41 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

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    This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

    Surface meshing algorithms optimized for efficient electromagnetic analysis

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    Jedna od najčešće korišćenih metoda za 3D elektromagnetsku (EM) analizu metalnih i dielektričnih struktura u frekvencijskom domenu jeste metoda momenata primenjena na rešavanje površinskih integralnih jednačina. Pokazuje se da korišćenje bilinearnih četvorougaonih umesto široko korišćenih trougaonih elemenata zajedno sa upotrebom funkcija bazisa višeg reda omogućava značajno povećanje efikasnosti ove metode. S obzirom da EM modelovanje najčešće počinje od odgovarajućeg CAD modela, nameće se potreba za razvojem metode za automatsku četvorougaonu segmentaciju CAD geometrije koja je pogodna za efikasnu EM analizu. U ovom radu će biti detaljno opisana nova metoda za segmentaciju CAD geometrije proizvoljnih topoloških i geometrijskih osobina, koja je optimizovana za efikasnu EM analizu. Nova metoda se sastoji od sledećih, međusobno gotovo nezavisnih, celina: Podela CAD površina na kvazi-planarne delove primenom nove heurističke metode. Kreiranje optimalne poligonske aproksimacije ivica kvazi-planarnih površina. Preslikavanje kvazi-planarnih površina na poligone u osnovnom domenu uz dodatne korekcije kako podele površina tako i poligonske aproksimacije ivica, u cilju dobijanja kvalitetne i usaglašene mreže. Finalna četvorougaona segmentacija ravnih poligona i njeno preslikavanje na odgovarajuće CAD površine. Pokazuje se da se metoda odlikuje velikom robusnošću i brzinom, a da kreirana mreža pokazuje sve poželjne osobine: usaglašenost, adaptivnost i anizotropiju elemenata. Tačnost geometrijskog, i posledično EM modelovanja, je potvrđena poređenjem rezultata simulacije sa analitičkim i merenim rezultatima. U radu je takođe pokazana superiornost nove metode nad široko korišćenom metodom baziranom na kreiranju kvalitetne mreže trouglova i njenoj konverziji u mrežu četvorouglova.Full-wave electromagnetic (EM) analysis of metallic and dielectric structures in the frequency domain is often performed using the method of moments (MoM) applied to surface integral equations (SIE). It is shown that using bilinear quads instead of widely used triangular mesh elements, together with higher order basis functions, significantly increases the efficiency of the method. Having in mind that EM modelling usually starts with an appropriate CAD model, there is a need for developing a fully automatic method for quadrilateral meshing of CAD geometry, which is optimized for efficient EM analysis. A new method for quadrilateral meshing of a CAD geometry of arbitrary topological and geometrical properies, optimized for efficient EM analysis, is described in this dissertation. The new method consists of the following, mutually almost independent, parts: Subdivision of all CAD faces into quasi-planar sub-faces using a new heuristic method. Calculation of optimal polygonal approximation of all edges. Mapping of the quasi-planar faces to polygons in the base domain, with additional corrections in subdivisions of the faces and polygonal approximation of the edges, to increase mesh quality while maintaining conforming mesh. Final quadrilateral meshing in the base domain and mapping the mesh to the CAD geometry. It is shown that the new method is very robust and fast, and that it creates conforming, adaptive, and anisotropic meshes. The accuracy of geometrical and, consequently, EM modeling is proven by comparing simulation results with analytically calculated and measurement results. The superiority of the method when compared with a widely used method based on the creation of a high-quality triangular mesh and its conversion to a quadrilateral mesh is also presented in the dissertation

    Generating patterns on clothing for seamless design

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    Symmetric patterns are used widely in clothing manufacture. However, the discontinuity of patterns at seams can disrupt the visual appeal of clothing. While it is possible to align patterns to conceal such pattern breaks, it is hard create a completely seamless garment in terms of pattern continuity. In this thesis, we explore computational methods to parameterize the clothing pieces relative to a pattern’s coordinate system to achieve pattern continuity over garments. We review previous work related to pattern alignment on clothing. We also review surface quadrangulation methods. With a suitable quadrangulation, we can map any planar pattern with fourfold rotations into each quad, and achieve a seamless design. With an understanding of previous work, we approached the problems from three angles. First, we mapped patterns with sixfold rotations onto clothing by triangulating the clothing pieces and ensuring consistency of triangle vertices on both sides of a seam. We also mapped patterns with fourfold rotations onto clothing by optimizing the shape of each clothing piece in the texture domain. Lastly, we performed quadrangulation guided by cross fields, and mapped fourfold pattern units into each quad. We assembled and simulated the texture mapped clothing in Blender to visualize the results

    Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

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    Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions is publicly available at https://github.com/mauriceweiler/MobiusCNN

    Retopology: a comprehensive study of current automation solutions from an artist’s workflow perspective

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    Dissertação de mestrado em Engenharia InformáticaTopology (the density, organization and flow of a 3D mesh’s connectivity) constrains the suitability of a 3D model for any given purpose, be it surface showcasing through renders, use in real-time engines, posing or animation. While some of these use cases might not have very strict topology requirements, others may demand optimized polygon counts for performance reasons, or even specific geometry distribution in order to take deformation directions into account. Many processes for creating 3D models such as sculpting try to make the user unaware of the inner workings of geometry, by providing flexible levels of surface detailing through dynamic geometry allocation. The resulting models have a dense, unorganized topology that is inefficient and unfit for most use cases, with the additional drawback of being hard to work with manually. Retopology is the process of providing a new topology to a model such as these, while maintaining the shape of its surface. It’s a technical and time-consuming process that clashes with the rest of the artist’s workflow, which is mainly composed of creative processes. While there’s abundant research in this area focusing on polygon distribution quality based on surface shape, artists are still left with no options but to resort to manual work when it comes to deformation-optimized topology. This document exposes this disconnect, along with a proposed framework that attempts to provide a more complete retopology solution for 3D artists. This framework combines traditional mesh extraction algorithms with adapting manually-made meshes in a pipeline that tries to understand the input on a higher level, in order to solve deficiencies that are present in current retopology tools. Our results are very positive, presenting an improvement over state of the art solutions, which could possibly steer discussion and research in this area to be more in line with the needs of 3D artists.A topologia (a densidade, organização e direções tomadas pela conectividade de uma mesh 3D) limita a adequação de um modelo 3D para um leque variado de usos, entre os quais, visualização da superfície através de renders, uso em motores real-time, poses ou animações. Embora muitos destes usos não possuam requerimentos de topologia muito rigorosos, outros podem exigir número de polígonos mais baixos por questões de performance, ou até distribuição de geometria específica para acomodar direções de deformação corretamente. Muitos processos de criação de modelos 3D, como escultura, permitem que o utilizador não esteja ciente do que se passa em termos de funcionamento da geometria por debaixo da utilização. Isto é conseguido oferecendo níveis de detalhe flexíveis, alocando geometria de forma dinâmica. Os modelos resultantes têm uma topologia densa e desorganizada, que é ineficiente e pouco apropriada para a maior parte dos casos de uso, com a desvantagem adicional de ser difícil de trabalhar com a mesma manualmente. A retopologia é o processo de gerar uma nova topologia para um modelo, ao mesmo tempo que se mantém a forma da superfície. É um processo técnico e demorado, que entra em conflito com o resto do fluxo de trabalho do artista, que é composto maioritariamente por processos artísticos. Apesar de haver investigação abundante nesta área focada na qualidade da distribuição de polígonos baseada na forma da superfície, os artistas continuam a ter de recorrer ao trabalho manual quando se trata de topologia otimizada para deformações. Este documento expõe esta divergência, propondo, em conjunto, uma framework que tenta oferecer uma solução mais completa para os artistas 3D. Esta framework combina algoritmos de extração de meshes tradicionais com adaptação de meshes feitas manualmente, numa pipeline que tenta compreender o input a um nível superior, resolvendo as deficiências presentes nas ferramentas de retopologia atuais. Os nossos resultados são bastante positivos, apresentando melhorias em relação a soluções de estado da arte, facto que poderá mudar o rumo da discussão e investigação neste campo, para melhor se adequar às necessidades dos artistas 3D

    Orthogonally Constrained Sparse Approximations with Applications to Geometry Processing

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    Compressed manifold modes are solutions to an optimisation problem involving the 1\ell_1 norm and the orthogonality condition XTMX=IX^TMX=I. Such functions can be used in geometry processing as a basis for the function space of a mesh and are related to the Laplacian eigenfunctions. Compressed manifold modes and other alternatives to the Laplacian eigenfunctions are all special cases of generalised manifold harmonics, introduced here as solutions to a more general problem. An important property of the Laplacian eigenfunctions is that they commute with isometry. A definition for isometry between meshes is given and it is proved that compressed manifold modes also commute with isometry. The requirements for generalised manifold harmonics to commute with isometry are explored. A variety of alternative basis functions are tested for their ability to reconstruct specific functions -- it is observed that the function type has more impact than the basis type. The bases are also tested for their ability to reconstruct functions transformed by functional map -- it is observed that some bases work better for different shape collections. The Stiefel manifold is given by the set of matrices XRn×kX \in \mathbb{R}^{n \times k} such that XTMX=IX^TMX = I, with M=IM=I. Properties and results are generalised for the MIM \neq I case. A sequential algorithm for optimisation on the generalised Stiefel manifold is given and applied to the calculation of compressed manifold modes. This involves a smoothing of the 1\ell_1 norm. Laplacian eigenfunctions can be approximated by solving an eigenproblem restricted to a subspace. It is proved that these restricted eigenfunctions also commute with isometry. Finally, a method for the approximation of compressed manifold modes is given. This combines the method of fast approximation of Laplacian eigenfunctions with the ADMM solution to the compressed manifold mode problem. A significant improvement is made to the speed of calculation
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