Abstract We develop an approach for hardware‐accelerated, high‐quality rendering of volume data using trivariate splines. The proposed quasi‐interpolating schemes are realtime reconstructions. The low total degrees provide several advantages for our GPU implementation. In particular, intersecting rays with spline isosurfaces for direct Phong illumination is performed by simple root finding algorithms (analytic and iterative), while the necessary normals result from blossoming. Since visualizations are on a fragment base, our renderer for isosurfaces includes an automatic level of detail. While we use well‐known spatial data structures in the CPU part of the algorithm for hierarchical view frustum culling and memory reduction, our GPU implementations have to take the highly complex structure of the splines into account. These include an appropriate organization of the data streams, i.e. we develop an advanced encoding scheme for the spline coefficients, as well as an implicit scheme for bounding geometry retrieval. In addition, we propose an elaborated clipping procedure to be performed in the fragment shader. These features essentially reduce bus traffic, memory consumption, and data access on the GPU leading to interactive frame rates for renderings of high visual quality. Compared with pure CPU implementations and existing GPU implementations for trivariate polynomials frame rates increase by factors between 10 and 100.

We present a novel GPU-based algorithm for high-quality rendering of bivariate spline surfaces. An essential difference to the known methods for rendering graph surfaces is that we use quartic smooth splines on triangulations rather than triangular meshes. Our rendering approach is direct in the sense that since we do not use an intermediate tessellation but rather compute ray-surface intersections (by solving quartic equations numerically) as well as surface normals (by using Bernstein-Bézier techniques) for Phong illumination on the GPU. Inaccurate shading and artifacts appearing for triangular tesselated surfaces are completely avoided. Level of detail is automatic since all computations are done on a per fragment basis. We compare three different (quasi-) interpolating schemes for uniformly sampled gridded data, which differ in the smoothness and the approximation properties of the splines. The results show that our hardware based renderer leads to visualizations (including texturing, multiple light sources, environment mapping, etc.) of highest quality.

We develop the first approach Tor interactive volume visualization based on a sophisticated rendering method of shear-warp type, wavelet data encoding techniques, and a trivariate spline model, which has been introduced recently. As a first step of our algorithm, we apply standard wavelet expansions to represent and decimate the given gridded three-dimensional data. Based on this data encoding, we give a sophisticated version of the shear-warp based volume rendering method. Our new algorithm visits each voxel only once taking advantage of the particular data organization of octrees. In addition, the hierarchies of the data guide the local (re)construction of the quadratic super-spline models, which we apply as a pure visualization tool. The low total degree of the polynomial pieces allows to numerically approximate the volume rendering integral efficiently. Since the coefficients of the splines are almost immediately available from the given data, Bernstein-Bezier techniques can be fully employed in our algorithms. In this way, we demonstrate that these models can be successfully applied to full volume rendering of hierarchically organized data. Our computational results show that (even when hierarchical approximations are used) the new approach leads to almost artifact-free visualizations of high quality for complicated and noise-contaminated volume data sets, while the computational effort is considerable low, i.e. our current implementation yields 1-2 frames per second for parallel perspective rendering a 2563 volume data set (using simple opacity transfer functions) in a 5122 view-port.

We propose a new approach to reconstruct nondiscrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition. We discuss the smoothness and approximation properties of our model and compare to alternative piecewise polynomial constructions. We observe, as a nonstandard phenomenon, that the derivatives of our splines yield optimal approximation order for smooth data, while the theoretical error of the values is nearly optimal due to the averaging rules. Our approach enables efficient reconstruction and visualization of the data. As the piecewise polynomials are of the lowest possible total degree two, we can efficiently determine exact ray intersections with an isosurface for ray-casting. Moreover, the optimal approximation properties of the derivatives allow us to simply sample the necessary gradients directly from the polynomial pieces of the splines. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.

We develop a new approach to reconstruct non-discrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition /spl Delta/. The approximating splines are determined in a natural and completely symmetric way by averaging local data samples, such that appropriate smoothness conditions are automatically satisfied. On each tetra-hedron of /spl Delta/ , the quasi-interpolating spline is a polynomial of total degree two which provides several advantages including efficient computation, evaluation and visualization of the model. We apply Bernstein-Bezier techniques well-known in CAGD to compute and evaluate the trivariate spline and its gradient. With this approach the volume data can be visualized efficiently e.g., with isosurface ray-casting. Along an arbitrary ray the splines are univariate, piecewise quadratics and thus the exact intersection for a prescribed isovalue can be easily determined in an analytic and exact way. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.

Presents an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a C/sup 1/-continuous bivariate cubic spline and our method offers optimal approximation order. Both local variation and nonuniform distribution of the data are taken into account by using local polynomial least squares approximations of varying degree. Since we only need to solve small linear systems and no triangulation of the scattered data points is required, the overall complexity of the algorithm is linear in the total number of points. Numerical examples dealing with several real-world scattered data sets with up to millions of points demonstrate the efficiency of our method. The resulting spline surface is of high visual quality and can be efficiently evaluated for rendering and modeling. In our implementation we achieve real-time frame rates for typical fly-through sequences and interactive frame rates for recomputing and rendering a locally modified spline surface.