 To je pravda, da me prejsem Giovanni Bellettini, sej member z UCTP, in zelo se pravda o vsej konturstv, in rekonstručenih soličnih vsej. Čakaj. Čakaj. Čakaj. Čakaj. Čakaj. Čakaj. Čakaj. SehemST, n endurancekla mi mi, obs valovAH nWelcome H externallime masnice. Čakaj 80 dal crawlingovo, dar sem se pogledTw svaril! ČakajAl mi? Čakaj. Čakaj. Čakaj. Čakaj, ... toward the tales will be writtenin this book three years ago. It was published by Springer with three co authors of mine, three colleagues. Ok. So let me explain the initial motivation. The motivation is the motivation from applied mathematics if you want, Or also from how the human brain tries to reconstruct a shape, a three-dimensional solid object, starting from a draw on the plane. So for instance here, in this plane, you see in this plane, there is a casp, presence of a casp here. And this casp comes as a projection locally of this fold, of this surface here, b-dimensional surface in three dimensions. Also this here, another projection. So the problem is, if I draw a sort of graph in one plane, suitable plane, why the human brain reconstruct almost immediately and very often a three-dimensional, uniquely essentially a three-dimensional solid object? Why is it the case and what's happening? So I will focus on the projection of a contour, say, on a graph in the plane, and then I will try to reconstruct the solid object, the scene. It is called the three-dimensional scene. Possible applications, as usually in the reconstruction of images, come from images from satellites or medical images, computer vision, image segmentation, and so on. I am not at the moment, I will not discuss really these applications. This is a talk focused on mathematics. One of the motivations that we had is the following interesting example. So you have this two-dimensional draw. And how is it possible that most of us, depending also on how far we are from this picture, but most of us are sort of reconstructing some triangle here, which is not there, but it is, in some sense, our brain automatically imagine a triangle here. So and the problem is still open here. Try to find a variational model, some energy to minimize, which is capable to reconstruct the minimizer of which could be, say, something which contains this triangle. This is a quite difficult problem in calculus of variation, I have to say. This is, these are one of the many examples of Kanica several years ago. Concerning these kind of problems, our starting point were an article by Nitzberg and David Manford in the 90. And almost immediately, three years later, it appeared a book on this, a book on computer science. And again, you have given a given grade level. Like this is a grade level. You can imagine a function, which is, say, one here. Corresponding to the black part, or to the gray part, and zero corresponding to the white part. In any case, you have given this grade level. And you want to find an action, functional, and energy to minimize, to be minimized, and energy defined on some sort of plane curves. And the minimizer of which should be the correct object that you imagine to, your brain is imagining. This is very vague, of course, because I don't want to, even to introduce the energy, I want to go to geometry. So this is just, again, a motivation for me. So suppose that you have defined this energy functional. This must depend on the order between the objects in the three-dimensional scene. For instance, in this case, we are imagining, say, a white triangle in front of three disks. Just in front of three disks. So there should be, in the functional, something which depends on the order of, in some way, on the order of the three-dimensional objects. And the minimizer should carry depth information on the order, saying which is the object in front and which is in back. So ne, John, I don't even understand the problem, actually. What is your? I want to, so suppose that I give you this. I would like to find an energy function defined on curves, set of curves, graphs, or something like that. The minimizer of which should contain, for instance, this, the boundary of this triangle. And the minimizer says, OK, exactly on this picture. Saying that, OK, a minimizer could be, the boundary of the minimizer could be a triangle. And there is also depth information, which says that this triangle is in front of three disks. This is the aim, one of the aims of this kind of papers of the mathematical literature. Sorry. So, oh, no, it's skipping the next slide. No. It is the one after, check it there, not your, it's not. No, no, oh, I didn't check this. Oh, there is an action, OK. Let me see. No, it's, OK. Let's go on from here. So, the minimal configuration is carrying, probably, a depth order, but these kind of models have a limitation, in various limitations, because the problem is quite intricate, I have to say. This is a very interesting model, but still there are some limitations. One of them is that this model enforces a global ordering on the various objects. So, you cannot, for instance, you cannot model or get into your minimizer, you cannot get an object, which is in front partially to another object, but at the same time, in another region, is behind the object. So, this is a situation which is excluded by the previous model. Again, also, one connected object, which is partially, I mean, this is an object which self-overlap, in some sense. Again, this kind of three-dimensional solid set cannot be obtained as a minimizer of such kind of model. So, in the effort of try to solve and try to include these as possible minimizers, we modified, we tried to modify the energy functional into another energy functional, which, in particular, is defined on possibly overlapping regions, taken to account to self-occlusions. So, self-occlusions, now, in a new model, and more refined, if you want, model, can be considered. Now, it turns out, after some variation analysis, that this new energy is defined on what we will call apparent contours. I will define what it is in apparent contours. Again, we are in the motivation here. So, no details. So, this was a motivation for our study on apparent contours. So, the motivation was trying to avoid the problem of self-overlapping in the Manford-Nitzberg model. And then we come out with a model defined on graphs, suitable graphs, which are apparent contours, and minimization of this function also can carry information on how to reconstruct hidden contours, and how to, which is the object behind, and which is the object in front of. This is an action that I don't want to write here. It is an action depending on several things, depending on the order, on some labeling, on the length and curvature of the graph, and so on. But this is technical, and I don't want to enter into the details. OK, just an example. Suppose that your grid level is this one, in the sense that you know that your grid level is jumping on this curve, this sort of ellipse, and this part here. So, presumably, the Nitzberg and Manford model produces, under certain assumption on the constants inside the model, produces such a minimizer. What is this minimizer? It is just, if you want, a solid ellipsoid in front of another solid object. This is not connected. See, you have two objects, just one in front of the other. But there is another possibility, which cannot be obtained, probably, with the Manford-Nitzberg model, which is this one. And this one is as a connected, just one solid object, which is a connected set, which is just the, this is the so-called apparent contour of a mushroom. OK? Now, you see, here there are some integral numbers. I will explain the meaning of this integral number, these integral numbers in a moment. So, this is just to say that if you have this object, you want to imagine, where does it come, which is the three-dimensional set, the two-dimensional draw of which is this. There could be this possibility, but also this one. And these two possibilities can be captured by the new model, while the first model here cannot capture this second reconstruction. So, this is all about trying to have a computer to recognize images. Yes, of course, of course. So, do you also keep in mind how to compare this with what we would recognize when we look at what humans would see? What do you mean to compare? I mean, do you, for example, I think we would all interpret the top one as the bottom one rather than the middle one. You would interpret it. Do you prefer this? No. You want to think, even if you think that it is one. It is a mushroom. Yeah. Well, that's not even part of the question. No, it is very difficult to say what it is natural here and what it is not natural. What I just can say is that I'm just enlarging the possible minimizers. That's it. And probably the answer to this question is sort of psychological and biological problem. Neuro-physiological problem, which. You like mushrooms, don't you? Yeah, for instance. OK, so now, this is just a motivation, so short introduction. So, let me go on. So, let me define something now. So, I have, suppose that I give you a three-dimensional object, this potato here. I call this E, always in the seminar. And the sigma is the topological boundary of E. Is a solid set. The boundary is a smooth object. Is a smooth surface. OK, then I project it. I choose the projection direction. I put my i here in the direction of minus infinity orthogonal to this plane. This is the so-called retinal plane. It is the plane when I project this solid object. And what I see in the projection, the apparent contour that I will define for you is this curve. In this case is just one curve on this plane, on this projection plane. And this obtained directly from this solid shape in a specific way, I will explain. Notice that this kind of curve here is singular, because you have here a transversal crossing here. And then you have two casps here, OK? So, casps and crossings are very important in this discussion. So, let me try now to define what is the apparent contour. So, suppose that you have given a three-dimensional smooth solid shape. And this is connected, but in general we can allow also non-connected objects. OK, suppose imagine this object to be transparent for a moment. So, this is the same object, but semi-transparent, if you want, just semi-transparent. And then I define what is this graph, which is called the apparent contour of this. So, what is it? So, you have to do the following. You take all points on the surface in three-dimension. So, this is a boundary, a boundary of solid set. This boundary has, there are points on this boundary containing the direction of projection. The projection is in this direction, OK? And all points of the surface, such that the tangent plane at that point contains this direction make a smooth curve on the surface itself, OK? For instance, it is clear that at this point the tangent plane is this, and this tangent plane contains the direction of projection. But this is not only the case for this part, as you see, but there are other points having this property, OK? So, this point give you a smooth curve on the surface, OK? Now, you project orthogonally this smooth curve on the surface, which is a very smooth curve without self-intersection. It's a perfect curve on the smooth surface. You project it on the retinal plane. So, you take a projection on a plane of this space curve. And when you project it, then you get singularities, OK? And which kind of singularities you get? In this specific case, you get, again, this corresponds to a transversal crossing. And this corresponds to a casp, another casp, another casp, another casp. And this is what you obtain. It is a complete graph having kind of singularities, in particular, it has casps and transversal crossings, OK? Now, it is interesting, and this picture shows an interesting fact, that this graph consists of two parts. There is one bold part. And there is another part, which is not bold. This one, this one, and this half circle here is not bold. What does it mean, bold and not bold? Well, the bold part is the visible part. And the non-bold part is the part, which is non-visible, in the sense that you see, you cannot see if this is semi-transparent. But if it is not semi-transparent, you don't see this part, because it's behind a fold of the surface, OK? So this graph, essentially, is divided into two. There is a visible part and, say, an invisible part, OK? Singularity of the visible part are t-junctions, like this. So there is a t-junction. So this terminates here and is occluded by this. And there is a terminal point here and a terminal point. These are singularities of the bold part of the graph. On the other hand, singularities of the complete graph are just transversal crossings. OK? OK, now I need to assign to this graph some label, because giving a graph only is not enough to reconstruct the dimensional shape. So now I want to endow the graph with some label, in particular, integral numbers. What is the meaning of this number here, 0, 2, and 4? We have 0, 2, and 4, and 4. These are even numbers, integrative even numbers, assigned to each region of the graph, or each connected component of the plane minus the graph itself, OK? So this 0, this 2, is simply what? Is simply the total number of folds that a light ray meet starting from the projection plane. So for instance, if I take a light ray corresponding to this point, I intersect the surface twice in the direction of the projection, OK? So you see, here you intersect four times. You have four times, total intersection of all your light ray, 2, 4, 2, 0. 0 is the external region. There are no intersections, OK? So this is just, this number, labeling is quite evident. It's a very easy labeling. And notice that the orientation of my graph, now I consider a graph oriented in which sense, in the sense that on the left, locally, I have the highest number, on the left I have 2 and 0, here on the left I have 4 and 2, et cetera. And notice that these are always even numbers, OK? Where F, I call this F, F depends on sigma. Sigma is the boundary of the solid shape. I call it F is a natural even number. And actually, where F is 0, this is called the background, actually this number here can be recovered simply by as a winding number. So it's not really something too deep. It is just twice the total winding number of a point with respect of the, you fix a point, you look at the winding number of the curve and you multiply by 2 and you recover this labeling here. So this is F. But what it is more important in order to uniquely reconstruct the three-dimensional shape is another number. So this F is assigned to regions. Now I assign another number D to arcs. Two arcs. Now, what is this D? So here 2, 0, and 4 is the old F, OK? So now I give you another information, is number D. So D is now assigned to this arc and to this arc, to this, to this, and to this. What is this D? D is the following. In words, D is the number of folds. So you take a point on this arc. This point comes from a unique point on the surface because this is a projection of a critical curve on the surface, as we told before, OK? Now you look at the number of folds. You start from the point on the surface and you look at the number of folds of the surface that you meet, a light ray meet, before arriving to this x, to this point here on the curve. So it is the number of folds of the surface anterior in front of the point here. So let me explain with this example. D equals 0, here you have no, this is the visible part. You don't have here any fold of the surface in front of this point. Now you take, for instance, a point here on this arc, this thin arc here. So this is a point, essentially, here. So this point is the projection of a unique point on this surface here. How many piece of surface are in front of this point when you start a light ray? You just have one. This part in front is just one layer and then there is the point. So here this one. If I go here, now I have two layers, two folds of the surface in front. So D is 2. And then it jumps to 0, here. This is really what matters. F is not so important, but D is extremely important. So you assume there's no light. No, this is a good question. Everything here in this seminar is under an assumption of stability, say, genericity. This stability is a delicate and fundamental issue in singularity theory. I don't want to enter this, but exactly one of the conclusion is that you don't have any kind of flat part in the projection direction. They are not totally independent. No, they are not totally independent. Let me go on, and I think I will answer immediately to your question. So is this clear? So what? Yes. OK, so look at the point here. I want to define the number D associated to this point, to this arc. Actually, it is constant on this arc, this number. So I have this point here. This point here is the projection of a unique point on your manifold sitting on a curve. And the tangent plane, this curve is defined as the points of the surface where the tangent plane contains the projection direction. So this pointing comes from a point here. Now I imagine a light ray starting from here and going to here. How many folds of the surface I am meeting before going to this? This one, and this is one. This is the number D. This is the definition. As I said, F is not so important. Suppose that I give you a set of curves, and I give you only F. F must be an even number. This is the background, so 0, necessarily. Then I have to jump always by 2, 2 and 4, for instance. But then this cannot identify the topology of this object in three dimensions. Because this could be, if you want, a large sphere behind and a small sphere in front, or a large sphere in front and a small sphere behind, or a small sphere inside a large sphere. So the number F is not enough. But now if I give you a labeling D, let me call D labeling from now on, F is not so important, but D is more important in this seminar. Could be connected, could not be. Yes, if I give you D and F compatible, I will reconstruct uniquely the topology. And therefore, I can tell you F. No, no, not just from here. From here, no, but from here, yes. So let me give you now not only F, but also D. OK, now I give you D as follows. For instance, F is this, 2, 4, and 0 as before. This is F, but D, this is 0 here, because necessarily this must be visible. So 0, 0, also on this. But suppose that I give you here D equal 2, for instance. So what does it mean, D equal 2? Well, this means simply that the small sphere is behind the large sphere. If D is 0, then this means simply that the small sphere is in front. And with this one, it means that there is a hole. It's a small sphere inside the big sphere. So this is a suggestion of the same. Maybe if I find a sufficient number of conditions, compatible conditions within F and D, then maybe I'm able to say that always this comes from some three-dimensional. This is one inverse. I have not understood. Yes? No, this cannot be a donut. This cannot be a donut, because a donut has a hole here. And therefore, that is visible part. So this is 0. This cannot be 4. This could be 0 here. So let me see that I try to go to explain that one of the question is the following. Locally, essentially, this is one of the pictures that happens at one singular point. So at the transversal crossing locally, we have this kind of situation. There is D, which is not jumping here, but this other D is jumping from D2 to D2 plus 2. Here D1 is constant. And you have that F, as always this structure, the minimum value is, say, here, this is the maximum value F plus 4, F plus 2, F plus 2. You have to imagine this sort of local three-dimensional object. The orientation, you see, is such that the highest number of intersection with the light rays on the left, like here and here. And you see that here D is not jumping, but D is jumping here. Because when you go in this direction, there is a couple of folds in front of the continuation of this arc. So you see D2 at the transversal crossing jumps of two units. And there is a constant here. Oh, sorry. There is a constant, which is this. This kind of inequality. This is fundamental. So locally, the situation is at a singular point of transversal cross type is this. We imagine it is this. And what happens at the casp? At the casp, you see, again, in this picture, we have this constraint here, F plus 2, when I am here F, and then when I'm here inside, local inside the casp, F plus 2. And then if I have this orientation, I could have also D plus 1 here and D. But this is just an example. You see, from this, this is visible. So D equals 0, and here D equals 1. So it jumps of one unit at another singular point at the casp. OK, this is sort of what we imagine to be true when we have a three-dimensional picture of a surface embedded in a tree. We always come to this kind of compatibility conditions within D and F. The point is, are these enough? Are these efficient? Notice that indeed, in the second action function I introduced by just discussed a little bit, if I define my action function on this kind of graphs with leveling, it turns out that these three objects, which are the datum of the problem, I can put on this object a compatible leveling. You see? And the possibility of putting here a compatible D and F on this configuration with the triangle is one indication that if we are able to set the parameter so that this becomes a minimizer, then we end up really with a triangle in front of the circles. So in the second action functional, the action function is defined on this kind of graphs. So it's a complicated domain. The function has a complicated domain, because the domain are graphs with levelings, and also something else. Now, coming to your question, Stefano, is the following. Notional stability is required everywhere here. This is a fundamental point in singularity theory. So for me, the reference book, I am not an expert in singularity theory, but I know a little bit about this book. And there there is everything. Oh, sorry, this is mappings. Mappings is a mistake. So everything we need to write the book was contained, essentially, in this book here. But as a byproduct of this kind of stability assumption that we will make and generosity of projections, stability of shapes, and so on, it turns out that these graphs, apparent contours, the singularity are just exactly only of these two types, transversal crossing and casps in finite number and t-junctions and terminal points, and no more than that. This is a consequence of these stability assumptions. Now, questions are the following. If I give you a plane graph with crossing and casps, is it the apparent contour of a 3D shape? So if I give you a plane graph with a compatible leveling, say? Or another question. When you have systematically analyzed all the possible compatible labels? It turns out that those two are essentially the only compatibility conditions you need to solve this problem here. So they are sufficient, necessary and sufficient. And then this concerns the complete graph. But if I want to, because in my picture concretely, I don't have a complete graph. I just see the visible part of the graph. So the final aim would be, I give you only the visible part. Then from the visible part, is there a way to construct a complete, apparent contour? If it is the case, after that, can we construct a three-dimensional shape coming from this complete graph? OK, this is what we are doing here. The possibilities are finite right now. On what? I mean, in the video now. No, I think that this is a good question. Maybe it's related to this second point. If you give you just only visible part, yes, there will be essentially, no, there will be at least one way to produce a complete graph. And not finite, not finitely many. OK. Now, so these are questions concerning graphs, just graph. Then there are more difficult questions. Suppose that you give you two apparent contours, very complicated, I don't know, 200 casps or 300 crossings. Can I say that these are coming from the same object? They reproduce the equivalent 3D shapes. This is the same question that appears in not theory. And even there is extremely difficult. OK, so this is a situation much more general than not theory. So it is even more difficult. In the class of equivalent meaning now that they are isotopic. Maybe I will define it what it means ambient isotopic later. In the class of equivalent shapes is there one, which is the simplest one. And is it possible to automatize this issue on a computer program? Maybe this is the most interesting part of the book. Are there several moves? Yes, yes, yes. I'm going very slowly, I'm sorry. So let's see examples just to understand. Suppose that I give you this. This is just F, there is no D. But it turns out, you can check, it is not possible to put a compatible D here. So this cannot, a posteriori, cannot be apparent contour of a three-dimensional shape. But if you slightly modify it, this with the same F, OK, there is another region here, 0, this actually you can prove, it's easy, it's easy that you can put a D here. And once you can put a D, there is a hope that this is coming from a three-dimensional object. Actually it's true. So this is impossible, this is impossible again. There is no D, but here there is D. So it's not so, OK, one should try by himself to find a D. Mascherina di macdi carnivali. I don't know how to explain this in English. It is a mask, this is a mask. This is coming from a mask. This is impossible, OK. OK, there is a first theorem, finally. The theorem says that if you give you an oriented plane graph with casps and crossings, and I give you a labeling, satisfying the compatibility conditions that I showed before, just only those, then there exists a smooth three-dimensional shape such that the apparent contour of the boundary of the shape is your G. And the D you get from the 3D shape is exactly your D you have given a priori. So this is an inverse problem. I give you a graph. I reconstruct the topology of a 3D shape, OK. We found some related reference. I think this is a sort of bibliography in computer vision, maybe. This kind of reference related to this problem here. What about uniqueness? So there is, it is possible to reconstruct a 3D shape. Is it unique? Yes, in some sense. It is unique up to transformation in the direction of the eye along each point monotonically. So you don't change the fibers. I mean, you have given any x, you have a transformation which is strictly increasing in this direction and moving continuously with respect to x. It is clear that you have a sphere, but you can modify the sphere like this. And you don't change the apparent contour. So this is the only thing that you can do, essentially. So this is a uniqueness result, OK. OK, the proof of this is a proof based on the cut and paste, essentially. If you want to go into details in this proof, it's not so easy. Maybe for a geometry, it's almost trivial, probably. For me, it's not. And working in the details, it's not so immediate. And let me mention that what we reconstruct is not the roundest, more natural in some sense, which I am not able to define. It is just an object, OK. It's essentially unique. But the most round is another problem, OK. And once we have reconstructed it to see what it is really, is it a torus? It is a torus inside another torus. It is to tori, like this, with the sphere inside. This can be very difficult in general. OK, there is now a program, which is free, which can be downloaded apparent up contour. It's called up contour, this program, which reconstructed topological structure automatically of this three-dimensional shape. And the number of connected components of the boundary and the one characteristic of each of connected component. And also information distinguishing between these kind of cases here. The program, I think it's this kind of program in the number of crossings. No, no, this is very quick. If you do this in a few seconds, you get it, the result is not. But we have not checked this with 100 vertices. Also, because giving to a computer, the structure of a graph is not so easy. Because now I tell you, giving as input your graph is a kind of piece of information, which is not easy to give to a computer. So proving with 100 of nodes, I don't know exactly. So OK, then there is by product. So suppose that they give you your graph and the labeling consistent, compatible. Now suppose that from this theorem, there is a unique essentially unique three-dimensional shape having this as an apparent contour. Then it turns out that the Euler characteristic of the boundary of the shape can be computed just from the apparent contour. So I give you the apparent contour. And from that, it is possible to deduce exactly the Euler characteristic, only from that. In particular, in a special case, when this surface is connected, then it is possible to deduce also the Euler characteristic of the solid set bounded by the surface and on the complement of the set bounded by the surface. All these can be found in this program up contour. We always look inside the stable and generic surfaces. So there cannot be flat, stricter increasing monotonic, stricter increasing must be at any point. OK, to show the difficulty of this kind of problems, let me tell you, let me make you this observation. Suppose that E0 is a noted solid torus like this. Suppose that E1 is a standard solid torus. Suppose that E2 is anti torus. So it's just a sphere, then you remove two disks and you do a gallery inside the sphere, a noted gallery like this. So these are three solid objects. It turns out that they have the same, that the surface of these have the same Euler characteristic. It turns out that also the interior and the exterior have the same Euler characteristic, but they are not equivalent. So this is an example that showed that, of course, knowing the Euler characteristic is not enough, even if you know the Euler characteristic of the interface of the inside and of the outside. So this says that, of course, you have to know to inspect a lot of invariance of this kind of objects, invariance in some sense, like in no theory, essentially. So other invariance of the apparent contour and more interesting of the 3D shape can be considered. Some of them are automatically obtained by the up-contour program, for instance, the fundamental group of the complement of the set, but also the fundamental group of the inside and the fundamental group of the surface itself. So another remark is the following. How to recognize the shape? How to say that this is a sphere when you have an apparent contour tremendously complicated, it turns out to be a sphere. What do you do? Well, again, as in no theory, there should be some way to simplify it, maybe, and to find a sort of elementary, class of elementary moves that modify locally the graph and simplify it, sitting inside the equivalence, isotope equivalence of the 3D shapes. I will explain a little bit about these moves, and this is maybe one of the most important features of this program, because applying these moves is not at all trivial. Now, the software code is devised in such a way to be insensitive to the particular embedded of the apparent contour in the plane. And as I was saying, it is designed in order to capture only orientation relative position of the ashencine topological structure of the apparent contour. You can always modify the apparent contour with an isotope of the plane, with a diffimorphizum, if you want. But this should not be important for the program. So OK. Now, what about the completion problem? Let us try again. Now let me focus on the visible part only of the graph. I want to understand maybe a class of sufficient conditions on the structure of this object in order to be visible part of some apparent contour. For instance, we realize that this cannot be visible part of anything, because this singular point, terminal point, cannot be, cannot terminate in the background. This is impossible. Also, this is impossible because it is impossible that to have a visible part, which has the background on the left, with this orientation. This is, again, impossible. One also realize that the emerging arc here, let me call this the emerging arc, must always lie on the right of the occluding arc. So this is allowed, but this is impossible, at least at the first side. And the theorem says that if I give you an oriented plate graph with t-junctions and terminal points only as singular points, so no casps, no crossings, but only t-junctions and terminal points, and suppose that the previous condition are satisfied, suppose that we do not fall in the example of the previous site, then this is visible contour of something. In the sense that there exists a labelled apparent contour which complete it in such a way that the given k is the visible part of g. So philosophically, we have the following. We start with only a visible part. Through this theorem, we reconstruct at least one complete apparent contour with a consistent labelling. After we have this, we can reconstruct a three-dimensional shape. This kind of property is exactly sufficient and necessary. Yes, you see, you realize that this is impossible, this is impossible, and these are only possible cases. That's it. And this is a characterization. This is the other way round. What I'm saying is the other way round. I say the other way round. So in some sense, we have solved the problem in computer vision in the sense that I can tell you in some sense when what I see on a two-dimensional picture, when this object is a visible part of an apparent contour which is coming from a three-dimensional shape. So I reconstructed a three-dimensional shape just only from the knowledge of part of the contour of the plane on the plane contour. So what else is there possible besides t-junction and terminal points, the principalness? No, the visible part. And that is always like that. Any visible part, always like that. This is due to genericity and stability assumptions. The proof of this theorem, this complexion theorem is constructive. And again, there is a part of the program which is called visible contour, which does this. And the proof is based on Morse description of the graphs. I don't want to enter this direction, but the Morse description of this object is fundamental in every description. Now, this is an important remark. When you do this proof, you have to prove this in any case. So you don't have a specific graph. You have any graph, and you have to prove you can do a complexion with a compatible labeling. This is a global problem, because you do this locally. Any t-junction becomes a crossing, you continue. Any terminal point becomes a casp. But then in the end, you have to close it with a compatible global labeling. This is the difficulty. And our proof produce one solution, but maybe it's not the best, and necessarily the best that the brain, the human brain, does. So the scope of this theorem is just to show that the hypotheses are sharp, and we are able to construct at least one complexion. Examples. So suppose that I give to the program just these two curves. Then the visible contour. This is the visible contour, and not oriented. Then what is the answer of the program visible contour in sufficient orienting information, because the internal circle cannot be implicit-oriented, because it can be oriented in both in two. This orientation is trivial, it's just like this. But this could be in two ways, and so the program cannot go on. Second case, suppose that I give you the same visible contour, but now I add the information that this is oriented like this. Now the clockwise orientation of the internal circle. And now suppose that the background is just only here, not here. So we impose f equals 0, only in the exterior region. The result in the construction cannot be a torus, because this is not a hole now, it's not a hole. So what is the solution of the program? The solution of the program is the following. The program adds a new circle, and the unique solution of that is this. What is this? Is a visible torus, the bold one, and there is a sphere behind. So this is what the program is producing, intermediate sphere behind. Another is the same visible part. And now you force f equals 0, also here, and the visible contour reconstruct the torus, seen from above. Again, same visible contour. Now let us orient this on the left. So if I orient this on the left, it means that f must be higher here. So this cannot be a hole, cannot be 0. f is 0 only here. So visible contour reconstruct the apparent contour. So there are no hidden arcs, as in example 2. But this is just a small sphere in front of a bigger sphere. This is what the program gives to you, just only from this information. Now, more interesting maybe now, suppose that I give to the program this visible contour here, and suppose that is in this region, I force this to have 0 f. So I suppose to be inside the background, this small e region here. So this is what visible contour produces, which is exactly our theoretical proof of the completion theorem. When we imagine a Morse line, our proof produces exactly this object. Now, as I said, this is not the most natural object that one could imagine. This is just one solution. But what is this? You see, this terminal point is becoming a cusp, and the completion, this is becoming a crossing, like this. And then everything goes down, and then you have to close it and put a consistent labeling. Is this possible? Yes, because this is a theorem. And what is this? Now, the program is able to tell you what is this. This is not clear. I mean, maybe it's clear, it's a torus. But the program is telling to you that after some command that you give, you simplify it with a list of moves that I will explain at the end. And what the program gives to you is the torus. So maybe we are a little bit happy about this. Now, suppose that you don't force F to be 0 here. Now, what is the solution? So you remove the marking of the small internal region as part of the background. So the background is only this one. You enforce your program, visible contour. Our proof, theoretically, is this one. It's a very complicated complexion where in the small circle there is an F, and the other labeling are the D. This is what we find in our proof. This is what visible contour is doing. And what is this? This is not clear at all. This must be simplified with the use of the moves. And visible contour, after using the moves that I will explain in a minute, must simplify the former 3D sphere, for instance, using the elementary moves. So this is actually is the apparent contour of a sphere, OK? Using the elementary moves that I will explain. So the above example show that it may be difficult to recognize your 3D shape. The topology of what is this is not clear. So it is necessary to find moves that simplify the apparent contours, remaining inside the equivalent shapes, ambient isotopic shapes. So this is one of the main problems in theory, of course. And in that case, these moves are three elementary moves and are called rademeister moves. So here also there are essentially six elementary moves. This situation is more rich. It's a little bit more complicated than not theory. And these moves have been implemented in this up contour. So there is a theorem here that for me maybe is one of the most difficult that says that two 3D shapes are equivalent if and only if you can pass from one apparent contour of the first one to the apparent contour of the second one only through a finite number of elementary moves. Like in not theory, exactly. And which kind of moves are these? These are essentially six moves. This is called t-move. Locally you are allowed to do this. This corresponds to the third move of rademeister. Then there are new moves. So this is, this collapse to casps. And also this collapse to casps. So this is called leap and big to big. This is, swallow tail is this one. This collapse to casps and destroy this swallow tail. The inverse, OK. There are also this one. And then the other, this k-type, k1, k0, k1, vk, k2, k1, et cetera. These are locally the move of type k. But essentially these are, the theorem as before says that these are only, the old, this set of moves is complete. So these are enough to describe isotopic shapes in 3D. These kind of moves on two planar graphs. So this kind of, OK. These are realizations in 3D taken from a paper of these authors in 206, which shows a sort of physical realization of the moves, OK. This is, I found this nice, sorry, producer. This is the reference. So now the proof of that theorem is, for me, is quite complicated. But it is based on classical and very important basic results of Whitney, Tom Arnold, and others on singularity theory, OK. The program up contour can manipulate these moves and can, in order to simplify and possibly recognize the object, OK. For instance, there is a rule, an action rule. If you ask this, the computer tell you which kind of, I can apply this move, this move, and this move here, here, and here. What do you want that they do? Start from here, and then he tries to go on, OK. For instance, what the computer is doing, if you give the computer this object, you apply moves k1b. So it is possible to push this part inside, keeping the two casps. And then there is a leap move, which collapse the casps together, and you come here, so this is sphere, OK. Now, if you think of a torus, if you look of a torus oblikli, just two minutes maybe, if you look from a torus, you look at torus oblikli, you see four casps. But if you look of a torus from above, for instance, you don't see any casps, you just see two circles, one inside the other, with the proper orientation. Now, essentially, this is interesting to say. It is always the case. So it says that if you have a smooth closed surface embedded in a tree, you can find a set of moves transforming it in such a way that the apparent contour has no casps, like for the torus. So maybe you increase a lot the number of crossings, but at least you can destroy all casps. This is a general theorem again. Is it all in your book? Yes, everything is there. So final remarks, sorry for the delay. We are now trying to understand a little bit better some other kind of invariance of 3D shapes. These are shapes more complicated than knots and links. So it's not so clear for us. We are trying to work on this. But we are going to three-dimensional topology, and I am not an expert really on that. So what I've said up to now does not include cubes, for instance. So, for instance, polyadrolipshits objects. We have plenty of no smooth objects here in this room. And the theory does not include that, unfortunately. There is a new singularity, y-junction in that case. And what about the discussion was concerned with embedded surface? What about immersed manifold, or even less than immersed, with not even immersed, with the singular point, the differential knot of maximum rank, et cetera. There is a lot of literature on that. Maybe everything has been done. So there are just few names on that. OK, that's it. Sorry for the delay. Thank you.