Smooth Surface Extraction from Unstructured Point-based Volume Data Using PDEs

• Published in: IEEE transactions on visualization and computer graphics Vol. 14; no. 6; pp. 1531 - 1546
• Main Authors: Rosenthal, P., Linsen, L.
• Format: Journal Article
• Language: English
• Published: United States IEEE 01-11-2008; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Artificial Intelligence; Convergence; Curvature; Data mining; Data visualization; Extraction; Fittings; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Imaging, Three-Dimensional - methods; Index Terms— Information retrieval; Isosurfaces; Jacobian matrices; level sets; Mathematical analysis; Mathematical models; Mesh generation; Partial differential equations; Pattern Recognition, Automated - methods; PDEs; point-based visualization; Rendering; Reproducibility of Results; Sensitivity and Specificity; Sensor systems; Studies; surface extraction; Surface fitting
• ISSN: 1077-2626; 1941-0506
• DOI: 10.1109/TVCG.2008.164

Smooth surface extraction using partial differential equations (PDEs) is a well-known and widely used technique for visualizing volume data. Existing approaches operate on gridded data and mainly on regular structured grids. When considering unstructured point-based volume data where sample points do not form regular patterns nor are they connected in any form, one would typically resample the data over a grid prior to applying the known PDE-based methods. We propose an approach that directly extracts smooth surfaces from unstructured point-based volume data without prior resampling or mesh generation. When operating on unstructured data one needs to quickly derive neighborhood information. The respective information is retrieved by partitioning the 3D domain into cells using a fed-tree and operating on its cells. We exploit neighborhood information to estimate gradients and mean curvature at every sample point using a four-dimensional least-squares fitting approach. Gradients and mean curvature are required for applying the chosen PDE-based method that combines hyperbolic advection to an isovalue of a given scalar field and mean curvature flow. Since we are using an explicit time-integration scheme, time steps and neighbor locations are bounded to ensure convergence of the process. To avoid small global time steps, one can use asynchronous local integration. We extract a smooth surface by successively fitting a smooth auxiliary function to the data set. This auxiliary function is initialized as a signed distance function. For each sample and for every time step we compute the respective gradient, the mean curvature, and a stable time step. With these informations the auxiliary function is manipulated using an explicit Euler time integration. The process successively continues with the next sample point in time. If the norm of the auxiliary function gradient in a sample exceeds a given threshold at some time, the auxiliary function is reinitialized to a signed distance function. After convergence of the evolvution, the resulting smooth surface is obtained by extracting the zero isosurface from the auxiliary function using direct isosurface extraction from unstructured point-based volume data and rendering the extracted surface using point-based rendering methods.