thanks first to me and and joe for for organising very interesting session which also gives us a the in to be prime uh so it's a will be talking to you about the reconstruction algorithms under sparsity constraints and so that kind of relates also to that's of activity going on in signal processing it a compressed sensing in give you some tuition uh in "'em" are i and by means nee since uh tomography um and the also i i mean in doing so i'll give you also kind of the art of for uh i turn to a reconstruction algorithms that use the uh uh regularization constraints uh by by the way of sparsity so uh first to a we i i i i i will just discuss a little the variational approach to image reconstruction will need to minimize so of some some uh a functional and and where we introduce some sparsity constraints then i will read uh read you some of the standard algorithm in particular uh it soft thresholding the uh uh the algorithm that's is stuff the fast version of that which is a a kind of multi step i'll grid which is gives you sort of the state of the arts coming from convex a analysis and then uh uh in this talk i i i i want to i discuss our variation of used star uh which is sort of you note a like to the problem which will use step adaptation to to gain uh some speed because the cool here really to get to the speed of the conventional in your taking is that have been use for for many years and then uh application to power L or i and and by mean this nest since the tomography so here is uh just the set so uh okay so we we have some object X that we'd like to to image for for example a proton density or some absorption map and then we have these scanner here that uh we will just assume we have a linear measurement system which is the the usual assumption in them are i X ray and so it's an integral operator but to since here we have a discrete eyes the problem so we can all represent body big fat matrix H that uh represents all our physics knowledge of of the problem and then uh at the outputs uh oops the yeah i here a see we have some modeling perfection and noise and and all that i i represent by oops by be uh maybe i should be problem like lawn knowledge knowledge that's professes where oh in the room uh okay so so uh uh a so here was saying we we have noise and so our imaging matching model here looks very easy just the this uh imaging matrix up like to the object that we like to recover plus some noise and then in the variational approach so you just set up this kind of uh uh that this criterion here where you have a data consistency requirements so we you you are kind of uh reconstructing X and then you are you're simulating your measurements and you would like this difference to to be very small spec to the true measurements and then here you it should use some regularization constraint and this is the arts of reconstruction here to to to just choose the the the right trigger is regularization and and and a classically here uh for example if we put a a quadratic term your was that's say sum but people operator here that it's that's tikhonov regularization but now uh i with compressed sensing actually all this was started by the observation that so many images natural images would represent was uh well was very few large group fusion is some basis in particular wavelet basis and and in fact this happens as well uh very significantly with medical images since so here's some "'em" are i image and and just the showing different transformations and and and and and so that you can get a pretty good uh a signal restitution but just keeping being a very small percentage of of large coefficients and and and so this this is really like used uh to promote uh i mean to to sort of justify the you the use of wavelets and so here then the idea ideas to use a regularization where you will impose a and one and that T uh a on the wavelet coefficients so here are would uh represent let's say might wavelet transform and one norm uh uh uh promotes sparsity and why it's good its convex so so that we have like some i guarantee you existence uh this oh a solution and also it will favour sparse solution okay so here is sort of uh the grandfather of of those uh a sparse a reconstruction written and and and so uh what do what you want to do is to minimize this okay you have this L two term and this sparsity constraint here i'm just as to me X is the representation in the wavelet basis and i'm using and augmented just the matrix that's sort of contents this a a wavelet transform followed by by by by by by uh i imaging device so then you have to very special case very easy completely classical also first if you put a day will zero just the square problem so it's a a text solution is one like calculation you get that uh and four she those i which may way a you at and rob you would not even want to compute that but then this sort of the grand father of our britains here which is but and a grid uh on on on to this data term and and and i mean this is sort of a you know the big the backbone of many iterative out and the i mean this works but the you have to two to stop it after a certain number of iterations but now the other interesting cases H is i you or you are like a wavelet transform in that case fact this problem you can by just going the wavelet domain uh doing a a a a a a soft thresholding and going now it soft thresholding algorithm which was the uh discovered well many people probably but uh feed yeah they do in no of act yeah uh where where among the very first here using your yeah formulation and so the a kind of a start if you were just to thing those two very three of remarkably uh this would solve this problem should look like to very difficult problem and then do a she the use them and two thousand four were able to sure that action prove convergence and so that that was uh a little revolution in the field although i'm say the output myth so maybe not so usable in that form and now if you just look at the structure actually the fixed points uh algorithm so uh what you have is a rule which is this a gradient this sense followed by based thresholding that will give you the new update and then there's this nice mathematics that guarantees convergence and fortunately it's very slow so that's been shown now that uh the cost will a decrease like one over and where and the number of iteration but then by uh building on many many years of full convex optimization then there was this proposition position in two thousand nine of fist yeah which is a like you control over the relaxation evaluation on this thing but essentially what it is it's it's a an updating dating a G that not only text uh uh uh uh built upon the previous estimate but takes two pretty a so and minus one and and and and then will a a construct a kind of vacation well car uh the correct uh update should be and then apply the is to out so to to just make it problem trying to solve and are remarkably this fist uh algorithm use you uh uh and square so it it's one or there faster or and and and and so gives you now a a a fast enough our to to to apply this kind of techniques so now what we have proposing here it's some variation and and so we call it fast the okay weighted iterative soft thresholding algorithm and and the idea is pretty simple so we had the step size or but not your yeah getting factor for for the great so we replace that by a diagonal matrix and actually we can also uh some flexibility on the a side of the regularization so with the slightly more channel regularisation matrix and you she uh the this is uh or or style with my are derived using uh max a uh uh uh and M a a a a a a a as a sort of constructing a a a quadratic bound on on the cost functional and and then all using a you know always trying to go and the bottom of this quadratic bound no if you have more flexibility here with introducing this matrix fact but you can do you can construct better a quadratic bound that re tailored to it to your problem and so hopefully this should allow us to go faster and then actually we we can do some have here and and so we obtain a now this uh this kind of uh in a quite the on the cost so no if you look at it superficially for mathematician doesn't look so much better because we still like what are and where one or and square but the important thing here we instead of having a in a norm we have weighted and actually this i for the call oh and now it means that clever engineering can really like from down those calls quite significant and and then can uh and that a be much fast so the application here i just want to a show you two examples by by the way i forgot to knowledge my cool all is okay and who the want to did all the work and and this is the first one matt i'm much should just can K on is working on product and more i actually in collaboration with the guy who invented the pollen and the rice so it's cuss was month from it T H three so it here the idea is just to to have the a bunch of colours here and for the soon or a little basics about "'em" or rice so M are i it's she we compute samples of of a a a a measure samples of the for yeah transform but if you have colours in multiple colour they also this sensitivity function so it's it's you are object X multiplied by a C a function that's a sense it to you but you of your cord and doing the for transform all all that and in power and the right you just put several core so you you can do the the measurement in parallel and hopefully this allows you to reconstruct images using fewer samples and so now here how can we exploit the structure of the problem so we need really to they those degrees of freedom that that left which is this diagonal weighting matrix so if you look at the pollen and more i what we have so for for each call here or you would have this sense T matrix which is just a a i uh a diagonal matrix followed by full you and now the problem here is this sensitivity can vary a not and on on you on your on your course and it's a pretty in a gene use and that makes the the problem not so well conditioned because for many years actually i are i would mean you and inverse for your transforms so with the unitary for vol oh rate in the problem for inverse a the imaging uh if inverse problem people but uh i mean with was that it becomes much uh uh less less well uh condition so now the idea of sensitivity based a precondition conditioning here uh okay so it's uh try to uh inverse he respect respect to to to the sensitivity just tried to get the better condition uh a system matrix uh in in this uh precondition them variables and so now if you think of it in the wavelet domain you could almost to the same but okay uh you just have to transpose your send i in the wavelet domain and an intuitive is just the way that is more less local so you would just i agree a sense C P T V T oh over the support or of the where let's and and and then get this kind of precondition E matrix here here and and and so that makes a a much better version after precondition version of the great so here are some examples i could also have shown you was real data are but here were just using the ship and look and don't because just the to you know compare the performance of the R so we had taking a a pretty realistic you a set up here with the uh for receiving corps some some noise here and and uh a certain number of of a radial a trajectory and where using uh a a reconstruction in the in the hard to domain using a two level haar transform and so this would be the traditional is to out so this chart soft thresholding and so this is the cost it but it would should oh of the cost function and this is the uh this is the the error with respect this solution actually exact sure optimization problem and so you see this kind of uh a typical uh it pollution of the classical uh i i mean the typical out with is not very fast and then if you do fist uh it's much five this is and square type of performance but now if you do this trick conditioning as as as we propose in this work we get more less the same slope but okay we get a or earlier and actually in practice this makes a huge difference because we may be happy here K with this kind of uh performance and so it can means we can get there uh at that much fun "'cause" as the number iteration and and same a here so maybe just to show you some uh uh uh examples of reconstruction so here have just showing you the evolution of of the solution was a number of iterations so that's the T star so it or too soft thresholding goes there very slowly fist fist close them very a much faster and if we have those appropriate weight we will there even faster so she here we chose a such that a uh we get yeah this quality here with this number of iterations so so we can a essentially the uh or or or here here's it what's okay to actually this is better quality of that so we can go go there for five this particular so why not yeah times that's sir and a a faster and and and anyway all those methods that uh optimising the same cost function okay okay so then these second application was but you wouldn't since tomography and so this is a a a a much harder inverse problem of this very very badly conditioned and hear what you have your you your you have some uh uh by hmmm hmmm this sense uh uh uh a Q markers within the specimen and and then a you are you i have like detector just the around the object of interest and here i mean you you have okay optical uh optical setup but i i i mean here the like this just if you and and in fact you have the diffusion equation so you can shall solve this diffusion equation using the green function and this will give you a a a a a system matrix and voice once to a a and near uh a model but three need badly condition because was green function so here the green functions actually look like that so it is it K like one over uh a a a normal hard to the power to and it happens be very well condition also so the idea of now is a by second call third row shot was try to you know use this flexibility of the precondition conditioning tried to try to you mode get a much better conditioning by just multiplying this by this diagonal matrix and so yeah the inside that if you few where a more this normal uh uh this a system matrix to make it like constant and energy it would produce a much better condition and and and so this this results in type of of set up for uh reconstruction of group and in this case of so here we have also some experiments use syntactic force just to demonstrate that it really works so we just have to for uh a by lee mean S and sources point sources so here we are imposing sparse state of because it's uh a very small sources here and here's a comparison of this three algorithms by the way also uh physically realistic modeling of different optical palm which is uh number of sources detectors et cetera also some noise and so here it's to a bit of course this slow the fist act was much faster and our i'll get was at the same the rate i the fist are but gets there much faster and in this case we have a ten fold you but essentially using the same out with just with this clever uh precondition E here are uh some is also here is the uh the source and and so this is the evolution of the standard i'll go so it has a no it takes a C is very badly conditioned so it's very very low it will take a long time to get there this is the fist a goes faster and this is the weighted fist or and and essentially here we also set it up so that you have have a correspondence your so it's a bit and fine or ten times is a lot oh for image so so then this makes it much more practical so that really gets me to the and here so the conclusion here uh in a principal we hope we can hope we get better quality taking advantage of of all those methods based on on sparsity so it has been demonstrated in the field that you get better instruction now we can also it might think speed and this is this a she's T Q was of different next generation strategy so the multi step update the adaptive weights and and so now present we we with a the factor so or even a little less of the best the you here taken we which are conjugate way and and so now it becomes very applicable now that when used for signal processing this you like it's not like you can our to requires you expert is to just find the right i i that this step and conditioning so that you algorithm we will work for us and finally generality because it's transpose able to many modalities as uh meal and was saying that X rate tomography more breathy action the lab we also use a uh working face thus the more graffiti also or S my process other modality so that's just you and thank you for your uh attention i have time for maybe one quick question just one if you can time i H i seven wavelet sparse in seven like total variation a their air you technique can be used X a total variation a construct yeah actually is uh a a a very good question which just the actually yeah i have a paper on that so that hopefully should come out in journal i the that so also uh uh there's a trick with wavelet uh a because where that's and not completely shift invariance so so uh two in fact if you just a use a a conventional wavelet and poke the valuation of the valuation works a little but but now if you allow for sure wavelet wavelet they something that's called cycle spinning we show that we can get as as good or even a little better than total the variation and it a in its much faster uh uh to a rate the way that you bed the total valuation where is all this a you do you are but of course people are also working on total variation and there also fast version of total variation but uh i i think it easier to get five our written uh with with the the sort of uh since this this formulation rather than and that is formulation but uh and and so the good news is this with way that we can it yeah that's with that oh so it's we have documented it with the M or i it's like my so once again things