Hello. We're going to start with some background in the area of planar differential geometry, the area of differential geometry and the understanding of curves on the plain. Before I do that, I want to provide some credit. I work a lot in the area of partial differential equations in image processing, and some time ago I wrote a book.. My good colleague and friend Ron Kimmel also wrote a book in the area and we used to give a lot of talks independently. I gave full semester classes, he gives full semester classes and sometimes we give together tutorials on this subject. A lot of the slides I am using during this week come from those tutorials and I want to thank Professor Kimmel for letting me share some of his slides and combining them with mine so that I can provide the topic of partial differential equations in this week. Thank you Professor Kimmel for that. now, let's talk about curves, so we can describe what do I mean by differential geometry of curves. So what's a curve on the plane? A curve on the plane will define a coordinate system, and this is for convenience. We have the X axis and the Y axis and then we define a curve parameterized by P, we called P the parameter. P runs between zero and one, and for every value of P, we get a coordinate in this axis, and a coordinate in this axis, and that's a coordinate of one point on the curve. So, basically, for example, for P equals 0.1, we get this point. Then for a different p like 0.7, we get another point, and so on. So I'm sure you get the idea. Basically a different values of p within zero and one give us different points on the curve. Now, if a curve is closed like here, what we have is that c at zero. Is equal to C and one. So it starts and ends basically on the same point, and that's a closed curve. Now you're very familiar with this concept. If maybe you're not so familiar with curves but for sure you are familiar with functions. So functions, you have a coordinate system, X and Y as here, and you have a function, that for every value of X there's one unique value of Y. That doesn't happen in normal curves. In that case your curve is parmitrized by this coordinate itself so we get c of x equal. The first coordinate is X by itself. That's the first coordinate, and the second is some function of it. So these are just functions that you are very familiar with. Again, the difference between a function and a general curve is that for each one of these coordinates, each one of the X's, there's only one Y, and that's why it's called a function. There's a, basically a unique value. So it's a particular case of a curve. We could think about curves also as a generalization of functions on the plane. Now, once we understand the curve, which is parameterized by this P, and we're going to discuss a lot about P in the next few minutes. There's a couple of very important concepts of curves on the plane, and two of them are the tangent and the normal. And also what we call the curvature, which we're going to explain in a second. Let's start from the tangent. Basically, we have a curve, and we can take the derivative of this curve. So this is where differential equations, differential geometry starts to appear. C_PP stands for the derivative of C taken by p. So its basically CP. It's just a simplifying notation of the derivative of C with respect to P, which is of course a vector, is the derivative of x with respect to p. C is a function of p, so I'm allowed to take the derivative of that, Y as a function of P. so this is what we have here and we normalize by its length, we want to get that unit length and that's actually very, very important. As we are going to see, there is a particular parametrization called S that provides us a tangent that has unit length. And we're going to use that a lot. And we're going to see how we go from P to S. But the basic idea is that we take a derivative of this curve. And you learn in your basic calculus class that the derivative of a function gives you a vector which is perpendicular, but which is sorry, it's tangent. To that function. In a second we are going to talk about perpendicular. So once again, the derivative of a function gives you a vector which is tangent to that function. So the derivative. Of this curve are vectors, which are tangent. And they don't have to have unit length, so we normalize them, and there is one particular parmituzation we are going to write s that will give us a unit length. And again we are going to see how we go from one to the other. Now what we are going to, now that we have the tangent, and in particular, bear with me for a second. We have the curve parameterized in such a way that the tangent, meaning the derivative, is always unit length. What do I mean by unit length? That the, basically the magnitude of the derivative vector is equal to one. Okay? Unit length. Let's assume that I can basically write a curve. Instead of a function of P, a function of a different parameter S that gives me unit length. Now length means, and I'm going to write this down and explain again, this means that the inner product of the vector with itself is equal one. If you don't remember what's the inner product between two vectors, is basically you multiply the first coordinate of vector one by the first coordinate of vector two, and to that you sum the product of the second coordinate of each vector. So it's X of the first times X of the second, plus Y of the first, times Y of the second. So that means that we have unit length. Let's take a derivative of this. So I'm going to take a derivative according to s in both sides. So this is the derivative. A constant is always unit length, so this becomes zero. The derivative of this is like any other product, it's the derivative of the first times the second plus the derivative of the second times the first but because these are equal then this is two times the inner product of the first derivative with the second derivative. It's like any other product, okay? The two I can eliminate, and this means that the first derivative, according to arc length, is perpendicular to the second derivative. Once again, just to remind you, two, the inner product of two vectors is equal to zero, if and only if one vector is perpendicular to the other. So from here, we get that CS, the first derivative, is perpendicular to the second derivative. Now the magnitude of the first derivative is one. The second derivative, we have it here, is perpendicular to the first one. All the time when we are taking derivatives according to s, the first is unit vector, the second is perpendicular to it. So this is the second derivative. Now it doesn't have to be a unit vector. Most of the time it won't. And we call the magnitude of that second derivative we call the curvature. This is the curvature. Okay, this is one definition of the curvature. It basically tells us how much the tangent is changing and it's very intuitive. If it changes a lot, it means that the curve is curving along there and has very high curvature. It it's not curving a lot, for example a straight line has zero curvature, so what's a tangent of a straight line? At every point it just changed colors. At every point the tangent is in the same direction. It's in the direction of the line. So the change of tangent, tangent is the first derivative, second derivative is the change. The change of the tangent is zero. So a planner, basically a straight line has zero curvature. What's a curvature of a circle? So we have to, basically again, draw the time change all the time. And then see how they're changing infinitesimally when I move from one point to the other. Those are derivatives. Derivatives means when you're very, very close how is the tangent changing? First of all, it's constant because the circle is all the same. So it has to be constant curvature. And then it's not hard to prove that the curvature is one over the radius. Okay, so the curvature is one over the radius of the circle. The larger the circle, the more curvature it has, the, the, the, sorry, the larger the circle, the more radius it has, one over the radius, the smaller the curvature. You could think about a straight line as a circle with infinite radius, very, very large radius and then one over infinite is zero, so basically has zero curvature. So there's a lot of new concepts in this very colorful slide. Let's go over them again. We have the tangent which is as we know from basic functional analysis, basic calculus, is basically the first derivative. We want to normalize it, we want to have unit length and there is a parametrization, we're going to see how we get to it, that basically gives us that the first derivative is always unit length. Always unit length we take a derivative of that concept and we get the second derivative which is always perpendicular to the first one only when we are talking derivatives taking derivatives according to s. And then that second derivative has a norm. Basic a demand to that vector and that's curvature. Curvature is the magnitude of the second derivative according to s. This is going to be called the arc length, as we are going to see in a second why. So we have s, the arc length. And we have. This is kappa, is the Greek letter kappa, which is the curvature. Now we have s and the curvature and we talk about their relationship between the curvature and the radius of a circle. These are actually the only two objects with constant. El cleanant curvature straight line zero a circle one over the radius. Now why are these guys s and curvature kappa. So important. Very important, because they're going to be preserved under transformations of the curve, under certain transformations. What kind of transformations can we have in curves? So, I have here a curve. Where's the curve? It's the boundary of my object. Now we are starting to understand why differential geometry is important, because curves are boundaries of objects and objects are present in images. Now, I can transform these in a number of ways, but always preserving it on the plane. I generally transform is, I take my coordinates, which are a vector and I multiply by a matrix and I translate and I get a new vector. And that's what called an define transform, if I don't put any limitations in this matrix, so I take the curve and with an define transform, I transform it into a new curve. Now, I can also just rotate my shape. Here it has the form, it has stretch in different dimensions in different directions. If I only want to translate and rotate, so I rotate and translate or I translate and rotate, that imposes some limitations in A and it's not hard to show that in order to only allow rotations and translations, the two vectors of A, so A is a two by two matrix. I have here a vector multiplied by a two by two matrix. Sorry, a vector that I get it by a matrix multiplying a vector. This has to have columns which are orthogonal, meaning the inner product means zero we just saw that in the previous slide. Inner product equals zero means orthogonal, and they also have to be unit vectors. So this is just not hard to prove, but important thing here is that you can do a rotation and translation that's called a Euclidean motion, or can you, or you can do a general transformation and that is called an affine motion and of course these are too or these are also related by an affine motion. Now there's a particular think in a fine motions that if they the determinant of this matrix is equal to one then basically those are called equal equi-affine motions, if the determinant is equal to one, what happens is that the area doesn't change. So you can stretch differently, in different directions. You can rotate, you can translate but the area is preserved. Of course if you only allow Euclidean motions, rotations and translations, the area is preserved, but if you allow. A fine transformation stretches there is none necessarily preserved but if the determinate of the transformation matrix is equal to one. Then it's not hard to show that the area of this object, so the area inside the curve or the object itself is preserved and those I call equi-affine motions, equi-affine transformations, and we're going to often just call them a affine and we're going to assume that we are talking about those that we get with the determinant equal to one. Now these are the type of linear transformations. We're not going to discuss non-linear. I can take X and square log Y, things like that we're not going to discuss. We're only discussing about multiplying by a matrix, so that means a linear transformation. So we have Euclidean and we have affine, normally equi-affine. So now that we know that we can rotate, stretch and do things like that, curves, meaning to shapes. Let's go back to, for a second, to curvature. We talk about these two guys. The arc length and the curvature. The arc length is the parametrization that gives us constant unit length tangents. The curvature is the magnitude of the second derivative according to the arc length. So if we go around the curve, around this shape, we have, for every point a given curvature and I'm drawing it here. So S as the parametrization goes around the shape and for every S I have a point in the curve and then I have a curvature. Now let's assume I do an Euclidean transformation. I rotate and translate my shape. Something very interesting happens. There are of course, corresponding points, this point got rotated and translated to this one and here is the beauty. There we have the same curvature, so the curvature here. And the curvature here, it's going to be exactly the same value. So, if you were to take the first derivative, second derivative, measured the magnitudes in both corresponding points, exactly the same value. That happens for all the corresponding points. So here we have another pair of corresponding points. We have the same curvature. Sometimes the curvature will be zero, sometimes will be negative. Every pair of corresponding points has exactly the same curvature. Every single point. So basically if we have two shapes, that are related by a rotation and translation meaning an Euclidean motion, then their curvatures are identical. Of course, you have to have a starting point, that is, the corresponding where is S equal zero. Once you have that, exactly the same curvature. So the curvature uniquely identifies the shape, actual rotation and a translation, and that's beautiful. That's a theorem that basically this curve uniquely identifies the curve a meaning the shape inside a curve up to a rotation and a translation. It actually has been shown these are beautiful results in transformational geometry that every other thing that doesn't change with rotations and translations. Every other differential thing is a function of S and kappa, of arc length and curvature. For example we could take the curvature and the derivative of the curvature, and that gives us another crazy curve on the plane. So for every curvature we, we look at, what's the derivative, how much is the curvature changing in that point? We do that for one shape. We do a rotation and translation. Voila. We get exactly the same value. And that's theorem in part, basically due to Cartan, and they're extensions of the theorem, some even more modern than these one's but it's the beauty of that, if you have the arc of S and the curvature Kappa, you have all what are called the differential invariance, functions that don't change when you basically do rotations and translations. So basically they uniquely identify objects. That up to rotation and translation. You cannot recover the rotation. They're invariant to them. But basically they uniquely identify those objects up to rotation and translation in Euclidean geometry. Now what happens with affine transformations? What happens when we don't just do rotations and translation, but we also allow basically the curve, the shape to stretch and with different stretches in different directions? Now unfortunately, or fortunately if you like math, not everything in the world of image processing is rotations and translation. So let's assume I have a shape like my hand. If I only do things like this parallel to the camera. I can model the motion of the object by a rotation. The moment I tilt it towards the camera, and then I do the projection of that on the camera, then this is not any longer just a rotation and a translation. And one of the ways to model this is with these fine transforms. So. Basically, if your object is relatively. On a plane relatively to the distance to the camera. Like here, for example, a drawing on a plane and we are looking, this is the camera. Us, is the camera. We are looking at kind of a plainer shape then there is affine transform. So it is a more general multiplication by a matrix once again because I'm not doing the rotation and the translations just in, in pilot place to the camera I'm doing them a bit tilted. And that can be model by an affine transform, a more general linear transform. So, we need to extend these basic concepts to basically allow for those affine transforms. remember an affine transform is a more generic, basically matrix. We have the x and y, but we multiply by a more general matrix that normally has unit determinant. We want to preserve areas just to make our life easier. We don't need this. We can do all this without, basically a unit determinant, but it's much easier if we assume this. So, when we were talking about rotations and translations, remember, we assumed that this column and this column were orthogonal. So, basically, they're in a product equal to zero, and they were also normalized to one. And those are rotations and then we have a translation. When we assume more generic things like here, just a unit determinant, those are affine transformed. And we have kind of this tilting that basically gives us stretchings in the different directions. So I want to define this arc length and curvature for the fine case, as well. Now in order to do that, let me just talk a bit more of what is parameterization. So I can have a curve parameterized by p as before and can have it marameterized by r by a different parameter. The curve will basically be the same. By R or by P the parameterization is kind of telling me how fast are we traveling the curve. Remember if you take the derivative of the curve according to p you get the tangent vector with a certain magnitude. If you take the derivative according to R you get the same tangent vector same direction with a different magnitude. If you remember from basic physics derivatives are velocity. So you're only telling me how fast are you traveling the curve. But you're going from the same point to the same point. So it's like you are driving the highway between two points. And the speed that you drive is the parametrization. But the highway is the curve. The highway doesn't change. So when we take functions geometric functions basically of our sorry I want one more let me come back. When we take geometric functions of curve, they have to be invariant to parameterizations. The distance between two points in the highway does not depend on the velocity I travel on that highway. So the geometric measurements that we take have to be invariant to the parameterization, and they also have to be invariant to basically the transformations that we want to do, either rotations or affine transformation. So if you take a map and you rotate the map, the distance between two cities has to remain the same distance, not just because I'm looking at the map differently, it changes. It has to be invariant to that. So let's use this concept to define the arc length S that we saw before. Let's just explicitly define it now and we are going to do the same for the affine case. But let's start with Euclidean, so basically, only allowing four rotations. So here is what we have, and this diagram is very good. Assume that you have a curve going through here. And you're going to take a very infinitesimal, very small portion of that curve. So I have, I can draw a coordinate system, like here and have a, very small movement in the X direction, and a very small movement in the Y direction. And then because of Pythagoras, this is basically X direction is perpendicular to the Y direction. So we have that this distance is the square root of the X direction distance and the Y direction distance, each one of them square. Simple Pythagoras. So now I'm allowing to multiply and divide by DP. P is my general parametrization. So I multiply and divide by a small change in P. Remember we're basically talking about the curve going around here. So I'm going to just draw that, I have a curve and I move a tiny bit X, a tiny bit Y and I measure how much I have moved there. So I multiply and divide and this curve is CP. So I multiply and divide by DP, just a small variation in the parameterization. And that's basically quality. And now, I basically move this DP in, so I get DP squared. Okay? I move it in, but I move it inside a square root, so I have to square it before, so I get this, okay? This is DP stays and this is exactly the definition of the Euclidean distance, the local, basically distance of the vector, just basically the definition of that. So we got that DS is simply DP. If we want to get the arc length S, it's basically the integral of that. Okay? So this is the way to go basically from. A generic parameterization P to the particular one S in such a way that the derivative now basically is unit length. Always the derivative will be unit length. It will be a unit length vector. And then the general distance of a curve, the total distance, distance of a curve is the integral of everything from P equals zero to P equal one of CPDP, this is the definition of the length. And if you basically replace what we have here is the integral of DS from zero to the length so when your parmitrizing by s your not going from zero to one your going from zero to the length. Basically once again. This tells us how to go from every parameterization to A, so from every speed of traveling between two cities, if you're basically integrating the magnitude of that speed. Along the way you're getting kind of a unit length of traveling. And that's what then lead us to basically. The Euclidean curvature when we took basically second derivative. So this is for Euclidean. Now Euclidean rotations and translations preserve length, and that's why I could use length here. Now, in the affine case or in particular in the equi-affine case, length is not preserve anymore. What's preserved? Area. So I can take areas. Now, in order to take areas, I need two vectors. One vector does not define an area, I need two. So, I'm basically going to take the first derivative. It's always a tangent V is going to be now my affine arclength. So the first derivative is not the unit vector. That's a characteristic only when you take derivatives according to S, to the Euclidean arc length. So I take a first derivative, and a second. Again, these two vectors not perpendicular in general. That's a property only when you take derivatives according to arc length. And then I basically define the, I look at the area and I'm going to look for the parameterization V that when I take first derivative. Second derivative this area is equal to one. I define it that way and basically you can do all the math. We take, the first derivative is a vector, and the second derivative is a vector, and we put them here as vectors in a 2 by 2 matrix. So remember, this is a vector, this is a vector, so we get the 2 by 2 matrix. It's of this form. It's basically. Let me write it down explicitly. X, derivative B, Y, derivative B, XVV, YVV. And that creates a matrix. And I'm going to ask for the determinate of this matrix the determinate of a matrix is the area of the parallelogram defined by these two vectors and now I'm giving you a new concept at, at. A, group of concepts that you might not be familiar with. But I am defining them on the way. So they, if I have a two x two matrix, this is a vector, this is a vector. The determinant of these is the area of the parallelogram between these two vectors, exactly what we have here. And I'm going to ask that to be equal to one. And remember for euclidean we ask the length to be equal to one here we ask for the area to be equal to one. I don't want to proof too many things to you but any parameterization p will give a new parameterization that holds this property following this integral. Okay? So that's a theorem. Just believe it for now. Now in particular, I can take p equal to the Euclidean arc length. That's allowed to do. So I can put cs and css. Now what are these two guys? This is unit length. Okay, unit length, and the second derivative in the case of the Euclidean arc length, is perpendicular to the first one, and it has basically length equal to curvature, Euclidean. So this is one, this is curvature. So, of course, this area is equal to curvature. So we get that V, the, a fine arc length, gets its. Obtain by the integral of the kregen curvature to the one third. The S. And therefore what we get is an extremely interesting relationship between the Euclidean arc length and defined arc length. So we have the Euclidean arc length coming from here and defined arc length and they're related by the Euclidean curvature. This is a beautiful relationship in differential geometry that relates Euclidean differential geometry with defined differential geometry. Now we have defined arc length, our next step is to find defined curvature. How do we do that in a very similar fashion as we did for the kleeant case if you remember in the kleeant case we started by basically doing the derivative. Of the property. That this was equal to one we say the length is constant and we took derivatives now we have to the area is constant when we use the fine arc length so here remember is a determinant of two vectors. And we're going to take the derivative. The derivative of the determinant is like any other product. We have to take the derivative of the first vector and then put the second and then the derivative of the second and put the first. So it's what we have here. The derivative of the determinant of these two vectors, cv and c, vv, is the determinant here, derivative of the first and we keep the second plus the derivative of the second and we keep the first. Now if we have a determinate with two identical columns that's zero so this basically is zero that's a very basic property of linear algebra. So we end up that the determinant here. Is equal to zero which means once again that these two vectors are parallel. When we took derivatives here, we basically had that. In their product to be zero, the two vectors have to be perpendicular. For the determinant to be zero, the two vectors have to be parallel. Another basic property again of linear algebra. So the first and the third derivative, when I'm taking derivatives according to defined arc length, these two vectors are parallel. So they are basically parallel. One for example goes in this direction. The other goes in this direction. The ratio between them. We define as defined curvature, the same way that we use the magnitude. of the second derivative, according to Euclidean arc length, we use this as Euclidean curvature, the ratio between these two parallel vectors is affine curvature. And it's not hard to show that this new, this affine curvature is an affine variant. What I mean is very simple if we take as we talk before, an affine transformation preserving area remember equi-affine. So we have a curve the boundary of a shape and we're taking a fine transformation. We compute the fine curvature of this shape as a function of the affine arc length of the shape. we do the same for the new curve. And voila, they're identical. So in the same way as we have Euclidean properties, with S and Kappa, Euclidean arc length and Euclidean curvature, we have them with the affine arc length and affine curvature. They're basically the same, and they completely identify the object, up to on a fine transform. So now you're an expert on planar differential geometry. You know the concept of arc length. It is particular parametrization that gives us some interesting property about the curve, like constant speed in the Euclidean case. or constant area in the affine case, and then from that, we got the curvature that is identical. For the Euclidean case, after rotations. For the affine case, after affine transforms. So those are very simple functions that uniquely identify shapes because they uniquely identify curves. Now our next step is basically, let's extend that a tiny bit to basically surfaces. And we're going to do that in the next video, and then you're going to have the basic tools of differential geometry to go and do the formations of curves and to do image processing and understand active contours. So I see you in the next video for that. Thank you.
