thank you
oh
a
all the slide
oh okay
so it's uh
meaning and
on
about fifteen slide
separating us from
go check
so
as all of it
uh this
uh work
that the uh a sparsity technique K in order to take the problem of uh
C reconstruction from a noisy data
it was down uh yeah or uh joint it was my piece two supervisors makes it out than then because
have ski
a snapshot of the resulting a technique
is a
at the uh the low part of the slide
yeah
after the
offline training is a to the previous uh work here that
chaining of the data is not offline
and then uh uh are we use a standard to construction all
uh algorithm such as uh filter the back projection
uh but the
before that a some uh noise reduction and that's sending on them in the sending domain is that perform
uh using a like S and the sparse representations
and finally we get there
finally
so that's a snapshot and uh i'm getting that the T
but this recall of some basic uh model of the in
so that that could
have a foot the big
we have uh
in this case it today
but no us slice
uh of a male had which uh
um i'm scanning with X
so each rate
is is um
has that some in shell uh an initial intensity of i zero four dollars per unit time
and uh as the ray travels to this so the body
it uh the the the four ones are sort the it's so that issue
oh the final number of of for we count
how just east estimate the line integrals was uh
attenuation map
and the this is actually there i don't transform to two-dimensional radon transform of the
of that initial map
and the what you're measuring uh are there
approximations of uh a line integrals up to uh look
well four
now in the a does it's can when when we want to keep this
number i as a little and i zero
keep it low
yeah we have a a deviation from the a correct ideal number of uh for tones
the be count
therefore the two measures measurements are
um model those so on the variables
instances of possible variables was that uh
uh a parameter that uh them that which is the bows the expectation of this variable
and their variance of
so the higher the
had had a that the time that the and number of that most count
uh the that's nice we
so we have here a
day they the trade between a
a a good image is want to get a and the
radiated and sick patients that we
the two will
have you free
try to improve they image them
oh
so
number of things can but can be done in order to reduce the does it's in the in this can
is first as to use algorithms which are which were
specifically designed to
acknowledges this a statistical model
uh so there is a
uh there is uh
a a map uh objective which is minimised
i directivity
and the usually is those algorithms are quite slow
uh despite by the fact that uh they do
yeah truly improve the performance of uh
such a basic uh are or assist few to by projection or
other similar pressed methods
which are currently
to really are used in there and the clinical ct "'em" is K
another way to reduce drastically there
uh the amount of eric's will deletion
he's not to eliminate the entire uh the entire had
but if you want to you look on at the small region on the in a a a a the
head
we can get the where was radiating on the on that region plus some
small additional amount of data
uh theoretically we mask ready at a i i'd to the whole had in order to
the recover you been one picks still because the there don't transform is not uh is not local but in
practice we can
yeah to by was much this radiation and you get a good image of some region
there are a
a special the algorithms to do that
we want to consider this scenario one where we do a scan that are had
and the you know it to improve the result of to the by projection
we perform some uh sign enormous asians some processing of a set run before uh
uh we can applied
so just uh
we called of your of there
images that we get along the way
uh from their ask and had we get a perfect sine gram
where are was uh
uh which is computed from
but on measurements was a log transform
then we have a a data dependent noise
which is at that because of a a little for don't count to the system
and from here we want to recover
the um
the the final image
okay now uh i'm we define the goal want to get to the method
that that via a attacking the problem
uh i see that the
some them them image was done to this that to the symbols here so this is uh
you uh almost people sign
um
you know that the you three i try to decompose natural signals in such a a
yeah yeah
yeah frames a to of let's or discrete con signs or the for transform
we have a rapid decay of the coefficient
so uh uh similar we want to
take a model which where we assume that only a few non-zero coefficients are needed to represent the signal well
here the signal for this case is the small uh uh quadratic page for the image
we put it the as a straight vector
and what that we want to represent that as uh the product
of uh matrix D
by the present a presentation i far
which she where the D is redundant so that i with a much longer
but we only use few nonzero uh a few if you comes from D
and the we both want a sparse vector L zero norm
measure the sparsity
but that the number of nonzero uh elements
and so that the residual at a
would be small
um based on this principle there is a noise reduction technique from uh for uh for standard the signal image
processing
developed by a lot and a on in uh and i that the something six
the define uh this objective function which contains three
basic turn
first time is uh is if you don't to term
uh which you compare is the noisy image F T today
was supposed to be and the
while which try to recover at
a second one uh uh
request that the
all the representations are a
uh
the J a the J runs over all the small patches in the an overlapping patches
and it J yeah operator extracts a small patch from have
and it here it is compared to
uh a sparse and coding
uh uh in in a form of the times i for G
so i the J is a sparse presentation
which she L zero norm is small
and also the residual
the difference a L two norm normal difference
is the
a required to be small
how this a
this equation is sold online after the
and noisy noisy image is a so the dictionary D
and this that the representations are
boast lower and only from the noise addiction
there is also a of and and uh uh
and of to procedure with training images
are you
so here a uh is what's a what's called a case the algorithm
we minimize for the second and that certain that
and third relevant turns for D and i'm five
uh there are two steps
to to do it
we optimize for are five and for the i directory
and to compute the odd us giving a a dictionary D
with perform what's called the sparse code
we want to find their the the sparse just are a
so
so uh under the condition that a threshold
but uh uh uh a different um
and norm of the residual
is below some threshold a epsilon J
this is done uh use and and that prior approximate and go algorithm
a pursuit algorithm such that uh a a a a uh orthogonal matching pursuit or P
or other uh you don't go
a second stage in the in the saturation is addiction of date
it does not relevant to might dog so i'll skip
finally any we have the both dictionary and they presentation
we can compute the that you image using the first and the sir
relevant terms for the image
the there is actually a closed form a to to solve this uh
equation so it is not quite quick
okay
and now this technique which is by the were quite efficient for noise reduction in image images
uh was used
but a couple of years ago by D appear sapiro
to uh to to
produce a reconstruction of an image from um
yeah from a C
so it is basically the same question is we so a minute ago
except the fidelity delta term
compare the noise assigning signing and you do that
and the image this sort image have which is uh
a transform by their at done transform
um
well first of should say that the uh this uh paper the shows some
very impressive to the results
a a a a on the yeah uh image which images to was uh region mention structure
and there a severe conditions of of of the partial data of the used very few
projections
uh but the
uh
in principle there are few
uh problems in this in this equations which we want
but to try and the
uh repair in a different uh set
so first of all know what is the
the the use the L two norm in the in the P do to to
which actually means that the assumption is that the noise
is a home a genie
a however do know that uh in the sinogram domain the main the noise does depend on the data
more more than would know exactly how does the we know the variance of the noise so
this can be used
and the um the second problem was that for that the term is
we actually want to get a good the
a a or uh a low error in the
and the image domain main is it images of
step of that we are
it decrease in the error or be in the signing gram the domain
and since the i don't transform is ill condition
a
this does not tell as much about what you image yeah
we are we're are seeing Q
the second problem is the
which is also a model but the also was that the we can not
surely obtain these coefficients you G
um S is the are related directly
to the um
to the uh thresholds i'd uh on J we don't really use
these new J's
but for each you for each batch we we need to know what is the expected that uh error energy
so that
to the put here the correct threshold
if we don't know the noise statistics and we do not know the noise statistics in the a ct image
domain
because there are uh after the reconstructions the noise is
quite complicated
we can not
compute these thresholds uh
a a quite right there are some estimation techniques but the the don't be a was uh uh a very
good result
so
we are trying to solve the problem
yeah shifting the different this their own where we do have a a few the back projection
not is the
the this concept does not use and any reconstruction the just of the in the a minimizer of this a
equation
and we do use and offline algorithm which you
does the provides learning was trained
so
uh
we
she from the image domain
but sparse coding was done or to the sending gram the mean
and a want to stick code the the pitch is of sending gram instead of the image
uh
so the
the panel to that the be are seeking for
is uh should should be in the image domain "'cause" are we can require for some
uh a nice properties um
of the image that they want to
we what we using an offline training stage which on the one hand to be yeah requires some training images
on a another hand
it makes the algorithm very fast because all the heavy work is done once the you just can or is
initiated
and then the in the reconstruction stage
it's
almost all most just just
almost as fast as the a F P P
so
the algorithm uses a set of uh a reference images high quality ct images
and the also uh corresponding glottal signing grams
should be applied
such that can be obtained the uh for instance use and using cadavers or find terms which can be scanned
without the
any any hesitation is to uh the radiation dose
okay so the algorithm goes as follows
we use the um
the the case we de algorithm to train
for for but very similar
equation is well before but the mix of extract the patches of sending gram and not of the image
except them from that is uh it is just the send question
uh equation uh except from the very
yeah important difference that we don't need a a different uh coefficients for uh for different presentations
here no it is it's
a weighted L two norm was the spatial matrix W able
detail on in uh okay and next slide
so this uh help us to normalize the noise over all the pitch
okay so we do and coding was a fixed the threshold Q
which is actually the size of page a number of pixels
uh a and the noise is normalized corresponding to so that this will work
okay now
and once once none of that we have a dictionary
a and the set of us uh that representations
is that the uh
help us though and it the produced in good sparse and coding for sending
uh one could stop here and use this dictionary to uh to improve the center of in the future
but think again that want the in the penalty to be in the image domain in here
we are uh we're not comparing to the
uh well actually not comparing to anything we just acquiring requiring that
for each patch that would be a good sparse code
so the second step is to take these representations
and the to make
uh the dictionary a better able
now
this uh
expression in in here is actually the or there is stored sending graph
i take this sparse encoded patches
the times i've the J
yeah return then to the my and a gram metrics
and finally the M inverse uh accounts for their but uh patches overlap
so this is the sign a gram
after i remove all the unnecessary
uh i necessary noise
and D is the
some field that is some construction algorithm
we wanted to be linear for this equations the be sort of the is but it can also be known
in near you have uh
i if this can be still so
so it can be the feel be projection or some other a
uh in your uh algorithm like that sounds like a the free inverse verse transform
i mean for the free uh algorithm for the
uh inverse to london
okay so a here D is uh is the linear function in terms of a of the data provided here
so of the
this L to more use the easily minimize for the using good
so you could you could writing
and
all this is an offline training which prepares as these two dictionaries D one and D two
and the in the in the second training stage we compares the reconstructed image
with the original one
and then
we use a a also a weighted to L two more
which she allows us to
demand specific things about the error that we are we are we are observing this is a construction error in
this
in this term
and um
and the matrix Q allows us to do some things which ill which i will also shown in a in
a couple of men
and meanwhile while how do i you use this the uh a train data
given then you noisy uh
uh uh a G till that
i idea compute the computer presentations using the sparse coding was the the same threshold Q
and the diction do you one
and then this presentations are used to encode
is the the a gram the restored sinogram a jury
okay this is the same for mil
finally when a have that uh the center gram i applied the
you the projection to to compute the the
now
uh what are the matrices W and matches
make matrix Q or was talking about
you know to build
uh
a good the note the not normalize is the the the error in the centre of the domain so that
all these the differences would the all the same
yes a yeah
a you need to buy you need one a energy i need to recall what are the statistics of the
of the noise in
so one using the their statistical model introduced the in the beginning
one can deduce
but that the the variance in each location of the sound of gram difference between the ideal sending a very
and the
measured one
is uh approximately uh a in verse to the true photon count
a and the and
we don't have it but we can
hey a use a good approximation by the mel for count
so when a
yeah multiply
by one over this the variance
i have a a uniform noise no in the in the uh
in the centre gone the so
is summing over the Q
yeah yeah as and the patch i
expect the energy to be just
Q
and therefore i can take this uh weight matrix as a diagonal matrix C a containing the initial photon count
in know to to uh produce a to do uh
correct sparsity decoding was a on thrash
and now as as uh a to the question of uh
what
kind of error measure i using
in fact to was this uh with this slide i'm more hoping for uh your help that the coming to
tell the something "'cause" the this is something come
i stumbled upon that the
wasn't quite considered in the literature
"'cause" not much of the supervised learning in the C reconstruction was done so far
i would like to think of uh and error measure
it which you can be designed using good a good quality reference image
which could we which which would help me to promote a a a good properties of the reconstructed image
for instance if i'm not looking at the had
i'm seeing a regions of bound and re in the regions of air
uh which are um
uh
not uh not necessary for my reconstruction if i'm interested only in the soft tissues there is a a great
to the dynamic range of C videos
he read is about on south than for five hundred
here it is a mine one thousand and i one plus minus third
so in the in the is in the special L by by design
i remove those regions
it can be seen
in this is a in the piecewise constant a
phantom here that
those regions are completely removed from the map then the
are not uh
i i don't i don't come then when a a a a a reduce my i all that the rest
of the image
also i would like to
to emphasise the uh that's the the the edges just of uh of small tissues
so that here all the you know an sis will be
uh
a good a good uh we'll build a good quality
so overall i E upper the i design such weighting map
and the maybe there are would other designs that can the proposed
and with the uh with respect to this map
i can obtain a visually better
uh images
uh um finishing in just one minute
all the just like the shows some uh um based some um
results
this is the piecewise uh
find tom is the um random L
a strong all about it
this is the reconstruction was standard fit the back projection
where it their uh of the parameters
uh a cut the frequency was optimal each known
uh this is compared to to our algorithm
and to double those you'll to but projection the result
which E which use
twice as much for tones
and the by the signal to noise ratio measurements
yeah white one can observe the these are
more is the same
also there are a some our results on clinical images this is a head section from a visible human uh
source
and this again is the egg F bp P algorithm which is
a little bit noisy
here i can uh recover the final fine details
much better
which she also use a a roughly just about what we can do was a double those
in the if to by projects
so in to summarise
we we can see that sparse presentations
can well already were you
uh for for computed tomography and the they can produce a very good results
white well taking not
much of the computational effort
and they can be easily incorporated in the
existing clinic kind as "'cause" it is just a matter of maybe replacing the soft
uh
so that's is it for now and thank you very much
recent to use the return my phone
and
question
no but
if a break dct
yeah oh
oh
you a balloon or to make you
X exposed to of a construct a and and and
just uh
and i still and to explain to uh
i mean cell has has a done
yeah
it's a reading can radiating a small neighborhood of the region of interest and only if few global projection so
that the low frequency can all source to
oh that there are good it's
one is a question you know come
you know i
and
yeah approach of just what comes to a are right
i of getting explain quite so it's not like
you ask if my approach can be used for for that uh race
a for uh construction from partial date
if if i only if i only measure that
a race sort through the region of interest if i can use the this technique
yeah
i i a i like should i yes because my technique is local i only i i i think local
in the sending them take a small patches and what our work on that
so uh even if a if the if i have a partial data by and their i
have some method of way
of of uh of dealing with a like and the extrapolation which is usually used
i can still uh
work on the available power data and the some uh
a a preprocessing in know though to get better or uh
uh but S now there before i does the
uh apply that that a algorithm so yeah you're right those things can be combined
okay okay you know mark
yeah we got all speak now more so on
point five
i