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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Sergio Escalera Guerrero i Cristina Palmero Cantariño, [en] The study of the gaze behavior using images is an ongoing research area of significant relevance in sociology and psychology for its participation in multiple cognitive processes. The use of Convolutional Neural Networks for their study is the standard option, but they need accurate gaze data in order to improve their estimation. The lack of gaze data in some populations, such as children or the elderly, makes the estimations of the devices that use this technology less accurate with them. In this project, and with the objective to solve this lack of available data, we propose a 3-camera system that will allow us to calculate the gaze vectors of any person in a simple, accurate and minimally invasive way, explaining step by step all the mathematical concepts involved in its calculation.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Juan Carlos Naranjo del Val, [en] In this work, we will study projective geometry mainly in the plane, beginning by giving structure of projective line to a conic so we can work with projectivities on conics. This is useful to prove theorems such as Desargues’ theorem for pencils of conics. We will also be studying metric projective geometry which will allow us to dicuss about circles, orthogonality, angles, etc. in the projective plane. This will be used to on one hand, study classical results such as Euler’s circle and, on the other hand, to prove a more advanced result, Steiner’s porism. To see this theorem, algebraic geometry is generally used but on this project we will study it under the projective geometry framework.

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Rosa Maria Miró-Roig, [en] Fixed $4 \leq d$ and a primitive $d$th root of unity $e$ we consider the ideal $I_{d}$ generated by all the $\mu$ monomials of degree $d$ invariant under the action of the diagonal matrix $M= Diag(1,e, e^{2},e^{3})$. We prove that $I_{d}$ is a monomial Galois Togliatti system ($GT$-system). We study the variety $F_{d}$ image of the Galois covering $\varphi_{Id}$ : $\mathbb{P}^{3}\rightarrow mathbb{P}^{\mu-1}$ with cyclic Galois group $\mathbb{Z}/d$ associated to $I_{d}$. We call this 3-dimensional variety $GT$-threefold. Finally, we demonstrate that the homogeneous ideal of $GT$-threefolds is a lattice ideal associated to a saturated partial character from $\mathbb{Z^\mu}$.

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Rosa Maria Miró-Roig, [en] In this work we address the minimality problem of GT-systems in three variables introduced in [8]. To study this problem, we consider an $N \times N$ generic sparse circulant matrix $M$ with only three non-zero entries per row: $x_0, x_a$ and $x_b$ . We consider $d _{(N;0,a,b)}$ (resp. $p_{( N;0,a,b)}$) the number of non-zero coefficients in the expansion of the determinant (resp. the permanent) of $M$. The minimality of a GT-system is translated to the equality between $d_{(N;0,a,b)}$ and $p_{(N;0,a,b)}$ with gcd $(a,b,N)=1$. We prove that this equality holds in some open cases giving rise to new minimality results.

Aquest treball de fi de grau consisteix en la creació d’un programa escrit en C++ fent ús de les llibreries OpenCV que ens permeti trobar la geometria epipolar d’un parell fotogramètric sense conèixer els paràmetres d’orientació de la càmera amb la que s’han realitzat les fotografies. Aquest programa permetrà realitzar aquest procés de manera automàtica introduint dues imatges. Primerament es realitza una detecció de punts d’interès en les dues imatges i desprès es troben les parelles de punts homòlogues, les quals posteriorment són refinades eliminant les que realment no són bones. Finalment, amb l’ajut de la matriu fonamental, obtenim la geometria epipolar de les imatges. This end-of-degree project consist in coding a program in C++ using the OpenCV libraries that allow us to find the epipolar geometry of a photogrammetric pair without knowing the orientation’s parameters of the camera with which the photos were taken. This program will allow us to perform this process automatically by entering two images. Firstly, there is a detection of keypoints in the two images and then the pairs of homologous points are found, which are subsequently refined, eliminating those that are not really good. Finally, with the help of the fundamental matrix, we obtain the epipolar geometry of the images.

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Eduardo Casas Alvero, [en] This project consists in a revision and extension of a classic result, Waring’s theorem, about the barycenter of the intersection points of two plane algebraic curves. The theorem arises from the study of the parts with highest degree of the equation of a curve, which are completely determined by the barycentric parallel lines of the groups of asymptotes. Among other consequences of Waring’s theorem we study a result, due to Chasles, about the barycenter of the contact points of parallel tangent lines to a plane curve.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Eduard Casas Alvero, Since before the Renaissance the rules for doing perspective drawings were known, obtaining representations of the reality just like a well placed observer’s eye would perceive it. We use our knowledge of extended projective geometry to study the basis of these rules. We obtain some of the basic properties of perspective drawings and learn how to construct simple examples. Finally, we apply this knowledge to a practical exercise, namely to study the perspective in some Canaletto’s paintings.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Juan Carlos Naranjo del Val i Oriol Pujol Vila, The capacity of retrieving an object from images taken from different positions, known as 3D reconstruction, is being used nowadays in fields like medicine and robotics. In this project, we will give an approach on what are its basis and how is mathematically grounded. Finally, we will give an implementation for carrying out calibrated and uncalibrated reconstructions.

Let $q$ be a prime power and $r=0,1\ldots, q-3$. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order $q$ by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of $(q-r)$-regular bipartite graphs of girth 6 and $q^2-rq-1$ vertices in each partite set. Moreover, in this work two Latin squares of order $q-1$ with entries belonging to $\{0,1,\ldots, q\}$, not necessarily the same, are defined to be quasi row-disjoint if and only if the Cartesian product of any two rows contains at most one pair $(x,x)$ with $x\ne 0$. Using these quasi row-disjoint Latin squares we find $(q-1)$-regular bipartite graphs of girth 6 with $q^2-q-2$ vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Juan Carlos Naranjo del Val, The history of projective geometry is a very complex one. Most of the more formal developments on the subject were made in the 19th century as a result of the movement away from the geometry of Euclid. If one digs a little deeper, however, one can see that the basic concepts upon which this branch of geometry is based can be traced back as far as the fourth century, where a theorem of Pappus of Alexandria appears as Proposition 139 of Book VII of the Mathematical Collection. These very early discoveries along with Euclid’s Elements are the building blocks for the foundations that were laid down by the projective geometers of the 17th century. It is here that the history of the subject becomes more interesting. Great strids were made in the 17th century, but for some reason projective geometry did not become popular among mathematicians until the 19th century. From this moment, very important results on this subject were made by great mathematicians as Max Noether or David Hilbert. In particular, the base of these notes is the study of the theory of plane algebraic curves. Willing to know more about the geometry behind the plane algebraic curves, I began to work with the Algebraic Curves of William Fulton [1]. Introducing myself with the algebraic sets and its ideals, and with its properties as well, I venture on the theory of intersection of plane algebraic curves, studying them on the affine plane and on the projective plane. To doing so, I had to apprehend so importants results such that the intersection number at points on curves, the Bézout’s Theorem or the Max Noether Fundamental Theorem. As an application, I proved some problems of the algebraic geometry, from the classics to the most contemporary, begining with the Pappu’s Theorem and ending with the addition on the Elliptic Law. Moreover, I state some ideas of plane algebraic curves from a more modern point of view, talking about the divisors on smooth curves and the concepts that derive from them.

The goal of this short note is to prove that any projective bundle $\mathbb{P} (\mathcal{E}) \rightarrow X$has a tilting bundle whose summands are line bundles whenever the same holds for $X$.

Treballs Finals de Grau de Matemàtiques de la Facultat de Matemàtiques de la Universitat de Barcelona, Any: 2012 , Director: Juan Carlos Naranjo del Val, This work, which consists in two separate parts, will attempt to build an equation for a non-singular plane projective quartic over an algebraically closed eld, namely C. In the course of this trail, algebric and geometric theory will be introduced and discussed here, taking special care in subjects such as Bézout's theorem, the divisor language, Riemann-Roch's theorem and some matters about symplectic algebra. This path will lead us to Steiner-Hesse's theorem which provides a way to write an equation for a non-singular quartic. Finally, we will use those methods in order to attempt the development of an equation for the Klein quartic. An appendix lies at the end of the work talking about a pair of questions which can be investigated parting from the subjects in the main text, and a brief historical background for the things shown on it.

In this article, some structures in the projective plane of order $q$ **image** are found which allow us to construct small $k$ **image** -regular balanced bipartite graphs of girth 6 for all $k\le q$ **image** . When $k=q$ **image** , the order of these $q$ **image** -regular graphs is $2(q^2-1)$ **image** ; and when $k\le q-1$ **image** , the order of these $k$ **image** -regular graphs is $2(qk-2)$ **image** . Moreover, the incidence matrix of a $k$ **image** -regular balanced bipartite graph of girth 6 having $2(qk-2)$ **image** vertices, where $k$ **image** is an integer and $q$ **image** is a prime power with $3\le k\le q-1$ **image** , is provided. These graphs improve upon the best known upper bounds for the number of vertices in regular graphs of girth 6. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 121–127 2011 © 2011 Wiley Periodicals, Inc.

Given two ellipses, one surrounding the other one, Poncelet introduced a map $P$ from the exterior one to itself by using the tangent lines to the interior ellipse. This procedure can be extended to any two smooth, nested and convex ovals and we call this type of maps Poncelet's maps. We recall what he proved around 1814 in the dynamical systems language: In the two ellipses case and when the rotation number of P is rational there exists a natural number such that P^n is the identity, or in other words, the Poncelet's map is conjugated to a rational rotation. In this paper we study general Poncelet's maps and give several examples of algebraic ovals where the corresponding Poncelet's map has a rational rotation number and it is not conjugated to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical tools., Comment: 14 pages; 5 figures on 9 EPS files

In this paper we examine the problem of compositional data from a different starting point. Chemical compositional data, as used in provenance studies on archaeological materials, will be approached from the measurement theory. The results will show, in a very intuitive way that chemical data can only be treated by using the approach developed for compositional data. It will be shown that compositional data analysis is a particular case in projective geometry, when the projective coordinates are in the positive orthant, and they have the properties of logarithmic interval metrics. Moreover, it will be shown that this approach can be extended to a very large number of applications, including shape analysis. This will be exemplified with a case study in architecture of Early Christian churches dated back to the 5th-7th centuries AD Geologische Vereinigung; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Càtedra Lluís Santaló d’Aplicacions de la Matemàtica; Generalitat de Catalunya, Departament d’Innovació, Universitats i Recerca; Ministerio de Educación y Ciencia; Ingenio 2010.

We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded.

We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded.

Projective homography sits at the heart of many problems in image registration. In addition to many methods for estimating the homography parameters (R.I. Hartley and A. Zisserman, 2000), analytical expressions to assess the accuracy of the transformation parameters have been proposed (A. Criminisi et al., 1999). We show that these expressions provide less accurate bounds than those based on the earlier results of Weng et al. (1989). The discrepancy becomes more critical in applications involving the integration of frame-to-frame homographies and their uncertainties, as in the reconstruction of terrain mosaics and the camera trajectory from flyover imagery. We demonstrate these issues through selected examples.

Tradicionalment, la reproducció del mon real se'ns ha mostrat a traves d'imatges planes. Aquestes imatges se solien materialitzar mitjançant pintures sobre tela o be amb dibuixos. Avui, per sort, encara podem veure pintures fetes a ma, tot i que la majoria d'imatges s'adquireixen mitjançant càmeres, i es mostren directament a una audiència, com en el cinema, la televisió o exposicions de fotografies, o be son processades per un sistema computeritzat per tal d'obtenir un resultat en particular. Aquests processaments s'apliquen en camps com en el control de qualitat industrial o be en la recerca mes puntera en intel·ligència artificial. Aplicant algorismes de processament de nivell mitja es poden obtenir imatges 3D a partir d'imatges 2D, utilitzant tècniques ben conegudes anomenades Shape From X, on X es el mètode per obtenir la tercera dimensió, i varia en funció de la tècnica que s'utilitza a tal nalitat. Tot i que l'evolució cap a la càmera 3D va començar en els 90, cal que les tècniques per obtenir les formes tridimensionals siguin mes i mes acurades. Les aplicacions dels escàners 3D han augmentat considerablement en els darrers anys, especialment en camps com el lleure, diagnosi/cirurgia assistida, robòtica, etc. Una de les tècniques mes utilitzades per obtenir informació 3D d'una escena, es la triangulació, i mes concretament, la utilització d'escàners laser tridimensionals. Des de la seva aparició formal en publicacions científiques al 1971 [SS71], hi ha hagut contribucions per solucionar problemes inherents com ara la disminució d'oclusions, millora de la precisió, velocitat d'adquisició, descripció de la forma, etc. Tots i cadascun dels mètodes per obtenir punts 3D d'una escena te associat un procés de calibració, i aquest procés juga un paper decisiu en el rendiment d'un dispositiu d'adquisició tridimensional. La nalitat d'aquesta tesi es la d'abordar el problema de l'adquisició de forma 3D, des d'un punt de vista total, reportant un estat de l'art sobre escàners laser basats en triangulació, provant el funcionament i rendiment de diferents sistemes, i fent aportacions per millorar la precisió en la detecció del feix laser, especialment en condicions adverses, i solucionant el problema de la calibració a partir de mètodes geomètrics projectius., Traditionally, the reproduction of the real world has been shown to us by means of at images. These images used to be materialised by means of paint on canvas, drawings or the like. Today, we still see hand made pictures, by fortune, although most of the images are acquired by cameras and they are either directly shown to an audience, like in the cinema, television or photographs, or they are processed by a computer system in order to obtain a particular result, like in industrial quality assurance or bleeding edge artificial intelligence research. Applying mid-level processing algorithms, 3D images can be obtained from 2D ones, using well known techniques called Shape From X, where X is the method for obtaining the 3rd dimension. While the evolution to the 3D camera begun in the 90s, the techniques for obtaining the most accurate 3D shape need to be continuously improving. The application of 3D scanners has spread signi cantly in the recent years, specially in elds like entertainment, assisted diagnosis/ surgery, robotics, etc. One of the most used techniques to obtain 3D information from a scene is triangulation, and more concretely, triangulationbased laser scanners. Since their formal appearance in scienti c publications, in 1971 [SS71], there have been contributions for solving inherent problems like occlusion avoidance, accuracy improvement, acquisition speed, shape description, etc. All of the methods for obtaining 3D points of a scene is accompained with a calibration procedure, and this procedure plays a decisive role in the performance of the acquisition device. The goal of this thesis is to provide a holistic approach to the problem of shape acquisition, giving a wide survey of triangulation laser scanners, testing the performance of di erent systems, and to give contributions for both improving acquisition accuracy under adverse conditions and solving the calibration problem. In addition, the calibration approach is based on previous works that used projective geometry to this end.

El objeto de esta nota es estudiar ciertas superficies desarrollables que contienen a una curva del espacio de 3 dimensiones y cuyas generatrices ocupan alguna posición especial respecto al tetraedro fundamental proyectivo unido a cada punto de la curva. Veremos de esta manera como se pueden caracterizar geométricamente ciertas curvas alabeadas cuyas curvaturas proyectivas cumplen determinadas condiciones

Consideremos una representación conforme de una superficie S sobre otra S' y sean P, P' dos puntos homólogos. El problema que vamos a estudiar es de naturaleza «local», es decir, consideramos únicamente la representación conforme entre dos entornos de los puntos P y P'. Sean π y π' los planos tangentes a ambas superficies en P y P' respectivamente. A cada curva C de S que pasa por P corresponde una curva C' de S' por P'; sea Q el centro de curvatura geodésica de C y Q´ el centro de curvatura geodésica de C que es un punto de π'. Al variar C, siempre pasando por P, tenemos así definida una correspondencia entre los planos π y π'. Puesto que cada punto Q de π es centro de curvatura geodésica de curvas de S que pasan por P y la transformación conforme es biunívoca, resulta que dicha correspondencia es también biunívoca y se extiende a todos los puntos de los planos π y π'. El teorema principal que vamos a demostrar es el siguiente: “La correspondencia Q – Q' entre los planos π y π' es una proyectividad” Let P, P' be two corresponding points in a conformal correspondence (local) between two twice differentiable surfaces S, S'. Π and π ' are the tangent planes at P and P' respectively. Each curve C of S pass through P corresponds a curve C 'of S' for P'; Q is the center of geodesic curvature of C and Q’; the center of geodesic curvature of C is a point of π '. By varying C, always passing through P, we have thus defined a correspondence between the planes π and π '. Since each point Q of π is the center of geodesic curvature of S curves that pass through P and the conformal transformation is one-one, is that such correspondence is also one-one and extends to all points of the planes π and π '. The main theorem we will prove it: "The correspondence Q →Q ' between the planes π and π' is a projective

In this note we describe the intersection of all quadric hypersurfaces containing a given linearly normal smooth projective curve of genus n and degree 2n + 1. Let C be an irreducible nonsingular curve of genus g, defined over an algebraically closed field of any characteristic. Let C be embedded in Pr by a complete linear system ILl. Saint-Donat [5] has proved that if deg L > 2g + 2 then the homogeneous ideal Ic of C C PT is generated by quadrics, and if deg L = 2g + 1 then Ic is generated by quadrics and cubics (see also Fujita [1]). In [2], Green and Lazarsfeld have announced the following result: In case deg L = 2g+ 1, Ic fails to be generated by quadrics if and only if C is hyperelliptic or L embeds C with a trisecant line, i.e., HO OC(L Kc) # 0, where Kc denotes the canonical divisor on C. In this note we describe the intersection of all quadric hypersurfaces passing through C C P' in the borderline situation deg L = 2g + 1. The main ingredient of the proof is a theorem of Castelnuovo on the postulation of points. A g9 on a curve is, by definition, a base-point free linear system of degree d and dimension 1. For the definition and properties of rational normal scrolls see [3]. Our result is the following. THEOREM. Let C C pn+l be a linearly normal smooth irreducible curve of genus n > 4 and degree 2n + 1. If W(C) denotes the intersection of all quadric hypersurfaces of pn+1 which contain C, then either W(C) consists of C plus (possibly) a line and finitely many isolated points, or W(C) is a rational normal scroll of dimension 2. In case W(C) is a scroll, one of the following situations occurs: (i) W(C) is smooth and C meets every fiber of W(C) at three points. C is trigonal and embedded by the linear system IKc + g9 . (ii) W(C) is a cone with vertex P, and C passes through P and meets every fiber of W(C) at P plus two other points. C is hyperelliptic and embedded by IP + ng' I. (iii) W(C) is smooth and C is a divisor in W(C) of class 2H + R, where H denotes a hyperplane and R a fiber of the ruling. In particular C is hyperelliptic, the g9 being given by restriction of the ruling of W(C). PROOF. Throughout this proof we will assume that W(C) is not the union of C and (possibly) a line plus finitely many points. Consequently, there exists a curve G C W(C), G # C, with degree of G > 2. G is allowed to be a pair of distinct lines. Pick two distinct general points Qi and Q2 on G, none of them on C. If G is a Received by the editors October 20, 1986. 1980 Mathematic8 Subject Classification (1985 Revision). Primary 14E25, 14H45, 14N05.

For each value of k ≥ 2, we determine the number pn of ways of dissecting a polygon in the projective plane into n subpolygons with k + 1 sides each. In particular, if k = 2 we recover a result of Edelman and Reiner (1997) on the number of triangulations of the MÄobius band having $\textrm{n}$ labelled points on its boundary. We also solve the problem when the polygon is dissected into subpolygons of arbitrary size. In each case, the associated generating function $\sum Pn^{{z}^{n}}$ is a rational function in $\textrm{z}$ and the corresponding generating function of plane polygon dissections. Finally, we obtain asymptotic estimates for the number of dissections of various kinds, and determine probability limit laws for natural parameters associated to triangulations and dissections.

This paper presents a Time-of-Flight (ToF) camera based system for hand motion and gesture tracking. A 27 degree of freedom (DOF) hand model is constructed and fleshed out by ellipsoids. This allows the synthesis of range images of the model through projective geometry. The hand pose is then tracked with a particle filter by statistically measuring the hypothetical pose against the ToF input image; where the inside/outside alignment of the hand pixels and the depth differences serve as classifying metrics. The high DOF tracking problem for the particle filter is addressed by reducing the high dimensionality of the joint angle space to a low dimensional space via Principal Component Analysis (PCA). The basis vectors are learned from a few basic model configurations and the transformations between these poses. This results in a system capable of practical hand tracking in a restricted gesture configuration space.

Projecte realitzat en col.laboració amb el centre Universität Karlsruhe Actualment la detecció de moviments i la segmentació segueixen sent problemàtics en el món de la visió, problemes que han de ser resolts pel grup dʼinvestigació de navegació visual. A l'hora de treballar amb vehicles en moviment el que es desitja és poder atraure l'atenció dels conductors davant d'objectes amb moviments independents (OMIs), perquè aquests objectes son indicadors de situacions potencialment perilloses. Per tant, és un repte el poder aconseguir detectar OMIs en base a captures de vídeo a partir de càmeres monoculars. Aquest algoritme ha estat dissenyat sota la condició de detecció de tots els moviments independents dʼuna escena, sense coneixement previ del moviment de la càmera, que en aquest cas està instal·lada en un vehicle, i de l'escenari (com per exemple, el pla de terra ground plane). Si es considera, per exemple, un vehicle conduït per un camí en mal estat on les oscil·lacions verticals de la càmera solen produir-se, no es pot assumir un moviment en un sol pla, per tant no seria correcte assumir un ground plane. A més a més, es vol implementar un sistema robust en condicions sorolloses. Per finalitzar, sʼha suposat una càmera calibrada, ja que no sʼha assumit una distancia focal que variï significativament.