Curvature estimation for meshes via algebraic quadric fitting

We introduce the novel method for estimation of mean and Gaussian curvature and several related quantities for polygonal meshes. The algebraic quadric fitting curvature (AQFC) is based on local approximation of the mesh vertices and associated normals by a quadratic surface. The quadric is computed as an implicit surface, so it minimizes algebraic distances and normal deviations from the approximated point-normal neighbourhood of the processed vertex. Its mean and Gaussian curvature estimate is then obtained as the respective curvature of its orthogonal projection onto the fitted quadratic surface. Experimental results for both sampled parametric surfaces and arbitrary meshes are provided. The proposed method AQFC approaches the true curvatures of the reference smooth surfaces with increasing density of sampling, regardless of its regularity. It is resilient to irregular sampling of the mesh, compared to the contemporary curvature estimators. In the case of arbitrary meshes, obtained from scanning, AQFC provides robust curvature estimation.Comment: 14 page

Improving Filtering for Computer Graphics

When drawing images onto a computer screen, the information in the scene is typically more detailed than can be displayed. Most objects, however, will not be close to the camera, so details have to be filtered out, or anti-aliased, when the objects are drawn on the screen. I describe new methods for filtering images and shapes with high fidelity while using computational resources as efficiently as possible. Vector graphics are everywhere, from drawing 3D polygons to 2D text and maps for navigation software. Because of its numerous applications, having a fast, high-quality rasterizer is important. I developed a method for analytically rasterizing shapes using wavelets. This approach allows me to produce accurate 2D rasterizations of images and 3D voxelizations of objects, which is the first step in 3D printing. I later improved my method to handle more filters. The resulting algorithm creates higher-quality images than commercial software such as Adobe Acrobat and is several times faster than the most highly optimized commercial products. The quality of texture filtering also has a dramatic impact on the quality of a rendered image. Textures are images that are applied to 3D surfaces, which typically cannot be mapped to the 2D space of an image without introducing distortions. For situations in which it is impossible to change the rendering pipeline, I developed a method for precomputing image filters over 3D surfaces. If I can also change the pipeline, I show that it is possible to improve the quality of texture sampling significantly in real-time rendering while using the same memory bandwidth as used in traditional methods

A recursive Taylor method for algebraic curves and surfaces

This paper examines recursive Taylor methods for multivariate polynomial evaluation over an interval, in the context of algebraic curve and surface plotting as a particular application representative of similar problems in CAGD. The modified affine arithmetic method (MAA), previously shown to be one of the best methods for polynomial evaluation over an interval, is used as a benchmark; experimental results show that a second order recursive Taylor method (i) achieves the same or better graphical quality compared to MAA when used for plotting, and (ii) needs fewer arithmetic operations in many cases. Furthermore, this method is simple and very easy to implement. We also consider which order of Taylor method is best to use, and propose that second order Taylor expansion is generally best. Finally, we briefly examine theoretically the relation between the Taylor method and the MAA method

Generating Distance Fields from Parametric Plane Curves

Distance fields have been presented as a general representation for both curves and surfaces [4]. Using space partitioning, adaptive distance fields (ADF) found their way into various applications, such as real-time font rendering [5]. Computing approximate distance fields for implicit representations and mesh objects received much attention. Parametric curves and surfaces, however, are usually not part of the discussion directly. There are several algorithms that can be used for their conversion into distance fields. However, most of these are based converting parametric representations to piecewise linear approximations [7]. This paper presents two algorithms to directly compute distance fields from arbitrary parametric plane curves. One method is based on the rasterization of general parametric curves, followed by a distance propagation using fast marching. The second proposed algorithm uses the differential geometric properties of the curve to generate simple geometric proxies, segments of osculating circles, that are used to approximate the distance from the original curve

Pressing: Smooth isosurfaces with flats from binary grids

We explore the automatic recovery of solids from their volumetric discretizations. In particular, we propose an approach, called Pressing, for smoothing isosurfaces extracted from binary volumes while recovering their large planar regions (flats). Pressing yields a surface that is guaranteed to contain the samples of the volume classified as interior and exclude those classified as exterior. It uses global optimization to identify flats and constrained bilaplacian smoothing to eliminate sharp features and high-frequencies from the rest of the isosurface. It recovers sharp edges between flat regions and between flat and smooth regions. Hence, the resulting isosurface is usually a much more accurate approximation of the original solid than isosurfaces produced by previously proposed approaches. Furthermore, the segmentation of the isosurface into flat and curved faces and the sharp/smooth labelling of their edges may be valuable for shape recognition, simplification, compression, and various reverse engineering and manufacturing applications.Postprint (published version

Renderização de curvas implícitas discretizadas no domínio da imagem

A representação gráfica de curvas implícitas continua a ser um tópico de investigação importante em computação gráfica e geometria computacional, e tem aplicações em várias áreas de interesse como sejam, por exemplo, representação de símbolos em tipografia digital, delimitação de contornos em imagem médica gerada por tomografia axial, bem como na definição de trajetórias para a simulação de movimento de personagens em animação computacional e jogos de vídeo. De forma sumária, pode dizer-se que esta dissertação propõe um algoritmo de pixelização de curvas implícitas que, ao que parece, não tem paralelo na literatura, a não ser nos algoritmos de rasterização de linhas retas e circunferências que incorporavam os sistemas gráficos primitivos, como por exemplo o algoritmo de Bresenham. De alguma maneira, o referido algoritmo de pixelização pode ser visto como uma generalização daqueles algoritmos primitivos no sentido de que se aplica a qualquer curva implícita, mesmo que ela apresente singularidades, pontos isolados, e outros pontos críticos.The graphical representation of implicit curves remains a major research topic in computer graphics and computational geometry, and has applications in several areas of interest such as, for example, representation of symbols in digital typography, delineation of contours in medical images generated by computerized axial tomography, as well as the definition of trajectories for the simulation of movement of characters in computer animation and video games. Briefly speaking, it can be said that this dissertation proposes a pixelization algorithm for implicit curves that apparently has no parallel in literature, except in the rasterization algorithms of straight lines and circles incorporated in primitive graphics systems, such as Bresenham's algorithm. Somehow, this algorithm pixelization can be seen as a generalization of those primitive algorithms in that it applies to any curve implied, even if it presents singularities, isolated points, and other critical points

Interactive ray tracing of arbitrary implicits with SIMD interval arithmetic

Journal ArticleWe present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions