Eurographics/ IEEE-VGTC Symposium on Visualization 2009 H.-C. Hege, I. Hotz, and T. Munzner (Guest Editors) Volume 28 (2009), Number 3 On Visualization and Reconstruction from Non-Uniform Point Sets using B-splines Erald Vuçini 1 and Torsten Möller 2 and M. Eduard Gröller 1 1 Vienna University of Technology, Austria 2 Simon Fraser University, BC, Canada Abstract In this paper we present a novel framework for the visualization and reconstruction from non-uniform point sets. We adopt a variational method for the reconstruction of 3D non-uniform data to a uniform grid of chosen reso- lution. We will extend this reconstruction to an efficient multi-resolution uniform representation of the underlying data. Our multi-resolution representation includes a traditional bottom-up approach and a novel top-down hi- erarchy for adaptive hierarchical reconstruction. Using a hybrid regularization functional we can improve the reconstruction results. Finally, we discuss further application scenarios and show rendering results to emphasize the effectiveness and quality of our proposed framework. By means of qualitative results and error comparisons we demonstrate superiority of our method compared to competing methods. Categories and Subject Descriptors (according to ACM CCS): Numerical Analysis [G.1.2]: Spline and piecewise polynomial approximation.—Image Processing and Computer Vision [I.4.5]: Reconstruction.—Computer Graph- ics [I.3.5]: Computational Geometry and Object Modeling.— 1. Introduction The traditional sources of volumetric data are simulations as well as data acquisition devices on uniform (Cartesian) lattices. In an effort to study larger and more complex prob- lems, there has been a move toward non-uniform data rep- resentations, since they offer a way of adapting the mea- sure location (or sample points) according to the importance (variance) of the data. Examples include a) simple data loss during data communication in sensor networks, b) Doppler measurements or other novel acquisition models (polar or spiral) for tomography and magnetic resonance imaging, c) adaptive and moving mesh approaches in mathematical sim- ulations in the physical sciences, and d) particle simulations. While the acquisition of data on non-uniform grids has become wide-spread, the available tools for processing, fil- tering, analysis, and rendering of data are most efficient on uniform representations. We do not make use of the explicit neighborhood information in non-uniform grids in this work and hence they are used like point-sets. There are two com- peting efforts to deal with non-uniform data: a) create novel and efficient tools that directly work on them, or b) convert the non-uniform representation into an efficient intermediate uniform representation and apply standard tools. Both ap- proaches have advantages and disadvantages. In this paper we make a contribution towards the latter approach. Among other things, this will allow us to better exploit the capabili- ties of modern GPUs. In Section 2 we will contrast these two approaches further and review alternative works. In order to find the best way to transform the non-uniform data onto a uniform grid, we first need to analyze the nature of the given data. One reason for non-uniformity is the abil- ity to capture different scales of information density (e.g. mathematical simulation of shock waves). Another reason for non-uniform data representations could come from im- precise measurement devices (e.g. ultrasound) or sparse rep- resentations (e.g. compressive sensing). While in the former case multi-resolution representations might be most suitable, in the latter case a single resolution representation might be all what is needed. Therefore, we develop some heuristics, based on a statistical analysis, to adapt to each scenario. In this paper we propose a uniform representation con- sisting of B-spline coefficients which define a C 2 continuous function across the whole volume. Our main contributions are: a) a statistical approach for selecting the resolution of reconstruction for non-uniform datasets (Section 4.1), b) a bottom-up multi-resolution pyramid (Section 4.2), c) a novel c  2009 The Author(s) Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.