Hello and welcome to Week 4 of our Image and Video Processing class. For the last few weeks, we have been very busy learning a lot of new concepts in image and video processing. This week, the materials are going to be a bit short. I think we all deserve a shorter week about after learning so much new material. But also, the main reason this week is going to be a bit short in material is because we're going to come back to some important examples of image restoration later when we talk about more advanced topics in the class of image processing. But we still need to learn the basics of image restoration. And we need to understand what is image restoration. So, let's just go straight into that. But before that, just for a second, what background do you need for the topic of this week? If you have knowledge about signal processing, and Fourier analysis, there are some parts of the week that you might benefit a bit more. But since it happened in the previous weeks, the material is going to be very self-contained so you should be able to understand the basic concepts with very limited background, just some background in Linear Algebra mostly. but again, Fourier analysis will be very helpful if you know it. In particular, for the topic of Weiner filtering that we are going to explain in, in a few hours or actually in a few minutes. So, let's just explain, what is image restoration? So, in image restoration in contrast with image enhancement, we are going to try to restore the image. We are going to have a model for the degradation and that's what we see here. This is a degradation model. Basically, we start with an image. The image goes through some degradation which is represented by this filter H. Examples of degradation, and we're going to discuss them in detail later on are, for example, blurring, defocus. When we take a picture that's out of focus, that's basically a degradation. Another example is what's called motion blur. When we take pictures of something that goes very fast, we actually see that as kind of blurry in our picture and that's basically another type of degradation. After the image has gone through degradation, there is noise added to it. For example, the sensors in cameras have noise, the sensors in magnetic resonance have noise. So, there is noise added to it and this is what we actually observe, g. Now, from g, we need to go back to f as close as possible as the original f, and that's why this is called restoration. We're going to try to invert this degradation and noise, in order to be able to produce an image which is as close as possible as the original image. So, this restored image, we want it to be as close as possible to the original image. That's a goal of restoration. We didn't have that task in image enhancement, we just wanted it to look better, to look sharper, to benefit, let's say, our visual perception of the image. The goal now is to actually restore the image. It's kind of a different goal, they're related, but it's kind of a different goal, okay? And some of the things that we saw the previous week are actually very useful for image restoration as well, for example, non-local means. You can show that under certain conditions, restores the image. So, what's the model that we have here? How was g produced? g, basically is a result of f going through the degradation. So, we have f (x,y) convolution with H (x,y). That's a degredation model and normally, it's assumed that is a linear operation. So, it's a convolution. And we add to it the noise. And most of the image and video processing literature leads with what's called additive noise. The noise is added to the image. That's not what happens in all real cameras in all real acquisition devices, for example. There is also multiplicative noise. Very often, were basically all these, basically the degradation gets multiplied by some noise. There are other types of noise that happens, but these are some, two of the most common. The most common of all is additive. Some literature, but much more limited, addresses multiplicative noise. One of the effects of multiplicative noise is that the amount of noise depends on the signal itself. So, if the pixel value is three, we're multiplying three, but if the pixel value is ten, we're multiplying ten, and then the amount of noise much, might be much larger. And with additive noise, that doesn't happen. So, it's basically kind of image-independent denoise. Now, there is a way to transform multiplicative noise into additive noise so we can enjoy all of the literature in additive noise and also work kind of with a small trick applied also to multiplicative noise. And I wonder if you know what trick can we use to do that. Maybe think for a second before I give you the answer. So, the question is very simple. Think for yourself if you know what to do with multiplicative noise so you transform it to additive noise. And let's think and I'll give you the answer very soon, okay? You just spend some time thinking. Maybe you go to the answer very fast. Maybe you go to the answer after a while or maybe you are waiting for me to give you the answer. And the idea is very simple. If you have multiplicative noise instead of additive noise, let me erase here, so I have everything to write down. So, if you are multiplying two numbers, a times b, I can take the logarithmic of those, of the product of those two numbers. And that's equal to the logarithmic of a plus the log of b. So, I have trans, if b was my noise, I basically transform it into additive noise. So, instead of working with the image, I'm going to work with the logarithmic version of the image and I'm going to work with the logarithmic component of the noise. So basically, you give me the degradation g, I take the log of g, and I treat everything that is now from now on as it was additive noise. When I finish recovering it, I basically have to invert this operation. And that's an exponential function. So, that's one of the reasons that a lot of the literature has basically concentrated on additive noise because of this trick. Now, this trick has its own problems in particular because if you don't really remove all the noise, the exponent can make it much, much larger. But it's an important trick that basically makes our additive noise, restoration techniques also applicable to multiplicative noise with some caveats and some limitations. So, having now a concept of what is image restoration, let's just talk in the next theory about noise. What type of noise can we have in an image? Where do they come from? So, see you in the next video to deal with that topic. Thank you.