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