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

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

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