throwing away all
the small coefficients
is a highly nonlinear operation as we just saw the set of signals
that you can compress is a highly nonlinear space
so why should we thank that just because we're doing the right thing
or even optimal thing here and the right thing or even the optimal thing here
that somehow they should be jointly optimal and in fact they're not
necessarily jointly
ok so the whole idea of compressive sensing is to try to cut out the
middleman
and and develop system that go directly from analog
data writer directly from from this high dimensional a light coming to the camera
straight to measurements strated new a new kinda measurements
there were gonna call compressive measurement for the fed up and we're
gonna build some new kinda camera say
white gonna come and and out the other end are gonna come measurements are
gonna make
and love them and you're gonna see or what we're trying to do rather not sit
well we're gonna
try and they were you see that we can do it and we're gonna try to put the number
measures
ms close to K as possible are rare let's remember what que was
que was that sorta intrinsic dimensionality up to
defect or register the bait number big wavelet coefficients a
you can't you can't take fewer measurement from that right at otherwise
he will is information
30 try to push ms caustic a.m. EDT 0.0 after the talk is to
show that in fact you can do this 82 attractive least a billion you actually
build real devices that the two-disc
any questions this point okay
so let's just to set this up left left a guy really
oversimplified view of what sampling yes so what a sampling
I fairplay I would argue is if is operating on
a signal by a linear operator by Elaine ear
matrix that is I basically the same as the identity
so hopefully you can get past the absolute reality of what I'm talking
about seeing how this is useful
just imagine that we multiply a factor actually this sparse
right so actors have liked and they have k 90 entries
procedures over and over for the rest the doc manually operate on it by an
identity matrix
K to get why while white wire just like sampling
lifelike sampling because look at each entry apply look at the first nooo is
the inner product and the first role of I
with all backs well what if the trial of i
it's all yours except the delta function so basically the multiplication of I
times axes really just
picking off sample objects everybody see that
okay so it can't reveal in fact dissident trivial
because this could be an analog signal living in a band live in a subspace
which cases in bisbee an operator which is essentially the UIDAI identity
operator
taking signals kms and dimensional single for that and I'm a subspace
increasing sample
I just on it going all that Kb for people who might be bothered
so there's two important questions that you can ask anytime you build a
system like this the first is information preservation
do am I taking measurements axe such that all the information act is
preserved NY
that seems important I i don't wanna lose information in the fencing princess
because if I lose information the fencing prices I can never get it back
rights up to do I preserve information obviously I preserve information
because why kovacs since like with exile is no information
K the second question is gather why I can I get ex back
that's the recovery problem can I recover signal
given its measurements course again in this case a trivial
since white ball sacks a trendy all operation will give me back X
right so as well as long as to motivate not death but more make it
very different problem K here this is what we're really doing in CS
it were not multiply by a big identity matrix ride it multiplying by a very
short
that matrix so now the number measurement is going to be much less
than an
so early and take him measurements and everybody should be able to see there's
no way
to arrange an identity matrix here so now
this question up information preservation and stable recovery
become open question there's no longer trivial
so this in the dimensionality reduction problem going from signals in
n dimensional space down to measurement in am
dimensional space right as a word
and Saul I so the first question we read look at now
information preservation can we preserve information and a course in general we
cannot
just anybody's had linear algebra realizes it fi
is if is too short so it has it does not help for a weak
doesn't have a break and answer have to lose information general what I mean by
loses information in general
is you give me an axe woke up like by I buy it to get a why
their infinitely many other far and their I
access they will get the exact same why
to take any back from the North Bay supply and added an axe
and you will get right now you get exactly what tell
so this is a problem why would we in general do not preserve information
so now the first really key hopefully raise with me now to part really key
track for the first key realization the first key realization that I'm not
interested in sensing arbitrary vectors
die interested in I'm only interested if anything data
that he had a sparse representation the data that has sparked
so now if I make the assumption that my ex is sparse the whole picture changes
because I remember back why is going to be the linear combination I'm just k
columns
a five-day and the K columns
correspond to the in this case the three columns correspond to be
locations up to 390 entries in my factor
X so in fact what looked like a really bad matrix like that
if in fact if you knew where but the these large
IOC officials have x4 it's a really nice matrix
it's a tall skinny matrix tall skinny matrices
no problem making those have a break and I could have like and
K no problem and why I am so affectively this is actually a really easy problem a
court in practice we're never gonna know
where the large coefficients are right if you did
you see that this would be a really I possible
and so this gives less a design problem okay so this is that
the critic all iraq design a problem of compressive sensing
and actually not that difficult the the design question and I want to buy a
matrix fi
that has the following property had the property
that if I take arbitrarily k columns out at matrix
K any K any k columns and I form
this matrix I want that to be up all rank matrix
meaning it should have a rank K
K should be a follow request tricks I can ask a little bit more
I don't want it to be for a week I actually want these columns that I Paul
out
I want this to be close to an orthogonal I want them to be close to orthogonal to
each other
K and this is I the so-called
I restricted I summitry property or the r.i.p
so I say into words we want to design a Phi
such that remembered its if like am by and
and we're gonna just that desire five facts that when I Paul
any k columns out arbitrarily
and I for more than 20 matrix those columns are gonna be
Coast orthogonal teacher
that's what I'm that's what I'm after so anybody have done
any coding theory I'll I will see that
know this %uh very is actually a difficult problem the non-trivial
problem
geometrically what is going on geometrically what is going on is
let's remember our setup sparse signals this is what it looked like it the more
elaborate picture
now I'm pretending I'm in our and instead are three
but we are in are and my back tre is
axe here's a single axis sparse which means it looked like a dimensional
subspace
member we said there's all these k dimensional subspaces aligned with the
coordinate axes
that our signal for line what that what we're doing when we project with this
far I
is we take I all-points in our
and then we map them down two points in RM
and since either linear subspaces almost linear subspaces just mapped a linear
subspace
each 100 subspaces it becomes a subspace here but we're cramming them down into a
lower dimensional space
right one for million a mental space maybe to 10,000 dimensional space for
everything it's getting closer together
and the problem with the fact they're going from high-dimensional 0 dimension
is that if i take 2 I've ever tried actors here arbitrary factors
they could easily mapped to the exact same factor here
that would be bad but what this restricted I summitry property says
is that as long as my signals are sparse
of x1 is a sparse signal annexed to is a sparse signal
then I will have essentially an orthogonal relationship
that I will preserve if x1 and x2 are certain debt Euclidean distance apart in
Rn
when a project into a lower dimensional space I will have basically they'll stay
the same distance apart
that's what this r.i.p property in church that if you have closed or
thought I'll call
if you're close to retire columns you will have close to this distance
preservation
which means information preservation because what will happen
have a picture two guys rights to mention two guys in front of our camera
erect a compressive measurements of them with our camera
the last thing we want it for two people to be confused we don't want to get the
same measurements out
from the set from different people and what this property assures us
is that the picture that these guys i Touring and Shannon are so far apart
they were guaranteed that their measurements are going to be about the
same distance apart
so they couldn't be confused is everybody with me
everybody's excited right sounds good right sounds exciting
sound like a good idea at the time right the problem is that this is hopeless
right designing a matrix as as property
I even bear up buying that a matrix has this property is np-hard problem and the
reason is
is that what are you doing you have but the to think a better find a matrix has
this night property you have to take this matrix
MI and matrix it to take a columns do it after the new year whatever you do to
figure out the rake and do that for every possible
selection okay out a man call it that in church K
and that grows exponentially and an rights and hopeless
okay so we are excited for how long thirty seconds in our depressed
right so what they hear really top what do we do
what anybody do I with that when things get tough we go to the russians
right I and in fact some Russian mathematicians cache includes Canessa
matters
actually have studied this problem and they found that while
this design problem is np-hard okay
they found that matrices and had this property are everywhere around us
they're all around it back
to take a random matrix this tour and m-commerce and in MATLAB
and you can show in fact they proved that went extraordinarily high
probability
that realization I cuz he matrix will have this property
just kinda mind-blowing right I
and again it's just this whole thing up house okayed np-hard problems
that looks a harkin can be sir solved by
randomized techniques and so what they found that there's a whole class is a
different random matrices that had just brought property
as long as the following very important for me a whole
as long as the number of cop role was in The Matrix
which is gonna correspond to the number of measurements we take
grows linearly NK so it's K Groza
sparsity grows we have to take more measurements
but only logarithmically in an
right so that means we can preserve information and signals
right where the resolution on the signals might be growing at a very very
good worker high-rate
but the number measurements were gonna need to take crows only at all I read
about that rate
which is a lot slower K this is really the key
the tempering that day to day lives right moving from linear increases with
an
number samples to or even quadratic to logarithmic
right la gas logs logs are good and defeated no cash in with bassist in
Leningrad Cowboys
your fan of Leningrad Cowboys so
just to summarize here what we what what what this is saying
is that we can we can preserve information and signals
by acquiring them for some kind of a randomized sensing
wheres that a fair polling we're still computing linear
measurements but the linear measurements each entra y
is a random set up wat we apply to our entire signal
and add up such randomized
data acquisition in some sense and we switch shoulder we talked about
there's no information loss as long as the number measurements at Leerink a
and logarithmic in an so that information preservation
then there is IOC
recovery right the recovery problem so we shall we dollars information going
this way
now we want to come back given the measure is why I doing X
and again if we look at his naively this is a hopeless problem
it's an ill-posed inverse problem there are infinitely many acts that when
multiplied by five why
I but again what we're gonna do it for an exploit the back
that we're not interested or recovering a lax were only interested in acquiring
a sparse
axe and now we're going to happen we're gonna explicitly use the
that that did just see her chin like geometric structure
to be able to recover signals case let's jump right
jump right into it left a bad question
yes
all we have all the talk will talk
what actually talk about that yeah very good point in
in general all this difference is a problem right because
first of all this is not a pseudo-random acres is a random matrix
so I do you even know them at matrix you used I
but let me just say that there are a whole host an actual talk about them
very much but there are a whole host of ways that you can have a matrix
that that I am I
have has the property that needs it is random enough
to make this theory work at least in practice
and yet is very amenable to computational storage
for example take up for a matrix or had a mark matrix
and to scramble a rows and columns with the permutation that
totally fine and that's a matrix you can you can hold the written an everyman all
is set to transform per meter data had a market for
right yes up
I if it to go see a matrix
it is a random real-valued matrix I again
you can imagine a referendum matrix is provably works as well
is possible if one matrix right and also works as well
yes could you repeat
because you can't you can actually go through this process
up taking of Elections okay columns that would take
a long time cell up
by observing is to run the choices can you
we are you be able to choose the closing up at away
all meaning design a better matrix yet
saw yeah selection can convert it some point yeah
sold designing yet so this is this is kinda but look at this
this matrix right it's it it's very after this is like Shannon called from
1948
k share according to their do we use calcium codebooks to
code anything these days right now people there's been
decades have work on finding the best called
and so wanna think people are doing our is trying to solve this kind of problem
I'll finding I really easy to build easy to deal with matrices
that have all these products have these nice properties but have the property
that
they have to the most orthogonal possible columns eccentric share
but these are very I it's very early there are there but very few
really good results in this area to that help
yeah that did the hard prop tell me will be solved with very hard
questions very good time for questions
okay yes
to find a better matrix very interesting question
yasser to at the question is that
question was could you imagine you could have a
algorithm of some kind for example simulated annealing
that would make a better matrix right and you would imagine you
for so there's a whole bunch of this is this good because alright at a time
again to talk anyways is a great research problem right what given
some more to add some a priori information about your signals you might
know more then it's just part
then why did bob at matrix for that problem
right to seize this is actually I open somewhat open problem is actually been
some work
here in San other work but its it sa certain things are known to certain
things are not know
absolutely there's all kind there's actually a big dark a prop project
that I cokie com which is the more you know about the data
should inform you to be a little better fine matrices right acceptor
at the result call this a doll is that
I enter exciting thing about what I'm talking about or the depressing thing
if you're an adaptive percent is that the f5 if you really are in a situation
where you assume
the minimum that you really don't know its case parts you don't know where that
the coefficients are
this is near-optimal Augusta radagast the matrix is
very close to optimal K but I did to press their citing
depending on whether you're a he rested her practical
a person okay soul rock reconstruction reconstruction is again
or give us a measure as you want to get ex back to let you geometry
so what's the geometry the problem on the geometry is step I have the No
space right by ethanol space and a dimensional space
is I and minus am so X with our regional sparse signal notice it's all on the
coordinate axes right it's
its it's a one sparse said in this case right that
when a tree in any other two entries are zero so that but when you ask yourself
what are all the axis
sucks up by actually cause why they will live on
done all space translated to the point tax
so this green hyperplane which is developed a mention in my fam is the fed
up all access such the
why up by actually cause why okay so the solution
given why live somewhere on this hyperplane everybody with me
K okay not a problem is
is infinitely many actors on the hyperplane Sol hideaway say what the
best one is the right one
and there's really that the only way to do it is that play favorites right to
say oh I have a sense of what I mean
by the best 1i want to pass tax from that
from that's at and the thing that everybody here always skid
you would fall back on would be least squares right we say
we saw this opposed inverse probably square I hope this
was unclear any every with mir okay
the least-squares what least great-great it basically want to solve the problem
at the top
we use an optimization we look for the minimum no armed
action let me know on energy axe all the actors
there when multiplied by five why and we make at our estimate why is as nice what
al again it beautiful
our all the stuff calls for formula in terms of the pseudo inverse
fast if I have some structure place like totally romp
kate is exactly the wrong thing to do for this problem so here is a
I sex sparse signal a blank one thousand I took random measurement
and Iran least squares in MATLAB and that's what I get
only thing you see is good phenomena at all you see is ringing
K the first thing you do is get her disappointment of
terrible but the other thing that you relied ACI think for a minute
is wait a minute compute the all too normal Bethany L to normal this
chris has a much smaller l2 nor it is the right answer
is the minimum energy signal that agrees with my measurements
so what is really just says it l2 is around metric
right l2 is the wrong way to say past so we need to change that to Lecce now to
I live here while too bad always forget here's why all too is bad
what is the geometry this problem the geometry this problem member with the
hyperplane
were all the solution before you have to live on a screen hyperplane
and what we're doing by minimizing the l2 norm is we're blowing up the l2 ball
and they are two ball is a sphere hyper sphere for blowing it up
Intel it touches the green hyperplane the point at which it touches
is the smallest energy signal that agrees with
wycoff iraq's right the problem as
West Bengal a bit hyperplane neglected to say with the angle
what the it's the No space in a random matrix
so you know what the angle as it's a random angle I don't know why it
straight up and down her straight horizontal
it is like a random angle for their just think you're a randomly angle thing is
your blood is red ball where are you gonna touch
you're gonna touch far away from the coordinate axes
just how we think about forty-five degree thing right you'll touch it 111
and that's exactly why this kind of thing not exactly what you see here
top l2 is bad not not you know that
in solving inverse problems bad Lt that the left
replace l2 so the obvious thing to do is replace it with
with just say let's what's called the l0 norm
which is just a fait les travel this entire no space
travel everywhere and less search for the sparse s tax
so instead of search for the acts with the smallest energy
well search for the one that's persist
K and this is sounds like a great idea in fact it is
great idea in fact that gives exactly the right answer
K because this is the sparse a signal that agrees that the random measurements
the problem is again I'd I want to get right up there
treating get too excited this is an np-complete algorithm to solve the
recovery
because again you have to go through all k dimensional subspaces event dimensions
to check F data sparse solution to
wycoff Iraq so again where excited and depressed
so the thing that did a bright blue turf field open
at starting in about 2004 was for super smart guys
who basically figured out you could take this optimization which is highly
non-contact
an np-complete and actually came back to fight
and the way to convict the fire at
is to replace n0 norm with the one norm
so what is the one or it's just take my vector take the magnitude of all I ask
of Asians not the magnitude squared
the magnitude and add them up totally natural kinda thanks not
same the energy but only natural so they actually show that if you do that
it will work now at work
probably work I it will also be doable in a polynomial-time algorithm
because this optimization problem here is essentially linear programming
which can be solved in pop and the geometry is actually quite interesting
because what you're doing now is it in searching for the men on 01
just about to solution is your blowing up a ball but now it's the 01 ball
to set up of signals with bounded l1 norm
and that if they dined right for and a casa he'd written right
blowing up and it is pointy Brescia poignant
right point and you go to higher dimensions you can't visualize it
but I get increasingly pointy as you go to higher and higher dimensions and so
it's no wonder
when you blow it up it is gonna pierce this randomly angle hyperplane all on a
coordinate axis
and that's really the crux of the proof even is to show that
that that's exactly what happened ya pierce's plane along
or close to one and he scored taxi K
Sol up this was actually a cover by Dan Brown to people look at their add up
in a div id she called it depends she knew this all on right that
now I estan Osher put it aline is the new al to
right no lapse tell us there from that joke
okay naaa any questions at this point
yessir
mean the rent them to use some party for the title fool
issue vision which theoretically would be a disk you pick a distribution for
the matrix
absolutely like II D Garcia
any other question
okay so this talk briefly about something nice properties that this
approach so I didn't talk about it but in fact it is stable
you can actually show that defied Nyquist sampling in recovery with think
interpolation
that this this acquisition recovery process is numerically
stable I it's also the most important thing is it's also a symmetrical
just like regular sensing but asymmetrical in the completely opposite
way
completely opposite way meaning but most conventional sense saying
users a star smart encoder I think if your digital camera
has a backpack for a NNJ pad and then a really dumb décor
read your laptop J pic decompression is very very simple
CS is exactly the opposite they were doing taking these randomized
measurements that are absolutely
dumb they don't know anything about that signal they're even looking at
and all work is I miss linear programming this computation
on their on the decoder on the recovery side
K there's also the matter nice properties related to
but added that these measurements have a fountain code like property which would
be very handy for transmitting them
they also have an encryption property which means for certain random matrices
that
there at least weekly cryptid and then there's a II didn't talk about it but in
fact these measurements will work
not just with signal sparse in the time domain canonically sparse meaning all
zeros
with some spikes but Ford Figo spars in any basis
DCT wavelet what have you
okay so let me let me just ask if I had two more questions and I can just
plowing into a couple
applications yeah them
back I can repeat feel like very
for image processing is it advantageous to perform the
um projection into let's see we've laid or something demean and then do the
very important point okay Sol absolutely not
so there's there's three things you there so so there's three even imagine
three kinda camera
I'm not track beat up on you but there's imagine three kinda
fencing fit you op frameworks
one would be acquire Nyquist sample of and then do I
at in digitally to wavelet transform right
when I'm mad another would be somehow optically compete wavelet coefficients
say and pick the biggest one right that
nice well I'm advocating is totally different than most to it it just take a
randomized measurement
and then use a priori the fact that you are you might be able to spar supply
that image in a wavelet basis to be able to recover
K what you're doing is actually a very important class a sensor that does that
but in that case you need to know for example that your signal is sparse
in wavelet you need to know that or hear you don't need to know that all
that to help yeah at very good that's a very good question
other questions and ladies at what time
because we started a little bit late and I could either stop
in like 23 minutes or I could go ten minute
what are the power to be say okok
to let let let leftist allies get the universality and a jump right to
compressive sensing
I so compressive sensing a grown-up right start in about 2004
so nicolás is now for right there is with this
compressive sensing t-shirt which you can purchase
here and go to bed par a floppy
I the other thing you can do it you can do it you can make a lot of money
i guess im compressed sensing cell so here's up painting by the artist stop
Gerhard Richter I from Germany
that I actually was a I worthless piece of art
and then we actually did an analysis and found that it I
yet up to his matrix actually satisfied restricted I summitry property
s4 pro 3.7 million dollars right show this is very very
profitable area II these days sawyer talk
up briefly about cameras and briefly about a/d converters
right in in like 10-15 minute tell the first-ever effect I
a camera idea that that Kevin Kelly at rights and I've been working on for a
number of years
of folks like Dennis Prager here at at at Delaware been working on similar
types of multiplexed
kinda cameras and the idea here is to
did there are certain situations where it is very inexpensive were easy to
multi-plex
light but very expensive to detect
light we'll talk about this I in a sec at
so this this particular camera pushes that to some kinda absurd limit
so I I just imagine a camera and a lands at the final a camera
light from the scene comes in hot through the lands
and then instead of focusing on top focal plane array of ir a half million
the little
I sensors you actually focus on a project on a track projector on a micro
mirroring
and it's the kind of microarray hot bat is
very likely in the projector is behind a screen right TI has about 50 percent
I love the market I and it basically a mic
at Bayfair memes device and he's bacterium size mir's there's a million
up number two million of them
can can clip one way or the other their binary
right 10 or 12 degrees one where ten o'clock recently
to now imagine you take the light and
you Lobao said of these mir's and then you buy out throughout the years to
another
plans and collect all the light at one the focal point with one thing all
detector K one single detector
and now imagine you put a random pattern on these mayors
she flipped happen one way and happened the other way
so we're gonna be collecting the light only problem to happen the mir that a
flip this way
the mir never put this way we throw the light away
so what are we why we built we built an optical computer
with built-in optical computer that takes the inner product
between aksar signal axe or at least a sample version a VAX
with a random pattern right
a random pattern of ones and zeros zeros is where we throw a light and one is
worries kept the light
because what we're doing is we're just adding up that some
I i've and all the white mayors in this case at that photo
flip the mir Emma times a.m. different random patterns
to build up a factor why and then use your favorite
linear programming either recovery technique to recover that the data
so to spatio-temporal multi-plex camera instead have
acquiring all the data that one instead over across the entire
send a pix of you multi-plex and acquire them over
overtime can say here's the very first
pictures that we talk here's the target years what you'd like to see
i sat 256 by 256 image and over here was the very first picture
this is taking just I I not 1300 randomized measurements and then running
a sparsity based recovery technique
and you see it kinda crappy but you can clearly see dr.
there is no alias saying and where fifty time supply quest
right so fifty time something I question we are able to recover
the the image now it's because it's simple right the very simple looking at
you for not happy with this crappy
picture this also shows that just like a lot of compression algorithms
there's a couple progressivity property if you're not happy with this just take
your measurements
take eight times more measurements you're still six times sup Nyquist
and you get a quick picture quality back which is pretty much as good as
up for a regular digital camera right for demonstrating again this
this idea that you can sub make which measure randomly
and be able to recover data yes
up
yes set put a random
random pattern on the red on I ramp at Amirs
allowed a light to come and collect the light at the focus quite eyes that you
know a/d converter
and stuff it in a vector switch the mir's
to a new pattern pattern number two stuff a number in there
so just basically go through time and patterns build up to like them vector
that clear yeah get very good point in this camera
the rates are slow enough like up to a hundred killer hurts that where
you know we can takes 20 bit samples if we want
brought talk I'll maybe a little bit later about
actually using this to with very few bit
like one even one bit per measurement yet could could point
up there is definitely a relationship
with a bunch of different kinda made a conversion
I'll on the outlets highlight II after the talk and
the pressure there's a relation by a weird one because it's randomized
right yes absolutely reconstructs
I in addition to my and samples from
detector I also need to store am and by and by a raise
board the position exactly in fact I have to stop it
worst case I would have to start m by n matrix
terrible right which is why in practice
first the law we never do that we always that at a very big most we in practice
we use our
I lay near shift register kinda thang and we just are the seed
so that one number that good and practice practice really really
we use permit asian-based techniques and use fast transforms
like TCT had a martyr but that's a very good question
restoring would be terrible okay I so why would anybody want to do this when
you can get you know you can buy a higher megapixel CCD now right
why would you ever want to do this and the reason is is when the measurements
are expensive
and I I could go into this a great like to hear some examples of bar
measurements are expensive
for example you want a bill a low light camera low light camera I mean really
low light
has to either be built using photomultiplier tubes or avalanche
photodiode
physically very large thanks
very large form of apartheid fist-sized K
that 500 bucks ok build me a megapixel Poe to multiply two camera
right very expensive very bulky they're actually Rd cameras like this
but they're third right and they cost millions of dollars
instead just for spatio-temporal multiplexing just bought one for a
multiplier to
at that focal point and now you can take a low-light photos
using this spatio-temporal multi-plex there's an example
up pretty up his target setting up the camera Kevin flat carry out all the
lights shutting the doors and just felt like coming under the door
being able to get a collar reaction all color wheel be able to get a pretty good
I very low light differ with be a few photons
a pretty good low-light images that particular target case let one example
of where
regular cameras would be very expensive very hard
another one infrared especially far infrared and even more care heard
millimeter-wave
very expensive caught I complicated a bill cameras
at those range but again here if you have a DMD or some other multiplexer
RI Mira rare at similar thing that can
deal infrared radiation then you can just swap out with a infrared photodiode
so here is a a picture I I love
nothing other than I are written in charcoal on a piece of paper
and this is so that that it up reasonable straw man I think which is
what is done today lot of scientific labs which is to take your one sensor
and just rafters Canada crash to
the image K that you wanna choir and if you see it's extremely crappy
these were normalized by you had the same amount of time to acquire
the image K another example would be hyperspectral
I get capped a hyperspectral data cubes a paste
wavelength is again extremely expensive and
here is where that day-to-day literally cut ten you can generate billions a box
on the hyperspectral data
so again just take out that photodiode put in a cheapo spectrometer
right five hundred dollars spectrometer and we're able adela really
I quite nice hyperspectral camera
again our favorite target and this is just a mosaic up to
that to all the different wavelengths with Fox collar and you see we pick up
our pollen then we start to pick up the greens and the rat on the nose and then
went on to the
infrared and this is basically at a 521 compression so I thought
makeup box all space wave-like cue
and we took two hundred thousand random measurement price on all straight put
the compression and that
for a feasibility of this approach working with the world's largest
makers have disco ball to make a disco ball camp
coming to club near you
not let me just ask as I only a 5-minute we get about to light one up to thanks
well as a good I love video the others we tie a titty
at like a our apt signals or serve ok
Choi and video okay show and our apt
a today's Wow
I some are professors hands with a dish
K thought but left it yeah well I mean to say that you
we were actually at video is extremely hard
extremely art but there's progress being made and at the end I talk about
good open problems video CES is a very good
open problem very difficult let's just talk briefly about a/d converters an
hour with
one-dimensional signals not to and
I think the striking thing that I learned well I
about AD conversion we're so used to exponential growth
in our world love technology right of Moore's Law
right that we double the power of processing every whatever 18 months or
so
right I well there's also a Moore's Law for 83 for analog to digital converters
in terms of like the bit depth and the sampling rate
but the doubling time right the doubling period is eight years
night 18 months right so there are all kinds of applications out there where
that that site you'd like to acquire data
and the you're just outside
dub that state-of-the-art what can be acquired
and and you may have to wait seventeen years Intel
ATT's catch up right you're able to acquire your data
which is kinda for a crazy right but in fact
they're all kinda problems where there is that %uh there is an inherent
sparsity in our data
and we can use CF to take today stated dr. ATT's
and bootstrap them to higher performance and today I'll talk about either
wider bandwidth and even as we just discovered
wider bigger deeper bit depth and that that that
canonical signal that's like da bass 12 to talk about if you're talking about
ad's is the frequency hopper as it's like a terrible signal
for my quest great ATT's at this signal that this hopped around a different
frequencies
if you didn't know what the frequency hopper you knew you would have to sample
it twice this panel with
right because it could be anywhere in there by at any given point in time
there is only one side when active
it is extremely sparse so we should be able to do better
Sol at the problem as is and I think I'd is good to talk a little bit like the
video problem
the problem is that you cannot implement a system like this
in analog hardware right because
what does this say right what has been a TD in a titty have been analog signal
stream by
and they take sample it take samples
and once the signals here it's gone right what I just said
a few seconds ago is gone write it for
path with the Paul the arrow of time so problem here is what is this measurement
this measurement is a linear combination of the entire signal from the beginning
of time to the end of time
right to these kinda matrices are totally out their der der
not cause all and their you know infinite in Berlin really both
directions so we needed something like this
we need something that has an effect not even really like that but we saw me it
is
not top what's up not close to top what's right is kinda like a
you could build a linear with some cash system and it definitely needs to be
cause also has to be movie role in the upper
upper quadrant right to hear light green ala
what has 30 there so the question is can you can you do this
right can you build these by matrices that actually had this r.i.p property
and I its kinda have Bob neat but you actually can
and and you can actually show that that that while
by itself might be so sparse
itself I did not have enough entry that it will not have the r.i.p
the really important thing is that a lot of that kinda signals that we want to
acquire
when aidid these are sparse in the for a domain
right there are sparse in time spike their sparse in frequency
and so the problem is actually not that the measurements are
a by operating on a sparse vector axe
but they're actually five operating on a for a transporter of a sparse
for EA up for a transfer matrix offering up for a corporation vector which is
sparse
and the neat thing yes is that you can actually Bell fis
right with rely on hardware fact that fight I'm so I
at this the product actually has the r.i.p
going Diana state and theorems in a second
that is actually possible and here's one system
and we caught the rain and the modulator and for people who can take a shit is
just a spread-spectrum
demodulator we take our input signal
and we don't sample it at the Nyquist rate week chip it at the Nyquist rate
we multiply at by a chipping sequence
hot tub random plus minus one sequence that what
at randomly at the Nyquist rate
K after about the next race so here's my actual picture here here's my input
signals to set aside way
I have multiplied by this chipping sequence case I get this
and they wanna do is on your desk to an integrated up
I'm gonna integrate and a sample at a very low frequency
why does this team like it should do anything interesting
while let much more not clear in the for a domain
my sign a waiver for a domain is just a spike
right my chipping sequence says it's a random square wave
is basically a random waveform in the for a domain
a random way for K well what happens when I'm not apply to sign away by this
chipping sequence
by the usual junior theorem I just basically convolved that spike with this
which can volley a spike do it is shipped
racer kit shipped to circulate ships s so I have
this basically thinking this I got called I haven't shifted
and I wanted by integrating dump do it's basically a low-pass filter and then I
sample
II preserve the information inside at little
that window okay to think about sup
if you stare at this for a second he should be able to see
that up I know this little piece just imagine you knew this little piece
you can go up here and find it right you can find a way to it that came from over
here
right that came from over here which means I had to a shifted this signal to
put that stuff and zero by a certain frequency
you should be able to unravel the frequency by knowing
right the coding sequence used to chip this making sense
and nine other hand hopefully except and you to play all the same
I machinery and you can actually show this is actually with that
what that I I matrix looks like okay you can actually show
that I I at like prove a theorem
that as a lot better if our signal is is very simple
so it had only kato loans per window of time
so it's very wide band with but I have kato loans
then the sampling rate that I need at that back in a 2d
is just a linear in the number of tones and only logarithmic in the Nyquist rate
tail at the core logarithmically in the Nyquist
for exact same formula that K lock further here and we prove a troll trop
a few years ago eyes exactly death get exactly this the only difference is
there's a square in the log
that's just cuz we can't prove a tight enough by owner but in fact if you do
this
empirically you actually see that the number of measurements you need is
really just about 1.7
times K number of tones times a lot
va Arvada Nyquist rate which is really not
that's that's very nice for small a small constant
so here's an actual example so here's a ganar frequency hopping signal
and this is the spectrogram let's just that time very expect from other
here is what we call the sparse program this parcel Graham
is just a silly name for the spectrogram computed from the top
a diet what comes out of this deathshot compressed sensing a titty converter
NEC that at twenty times subject was measurement measurement
we definitely have a few artifacts here due to for leakage
up from that recovery algorithm but still all the tones are there
and really pretty easy to find right here's a better example
I'd like real data this is like a again some
neither actually FM tones I in in a in a pub
pretty wide up white noise background with real date and you can see we can do
a pretty good job pulling those
tones out and is again twenty times Supply Co
right well the reason why this is actually quite
potentially useful is that this is the interest if education at this
this this chart are that this this progression I love
upward law with respect ADD is right that kinda alluded to
the x-axis is the login a sampling rate
case a lot of the sampling rate so I few
if you increase affect you as we increase the well
i think is clear live sampling rate and up here is the number up the bit depth
up in a titty
so for people over to the famous walled in chart this is basically a reform
adding up the act
and what white wall and basically found is that there is
up i lying here these is actually really ADC data
I was just looking up part numbers missing why today
rip you to sample at and what county bit effective number better they claim that
they get
and you can see that there's a very clear line here that as you want it
sample faster you have to have fewer bit
right as a sample faster you need to have fewer bits not only that
their I this line is a line of Thai about that moves that I various law or
eight
like this eight-year doubling up this you know this blindly
okay so what is CS tell us k well we did action a brand new paper
I that actually will sorry with I that actually tells us
the point at the very top there that the this CF putting the CSS down with this
this this Randy Michel air in front of an AGC does not in any way destroyed a
dynamic range
it does not hurt the dynamic range
which means we can actually look at and I just realized I told the jury this
chart run:
because I didn't really show in a game here that flourish last
fiction in real time
how about that at
the truth every up yeah
so the poignant is that since we don't destroy
I that the since we don't destroy the dynamic range with our hot putting the
fan likes estimate from we can actually show
that since we do don't change the dynamic range
right but we do Inc up decrease the sampling rate
we can actually get apart which will sample and a greater bit depth
which means that we're able to effectively increase both the sampling
rate and a bit depth have a titty converters by putting these little
analog systems in front I hope that was clear
but it was actually something quite surprising to us we thought that we
would be able to move along this line
but in fact we can show now that we can actually push about the line and get
past state-of-the-art performance using these are
compress compressed sensing idea okay I think I'll
and and say that what we've been talking about today is
applying these idea to buy dimensionality reduction to show that we
can do to thanks to show that we can
preserve information and signal through random measurements
but also show that we contracted billion stable e recover
signals back from those randomized measurement and a and the reason why it
works is because I some beautiful
Matt geometric result in in mathematics say
randomized measurements are able to preserve information it did for any kind
of signal has a a concise geometrical structure
we talk a little bit about that math we talked a little bit about
I the building some ABC's This Way to go into radios for adult talk all day about
cameras
and and whatever I think very interesting open problems
hot date did that we're trying to work on
is to actually I work directly on
up a processing signals and images by working directly with the compressor
measurement
rattan and do %ah some kinda %ah recovery
so hopefully this was you stole a few
even know a lot about CS I already I think it should be clear that there's
all kind the links with
a whole bunch of different areas in information theory machine learning
I inference algorithms and and and and also the potential %uh really designing
and building radically new kinds of
hardware right now kinda sensors I using these techniques
I think of hand and let me just point you to that
that I repository RCS well us
up papers at Rice University there for two reasons I bring this up
the first is that if you're interested in learning more about this area
this is a good place to go because it has officially all the papers that have
been published
in this area and the second is that I really invite you
II as you're writing papers to please make sure that you are also contributing
them to the site up so that we can really I
try to continue this experiment and I open site
okay thank you very much Piper
I'll a very few questions saying that we have a reception afterwards and coupon
haul
betsy if there's a some very
few questions and then we can week in the
help from further discussions at the reception probably got trawler
11 last night he
compressed domain yes at now so what do you think is there's going to be like up
whole new like these ppl rebuilding the compressor mean AP Attorney theory
up machine learning theory although like Lee Dr
our union signal processing theory but now in the compressed domain are we
going there are were gettin yeah I'm asking could could question so
I think there's 12:32 answers 11 P
I we can look at what I think we should be looking at it
poor earning what we do now filtering
processing Dino raising machine learning
all that stuff porting those ideas so that week those concepts
into some ok other cable operators
they would operate on and compressive measurements that would have
back to the same result right so that since the thermal parting thing
the other thing we also need to be looking at it could there be
entirely new ways to process signals by doing things
yeah I to compress a measurement and I think that
example here would be you went when when
wavelet transforms came around people intel the Intel wavelengths and all
wavelet thresholding thinking on people and realized
how powerful certain kinds love nonlinear algorithm could be four
but so simple but so powerful up for doing processing on imagery and
different kinds a single
so could be that this inspires us to develop entirely different
kinda valor yeah question
okay well join me again in the way
thinking richer for wonderful talk thank you
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