Computing the minimum distance between a point and a NURBS curve

International audienceA new method is presented for computing the minimum distance between a point and a NURBS curve. It utilizes a circular clipping technique to eliminate the curve parts outside a circle with the test point as its center point. The radius of the elimination circle becomes smaller and smaller during the subdivision process. A simple condition for terminating the subdivision process is provided, which leads to very few subdivision steps in the new method. Examples are shown to illustrate the efficiency and robustness of the new method

Computing the minimum distance between two Bézier curves

International audienceA sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method

BoolSurf: Boolean Operations on Surfaces

We port Boolean set operations between 2D shapes to surfaces of any genus, with any number of open boundaries. We combine shapes bounded by sets of freely intersecting loops, consisting of geodesic lines and cubic Bézier splines lying on a surface. We compute the arrangement of shapes directly on the surface and assign integer labels to the cells of such arrangement. Differently from the Euclidean case, some arrangements on a manifold may be inconsistent. We detect inconsistent arrangements and help the user to resolve them. Also, we extend to the manifold setting recent work on Boundary-Sampled Halfspaces, thus supporting operations more general than standard Booleans, which are well defined on inconsistent arrangements, too. Our implementation discretizes the input shapes into polylines at an arbitrary resolution, independent of the level of resolution of the underlying mesh. We resolve the arrangement inside each triangle of the mesh independently and combine the results to reconstruct both the boundaries and the interior of each cell in the arrangement. We reconstruct the control points of curves bounding cells, in order to free the result from discretization and provide an output in vector format. We support interactive usage, editing shapes consisting up to 100k line segments on meshes of up to 1M triangles

On the intersection and derivative bound estimation for triangular Bézier surfaces

近年来，在CAGD和CAD中，三角Bézier曲面受到越来越广泛关注.它在表示拓扑结构复杂的几何模型时比张量积型Bézier曲面更灵活.因此，它更适合作为复杂几何体的拟合、插值离散数据点的工具.本文我们以三角Bézier曲面为研究对象，针对三角Bézier曲面在求交问题及其有理形式在低阶导矢界的改进两大方面的应用进行了研究.主要成果如下: 1.提出了implicitBézierlinearclipping(IBLC)算法来解决两平面Bézier曲线的交.在研究射线与三角Bézier曲面的求交方法之前，我们先研究了两Bézier曲线的求交，希望得到一些启发.IBLC算法利用了Bézier曲线的...In recent decades, triangular Bézier representations have attracted widely attention in the fields such as CAGD and CAD. They are a topologically more flexible modeling tool than tensor products. Hence, they are considered to be more suitable for representing models with complex geometries and fitting or interpolating scattered data points. In this dissertation, our objective is to study triangula...学位：理学博士院系专业：数学科学学院_计算数学学号：1902013015416

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the problem of finding the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to undesirably conservative results in many applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the $n^\text{th}$-degree polynomial curve with largest convex hull enclosed in a given $n$-simplex. Then, we present a formulation that is \emph{independent} of the $n$-simplex or $n^{\text{th}}$-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any $n\in\mathbb{N}$ and prove numerical global optimality for $n=1,2,3$. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-degree polynomial curve, the MINVO basis is able to obtain an enclosing simplex whose volume is $2.36$ and $254.9$ times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When $n=7$, these ratios increase to $902.7$ and $2.997\cdot10^{21}$, respectively.Comment: 25 pages, 16 figure

Infinite Procedural Infrastructured World Generation

Käesolev magistritöö kirjeldab uudset algoritmi lõpmatu deterministliku maailma genereerimiseks koos üldlevinud tehislike ja looduslike struktuuridega, mis leiduvad parasvöötme asustatud piirkondades. Kuna lõpmatuid maailmu tuleb genereerida jupikaupa ning ilma ühegi naabruses oleva tüki olemasoluta, on genereeritavate struktuuride järjepidev ja deterministlik genereerimine keeruline. Kirjeldatav algoritm kasutab eksponentsiaalse genereerimise metoodikat, mis võimaldab genereerida erineva suurusega struktuure alates liiklusmärkidest kuni pikkade jõgedeni. Algoritm genereerib erinevat tüüpi lõpmatuid teede võrgustikke, nimedega linnu ja külasid, elektriliine ja levinumaid liiklusmärke nagu kiiruspiirangud ja suunamärgid. Asulatest väljaspool olev maastik genereeritakse kolme üldlevinud maakasutuse kategooria vahel – metsandus, viljakasvatus ning looduskaitsealad. Lisaks kirjeldatavale algoritmile antakse ülevaade eelnevast teadustööst lõpmatu protseduurilise maailma genereerimise valdkonnas ning kirjeldatakse edasiarendusvõimalusi lõpmatute asustatud maailmade genereerimiseks.This Master's thesis describes and provides an implementation of a novel algorithm for generating infinite deterministic worlds with both man-made and natural features commonly found in the civilized regions of the temperate climate zone. Considering that infinite worlds have to be generated in a piecewise manner without any of the neighbouring pieces necessarily existing, ensuring continuity and deterministic results for the generation of such features can be challenging. The algorithm uses an exponential generation technique, which enables the generation of varying sized features from traffic signs to rivers. The algorithm generates infinite road networks of different tiers, named cities and villages, power lines between them and common traffic signs like speed limits and navigation signs. Rural areas are generated based on three types of land usage – forestry, cultivation of crops and untouched nature reserves. The thesis also gives an overview of the previous work in the field of procedural world generation and proposes multiple new ideas for further expansion of infinite infrastructured terrain generation

Interactive real-time three-dimensional visualisation of virtual textiles

Virtual textile databases provide a cost-efficient alternative to the use of existing hardcover sample catalogues. By taking advantage of the high performance features offered by the latest generation of programmable graphics accelerator boards, it is possible to combine photometric stereo methods with 3D visualisation methods to implement a virtual textile database. In this thesis, we investigate and combine rotation invariant texture retrieval with interactive visualisation techniques. We use a 3D surface representation that is a generic data representation that allows us to combine real-time interactive 3D visualisation methods with present day texture retrieval methods. We begin by investigating the most suitable data format for the 3D surface representation and identify relief-mapping combined with Bézier surfaces as the most suitable 3D surface representations for our needs, and go on to describe how these representation can be combined for real-time rendering. We then investigate ten different methods of implementing rotation invariant texture retrieval using feature vectors. These results show that first order statistics in the form of histogram data are very effective for discriminating colour albedo information, while rotation invariant gradient maps are effective for distinguishing between different types of micro-geometry using either first or second order statistics.Engineering and physical Sciences Research (EPSRC

Towards a High Quality Real-Time Graphics Pipeline

Modern graphics hardware pipelines create photorealistic images with high geometric complexity in real time. The quality is constantly improving and advanced techniques from feature film visual effects, such as high dynamic range images and support for higher-order surface primitives, have recently been adopted. Visual effect techniques have large computational costs and significant memory bandwidth usage. In this thesis, we identify three problem areas and propose new algorithms that increase the performance of a set of computer graphics techniques. Our main focus is on efficient algorithms for the real-time graphics pipeline, but parts of our research are equally applicable to offline rendering. Our first focus is texture compression, which is a technique to reduce the memory bandwidth usage. The core idea is to store images in small compressed blocks which are sent over the memory bus and are decompressed on-the-fly when accessed. We present compression algorithms for two types of texture formats. High dynamic range images capture environment lighting with luminance differences over a wide intensity range. Normal maps store perturbation vectors for local surface normals, and give the illusion of high geometric surface detail. Our compression formats are tailored to these texture types and have compression ratios of 6:1, high visual fidelity, and low-cost decompression logic. Our second focus is tessellation culling. Culling is a commonly used technique in computer graphics for removing work that does not contribute to the final image, such as completely hidden geometry. By discarding rendering primitives from further processing, substantial arithmetic computations and memory bandwidth can be saved. Modern graphics processing units include flexible tessellation stages, where rendering primitives are subdivided for increased geometric detail. Images with highly detailed models can be synthesized, but the incurred cost is significant. We have devised a simple remapping technique that allowsfor better tessellation distribution in screen space. Furthermore, we present programmable tessellation culling, where bounding volumes for displaced geometry are computed and used to conservatively test if a primitive can be discarded before tessellation. We introduce a general tessellation culling framework, and an optimized algorithm for rendering of displaced Bézier patches, which is expected to be a common use case for graphics hardware tessellation. Our third and final focus is forward-looking, and relates to efficient algorithms for stochastic rasterization, a rendering technique where camera effects such as depth of field and motion blur can be faithfully simulated. We extend a graphics pipeline with stochastic rasterization in spatio-temporal space and show that stochastic motion blur can be rendered with rather modest pipeline modifications. Furthermore, backface culling algorithms for motion blur and depth of field rendering are presented, which are directly applicable to stochastic rasterization. Hopefully, our work in this field brings us closer to high quality real-time stochastic rendering

Efficient and High-Quality Rendering of Higher-Order Geometric Data Representations

Computer-Aided Design (CAD) bezeichnet den Entwurf industrieller Produkte mit Hilfe von virtuellen 3D Modellen. Ein CAD-Modell besteht aus parametrischen Kurven und Flächen, in den meisten Fällen non-uniform rational B-Splines (NURBS). Diese mathematische Beschreibung wird ebenfalls zur Analyse, Optimierung und Präsentation des Modells verwendet. In jeder dieser Entwicklungsphasen wird eine unterschiedliche visuelle Darstellung benötigt, um den entsprechenden Nutzern ein geeignetes Feedback zu geben. Designer bevorzugen beispielsweise illustrative oder realistische Darstellungen, Ingenieure benötigen eine verständliche Visualisierung der Simulationsergebnisse, während eine immersive 3D Darstellung bei einer Benutzbarkeitsanalyse oder der Designauswahl hilfreich sein kann. Die interaktive Darstellung von NURBS-Modellen und -Simulationsdaten ist jedoch aufgrund des hohen Rechenaufwandes und der eingeschränkten Hardwareunterstützung eine große Herausforderung. Diese Arbeit stellt vier neuartige Verfahren vor, welche sich mit der interaktiven Darstellung von NURBS-Modellen und Simulationensdaten befassen. Die vorgestellten Algorithmen nutzen neue Fähigkeiten aktueller Grafikkarten aus, um den Stand der Technik bezüglich Qualität, Effizienz und Darstellungsgeschwindigkeit zu verbessern. Zwei dieser Verfahren befassen sich mit der direkten Darstellung der parametrischen Beschreibung ohne Approximationen oder zeitaufwändige Vorberechnungen. Die dabei vorgestellten Datenstrukturen und Algorithmen ermöglichen die effiziente Unterteilung, Klassifizierung, Tessellierung und Darstellung getrimmter NURBS-Flächen und einen interaktiven Ray-Casting-Algorithmus für die Isoflächenvisualisierung von NURBSbasierten isogeometrischen Analysen. Die weiteren zwei Verfahren beschreiben zum einen das vielseitige Konzept der programmierbaren Transparenz für illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle und zum anderen eine neue hybride Methode zur Reprojektion halbtransparenter und undurchsichtiger Bildinformation für die Beschleunigung der Erzeugung von stereoskopischen Bildpaaren. Die beiden letztgenannten Ansätze basieren auf rasterisierter Geometrie und sind somit ebenfalls für normale Dreiecksmodelle anwendbar, wodurch die Arbeiten auch einen wichtigen Beitrag in den Bereichen der Computergrafik und der virtuellen Realität darstellen. Die Auswertung der Arbeit wurde mit großen, realen NURBS-Datensätzen durchgeführt. Die Resultate zeigen, dass die direkte Darstellung auf Grundlage der parametrischen Beschreibung mit interaktiven Bildwiederholraten und in subpixelgenauer Qualität möglich ist. Die Einführung programmierbarer Transparenz ermöglicht zudem die Umsetzung kollaborativer 3D Interaktionstechniken für die Exploration der Modelle in virtuellenUmgebungen sowie illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle. Die Erzeugung stereoskopischer Bildpaare für die interaktive Visualisierung auf 3D Displays konnte beschleunigt werden. Diese messbare Verbesserung wurde zudem im Rahmen einer Nutzerstudie als wahrnehmbar und vorteilhaft befunden.In computer-aided design (CAD), industrial products are designed using a virtual 3D model. A CAD model typically consists of curves and surfaces in a parametric representation, in most cases, non-uniform rational B-splines (NURBS). The same representation is also used for the analysis, optimization and presentation of the model. In each phase of this process, different visualizations are required to provide an appropriate user feedback. Designers work with illustrative and realistic renderings, engineers need a comprehensible visualization of the simulation results, and usability studies or product presentations benefit from using a 3D display. However, the interactive visualization of NURBS models and corresponding physical simulations is a challenging task because of the computational complexity and the limited graphics hardware support. This thesis proposes four novel rendering approaches that improve the interactive visualization of CAD models and their analysis. The presented algorithms exploit latest graphics hardware capabilities to advance the state-of-the-art in terms of quality, efficiency and performance. In particular, two approaches describe the direct rendering of the parametric representation without precomputed approximations and timeconsuming pre-processing steps. New data structures and algorithms are presented for the efficient partition, classification, tessellation, and rendering of trimmed NURBS surfaces as well as the first direct isosurface ray-casting approach for NURBS-based isogeometric analysis. The other two approaches introduce the versatile concept of programmable order-independent semi-transparency for the illustrative and comprehensible visualization of depth-complex CAD models, and a novel method for the hybrid reprojection of opaque and semi-transparent image information to accelerate stereoscopic rendering. Both approaches are also applicable to standard polygonal geometry which contributes to the computer graphics and virtual reality research communities. The evaluation is based on real-world NURBS-based models and simulation data. The results show that rendering can be performed directly on the underlying parametric representation with interactive frame rates and subpixel-precise image results. The computational costs of additional visualization effects, such as semi-transparency and stereoscopic rendering, are reduced to maintain interactive frame rates. The benefit of this performance gain was confirmed by quantitative measurements and a pilot user study

Curriculum Proximal Policy Optimization with Stage-Decaying Clipping for Self-Driving at Unsignalized Intersections

Unsignalized intersections are typically considered as one of the most representative and challenging scenarios for self-driving vehicles. To tackle autonomous driving problems in such scenarios, this paper proposes a curriculum proximal policy optimization (CPPO) framework with stage-decaying clipping. By adjusting the clipping parameter during different stages of training through proximal policy optimization (PPO), the vehicle can first rapidly search for an approximate optimal policy or its neighborhood with a large parameter, and then converges to the optimal policy with a small one. Particularly, the stage-based curriculum learning technology is incorporated into the proposed framework to improve the generalization performance and further accelerate the training process. Moreover, the reward function is specially designed in view of different curriculum settings. A series of comparative experiments are conducted in intersection-crossing scenarios with bi-lane carriageways to verify the effectiveness of the proposed CPPO method. The results show that the proposed approach demonstrates better adaptiveness to different dynamic and complex environments, as well as faster training speed over baseline methods.Comment: 7 pages, 4 figure