Convolve, let's say, with the Gaussian function with zero mean uncertain
variance, or standard deviation. And we already saw the general function
of a Gaussian. Basically, this exponential with a
quadratic exponential function. We saw it when we were discussing noise.
Basically, it blurs my image. Now if you know that there is a Gaussian
function that is affecting your observation.
So this would be my observation. This would be my G.
If you know that G is the result of F. With a Gaussian filter.
And you also know that the F was actually a dot.
A delta function. Just a point, that in that case for
example, that very trivial case, you can actually estimate the blurring.
Actually, not very hard because a simple property of the convolution says that
basically if F is the delta function that is convolved, with G.
Then we get G, so basically this is actually the Gaussian. So a very simple
case that we can actually estimate our H function for the particular case,
that we have a Gaussian filtering. Now,
this might sound a bit ridiculous, as first.
I say, how do I know that there was a dot, that it was a delta function in my
image? But actually it is not so ridiculous,
because we can use this to calibrate a system.
If we know that our camera is blurring. Then I can put artificially an image that
has a delta function and then I estimate my blurring.
So this is very useful to, to calibration to do estimation of how the camera is
behaving. Very important.
Now I'm going to tell you something else that will further enhance the knowledge
and the concept why this is a hard function.
A hard function to estimate. If convolution G is equal to G
convolution F. So you might wonder is F my original
image? So was my original image a delta?
Am I blaring a Gaussian? Or was my original image a Gauissian?
Am I blaring a delta, which means no blaring.
So there is a lot of basic ambiguity here that we don't' know which one is what,
and that is yet another challenge that we have when we are trying to estimate the
filtering function. And normally what you do is you assume
models for the image and models for the filtering.
But this is just to illustrate how hard this problem is.
So this is an example where we can estimate.
Gaussian actually as I say is very important for blaring.
It's also a model that is very frequently used for turbulence.
So here is an image which is courtesy of, of, of NASA,
and it appears in the book that we are using so far.
And here we have basically the same image under different levels of turbulence,
from more turbulence to less turbulence. So at least you can observe here the
blurring effect and very often turbulence is modelled with a Gaussian filter.
So it's very, very important, we discussed that Gaussian noise
distributions are very important mathematically although they don't appear
really, in real physical scenarios very often.
Gaussian smoothing, Gaussian filtering actually is a very realistic model.
For a lot of things that are happening in physical cameras.
The other type of blurring that appears a lot is what's called motion blurring.
And the basic idea of motion blurring is that when you're taking a picture if the
object is moving. Pictures are taken by integrating the
light that comes into the sensor. So if the object is moving during your
integration time, basically you get a blurry picture.
And that's illustrated here before I write down the formulas.
So in this case, it's basically a simulation of instead of moving the
object, we move the camera. And the basic idea is, going back to our
GF functions, basically your G function that we are observing is actually the
integral from zero to T. That's a time of integration of your
camera, of your image at X. But the point X is moving, so I can call
that function of motion, let's say X of T,
to write down that basically it's moving, and it's also moving in the Y direction,
Y of T. And I'm integrating over T.
Okay? So basically, what I'm doing here is I'm
not taking one image. It's like I've taken multiple images and
I'm adding them, and what I see is the result.
This actually was simulated by taking this image here,
shifting it, let's say, to the right, just by a tiny bit. Shifting it again,
shifting it again, and then adding that. That's what this operation is doing, is
basically adding images on its own shifts.
And, this is how we're observation. Now, if we take, if we do the math and we
take the Fourier transform in both sides, and we use properties of the Fourier
transform. Then we can also write this in the form
that we have seen before, G equal H,
F. So, also this translation is actually a
filter, and I am not going to do that but you're more than welcome to do that as an
exercise, a very simple exercise, if you are familiar with basic concepts
in Fourier transform, you basically just as I say, take the Fourier in both sides,
and use properties of, of translation. And you get this filter, which of course
will depend on the velocities. Okay, so you could imagine that if the
object is not moving at all, there is no blurring.
If the object is moving a lot, there is much more blurring.
So, this filter will depend on these velocities.
And once again the question will be, just to show how hard it is, how do I estimate
from this? Both my image and my blurring effects.
Certainly not a trivial thing to do. Even if I knew that the result is a
result of motion blurring. The, the concept is that, from the sum of
multiple images, you have to estimate one.
Okay? So that's certainly not a trivial
concept. If I give you the number five, and I tell
you it's the sum of two numbers, there is no way for you to know which two
numbers I added unless I give you more information about that.
We're going to discuss this more when we talk in a few weeks more advanced topics
in image restoration. When we come back to this topic you'll
see different tools like sparsmalling and to expect that.
This is to illustrate that is a hard problem, and an extremely important
problem. Simple models like Gaussian de-blurring
or motion, or, Gaussian blurring, sorry. The blurring is the operation of, of
inverting, and motion blur are very, very important.
And we need to be able to invert them to go back to our original very, very sharp
image. What we're going to do next, is
basically, we're going to see one way of doing that,
and that's called Wiener filtering. There's one way of inverting a filter and
inverting noise, and we're going to see that in the next
video. See you soon.
Thank you.