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