Generating Hierarchical Non-Uniform Sparse Noise for Tensor Field Visualization Louis Feng, Ingrid Hotz, Bernd Hamann, and Kenneth I. Joy Institute for Data Analysis and Visualization (IDAV) Department of Computer Science University of California, Davis, 95616, USA E-mail: zfeng, ihotz @ucdavis.edu, hamann, joy @cs.ucdavis.edu Abstract Noise textures have various applications in com- puter graphics and visualization. They have been used extensively in line integral convolution (LIC) and texture generation. Many applications only uti- lize size and shape of the noise as free parame- ters and have not taken advantage of varying dis- tributions of the noise. This paper describes a new technique for generating non-uniformly distributed sparse noise with local control. Sparse noise gen- erated with this technique leads to a Poisson-disk distribution. Based on this technique, a method to generate sequences of hierarchical sparse noise suitable for continuous animation is presented. By varying size and density, sparse noise is quite appro- priate for visualizing data sets with non-uniformity, where some information can be encoded in the dis- tribution of the sparse noise. The method is effi- cient for the generation of two-dimensional as well as three-dimensional textures. We have applied this method to visualize compression and expansion in tensor fields. 1 Introduction Tensor fields play an important role in many areas of engineering and physics. Due to their high di- mensionality it is not easy to understand and inter- pret these fields. Therefore, it is important to vi- sualize the data in a way that represents the phys- ical meaning of the tensor field. In our applica- tion, we focus on stress and strain tensor fields with their main features as compression and expansion. We use a texture-based method [6]. The texture is aligned to the eigenvector fields. The eigenval- ues are encoded by the free parameters of the tex- ture. To improve the impression of compression and stretching we animate this process. This approach requires one to control parameters locally, and for continuous animation we need a sequence of hier- archical sparse noise input images as the basis for texture generation. Noise generation and sampling are closely re- lated. Both are concerned with the placement of samples and their distributions. Many sampling al- gorithms, such as jittering, are often directly used to generate noise textures. Various techniques for generating samples and their properties have been explored extensively in the area of sampling and re- construction theory to avoid aliasing problems. For a survey on sampling techniques, we refer to [3, 15]. A Poisson-disk sampling pattern is commonly preferred over ordinary random patterns due to its effectiveness on handling aliasing. One sim- ple approach to generate random samples having a Poisson-disk distribution is the dart throwing algo- rithm. It mimics the stochastic Poisson-disk process by successively adding random points to a point set. New points are accepted if no other point is inside the Poisson disk. Unfortunately, this process is not guaranteed to terminate. Some representative work on generating samples with Poisson-disk dis- tributions include Mitchell’s best candidate algo- rithm [10], and McCool and Fiume’s decreasing ra- dius algorithm [8]. While both algorithms are more efficient than the dart throwing algorithm, they are still expensive for generating large number of sam- ples. Mitchell also presented a more efficient point- diffusion algorithm [9], but the results are subopti- mal and lack of ways to control the distribution of the samples. Some other types of random pattern genera- tion algorithms are based on Llloyd’s relaxation method [7] for generating digital halftone images VMV 2004 Stanford, USA, November 16–18, 2004