Hello, and welcome back. We're going to be talking now about Wiener Filter. So we have to remember our basic degradation model that we see here. We have again, a filter basically blurring motion blur the focus as we have thought. And we have the noise and we discuss a lot about noise. And then we observe this and we want to reconstruct an image from our observation. And we're going to design a filter here called Wiener Flitering. That basically is going to help us to get a very, very good restoration under certain conditions. The basic idea of Wiener Filtering is that we want to minimize the mean square root between the reconstruction and the original signal. So I'm going to write the mean square over. It's basically the expectation of the square of the error between the original signal and the reconstructed signal. And we have talk a lot about this type of error in the past. We have to square it and here it is so it's the expectation of the square of the error. Now you may wonder but I don't know the original sigma. That's the whole point of doing restoration it's just getting back to something which is like the original signal without knowing it. But we have an expectation here. So don't need to know the original signal. We just need to know some statistical characterization of the original signal. Now it's not hard to show that the filter to put here that will minimize this. We'll get the following form so basically the Fourier. Transform of the reconstruction of the reconstructed signal will be, and I am going to write it down and explain what I am writing. It's the conjugate of the filter, a complex conjugate of the Fourier transform of the filter that we have estimated as we have described in the previous videos, divided by basically the magnitude of this filter that we have estimated. So this is the degradation filter, plus the power spectrum, and don't worry, if you don't know exactly what that means or if you forgot I'm going to remind you in a second. The power spectrum of the noise, divided by the power spectrum, of the signal, and all these multiplying our observation. So, this is the Wiener Filter. It's a filter that is multiplying our observation to obtain the reconstruction and of course, we have to invert this, the inverse Fourier to get the basic, the estimation. So once again, what is this doing? This is minimizing the mean square error. I have to explain to you or remind you, what are these concepts? But before that, if you want to know how do I get from wanting to minimize the mean square error to this expression is actually not hard. You can find a folderization for example in this Wikipedia page. It's truly not hard. You just write almost they say you write the definition of the mean square error, the expectation, and we know that we need to take derivatives when we are trying to minimize the function. I knew very fast I'm going to go to this. So let's just explain again this expression. Here is once again the complex conjugate of your Fourier response of the degradation filter. Here is basically the magnitude square nothing has than that and here as I said this is the power spectrum so, of the noise and the power spectrum of your original signal or the signal class. The power spectrum is basically the magnitude of the fourier transform, of the correlation function. So you have F. F has a correlation function. You take the Fourier transform of that. And you compute the magnitude of that, and that's the power spectrum of the signal, and the same for the noise. You take the noise, you compute a correlation of the noise, you take the fully transformed. You compute the magnitude, that's the power spectrum so you don't need to know the function, you need to know the noise, the exact function or the exact noise. You need to know some statistical characterization of that. So this is the ideal Wiener Filter and of course you need to know this in order to be able to actually implement it. So most of the time there's a very simple trick that is done instead of estimating this. This is basically replaced and I'm just going to use a different color. This, all this expression, which is difficult to estimate unless you know a lot of statistical characterizations of your signal. This is basically replaced by a constant. Pretty surprising, isn't it? A very complicated thing is replaced by a constant. So, I don't use this anymore, I basically go and replace this by a constant. And then, all I have to estimate is one number. Assuming of course that I have already estimated the, degradation. The basic, the operation that produced a degradation like a Gaussian or like, the filter that produces motion blur. So very simple operation. A very simple filter, with only one unknown, that can be estimated in a number of ways. But we are not going to discuss that, now. It's basically similar to the way of that we needed to estimate the variance of the Gaussian or how much motion blur there is in an image. It looks very similar, to the basic to the inverse filter. Remember in the inverse filter, all that we need is we actually took the degraded image, meaning the observation and we divided by basically the Fourier transform of H, basically this H that we see here. And here is slightly different, similar but different. And that slight difference is critical as we're going to see in the next examples. So let's just look at these images. We have seen this image before. This is the original image, and this is basically the result of turbulence on that image. And as we know and we discussed before, this model basically with a Gaussian filter. This is the result of inverse filtering, so we basically take the gaussian filter, we take the fourier transform and we divide this image, basically it's fourier transform, we divide it by the fourier transform of the Gaussian and then we reconstruct. So this is nothing else done, taking this image, taking its Fourier and dividing it, but by the Fourier Transform of a Gaussian. And then inverting the Fourier of the result. And is, I would say, quite a bad result and we know why. We're dividing, although the Gaussian memory goes to zero, it gets very small values. And that basically is enhancing a lot, and is making our reconstruction very, very nonstable. On the other hand, this is the result of the Wiener filtering. I think its outstanding result so its basically what we just saw in the previous slide we basically the filter is to have been one over the Gaussian as here, the filter basically is the conjugate over H^2 plus the constant. That's the filter that we apply to this image. We don't apply 1 over the Gaussian, or 1 / H. We apply this filter and we get that significantly better result. And the result here is illustrated with basically almost trial and error. We basically try the best possible K knowing the original, it's just for better logical reasons basically to show you how great we can get the result with Wiener filtering if we estimate this constant. So to show you that this is not, just not all one example that it can degrade, let's just show a second example of an image that we have seen before. Here we have an image. It's based in the book, it's the cover of the book that we have been using but just in black and white. It has motion. We have seen that already, so it has motion blur. And then we add to it noise. And then we apply, we know exactly a motion blur so we computed exactly the filter and we apply inverse filter and this is what we get. So once again going from here to here is the result of taking this image and filtering with an inverse filter, 1 / H. And I'm not including the problems that might happen because I need to estimate age, I'm actually putting the exact age that I used to create this image and the result is close to a disaster basically for the same reasons. As we saw in the previous example. This is the result of Wiener filtering, okay? Once again, this is what we are applying here and again we use the best constant that we could use against for Pedagogic reasons to just rate the power of the inner filtering. Now, you probably see here the blur and you might see also the noise. Here we actually take one order of magnitude less noise. Same blur same motion blur and again we apply the inverse filter and we apply the Weiner. And again this is really, really bad. Basically what we are recovering here is the motion. We are not able to recover anything else but Weiner filtering is doing outstanding job. If we reduce the noise even further, actually five orders of magnitude. I don't think you can see any noise. If you compare this to this, basically there is, the noise is so small, that basically the dominant degradation here is the motion blur. This is the inverse filter. Now, it's much better, but it's still not great. Again, this is nothing else than an inverse filter. but applied, to the, to the blur and noisy image. And this is the result of Wiener filter. So this column is always inverse. This column is always Wiener filtering. And just different levels of noise here. I think we've got the perfect result. Why it's perfect because as I say in this exampl, in this kind of artificial Pedagogic example we didn't have to estimate H, we know exactly H and also we estimated the best possible K. But this illustrates how this filter is extremely powerful once we estimate the degradation filter or the motion blur in this case, and the constant. And still, filter inverse filter. Sorry, here even if we know exactly H, it just doesn't do a good job. So I hope I convinced you that Wiener filtering is a very simple operation, and it can be extremely, extremely powerful. So by this we conclude Wiener filtering and in the next video we are going to just basically conclude the unit on restoration. Thank you very much.
