Speech Transcript - SPARSITY-BASED SINOGRAM DENOISING FOR LOW-DOSE COMPUTED TOMOGRAPHY

0:00:19	thank you
0:00:28	oh
0:00:35	a
0:00:35	all the slide
0:00:42	oh okay
0:00:43	so it's uh
0:00:44	meaning and
0:00:45	on
0:00:46	about fifteen slide
0:00:47	separating us from
0:00:49	go check
0:00:50	so
0:00:50	as all of it
0:00:52	uh this
0:00:53	uh work
0:00:54	that the uh a sparsity technique K in order to take the problem of uh
0:00:59	C reconstruction from a noisy data
0:01:02	it was down uh yeah or uh joint it was my piece two supervisors makes it out than then because
0:01:08	have ski
0:01:10	a snapshot of the resulting a technique
0:01:13	is a
0:01:14	at the uh the low part of the slide
0:01:17	yeah
0:01:18	after the
0:01:19	offline training is a to the previous uh work here that
0:01:24	chaining of the data is not offline
0:01:26	and then uh uh are we use a standard to construction all
0:01:30	uh algorithm such as uh filter the back projection
0:01:33	uh but the
0:01:34	before that a some uh noise reduction and that's sending on them in the sending domain is that perform
0:01:40	uh using a like S and the sparse representations
0:01:44	and finally we get there
0:01:46	finally
0:01:47	so that's a snapshot and uh i'm getting that the T
0:01:52	but this recall of some basic uh model of the in
0:01:56	so that that could
0:01:57	have a foot the big
0:01:58	we have uh
0:02:00	in this case it today
0:02:02	but no us slice
0:02:04	uh of a male had which uh
0:02:06	um i'm scanning with X
0:02:08	so each rate
0:02:10	is is um
0:02:11	has that some in shell uh an initial intensity of i zero four dollars per unit time
0:02:17	and uh as the ray travels to this so the body
0:02:20	it uh the the the four ones are sort the it's so that issue
0:02:25	oh the final number of of for we count
0:02:27	how just east estimate the line integrals was uh
0:02:30	attenuation map
0:02:32	and the this is actually there i don't transform to two-dimensional radon transform of the
0:02:36	of that initial map
0:02:38	and the what you're measuring uh are there
0:02:41	approximations of uh a line integrals up to uh look
0:02:45	well four
0:02:47	now in the a does it's can when when we want to keep this
0:02:51	number i as a little and i zero
0:02:54	keep it low
0:02:55	yeah we have a a deviation from the a correct ideal number of uh for tones
0:03:00	the be count
0:03:01	therefore the two measures measurements are
0:03:04	um model those so on the variables
0:03:07	instances of possible variables was that uh
0:03:10	uh a parameter that uh them that which is the bows the expectation of this variable
0:03:15	and their variance of
0:03:16	so the higher the
0:03:18	had had a that the time that the and number of that most count
0:03:22	uh the that's nice we
0:03:24	so we have here a
0:03:26	day they the trade between a
0:03:29	a a good image is want to get a and the
0:03:32	radiated and sick patients that we
0:03:34	the two will
0:03:34	have you free
0:03:36	try to improve they image them
0:03:40	oh
0:03:41	so
0:03:42	number of things can but can be done in order to reduce the does it's in the in this can
0:03:46	is first as to use algorithms which are which were
0:03:49	specifically designed to
0:03:51	acknowledges this a statistical model
0:03:54	uh so there is a
0:03:56	uh there is uh
0:03:58	a a map uh objective which is minimised
0:04:01	i directivity
0:04:02	and the usually is those algorithms are quite slow
0:04:05	uh despite by the fact that uh they do
0:04:08	yeah truly improve the performance of uh
0:04:11	such a basic uh are or assist few to by projection or
0:04:15	other similar pressed methods
0:04:17	which are currently
0:04:18	to really are used in there and the clinical ct "'em" is K
0:04:23	another way to reduce drastically there
0:04:27	uh the amount of eric's will deletion
0:04:29	he's not to eliminate the entire uh the entire had
0:04:33	but if you want to you look on at the small region on the in a a a a the
0:04:37	head
0:04:37	we can get the where was radiating on the on that region plus some
0:04:42	small additional amount of data
0:04:45	uh theoretically we mask ready at a i i'd to the whole had in order to
0:04:49	the recover you been one picks still because the there don't transform is not uh is not local but in
0:04:55	practice we can
0:04:56	yeah to by was much this radiation and you get a good image of some region
0:05:01	there are a
0:05:02	a special the algorithms to do that
0:05:05	we want to consider this scenario one where we do a scan that are had
0:05:09	and the you know it to improve the result of to the by projection
0:05:14	we perform some uh sign enormous asians some processing of a set run before uh
0:05:19	uh we can applied
0:05:22	so just uh
0:05:24	we called of your of there
0:05:25	images that we get along the way
0:05:27	uh from their ask and had we get a perfect sine gram
0:05:31	where are was uh
0:05:33	uh which is computed from
0:05:35	but on measurements was a log transform
0:05:37	then we have a a data dependent noise
0:05:40	which is at that because of a a little for don't count to the system
0:05:44	and from here we want to recover
0:05:46	the um
0:05:48	the the final image
0:05:52	okay now uh i'm we define the goal want to get to the method
0:05:56	that that via a attacking the problem
0:05:59	uh i see that the
0:06:02	some them them image was done to this that to the symbols here so this is uh
0:06:06	you uh almost people sign
0:06:10	um
0:06:11	you know that the you three i try to decompose natural signals in such a a
0:06:15	yeah yeah
0:06:16	yeah frames a to of let's or discrete con signs or the for transform
0:06:21	we have a rapid decay of the coefficient
0:06:24	so uh uh similar we want to
0:06:26	take a model which where we assume that only a few non-zero coefficients are needed to represent the signal well
0:06:33	here the signal for this case is the small uh uh quadratic page for the image
0:06:38	we put it the as a straight vector
0:06:40	and what that we want to represent that as uh the product
0:06:43	of uh matrix D
0:06:44	by the present a presentation i far
0:06:47	which she where the D is redundant so that i with a much longer
0:06:50	but we only use few nonzero uh a few if you comes from D
0:06:55	and the we both want a sparse vector L zero norm
0:06:59	measure the sparsity
0:07:01	but that the number of nonzero uh elements
0:07:03	and so that the residual at a
0:07:06	would be small
0:07:11	um based on this principle there is a noise reduction technique from uh for uh for standard the signal image
0:07:18	processing
0:07:19	developed by a lot and a on in uh and i that the something six
0:07:23	the define uh this objective function which contains three
0:07:27	basic turn
0:07:29	first time is uh is if you don't to term
0:07:32	uh which you compare is the noisy image F T today
0:07:36	was supposed to be and the
0:07:38	while which try to recover at
0:07:40	a second one uh uh
0:07:42	request that the
0:07:43	all the representations are a
0:07:46	uh
0:07:47	the J a the J runs over all the small patches in the an overlapping patches
0:07:52	and it J yeah operator extracts a small patch from have
0:07:55	and it here it is compared to
0:07:58	uh a sparse and coding
0:08:00	uh uh in in a form of the times i for G
0:08:03	so i the J is a sparse presentation
0:08:06	which she L zero norm is small
0:08:08	and also the residual
0:08:10	the difference a L two norm normal difference
0:08:12	is the
0:08:13	a required to be small
0:08:15	how this a
0:08:16	this equation is sold online after the
0:08:19	and noisy noisy image is a so the dictionary D
0:08:22	and this that the representations are
0:08:25	boast lower and only from the noise addiction
0:08:27	there is also a of and and uh uh
0:08:30	and of to procedure with training images
0:08:33	are you
0:08:34	so here a uh is what's a what's called a case the algorithm
0:08:38	we minimize for the second and that certain that
0:08:41	and third relevant turns for D and i'm five
0:08:44	uh there are two steps
0:08:45	to to do it
0:08:47	we optimize for are five and for the i directory
0:08:51	and to compute the odd us giving a a dictionary D
0:08:54	with perform what's called the sparse code
0:08:57	we want to find their the the sparse just are a
0:09:00	so
0:09:01	so uh under the condition that a threshold
0:09:04	but uh uh uh a different um
0:09:06	and norm of the residual
0:09:07	is below some threshold a epsilon J
0:09:10	this is done uh use and and that prior approximate and go algorithm
0:09:14	a pursuit algorithm such that uh a a a a uh orthogonal matching pursuit or P
0:09:19	or other uh you don't go
0:09:21	a second stage in the in the saturation is addiction of date
0:09:25	it does not relevant to might dog so i'll skip
0:09:29	finally any we have the both dictionary and they presentation
0:09:32	we can compute the that you image using the first and the sir
0:09:36	relevant terms for the image
0:09:38	the there is actually a closed form a to to solve this uh
0:09:42	equation so it is not quite quick
0:09:47	okay
0:09:48	and now this technique which is by the were quite efficient for noise reduction in image images
0:09:53	uh was used
0:09:54	but a couple of years ago by D appear sapiro
0:09:57	to uh to to
0:10:00	produce a reconstruction of an image from um
0:10:03	yeah from a C
0:10:04	so it is basically the same question is we so a minute ago
0:10:08	except the fidelity delta term
0:10:09	compare the noise assigning signing and you do that
0:10:12	and the image this sort image have which is uh
0:10:16	a transform by their at done transform
0:10:19	um
0:10:20	well first of should say that the uh this uh paper the shows some
0:10:25	very impressive to the results
0:10:27	a a a a on the yeah uh image which images to was uh region mention structure
0:10:31	and there a severe conditions of of of the partial data of the used very few
0:10:36	projections
0:10:37	uh but the
0:10:39	uh
0:10:40	in principle there are few
0:10:42	uh problems in this in this equations which we want
0:10:46	but to try and the
0:10:48	uh repair in a different uh set
0:10:50	so first of all know what is the
0:10:52	the the use the L two norm in the in the P do to to
0:10:56	which actually means that the assumption is that the noise
0:10:59	is a home a genie
0:11:01	a however do know that uh in the sinogram domain the main the noise does depend on the data
0:11:06	more more than would know exactly how does the we know the variance of the noise so
0:11:11	this can be used
0:11:13	and the um the second problem was that for that the term is
0:11:16	we actually want to get a good the
0:11:19	a a or uh a low error in the
0:11:21	and the image domain main is it images of
0:11:24	step of that we are
0:11:25	it decrease in the error or be in the signing gram the domain
0:11:28	and since the i don't transform is ill condition
0:11:31	a
0:11:32	this does not tell as much about what you image yeah
0:11:35	we are we're are seeing Q
0:11:37	the second problem is the
0:11:40	which is also a model but the also was that the we can not
0:11:43	surely obtain these coefficients you G
0:11:46	um S is the are related directly
0:11:50	to the um
0:11:51	to the uh thresholds i'd uh on J we don't really use
0:11:55	these new J's
0:11:56	but for each you for each batch we we need to know what is the expected that uh error energy
0:12:02	so that
0:12:03	to the put here the correct threshold
0:12:06	if we don't know the noise statistics and we do not know the noise statistics in the a ct image
0:12:10	domain
0:12:11	because there are uh after the reconstructions the noise is
0:12:15	quite complicated
0:12:16	we can not
0:12:17	compute these thresholds uh
0:12:19	a a quite right there are some estimation techniques but the the don't be a was uh uh a very
0:12:24	good result
0:12:26	so
0:12:27	we are trying to solve the problem
0:12:30	yeah shifting the different this their own where we do have a a few the back projection
0:12:35	not is the
0:12:35	the this concept does not use and any reconstruction the just of the in the a minimizer of this a
0:12:41	equation
0:12:42	and we do use and offline algorithm which you
0:12:45	does the provides learning was trained
0:12:49	so
0:12:50	uh
0:12:51	we
0:12:52	she from the image domain
0:12:54	but sparse coding was done or to the sending gram the mean
0:12:57	and a want to stick code the the pitch is of sending gram instead of the image
0:13:01	uh
0:13:02	so the
0:13:04	the panel to that the be are seeking for
0:13:07	is uh should should be in the image domain "'cause" are we can require for some
0:13:11	uh a nice properties um
0:13:13	of the image that they want to
0:13:16	we what we using an offline training stage which on the one hand to be yeah requires some training images
0:13:21	on a another hand
0:13:23	it makes the algorithm very fast because all the heavy work is done once the you just can or is
0:13:28	initiated
0:13:29	and then the in the reconstruction stage
0:13:31	it's
0:13:32	almost all most just just
0:13:34	almost as fast as the a F P P
0:13:38	so
0:13:38	the algorithm uses a set of uh a reference images high quality ct images
0:13:43	and the also uh corresponding glottal signing grams
0:13:47	should be applied
0:13:49	such that can be obtained the uh for instance use and using cadavers or find terms which can be scanned
0:13:54	without the
0:13:56	any any hesitation is to uh the radiation dose
0:14:00	okay so the algorithm goes as follows
0:14:03	we use the um
0:14:05	the the case we de algorithm to train
0:14:08	for for but very similar
0:14:10	equation is well before but the mix of extract the patches of sending gram and not of the image
0:14:17	except them from that is uh it is just the send question
0:14:20	uh equation uh except from the very
0:14:23	yeah important difference that we don't need a a different uh coefficients for uh for different presentations
0:14:30	here no it is it's
0:14:31	a weighted L two norm was the spatial matrix W able
0:14:35	detail on in uh okay and next slide
0:14:37	so this uh help us to normalize the noise over all the pitch
0:14:43	okay so we do and coding was a fixed the threshold Q
0:14:47	which is actually the size of page a number of pixels
0:14:51	uh a and the noise is normalized corresponding to so that this will work
0:14:55	okay now
0:14:57	and once once none of that we have a dictionary
0:15:00	a and the set of us uh that representations
0:15:03	is that the uh
0:15:04	help us though and it the produced in good sparse and coding for sending
0:15:09	uh one could stop here and use this dictionary to uh to improve the center of in the future
0:15:16	but think again that want the in the penalty to be in the image domain in here
0:15:20	we are uh we're not comparing to the
0:15:23	uh well actually not comparing to anything we just acquiring requiring that
0:15:28	for each patch that would be a good sparse code
0:15:31	so the second step is to take these representations
0:15:34	and the to make
0:15:36	uh the dictionary a better able
0:15:38	now
0:15:39	this uh
0:15:41	expression in in here is actually the or there is stored sending graph
0:15:44	i take this sparse encoded patches
0:15:47	the times i've the J
0:15:49	yeah return then to the my and a gram metrics
0:15:52	and finally the M inverse uh accounts for their but uh patches overlap
0:15:57	so this is the sign a gram
0:15:59	after i remove all the unnecessary
0:16:02	uh i necessary noise
0:16:04	and D is the
0:16:06	some field that is some construction algorithm
0:16:09	we wanted to be linear for this equations the be sort of the is but it can also be known
0:16:14	in near you have uh
0:16:16	i if this can be still so
0:16:18	so it can be the feel be projection or some other a
0:16:22	uh in your uh algorithm like that sounds like a the free inverse verse transform
0:16:28	i mean for the free uh algorithm for the
0:16:30	uh inverse to london
0:16:33	okay so a here D is uh is the linear function in terms of a of the data provided here
0:16:40	so of the
0:16:41	this L to more use the easily minimize for the using good
0:16:45	so you could you could writing
0:16:48	and
0:16:49	all this is an offline training which prepares as these two dictionaries D one and D two
0:16:54	and the in the in the second training stage we compares the reconstructed image
0:16:59	with the original one
0:17:00	and then
0:17:02	we use a a also a weighted to L two more
0:17:05	which she allows us to
0:17:07	demand specific things about the error that we are we are we are observing this is a construction error in
0:17:12	this
0:17:13	in this term
0:17:14	and um
0:17:15	and the matrix Q allows us to do some things which ill which i will also shown in a in
0:17:20	a couple of men
0:17:23	and meanwhile while how do i you use this the uh a train data
0:17:28	given then you noisy uh
0:17:30	uh uh a G till that
0:17:32	i idea compute the computer presentations using the sparse coding was the the same threshold Q
0:17:38	and the diction do you one
0:17:40	and then this presentations are used to encode
0:17:44	is the the a gram the restored sinogram a jury
0:17:47	okay this is the same for mil
0:17:50	finally when a have that uh the center gram i applied the
0:17:53	you the projection to to compute the the
0:17:58	now
0:17:58	uh what are the matrices W and matches
0:18:02	make matrix Q or was talking about
0:18:04	you know to build
0:18:05	uh
0:18:06	a good the note the not normalize is the the the error in the centre of the domain so that
0:18:11	all these the differences would the all the same
0:18:14	yes a yeah
0:18:16	a you need to buy you need one a energy i need to recall what are the statistics of the
0:18:21	of the noise in
0:18:23	so one using the their statistical model introduced the in the beginning
0:18:28	one can deduce
0:18:29	but that the the variance in each location of the sound of gram difference between the ideal sending a very
0:18:35	and the
0:18:36	measured one
0:18:37	is uh approximately uh a in verse to the true photon count
0:18:42	a and the and
0:18:44	we don't have it but we can
0:18:45	hey a use a good approximation by the mel for count
0:18:50	so when a
0:18:51	yeah multiply
0:18:52	by one over this the variance
0:18:55	i have a a uniform noise no in the in the uh
0:18:58	in the centre gone the so
0:19:00	is summing over the Q
0:19:02	yeah yeah as and the patch i
0:19:04	expect the energy to be just
0:19:06	Q
0:19:07	and therefore i can take this uh weight matrix as a diagonal matrix C a containing the initial photon count
0:19:14	in know to to uh produce a to do uh
0:19:17	correct sparsity decoding was a on thrash
0:19:22	and now as as uh a to the question of uh
0:19:26	what
0:19:26	kind of error measure i using
0:19:28	in fact to was this uh with this slide i'm more hoping for uh your help that the coming to
0:19:34	tell the something "'cause" the this is something come
0:19:37	i stumbled upon that the
0:19:39	wasn't quite considered in the literature
0:19:41	"'cause" not much of the supervised learning in the C reconstruction was done so far
0:19:46	i would like to think of uh and error measure
0:19:50	it which you can be designed using good a good quality reference image
0:19:53	which could we which which would help me to promote a a a good properties of the reconstructed image
0:19:59	for instance if i'm not looking at the had
0:20:02	i'm seeing a regions of bound and re in the regions of air
0:20:06	uh which are um
0:20:07	uh
0:20:08	not uh not necessary for my reconstruction if i'm interested only in the soft tissues there is a a great
0:20:14	to the dynamic range of C videos
0:20:16	he read is about on south than for five hundred
0:20:19	here it is a mine one thousand and i one plus minus third
0:20:23	so in the in the is in the special L by by design
0:20:27	i remove those regions
0:20:29	it can be seen
0:20:30	in this is a in the piecewise constant a
0:20:34	phantom here that
0:20:35	those regions are completely removed from the map then the
0:20:39	are not uh
0:20:40	i i don't i don't come then when a a a a a reduce my i all that the rest
0:20:45	of the image
0:20:46	also i would like to
0:20:48	to emphasise the uh that's the the the edges just of uh of small tissues
0:20:55	so that here all the you know an sis will be
0:20:57	uh
0:20:58	a good a good uh we'll build a good quality
0:21:02	so overall i E upper the i design such weighting map
0:21:05	and the maybe there are would other designs that can the proposed
0:21:09	and with the uh with respect to this map
0:21:11	i can obtain a visually better
0:21:14	uh images
0:21:16	uh um finishing in just one minute
0:21:18	all the just like the shows some uh um based some um
0:21:21	results
0:21:22	this is the piecewise uh
0:21:24	find tom is the um random L
0:21:26	a strong all about it
0:21:28	this is the reconstruction was standard fit the back projection
0:21:32	where it their uh of the parameters
0:21:34	uh a cut the frequency was optimal each known
0:21:37	uh this is compared to to our algorithm
0:21:41	and to double those you'll to but projection the result
0:21:45	which E which use
0:21:46	twice as much for tones
0:21:48	and the by the signal to noise ratio measurements
0:21:51	yeah white one can observe the these are
0:21:54	more is the same
0:21:55	also there are a some our results on clinical images this is a head section from a visible human uh
0:22:01	source
0:22:02	and this again is the egg F bp P algorithm which is
0:22:06	a little bit noisy
0:22:07	here i can uh recover the final fine details
0:22:11	much better
0:22:12	which she also use a a roughly just about what we can do was a double those
0:22:16	in the if to by projects
0:22:19	so in to summarise
0:22:21	we we can see that sparse presentations
0:22:23	can well already were you
0:22:26	uh for for computed tomography and the they can produce a very good results
0:22:30	white well taking not
0:22:31	much of the computational effort
0:22:34	and they can be easily incorporated in the
0:22:36	existing clinic kind as "'cause" it is just a matter of maybe replacing the soft
0:22:42	uh
0:22:43	so that's is it for now and thank you very much
0:22:50	recent to use the return my phone
0:22:54	and
0:22:55	question
0:23:00	no but
0:23:01	if a break dct
0:23:05	yeah oh
0:23:06	oh
0:23:07	you a balloon or to make you
0:23:13	X exposed to of a construct a and and and
0:23:20	just uh
0:23:21	and i still and to explain to uh
0:23:27	i mean cell has has a done
0:23:30	yeah
0:23:30	it's a reading can radiating a small neighborhood of the region of interest and only if few global projection so
0:23:36	that the low frequency can all source to
0:23:40	oh that there are good it's
0:23:42	one is a question you know come
0:23:47	you know i
0:23:49	and
0:23:49	yeah approach of just what comes to a are right
0:23:53	i of getting explain quite so it's not like
0:23:59	you ask if my approach can be used for for that uh race
0:24:03	a for uh construction from partial date
0:24:06	if if i only if i only measure that
0:24:08	a race sort through the region of interest if i can use the this technique
0:24:15	yeah
0:24:17	i i a i like should i yes because my technique is local i only i i i think local
0:24:23	in the sending them take a small patches and what our work on that
0:24:27	so uh even if a if the if i have a partial data by and their i
0:24:31	have some method of way
0:24:33	of of uh of dealing with a like and the extrapolation which is usually used
0:24:37	i can still uh
0:24:38	work on the available power data and the some uh
0:24:42	a a preprocessing in know though to get better or uh
0:24:45	uh but S now there before i does the
0:24:48	uh apply that that a algorithm so yeah you're right those things can be combined
0:24:53	okay okay you know mark
0:24:57	yeah we got all speak now more so on
0:25:02	point five
0:25:04	i

SPARSITY-BASED SINOGRAM DENOISING FOR LOW-DOSE COMPUTED TOMOGRAPHY

Medical Imaging

Presented by: Joseph Shtok, Author(s): Joseph Shtok, Michael Elad, Michael Zibulevsky, Technion / Israel Institute of Technology, Israel