Hello and welcome back. One of the fun things of this week is that we are learning a lot of background material. While the background material is important for image processing, it's important for many other disciplines. For example, in the previous video we learned about level sets. Level sets are important in image processing to the curve evolution like active contours as we have seen numerous examples last week. We saw a few this week and we are going to see more even in the next coming videos. Another topic that is very important in mathematics and is also important in image processing is the calculus of variations. And that's what we're going to be talking right now in this video. And then in the next video we're going to see an example of that for image denoiseing and image enhancement. So what is calculus of variations? The idea is very simple, because it's an extension of finding extrema of functions. But now we are going to be finding extrema of functionals. And what do I mean by that? We're going to have images like you. And. It's derivatives. So this is going to be an image for us. And we're going to be looking at minimizing a function of the image. Might be the image is square, might be the gradient of the image, might be the Laplacian of the image. So we're going to be trying to find what's the image that minimizes that function in the same way that we tried to find, what's the point that minimizes a given function. For example, if I draw this function then you say oh, this is the point that minimizes it. But here, U is by itself what we are looking for. So we are not looking for a coordinate, we are looking for a function that minimizes this. This is going to be very clear even in the next slide with an example. And we are going to make all the connection of the minimization of function X with the minimization of functions. Now, we're going to discuss in a couple of minutes that a condition for U to be a minimizer of this is to hold. This equation which is called the Euler-Lagrange equation and this is the type of partial differential equation that we are going to be solving in image processing. Let me illustrate a bit more about this and then how we came to this equation which is completely parallel to the minimization of regular functions that we are away from regular standards in calculus. So let's just illustrate, first of all, what do I mean by a functional. Let's assume that I'm giving two points. And I am asking you what's the curve connecting these two points that has the minimum length. So I am asking you for the function of that curve, the minimizer is going to be a function, not a point. The length of. Any curve, absolutely. Any curve connecting these two points is basically written here. The curve is parameterized as X Ux, we know about that, that's the particular case of that parameterization for basically any curve that goes. And connect those points. the length is. As we know, the derivative of the first coordinate squared so that's one squared derivative of the second coordinate squared. So it's this function that we are going to try to optimize. We already know what a function that minimizes the length between two points. It's a straight line. But let's see how we get that. You see calculus offers variations, so I'm trying to find U connecting these two points. This is the condition for a function to be a minimizer of this function. So it's that integral over a function of functions. We haven't arrived at it yet, but we're going to discuss it in a few slides that this is what happened. So F is square root of one plus UX squared. So if you just do the numbers and takes, you take this and do the derivative according to U minus the derivative that is written here, you get this expression. So this expression is nothing else than applying this to f as written here. Now this equal to zero is a condition for the particular U we are looking for to solve the problem that we are searching for, meaning the minimizer of the length connecting these two points. Now we have this equals zero, the denominator doesn't matter. In when we have eight over B equal zero when its to be equal zero is U, so U double derivative equal zero, that means that the first derivative is constant and then is that the function is 8X plus B that's a straight line. And A and B are found by what's called the boundary conditions. We know that it has to go through these points. So if I replace here X by a zero, I have to get this point and if I replace by X one, I have to get this point and in that way, I get A and B. So we see that for length, the function that basically solves the Euler-Lagrange, and that's a necessary condition, is a straight line. So basically, if we take this curve here, it will have certain length. But actually, the one that minimizes is not this because this is not a straight line. The one that minimizes is this one. So, a function to be a minimizer has to haul the Euler-Larange equation. Why is that important, and how do we get to the Euler-Lagrange equation? Let us go for a second back to regular functions, and how we find the minima of a regular function. So you might not remember exactly how we do that. But you do remember the result, which is going to explain next. The basic idea is, you have a function. And you have the function around a certain point. And you do a small perturbation of that function. And you take the derivative according to that perturbation. And the condition for a point to be a maximum or a minimum is that when you do any perturbation, the derivative is equal to zero. If you compute the derivative according to the permutation, you get this, and this has to hold for every n. So the derivative of the function for every end has to be equal zero which means that this has to be zero, and we know that. We know that the condition of a point to be an extreme of a function is that it's derivative have to be equal to zero. Now why is that so important? We know that but why is that so important? Because then I can write this differential equation. Xt so I can make X change in the direction of minus the derivative and that basically says that you start from a point and your next point, because X is changing in time, it just, you move a bit in the direction of the derivative of the function. Remember, derivative is tangent so you move a bit to the next spot. This is how you move. Your next point is this point minus a tiny step in the direction of the derivative. That's what this equation is telling us. Because if we were to discretize this equation, we get that, that. X. We just saw the result there. We had the X at T plus Delta T minus X at T. Divided by delta T, that's a [INAUDIBLE] equation of this. This expression is equal to minus fx and that's what we have here so we go slowly. In the direction of the derivative of X until we get to this point. Let's just see that again. We go one step and then we will go another step and another step until basically we get to the steady state. In the steady state, we have the XT equals zero. Nothing is changing anymore and, therefore, this. Is = to zero, which is our condition. So once again, we know that the condition for a point to be a minimal of a function. Is it has to be the derivative = to zero. So we take a point. I'm going to just show that to you again We take a point and we move it in the direction of the derivative and then slowly we are moving it in the direction of the derivative and we are moving it again until it doesn't move anymore. When it doesn't move anymore. This is equal to zero, which is the condition for it to be a minimum. So we arrive to a point that is in this case a minimum. So this concept extends to functionals, in the same fashion. Very simple again. We start from a functional, and then we're going to ask u to be in extrema, so we do a perturbation. So, u is here, the yellow curve. It's basically our original function now. We do a perturbation. Now, the perturbation, now, is another function. Because we're not talking about points. We're talking about entire functions. So this is the perturbation. And then u tilda is the sum of my function and the perturbation. And what I want is when I replace U by U tilde, I replace it by the perturbation and I take the derivative, according to the perturbation, has to be equal to zero, the same way than when I did the point perturbation for functions, I got zero. If I do a function perturbation here, I have to get zero. From this, after a few lines of just taking derivatives, we get the Euler-Lagrange equation. So that's how we get basically the Euler-Lagrange equation. Doing the derivative according to the [INAUDIBLE] that we are using. Once you have basically the Euler-Lagrange equation, you can do the same that we did for functions. You can move your function in the direction of the Euler-Lagrange equation until steady state, when you get to state, you have solved your Euler-Lagrange equation. So let's just illustrate that here. Look what happened. I want to put that, basically. I want to do that again, okay? So you're going to basically move until you get to the steady state. When you get to steady state, basically we've got zero. So let us recap what we just saw. In regular calculus, we basically are moving a point, not a curve, removing a point in the direction of minus the derivative and basically we get, the solution which is the minima. In the case of the calculus of variations, we move, we move the function in the direction of the Euler-Lagrange basically again the derivative for the perturbation. We do that all the way to stay state, okay? And just do that again. We do it all the way to steady state and at the point of steady state basically this became zero. And we have solved the Euler-Lagrange which is the condition, basically, for getting an extreme amount in this case a minima. We can change the signs and get the maxima. So we are going to see, in the next video, how we can take you to be an image. We'll do some interesting function of F. And get, for example. Blurring the image or get image noisy. So once again, our unknown is the image that basically optimizes this function. And we get the complete analogy between regular calculus and calculus of variations and as I say for [INAUDIBLE] and even more for calculus of variations, these type of techniques go way beyond image processing. What image processing did, and in particular the area of partial differential equations, is to borrow these techniques from continuous mathematics into this area. Once again, I'm going to show you in the next video an example of a function that is very useful for image processing. I see you in the next video. Thank you very much.
