Let us now discuss the different types of noise that we can have in an image or a video. normally, the way we do that is we basically define what's the probability of the noise taking certain value. Remember, we are talking about adding noise to the image as we discussed in the previous video. So let's just start with Gaussian noise and I'm going to write down the formula for it. So the probability of a certain value, this is a function that you have probably seen a large number of times already, is 1 over square root of 2 pi sigma. That's basically, we are getting, they are the standard deviation of the noise. The exponent of minus c minus average. We see it here, the average of the noise, square, that's very important, divided by 2 sigma square, the variants of the noise. So this is the formula for Gaussian noise and we see it's function here. Now, you might ask yourself, are there any real systems than, let's say when I take an image, what I see is additive Gaussian noise? And unfortunately it's not, actually there's not real physical systems that produce Gaussian noise. So, why is this such an important noise? This is probably the most used noise model in most of image and video processing systems for a couple of reasons. One of them, because mathematically, is very easy to work with and we are going to see that later during this, this few weeks of, of image and video processing classes. And also, because it's a good approximation to other types of noise, especially when we look at small regions of the image or small region of pixel values. So it's very powerful and you're going to see most of the literature in image and video processing addressing Gaussian noise. And once again, here, we have the mean, the average, and here we have the variance of, of the noise. Okay? So that's Gaussian. Now, let's just talk about Rayliegh noise. In contrast with Gaussian, this actually does appear in real physical systems and let's just write a formula for it. It's a big longer, but bear with me. So the probability of a certain value. We're going to decompose it in two parts. One is 2 / b and there is a few parameters as we see here, 2 / b, z - a, e minus, z - a^2 as we have for the Gaussian, over b. Now, that's the value that the function assumes if z is greater or equal than a and is actually 0 if z is less or equal pictures, basically less than a. Okay? So, it looks like a complicated formula. It has this quadratic exponential, the same as in the Gaussian, it's multiplied by this and there's two parameters, b and a. And let me write down two important things before I describe you examples of the Rayleigh, Rayliegh noise. One is the average, it's not hard to compute, that the average of this is a plus square root of pi b over four. So that's the average of the signal if we have that type of noise. And the variance of the signal, of the sigma, of the signal I apologize, is b(4 - pi) / 4. Okay? So these are for you, just formulas, and you know, simple models of the noise. Now, this noise normally is used to model, for example, noise in certain areas of, of magnetic resonant imaging. So, in contrast with the Gaussian noise that is extremely good for us to basically do some math and do some nice restoration, but it's not really appearing in real physical devices. This is a good model for real physical devices. As I say, for example, the magnetic resonance is also used in under water imaging. So there are many different disciplines, that basically the Rayleigh is a very, very good model for, for that, for, for, basically the noise appearing in those devices. Let's just, you have the gamma noise here, the formula. Let's just talk about exponential noise. In the exponential noise, so once again, I just start by writing the formula. The probability of z of basically a certain value of the noise is once again, and this is going to be very important in a second, e to the, is a, e to the minus az when z is positive, 0 when z is negative. So, basically, the Rayleigh noise was nonsymmetric, but the exponential, basically, there was no noise. Basically, there is no negative values of, of noise and that's a very important property. Once again, the average, not hard to compute, is 1 / a, and then sigma square is 1 / a^2. And you can have different numbers here depending on your scaling, a is just a scaling. So this is the exponential noise. Another popular noise, but I'm going to discuss the, when, can we see this exponential noise? Actually, when I describe next the uniform noise. So let me just first write the formulas for the uniform noise, ask you a couple of questions about that, and then explain worries that we actually see this type of, of noise. So let's just write the forumula for the uniform noise, the uniform noise is basically saying that there is noise of exactly the same value between these two. No noise outside and once you are inside, all the noise has basically is uniform, as the, as the name says. So the probability is once again 1 / b-a if you are in this interval. So basically, a less wqual than c, less equal than b. 0 if you're outside of the interval. So I'm going to write other. Basically, you're outside of the interval. So, uniform inside the interval 0 outside of the interval. So let me ask you a question. What's the average for this noise? So look at this picture or if you want, do the math on the side and tell me, what's the average of this noise. Okay? I'm going to give you, in the quiz, a number of possibilities. And I'm going to respond to that in just one second. So what's the average here? Very easy to compute. The average here is a + b / 2. So that's the average. So if I'm having uniform noise, that's the average of my noise. Okay? And how much is sigma squared? You're going to have that as one of your homework, as one of your quizzes for, for week 4. Now, when do we see this type of noise? For example, this is a very good model. Actually, both of them are good models for quantization noise. Okay? So, if you remember, in the, mostly in the first week, but also in the second week, when we discussed image compression, we talked about quantization. And the basic idea in quantization is that we had a range of values and we represented it by a quantized value, let's say, the point in the middle. So when you actually obtain this value, you actually represent it with a point in the middle, so you're making this noise. If you get this value, you are making this noise. And that type of noise very often is modeled as uniform noise inside this interval. There's no noise outside that interval, because when you're quantizing, you define an interval, and then you define, basically, the representative of that interval. So uniform noise is a model for quantization noise and that teaches us something new, a new concept. Noise sometimes comes from the device. Let's say, the sensors in our cameras. Noise, sometimes comes because of an operation that we do to images. And beautiful noise is a good model to this quantization operation. Exponential noise sometimes is also used as a good model for this quantisation, but it's also the type of noise or error when we do predictive coding. Remember, in predictive coding, when we're trying to predict a pixel, let's say, from it's surrounding pixels. So, what's the value of this pixel if I tell you all this? That's what we did for predictive coding. The error of that prediction is normally more than as an exponential error or an exponential noise. For example, the JPEG or less standard, which is basically used in the Mars Rover uses this model for the error of the predictive column. So very interesting these are really, really useful, before I go to this we see these are useful to model for example operations that we do in images. This is very useful to model real sensor noise, this is a very useful mathematical model to develop algorithms, so each one of them has it's own features and values. Lastly, I want to talk about impulse noise and I can write down the formulas for these. But let me explain it with a picture, it's much easier. The basic idea on, on, on these type of noise, which is also called salt and pepper noise, is that with certain probability you change the pixel completely to a new value. And with certain other probability, you change it to a different value. For example, we start that I go over the image and with certain probability I'd change the pixel, let's say to white, and with other probability I change the pixel, let's say to black and that's why it's called salt and pepper. If I changed it to white that's called salt, and if I changed it to black, that's called pepper. So we go over the image and we change pixels with a given probability. It's very different than these types of noise. The Gaussian noise will affect every pixel in the image. The salt and pepper will affect some more, some not, but when it effects, it effects all of them in the same, in the same fashion. Turns them either white or turns them black with a different probabilities, so it's a different characteristic. You can think about this noise, for example, a noise to model when the probability is very low, actually sensors defect. If one part one pixel in your camera is actually burn for example you don't get signal there. So that's kind of a pepper noise, it might be black so with, hopefully, very low probability, if your camera is a good camera, you get a pixel which is black and that's a pepper noise. Okay? So these are very, very useful noise that we can see. Now, what we're going to see next is how do we estimate this types of noise that's going to be very important. It's going to be the topic of one of the forthcoming videos. See you next. Thank you.
