um i will play a will explain and the second what a Q as complex as

uh this is the work of uh

my former student a a sundry high low grade it in two thousand ten then he would have laughed to

present uh this work but if we can a father last week

and uh i let that count as an Q

um so it's code advice the by michael a the yeah not much of it sure go from carnegie mellon

university and i was was at carnegie mellon and died recently moved to

you cage thirty

um so the word is a concern with you C G signals

so electrocardiogram signals

and uh you as an example of a measure D C G signal from some data base

oh you see it's a periodic repetition

of um

of

sort of elements that we to each other but with slight variation

and so if you take out one period

then and text books it's typically described like this so this is it is an idiot idealistic representation of that

period

and the cardiologist they divide this into into pieces

um with you see here is a T are interval and S T interval and in the middle of the

one that contains the peak that's stick Q R S and of a Q or as complex

for yeah i repeated it and um

the middle part that's the your S

complex of purist's interval and that's what we are concern

and uh

here's sort of a an extract version of it because you it's kind of a

which so this is roughly how it looks like

uh and it turns out to as

you know cardiologist tell us this as the uh most important part of this uh of the signal

and it's used to the for a variety of purposes and in in diagnosed

so

so cardiologist look at this part to determine whether to and you problems with a hard read them or two

detect certain D Cs

a typical duration of that part it's hundred milliseconds or a tenth of a second

and um you know you may imagine many C G signals are recorded and their stored in a database and

if you have millions of patients so

uh a and and and many many seconds stand this amounts to you to amount of data so that's why

people have started to study the problem of compressing these Q R S comp

because you one store it

but you want to reduce the amount of data

that it takes

so of course when you start compressing it you introduce an arrow and you don't want T error to be

so large that the diagnostics doesn't work in

so all goal is

is an essence since to

what we what we have in doing is present a method to

to compress

the

Q are as complex

while maintaining sufficient quality

that's and prior work on this problem because it's so it's relevant and that there was some early work by

certain more and others

from at one

and take compressed to Q are as complex by extracting certain feature

that they did reduce the dimensionality

uh a late later this work was subsumed by by this is uh a line of work

which uh which used or her meat functions to to describe these Q S comp

in essence taking that

part of the uh

taking the cure as complex and expanding it as a linear combination of a mute function

and the reason why i meet function came into play was not some

the mathematical theory that says a limited functions adjust the functions you want to use here

it was just because of the

similarity in shape it's just the timit function somehow have system looks similar

then together as a cue or as complex and that's why people started doing

our approach also is continuous this line of work and um meaning we take that you are as complex and

we

we expanded into a basis of discrete met function

and that this is a somewhat different than the prior work and and i weeks i will explain how

the first some background whatever you'd function is to understand to meet function we first have to look at the

close "'cause" in the army polynomial

i mean polynomials is a is a sequence of polynomials there's one for every degree

so the first ones are

one and Z uh so here you see for degree zero it's one and then four minus one it's two

at zero

and then you have a two term occur

and that because if defines in um

H L S degree at it's easy to see that

um and uh

because it's a two term occurrence that's a theory that says this put enormous all of can no with respect

to some weight function

if you take out this weight functions so you divided of

you get functions that are

truly orthogonal

without any weight function and these other of meat function

and don't you see that

the meat functions it's are in essence the polynomial

but no you have to store in front of it and you see there's to E to the minus least

square

and that just usually make sure that these functions tk very quickly as you go to infinity

and so are just a few examples of the first three or meet functions

and uh and you see that uh

that they have a support that this the entire realign but they D K very quickly because of the each

of the minus T square

and he you see a little bit you know that

good argue some similarity to Q R S K

that's why people start

and so just the orthogonality condition there exactly orthogonal on the real line

but because they D K so quickly you can an essence assigned to them

an interval of support and say the orthogonal and that interval approximate just because they are very close to zero

outside

new see there's also does magic factor uh my trigger parameters mar

and that's just stretch as

allows you to stretch these functions and that maintains the of net

so

so the previous work used is for me functions to to express these Q as complex as because of hour

that before it was because of visual similarity and the other worked as follows

you have a cure or as complex

so that one has a certain supports so first you stretch their meet functions to have the same support

then you pick the first i don't know however many you one fifteen seventeen a meet functions and you project

you Q as complex on to them by just computing the scalar product because they also but now

so compute D seattle L coefficients and the more of those you have the better you represent present you see

that's the idea

and then you keep the first so many

fifty

that's a use stop somewhere because you don't want to a and seen it many

and that's what used store

and then if you want to reconstruct you take these fifteen coefficients out of your data base and he the

reconstruction for

and you get an approximation of the original

so that was the approach take

now there a problem if you want to implement this yeah this stage you have to compute these integrals and

in the computer you cannot do that right you have to is tie some so they went on and discrete

times that

and if you discrete ties an integral

you use a credit sure rule

and the at it picked the standard one out of a book

and that this as the credits so the whole than an essence what you do is

you um

you sample at you a function at certain point

then you multiply by the length of interval

or to see that the usually

you have the interval of support

and i normalized for minus one to one

you divided in this case into a touch of seven just randomly

in to seven inch of and now you have to pick one

sampling

location in each of these intervals and what they did naturally actually the chose an actually actually spaced sampling because

it just a natural thing to

and then an that since this is the computation you perform you have your samples

at exactly space point

and then you have this

mar you multi but this transform matrix and you C these other meat

polynomials evaluated at actually distant point

and if you want to reconstruct a and you get the code fish

and if you want to reconstruct a

you uh you have to use the inverse trend

that's fine but there were a a a a few problems with the approach the first is this transform is

not orthogonal

um and typically be like to have orthogonal transforms for for a variety of three

signal processing

and then for this kind of transform there that that is not really a very efficient al

in other words it's a kind of really comes and

you have to do it by definition and that can be very expense

so so we built on this approach but

but uh changed something namely we change the uh the sampling points because

this theory of orthogonal polynomial

suggests jess explains that if you sample the actress space are not the natural sampling

but in fact

uh the sampling points are the remote

the proper sampling points should be the roots of of of the and i meet me norm and so i

in our case seven

uh a here again for price of seven you divide them to seven sub intervals you do not want to

sample at these is space points but you want to pick the seventh for me polynomial

pick its zeros

and these are sort of mathematically than natural sampling

and that's why be that interested to see whether these are the actual sampling points so the big you can

also get better result

and so you see this is here you see it not to space

and now you have a slightly different version of transform and reconstruction of the transform yeah the sampling points to

are not decrease space

and all the transform looks like this and these are the in essence the zeros of the of the and

t-norm polynomial

and the theory of uh are me in most tells us now that this transform some orthogonal so it you

want to reconstruct you can do would with the

with the trends

what's orthogonal

and um

um

and and just the second part is that actually it's non that for these transforms you have fast out

so to test

uh

to to test this approach to the uh uh uh for the icassp paper to time presenting here we took

uh we took a

right now a relatively small experiment taking twenty nine Q or as complex of from three different uh uh is

C G signal

an be compared to a be computed the compression ability

and we compared to the previous

or meet

approach

and then we through and also the dft and D T just a

yeah because these are the more natural candy that's when you can

so in essence dct based compression just like an image process

a do a dct transform to keep only the first five or six

but visions

and that that we tried that also

and and here the results

um and on the x-axis you C

how many coefficients we cap

and because once your as complex has twenty seven samples and this particular example of a few people all the

if you do a transform and then you keep all the twenty seven coefficient

you get perfect reconstruction euro error

on the Y is the arrow

and the fewer coefficients you keep the large as the error match

a now you get these are rate of fronts uh sort of these curve

and then depending on the area you can choose how many coefficients you key

and every line of the different algorithm and and our new one is the red one

not the question is what arrow can you tolerate

and i and after the icassp paper we started talking to a cardiologist and we had to a cardiologist go

a reconstructed signals

and then since the result was that ten percent is good

and more is not good

so in since you wanna cut here

and then you see that for ten percent year

the red one is a bit better than the previous one so if you if you quantify that

it's again about fifteen twenty percent and the compression ratio that achieve as a factor of five

so used store you transform and because to store only like five coefficients

before you had twenty seven of sec

and uh just a visual confirmation that ten percent

is reasonable and twenty percent not

so the black

line is a a is the a your as complex

as a obtained uh as in the database

and the the red one is the reconstructed one ten percent error uh and a twenty one of the blue

one is with twenty percent

you see the blue one is already quite

different

but really the gold standard is not

a a picture like this but what are we real

a practitioner say

so uh

after the get but we continued with the larger experiment where one a brief you show the results because twenty

nine or as complex as a

relatively small sample so

we went into the timit database and B now hundred sixty eight leads so hundred sixty at C E chick

uh is C G signals

and we two coats five fifteen a hundred of these company

and then we also compared to the to the wavelet transform because that's also a natural candidate for

for compression

um and you the results

and you see that uh

we have here different arrows

because we are a established that more than ten percent is not really acceptable you only need really to look

at ten percent error

and uh you see here the new algorithm which is a a a minor

only a minor modification of the of the one from from this icassp paper since the same one

this is the previous one dft based dct based and dwt bay

and uh you see always two numbers

so the first number is the compression racial by what's factor could you compress the data

and you see that all and of large scale experiment we get about a factor of five

oh that stuff the

a the full to reduction of the amount of data

and the the previous algorithm actually on that's larger or uh dataset of worse than on that smaller ones of

the smaller one was apparently nice for this previous gore

for now here the compression ratio is only three point five

so that sort of an improvement of uh

fifty percent

um and it turns out actually that the dft D T and D W

dwt T

a a compression to also reasonable candy that

uh actually some perform better than their meat based one

but the or like in the range around for

oh here

so

yes we

we better and you what you've seen the parent as this

is how many coefficients you need to keep

oh

you have a a high compression ratio you only need to keep few

coefficients efficient if you have a low one you need to keep more

or the method seems to to work

okay so my conclusion um i i i the method

for the compression of cure S complex as

using discrete ties to meet function

and it's really a could argue only a small sort of incremental step from the prior work which already used

continuous or meat function then using the quadrature rule

um but what's really interesting here arguably is that

that non could distance sampling performs better than a distance a

that's kind of nice because

the mathematics of these or meet functions suggests these as natural sampling points

i i this functions just have those points as an actual sampling

that's actually the reason why we studied the problem because we started actually

um

see approach to signal processing based on orthogonal polynomials and then we were looking for application and this is one

of the applications we found

but interesting thing is really that non equidistant sampling you can make say

or if you another what's if you deploy it this into real but application maybe be would immediately sampled those

C C G signals at those points

rather than actually distantly as it's of course done right now and

a so we need to actually to interpolate

to find the value

at these uh

these other

and uh and compared to the previous methods that's and also to compare to reasonable uh candy that's based on

wavelets some cosine transform

we achieve a a compression improve compression rates of like twenty to forty percent on no

about

a some obvious future works one can do and some of these things we already uh where we have been

doing in the last three like

you know better adjusting those to stretch factor because they you can still do some of fine tuning

um

there's the question of efficient implementations and be also that some results on having

a fast

a faster out with for this from me try

then of course the question is

can you use a similar approach to to other to the other segments of these C G or not to

Q as complex but but the other

and that concludes my talk thank you very much

which work

yeah

how we choose to segment is that um in sense if you look at the support of the Q or

as complex so this my depends on the support of related to the support

and so what i say did he looked at the uh interval of support of the cure as signal

and that narrows down the choices of sigma very much to like a small interval

and then in there you just does a little bit of trial and error

and uh because typically only need to do that once for a neat

and then you can keep the same signal for your entire lead of Q as complex it's which means you

you can actually invest computation time to really fine tune it uh you will not and of i

oh so no not a terribly sophisticated approach but the let's say works in correct

oh

exactly yeah oh what in um

uh we do but i mean you get those in from the hard be functions or

um always always the non-uniform sampling i mean what are do something instants the sampling points if you want um

if you um

that's a stick you are as complex has twenty seven samples

and uh that so in the beginning you would assume then you take the first twenty seven how me polynomials

uh meaning from to be put in a meat functions from number zero to number twenty six

and the sampling points are

the zeros of the twenty seven

one

okay and that seems like a we at random choice but if you study mathematical books on orthogonal polynomials

then they explain to use that if you choose to this is a general thing more for form of is

that if you choose if you use this approach to choosing the sampling points

the trends some will become orthogonal

okay that doesn't necessarily say that it would

perform better for compression but it turned out that actually it it

but that that's the

oh

hmmm

yeah

the

yeah

we have no no no

oh no insight here

it's really i think that is something that you can only find out a group

course

but Q as complex clearly has a particular shape

and the question as the basis function that correspond to this

but the different forms

which can approximate shape better

and and uh

yeah no no in so there's not nice biological a model

of the heart let's say that would explain to you

why what function of the natural one so we have absolutely no inside it's some period

in fact it was a surprise to us that

and many cases to just using the dct was better than the previous method on a meat functions and that

maybe the authors didn't

try

no problem

interesting so