Anisotropic Noise Samples

Abstract

We present a practical approach to generate stochastic anisotropic samples with Poisson-disk characteristic over a two-dimensional domain. In contrast to isotropic samples, we understand anisotropic samples as non-overlapping ellipses whose size and density match a given anisotropic metric. Anisotropic noise samples are useful for many visualization and graphics applications. The spot samples can be used as input for texture generation, e.g., line integral convolution (LIC), but can also be used directly for visualization. The definition of the spot samples using a metric tensor makes them especially suitable for the visualization of tensor fields that can be translated into a metric. Our work combines ideas from sampling theory and mesh generation. To generate these samples with the desired properties we construct a first set of non-overlapping ellipses whose distribution closely matches the underlying metric. This set of samples is used as input for a generalized anisotropic Lloyd relaxation to distribute noise samples more evenly. Instead of computing the Voronoi tessellation explicitly, we introduce a discrete approach which combines the Voronoi cell and centroid computation in one step. Our method supports automatic packing of the elliptical samples, resulting in textures similar to those generated by anisotropic reaction-diffusion methods. We use Fourier analysis tools for quality measurement of uniformly distributed samples. The resulting samples have nice sampling properties, for example, they satisfy a blue noise property where low frequencies in the power spectrum are reduced to a minimum.

Dense Glyph Sampling for Visualization

“…For visualization purposes the main step is the definition of the metric, ensuring that it incorporates the most important features of the data. A further analysis of the results in frequency space can be found in [3].…”

“…Their work is built on a particle approach simulating attractive and repulsive forces. This work is an extension to our recent work for the generation of anisotropic noise samples [3] by adding a control over the alignment of the ellipses.…”

“…Therefore, we define a localized version of the Voronoi cells considering only the part containing the sample point. For more details we refer to [3]. We define V or r (P i ) = {P ∈ D|i ∈ I P and d(P i , P ) ≤ d(P j , P ) for all j ∈ I P with i = j,…”

“…Anisotropic spot samples with certain characteristics, such as spatially varying density and size, have many applications in visualization and computer graphics ranging from glyph rendering and texture generation for visualization purposes [14,8,13,11,12,3], digital halftoning [19,15,25] to mesh generation [2,16,22]. While some of the desirable properties of the samples are similar across applications, the goals and appropriate sampling strategies are problem-dependent.…”

Eigenvector-based Interpolation and Segmentation of 2D Tensor Fields

“…Commonly used glyphs are, e.g., ellipsoids, Haber glyphs [7], or superquadrics [9], with improved perceptional properties. Different placement strategies are used to maximize the information displayed per image [6,11]. While glyphs are appropriate for displaying single tensors, they are limited to low resolution and fail to give insight into the structure of the entire field.…”