thank you um good afternoon um so one of the problem that we tried to fall an image processing is a a is of some data that have been exposed to geometric transformations for example we might want to reduce is such data are or or classify them in a transformation invariant like one of a common approach for um be with such problems is the use all my fault models so in this work we have a um concentrated on that transformation manifold and it at transformation manifold in that family a little images that are generated by a in certain set of a geometric transformations to reference better for example if you take the structure ten P we do not its transformation manifold by P here and uh so we assume that this is an an picks out much and the mind fold is also a subset of R and this case each each which on this the transformation manifold is a geometric a transformed version of P and we define just transformation by a parameter vector or that in the parameter space and just that uh this uh a lot that house as the type of the geometric transformation for instance uh it could be any combination of two D to transformations like rotation and translation scale change one can find also for example so our for as in this work is uh the following we assume that we in one a set of uh geometric metric "'cause" transform observations of a signal type like uh a five digit illustration a from the observations we are trying to construct a it pattern transformation manifold so in part we want to find a pattern P such that the transformation manifold of P uh represents about this state that so it's like a extra fitting problem but if it and when you to the data instead of like so that problem is the to find a spectral P so um this kind of a framework has some meaning that applications for the modeling and the registration of the input data on including is also possible because we will be finding the pattern P in terms of uh some parametric at so it's also used to called the input data and like and also another ad don't use that we provide an unknown not to the model for our money false so that we can since to sides we can generate a new data on the manifold and this makes it possible to compute exactly at distance bit mean it's estimates and construct a old so this can be a a time as some classification settings for instance if we are given that test image some geometric transformation if you want to class by we just need to compute its distance to the uh of the transformation manifold so for a so that all i was first uh try to form like the problem than i will describe a solution that of the based on computing a representative pattern P uh with the greedy out to great really by selecting some atoms from a parametric dictionary so uh here's a show the manifold is not the by a and he and each image on the manifold of this from by you um that P it means a and the pattern P and uh i applied it to from some which the on it and we denote are uh input them just by you why this R uh uh this gone for some geometric transformations and what we are trying to do is find a common reference pattern P and model be um input points uh uh is transformation of this common pattern P plus some uh ever try and this error time you i shows the deviation of the image you i from the construct mind for and uh we assume that you know the type of transformations for instance you know bidders rotation translation scale that's a drought but still we to re just their input that the that means we need to compute a vector along the i for each of input image and then we use this idea phone construct thing P is a combination of some uh i so P equals the sum of atoms a J base of by this collection of C J and we also assume that use that sums come from a parametric dictionary that means each atom in a dictionary is a a geometrically transformed version of an an i'm not function so mother functions from by five here this is a a a a a marshal so the geometric transformation and some possible a little uh some examples for this on and will uh a generating mother function could be a process cost and motor function or an isotropic refinement but or functions from by a and R and here you see some at some that are um the i form house thousand motor function to some geometric transformation and um here is the formation of this month for fitting problem so we like to minimize the total distance of our input images to construct the money full we shall we by E and and we want to we would like to uh it she'll just by picking a subset all the atoms in the dictionary slow not P us these A J that comes a G R and also optimized the for options of these atoms such that this total distance that are he is mean the uh and you know next case of this uh read out from that we propose so we first so choose arbitrarily and that to mean the dictionary a suitable one and then be set that part pattern P uh and then we compute the projection of are input images on the money and then here the main loop now all uh at each iteration we select and at some at a and the coefficients C such that we reduce the errors and then we at this at some our pattern so this this based on my fault and an now the money for that it is a very compute the projections of are uh input image of on them i if what and then we continue this loop and till the the data approximation error is minimal and now how to be a of the minimisation of this error are still i'm fortunes as error has a complicated the panels on the at and option and is for the following reason uh let's imagine that we are now in the j-th iterations of the already have a computer this manifold and P J A lines one and so if you take an input image you why i mean its projection smile of that's already compute so we know the parameter vector or but i corresponding i mean were when the a minor followed by adding and you want it's projection point no change and most probably will correspond to a parameter vector number i pride which is a a different from um by and we don't know what this number by prime will be uh but if we right down the total distance used in that it depends on this uh a will real new you of the parameter vector by prior so that's this it's not use it to um minimize directly this uh error E so we uh defined an approximation you have of of know instead of we minimize this you and then what is the C had it is just the sum of the kind and this distance a little imp point to the new my fall and and time and this as as as follows we had a new manifold now and we obtain a first order approximation of this money there on the projection points that are already or and then the change in the sense of you i for this manifold is just the this distance between you Y and a uh a first order approximation so uh actually be do something pretty straightforward to minimize the that we just to each of the atoms of addiction or one by one and for each at an we find we compute the optimum options see that minimize the stereo tab and if we you right this you had as a function of C um we see that is a it's in the form of a racial function that means this function at a i and G I's are on my meals of C so in general um such a function has several local minima and it where we can seen in practice a experiments we have seen that it is also in most most of the time is possible to minimise you that just by a simple a and the sound out or two is not that extreme complicated function in practice so um we try each at and can compute all the local options uh and then in of all the atoms if we the best one that you small star then we add the this at some to the new cut and uh by its uh optimal corruption and you repeat the use of course so now um some experiments for some on for and a later on and in this experiment we use a transformation model of of uh we use the transformation manifold model of the mansion three so we have uh rotation and then it would be to two national translation uh so we can generate a the syntactical path and by adding some loss in and a and are i don't and uh so we construct a different data sets from this at some uh each dataset consists of some random geometric transformations of this the synthetic that pattern and you have a four out to each data of that that uh it it is a uh gaussian noise with for noise variances for sports data set and we use that dictionary consisting of some cost in them the R so um here you see the data approximation error or lot that just like the noise variance so i approximation error is the total squared distance of input images the computed my is see that it's uh it is it has a a linear variation like to noise variance which is an expected result uh uh have are if you pay attention here does just line doesn't pass from the origin so this actually re we'll the error of the algorithm and there are two main source of though uh for this error of all is that use a grid out them and it doesn't have an optimal performance T and secondly we use a dictionary of fine size that's the discrete or this also introduce some there and uh experiment sometime in it this time we use the four dimensional transformation model because we also have a you changed um in the um a as and is they are uh we use a hundred to the geometric to transforms hundred five and use a similar dictionary so on the left you see some of the sound of they in the experiment and on the right so uh you see the patch that we obtain the twenty four at so it looks like a five digit that sure about the characters digits five um despite the variation the they does that and also uh some uh for some numerical comparison we have compared to some rec approach and we have use this error measure a measure which is a the data approximation error so in the first to uh a reference is that have again computed progressive approximations of the uh are designed so in the first one we have applied matching force on a typical are in the data that the average are here and we have chosen it to be the input data out it close as to the centroid of all and i say J and uh in the second one we have applied simultaneous matching pursuit on or a line to achieve that sparse uh find i and we don't and finally as order approach like everyone provide a comparison between our method and uh classical manifold learning and it doesn't on that in some of the typical manifold learning algorithms they make use of the assumption that data has a local in your be or on the mind so we just uh a compute the this uh a local linear manifold approximation error is the sum of these E i one E i is uh the distance between a point you Y and the plane thing from the nearest neighbor oh um you see that are lots here are so the move of is the transformation invariant matching proof of word that we have proposed so we get the best or performance um we see that the red of corresponds to matching pursuit on average but a if i and it's as that okay so to do that and the data that that that for all uh you know like you're i that that that would be a lot but is it or not and this station is and that the one time and the patterns are um when we have applied simultaneous a a sparse estimation of that such P and finally some experiments on face image this time this time at high dimensional the because we have an an isotropic scaling and we have used some um face images of the same subject but we also uh i had some but uh in the data set and some variation of facial expression that we don't not model things like uh a facial expression variations but these things are are rather close there that the source of the deviation from the computed manifold and uh uh here on the right so the that some they like can from the data set of on the right to face them me that we have computed so it looks more or less like the phase of the same person there is also some kind of averaging and facial expression and uh we you have a doubt that all lesions and um if you look at the error loss we see here that so okay K even if is still get the best error for from a uh we can see here that the and and in is some people's the perform about this is because the number of variation then the face image of the same person are what's smaller and compared to the micro variation the hand it that typical people pattern of the data set like to approximate that all patterns i mean there and if you look at this uh that line as locally in or approximation or is pretty i and very for this is that the data uh do we have just use thirty five of just so the data is sparse the sample on the my fault the local linearity assumption that hold the anymore you had to so um to a little bit have present presented the method to the for transformation and rent sparse approximation of a set of signals we are we have built a representative pattern with the grid out some by a parametric atom selection and the complexity of the matter a method that we propose a changes linearly with respect to the number of atoms in the dictionary as a linear with respect to the number of images and the input that and it has a corn of the panels on the notion of the mind for the image resolution a there are um we have shown in another work that uh under some assumptions on the transformation model and also the structure of the dictionary we can it cheap a joint optimization of the at parameters and uh the functions C so in this case uh we optimize on the continuous dictionary might of fall rather than a um fixed dictionary a speech uh at samples and in this case uh we get rid of just for star here we don't have a uh a the depends a number of because the local jurisdiction so um is a final remark a um are right can related to to as in general one is sparse signal approximation of and the other is a all learning so um what's that we gained over sparse signal approximation at like and P S and E it is that we H you uh in a variance to geometric transformations of the data of you you we use a transformation manifold model on the other hand the and on to as we have over classical month learning algorithms are the following a first of all we provide an article model for the data and that has a nice properties like a it's the french it's move it is also used to call the take the parametric atoms it a L the end generation need they on the manifold and finally it has that it can still work if uh the something of that database sparse whereas as um many need fall that work and would require a much source oh so uh that's all and take you very much for function thank you as a first i yeah that's it the best to extract that um for time read actually what we do is we on minimize mean Z V in one as an approximation yeah so we do a oh so at each iteration okay we minimize the if that's that's T but as E that is not equal to you that's one reason the second reason um so it and if you mean my is this is one of the projection points change the forty two reasons uh a menu do this optimization thing you want to a guarantee that you will reduced so but we do in practice of that uh okay so we try this pick the best that some of we want to the project and than on be check if there are set of it just um we are fine accounting a if the error you don't the green the we try and reckon at them like don't to pick the best one but pick the second best one and then tried a but we we all well of course they are able to uh but a set up a date only if the error is it just so since the V we reduced the error E he for sure and in each iteration and so uh it has a lower bound and that it has to converge at some point um oh for situation what do you we exploit yeah i i that um i i think in whatever may you define fine of i mean whatever kind of transformation you can there i think as long as a um you did find this error E and this like like double distance of them but that you for it to the degree that each iteration um yeah so if you use degrees and a function that is lower bound that uh it means that a test to code word after a while is monotonically decreasing function you as a in i seven it depends on also a should be you have to to be used for the the dictionary you you need a note the is actually that it to to play even if you do the meeting you try to would yeah like your that that's it so um so that it's a question about dictionary learning i guess um we have a on anything like a um i mean doing something like a C you like case we get to optimize that one reason for this is that we really would like to to to been a parametric forms of all uh a we need them to be differentiable function because we're talking about ten just to the might so they just can't be an arbitrary function so that than this a but uh this think that i have mentioned here uh finally this the for all that kind of such as this field of addiction learning because here we have a a dictionary of money for and not what we do is you optimized a on the big show mind fall that are you optimise the parameters of the atoms this is can related to a lot but we i consider a differentiable uh at like a in a and i to be any the french on article function so it's gender can that's yeah but is not learn from the they are no we with that yeah as wish to and that was to uh you said that yeah actually used to as the fact that the sparse approach but the a D do actually they will go explicitly use is the constraint in your uh optimization so yeah and question is there a house as is it depending on your T and know how were how do you think this into account the in to you are uh uh optimization problem we have introduced the that and L one norm or or or or no we don't take it this like to that hmmm you oh sparse sparse the they'd sure talking about is and which main so here and uh we there's sparsity in a a times of these dictionary atoms that we use so we have to um version yeah so here uh uh you have J O D is that some so if K is much smaller and and number of cells that you have and the in which is done this pattern P is sparse in this domain just consisting of or a or a an hour and so um the made that you stick here is that look okay you can do that like okay and not take uh fifty atoms i keep the best fifty the atoms and um use yeah head and it's parsons and okay approximation yes okay as question a you've not that again a no and