so uh

this is the joint work of a

we my phd student "'cause" to the money i come from poland something using university

of technology uh include need to this is uh in the self of a whole

and uh the topic of the paper is uh and it'll bit uh

uh similar to the previous one because it is with a uh actually with convolutions

so it's is uh about a bi lateral filter uh which is a actually

based on the convolution so uh the outline first so i would like to say

a few words about uh the bilateral uh filter then uh the problem of impulsive

noise this uh this is the problem rather than interested in

then the proposed solution

i on modification or extension ordered normalization and some uh experimental uh results

so uh

the uh bilateral a filter is a

is uh

is uh

very popular it was uh in the literature uh it is uh mostly people mostly

us right that it was uh introduced by too much C and monthly G also

be uh

so

so this is this is the uh the paper which is uh actually uh very

often cited but in fact uh it was um no before and it was described

by us me uh in uh in and approach which is called susan and also

uh this is this uh this uh it is available on the internet and it

was also uh described by yellow sloughs P uh in a in about the old

book but uh the destruction between that was so i was known uh

more than uh ten years uh before so uh

what is the bilateral a filter

but actually this is a convolution

uh we have a window which are which can be can be quite large

uh and we assign some weights to the pixels which are uh inside uh of

the of the filtering window would have a central pixel which is the uh the

pixel that we would like to modify we have uh the window uh many pixels

and we have uh some weights so we can uh call this pixel X

and so we have a weight X uh why are the pixels uh which are

we can say in a neighborhood relation

so uh we uh multiply the uh intensities uh of the pixels in uh basically

uh image for example are we the weights and then i would invite uh

uh this uh some by the sum of the weights

so uh the weights is a quite important a part of the bilateral filter oh

of course we should have a tool weights because of the name of this uh

filter one hour wait is uh

uh expressing the distance the topological distance

uh on the image than the main be between centre and all the pixels so

uh we have here of a small distance and here those pixels are at large

uh distances so this is a topological distance and so a people use uh gaussian

like uh functions so we have a smoothing factor here

and uh the uh second solo part uh is also a cost and like but

uh here in the number need to we have differences in basically considers

so if uh that topics as which are quite similar then the weight will be

large if they are uh quite different then and the we would be small

and those two weights are multiplied and the result is this uh weights W

so uh yeah we have a example uh a small part of the image we

can uh say this is the filtering window so we have uh the distances and

we can uh make uh and that's a amount of the differences uh absolute differences

uh between the central pixel and the neighbours for example this is to say uh

or is a six nodes

i six

so this is one hundred sixteen then so we have a map of uh of

the absolute differences so uh we just need uh to take into account this distance

this part and the absolute differences that we basically the problem to uh and of

course uh we have to choose uh the uh smoothing factors which are actually uh

parameters of the bilateral filter

so uh

we can uh see that this bilateral filtering is quite similar to the convolution with

a gaussian and uh

because the gaussian kernel actually assigns weights to pixels uh um

which are in the neighborhood relation in a large window with the central one so

this is uh this is like in this uh bilateral filter here uh for the

center a pixel we have a large weight and the weights are decreasing yeah so

we can hear a noisy uh step or edge

and then

if we imagine that we have a pixel on this side of these edge

then a discussion uh function is modified by the structure of the of the image

inside of the of the filtering window and here the difference is uh in intensity

a small so uh the structure of the gaussian uh function will be presented but

here the differences are large and that's why the advantages of the gaussian uh function

will be decreased so actually a instead of such a nice uh no noticeable gonna

we have a kernel uh which is modified by the structure of the of the

uh image of uh of the of the uh part of the image which is

contained in the filtering window so uh

as a result this uh this uh this uh account no will be is a

taking uh values we've large weights from this side and small uh ways which will

be assigned uh to this uh side and that the result will be a small

image uh but with a result uh edges there would be no blurring of the

edges

and this is uh the structure of the bilateral filter they're of course um an

extension so uh this of this kind of filtering is quite popular people are uh

developing uh

two lateral filters and so on and modified and that would like to present a

modification before one or more example oh have a uh artificial uh image just uh

of us were and say we have a point here so this is this is

the gaussian uh like function but what defined by the by the uh like the

structure here the differences a lot so uh they are almost zeros and if we

and if we add some noise this uh kind of will be modified you can

see some sports some dot sports uh this is because uh some of the pixels

for example here or here they are quite different uh like this pixel in the

cone so this is uh actually

remove the that we know and perform the convolution with a with a kernel for

which is dependent on structure of the image

so uh now a impulsive noise

and the colour images

the modification um or extension of the bilateral filter to color images is uh is

quite simple because uh the topological distance uh system defined in the same way but

uh the difference is uh in uh intensities and can be can be huh

change or substituted by uh distances in the colour space so uh we can we

can easily modify uh the bilateral filter uh to the uh color uh images

so uh color images are

quite often uh corrupted by impulsive noise so we kind of uh here an example

this is the land line image with uh which was uh

uh polluted by uh artificial uh at efficiently uh modeled and noise we can see

some pixels uh depends on the intensity of the noise which are corrupting the image

and

for example uh let's cut not X the

uh that's cover uh

look at such a situation uh for example we have a gray scale uh image

that's for simplicity and all the values for the here are the same so maybe

one hundred

and such a small part of an image is corrupted by impulsive noise so for

example the uh white pixels

oh we can also uh make a the colour uh example but uh that be

easier to just to look at this is uh this one so uh this is

the white pixel we are calculating uh the similarities and it turns out that the

similarity between the two hundred fifty five and uh let's say one hundred uh is

low and the similarity between uh the central pixel images corrupted and pixels which are

in the filtering window which are also corrupted is quite large

so in fact the weighted sum up to one which means that uh when we

perform this uh multiplication and so on this noisy pixel will be preserved and this

is a common drawback of the bilateral filter and also a bit for filtering designs

like uh isotropic diffusion and many others because uh if pixels are the same uh

in the in the filtering window uh mostly uh the corrupted pixel is of these

are

so

uh

we would like uh we wanted uh to improve the situation and then we show

uh away how uh this bilateral filter can be improved a first we introduce digital

puffs so in a result of se in eight directions for example would have a

uh small filtering window uh is one hundred can as before the differences and we

can we can calculate sparse

uh joining uh the pixels for example this one this one uh free pixels the

are depicted in this uh example

and we can we can calculate digital buffs

which uh have some cost

the cost of a digital path joining us central pixel with the pixel uh in

the filtering window used uh defined as the sum of the absolute differences between the

successive steps

so uh when i have uh above we've all uh what the pixels are the

same then the cost of the power will be zero because the don't know changes

but uh if the uh big changes then and uh the cost is high and

for every pixel

oh we need to find an optimal power optimal means and that the cost is

medium

so uh we are looking for minimum laughs

and

this is uh

this is

uh different uh done in the uh case of bilateral filter because in the bilateral

filter we are comparing on it on a peaks as well not looking at the

pixels which uh between the we can say a would jump from the center the

would be is that uh the pixels which are inside of the filtering window not

considering uh the structure of the and here we have to consider uh the neighbourhood

the close neighborhood of the central pixel and so on so for every pixel would

have to find the above are so uh is a lot of intensities and these

are these are the differences

for the uh for finding the parts uh we are using the extra uh i'll

go with uh which is uh quite uh quite fast

and

using this uh extra et al gore of four grayscale images the cost is defined

us uh is the sum of data from the uh intensities of successive pixels

and we have a uh this the same structure but

the uh weighting coefficient

depends only on the cost

so uh there is no actually this is not the bilateral filter this is just

a filter which uh which uh takes into account the cost of the uh of

uh of that pops linking the that the pixels so for every pixel there is

the but there is a cost and we calculate that the cost and use them

us weighting coefficients

so for maybe just a for color images uh we substitute this uh instead of

uh intensities uh we take uh the color uh the distance in a between color

space for example of to be uh we made uh the uh calculations uh using

rgb but in fact the uh

you can use a suitable uh colour space uh we didn't check uh how it

looks like we've other color spaces but uh even in the simplest uh rgb color

space uh the results are promising and also we did a show uh in a

moment so uh i will uh give uh some examples are using the three uh

test color images and for the evaluation of the restoration quality

oh we used the psnr which is uh which is uh

uh using the and these word error uh for a color images we just uh

at the differences in each channel and divided by the number of pixels and psnr

is a good uh objective uh quality measure because it correlates the uh

in my opinion wide world with on the objective uh coefficients are the correlation with

subjective uh of expression is uh maybe not so uh so good but uh use

the noise generally uh regarded as a reliable uh coefficient and some plots this is

that this is the bilateral filter

as you know maybe remember their uh to uh coefficients uh two parameters were required

for the uh topological distance and for the intensity uh a lot and as you

can see when we have a gaussian noise this is segment and sigma twenty for

the um the choice of the part this is uh not so straightforward so uh

mostly people uh don't write about uh choosing the parameter C yeah uh to it

mostly uh

set the parameters so that uh the images look nice but in fact uh choosing

the players the parameter that is uh a little bit difficult uh with the uh

if the images uh corrupted by a high intensity noise uh then it is easier

and when we have uh images corrupted with guys for the gaussian noise

and additionally uh impulsive noise is added

then i the situation is better so that the choice of that of the optimal

of parameters is uh

is uh easier so um we are uh actually uh i'm interested in this case

this is the mixed uh a gaussian and impulsive noise because it is uh quite

quite uh quite uh difficult to uh that's uh press

uh because uh

it is switched to deal with impulsive noise uh they are not uh quite good

at the gaussians and this is uh that this was proposed five by five that

this nine by nine window uh it appears that um nine by nine five by

five is enough so we don't need a very large uh got a kernels and

this on some plots uh showing that our uh filter so only one parameter is

uh needed uh it is denoted as H uses gaussian noise and this is the

mixed noise so as you can see a we can take for example two hundred

as a as a default value for a with the images and is uh my

one and some results

uh we compared it with uh some uh well no uh filters

uh the interesting thing uh in the gaussian noise uh is that uh our filter

was not was not matter

than they know not a non-local means uh i don't have at the time and

to talk about it but i didn't know local means non-local means uh is uh

i think the best uh filter uh available one of the best maybe this is

an anisotropic diffusion vector median filter bilateral filter by five by five modified this is

the modification to use a five by five nine by nine

so a non-local means uh is the wiener for the gaussian noise

oh our uh all the filter is a has low efficiency but our filter uh

works good uh especially for high noise intensity for the for the mixed noise case

when you look at the examples other here so you can clearly see that uh

this is uh obviously uh it gives much better results

so uh usually a i think is i think is that you can you can

see that uh that the uh the result is that looks better than me

the uh there are no sell so many spectacles

uh

this is the land i image what is uh what is a disturbing thus some

clusters of pixels which are very close uh

uh to the center uh and we will uh result the problem uh in the

future work so as a conclusion this is a

and modification of the bilateral filter

uh only one parameter uh is uh you did uh

and then being a and uh we will uh we solve the but the problem

of uh neighboring pixels which are uh corrupted

so this will be a result in future work a given much for your attention

you have any questions

uh the computation time use a comparable uh we've the bilateral filter because uh we

only need uh to find the parts

we don't need to calculate the distances which is uh

which can be uh done directly but it can be also that uh by uh

some uh fast objects techniques

or about actually the times are more or less uh the scene

so i cannot say this is much faster or much slower well as the C

uh yeah

yeah right

thank you