Analytic Anti-Aliasing of Linear Functions on Polytopes

Mathematical Notes of the Academy of Sciences of the USSR, 1975

ArXiv, 2021

Interpolation is a fundamental technique in scientific computing and is at the heart of many scientific visualization techniques. There is usually a trade-off between the approximation capabilities of an interpolation scheme and its evaluation efficiency. For many applications, it is important for a user to be able to navigate their data in real time. In practice, the evaluation efficiency (or speed) outweighs any incremental improvements in reconstruction fidelity. In this two-part work, we first analyze from a general standpoint the use of compact piece-wise polynomial basis functions to efficiently interpolate data that is sampled on a lattice. In the sequel, we detail how we generate efficient implementations via automatic code generation on both CPU and GPU architectures. Specifically, in this paper, we propose a general framework that can produce a fast evaluation scheme by analyzing the algebro-geometric structure of the convolution sum for a given lattice and basis function ...

Figure 1: An input coarse mesh (left) is subjected to tessellation using Curved PN Triangles (middle) and Phong Tessellation (right).

Since the last decade, graphics processing units (GPU) have dominated the world of interactive computer graphics and visualization technology. Over the years, sophisticated tools and programming interfaces (like OpenGL, DirectX API) greatly facilitated the work of developers because these frameworks provide all fundamental visualization algorithms. The research target is to develop a traditional CPU-based pure software rasterizer. Although currently it has a lot of drawbacks compared with GPUs, the modern multi-core systems and the diversity of the platforms may require such implementations. In this paper, we are dealing with triangle rasterization, which is the cornerstone of rendering process. New model optimization as well as the improvement and combination of existing techniques have been presented, which dramatically improve performance of the well-known half-space rasterization algorithm. The presented techniques become applicable in areas such as graphics editors and even computer games.

IEEE Transactions on Visualization and Computer Graphics, 2005

Various acquisition, analysis, visualization and compression approaches sample surfaces of 3D shapes in a uniform fashion, without any attempt to align the samples with sharp edges or to adapt the sampling density to the surface curvature. Consequently, triangle meshes that interpolate these samples usually chamfer sharp features and exhibit a relatively large error in their vicinity. We present two new filters that improve the quality of these re-sampled models. EdgeSharpener restores the sharp edges by splitting the chamfer edges and forcing the new vertices to lie on intersections of planes extending the smooth surfaces incident upon these chamfers. Bender refines the resulting triangle mesh using an interpolating subdivision scheme that preserves the sharpness of the recovered sharp edges while bending their polyline approximations into smooth curves. A combined Sharpen&Bend post-processing significantly reduces the error produced by feature-insensitive sampling processes. For example, we have observed that the mean-squared distortion introduced by the SwingWrapper remeshing-based compressor can often be reduced by 80% executing EdgeSharpener alone after decompression. For models with curved regions, this error may be further reduced by an additional 60% if we follow the EdgeSharpening phase by Bender.

Graphical Models and Image Processing, 1999

We present an unstructured triangular mesh generation algorithm that approximates a set of mutually nonintersecting simple trimmed polynomial parametric surface patches within a user specified geometric tolerance. The proposed method uses numerically robust interval geometric representations/computations and also addresses the problem of topological consistency (homeomorphism) between the exact geometry and its approximation. Those are among the most important outstanding issues in geometry approximation problems. We also extract important differential geometric features of input geometry for use in the approximation. Our surface tessellation algorithm is based on the unstructured Delaunay mesh approach which leads to an efficient adaptive triangulation. A robust decision criterion is introduced to prevent possible failures in the conventional Delaunay triangulation. To satisfy the prescribed geometric tolerance, an adaptive node insertion algorithm is employed and furthermore, an efficient method to compute a tight upper bound of the approximation error is proposed. Unstructured triangular meshes for free-form surfaces frequently involve triangles with high aspect ratio and, accordingly, result in ill-conditioned meshing. Our proposed algorithm constructs 2D triangulation domains which sufficiently preserve the shape of triangles when mapped into 3D space and, furthermore, the algorithm provides an efficient method that explicitly controls the aspect ratio of the triangular elements.

Proceedings Computer Animation '98 (Cat. No.98EX169), 1998

Smoothing techniques are essential in providing high quality renderings out of polygonal objects that are described with a minimal amount of geometrical information. In order to remove the "polygonal" aspect of rough polygonal meshes, several techniques are available, such as shading or interpolation techniques.

2015

In a virtual sculpting environment, we manipulate objects as a set of volume elements (voxels). In order to start the sculpture process from a polygonal object, we have to discretize this object as a set of voxels. This step is called voxelization. Several voxelization methods have already been proposed, but none matches all of our criteria. In this paper, we propose a practical approach that handles these criteria, based on an optimized ray casting. The method provides a voxelization of the inner space of a polygonal object, and not only of its surface. It is designed to work with closed objects which may contain holes or tunnels. Even if our goal is not real time conversion, it is fast enough to provide voxelization of objects made of several thousands of triangles within few seconds. As voxelization is a 3D sampling process, it entails aliasing problems. Our method allows a reduction of aliasing by using a low cost oversampling in order to compute grey scale values for the voxels...

Electronics Letters, 2007

Closed-form solutions for lowpass filtering of 3D digital geometry are presented. Conventional signal processing is difficult to apply since 3D digital geometry does not usually satisfy the prerequisites of regular sampling and global parameterisation. Representing digital geometry as a level set of a 3D scalar field fitted with Gaussians and polynomials, derived are closed-form solutions of convolution with Gaussian smoothing kernels and lowpass filtering without the pitfalls of other approaches are analytically achieved.

2018

Image convolution with a filtering mask is at the base of several image analysis operations. This is motivated by Mathematical foundations and by the straightforward way the discrete convolution can be computed on a grid-like domain. Extending the convolution operation to the mesh manifold support is a challenging task due to the irregular structure of the mesh connections. In this paper, we propose a computational framework that allows convolutional operations on the mesh. This relies on the idea of ordering the facets of the mesh so that a shift-like operation can be derived. Experiments have been performed with several filter masks (Sobel, Gabor, etc.) showing state-of-the-art results in 3D relief patterns retrieval on the SHREC'17 dataset. We also provide evidence that the proposed framework can enable convolution and pooling-like operations as can be needed for extending Convolutional Neural Networks to 3D meshes.