Hardware‐Accelerated, High‐Quality Rendering Based on Trivariate Splines Approximating Volume Data

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.

GPU-accelerated 2D point cloud visualization using smooth splines for visual analytics applications

“…While [Loop and Blinn 2006] use the BB-form of the polynomial pieces, [Kalbe and Zeilfelder 2008] extend this approach to the visualization of trivariate splines, first taking their complex structure into account.…”

Interactive isosurfaces with quadratic C1splines on truncated octahedral partitions

“…Then, we introduce the new spline in Section 3, i.e., describe the underlying tetrahedral partition and give formulas defining the coefficients of the polynomial pieces. In Section 4 we briefly describe the visualization algorithm which is adapted from our earlier work 5,6 to the new partition. Finally, we compare the visual quality obtained from our method with standard trilinear interpolation as well as cubic C 1 splines on type-6 tetrahedral partitions, and analyze its performance.…”

“…Cubic C 1 splines on tetrahedral partitions have been proposed by Sorokina and Zeilfelder, 4 and a GPU visualization algorithm for isosurface ray casting based on cubic C 1 and quadratic super splines has been given. 5,6 Both splines are defined on the so-called type-6 tetrahedral partition where each data cube is split into 24 congruent tetrahedra. Another closely related approach is given by Kloetzli et al,7 where cubic C 0 splines are constructed on arbitrary tetrahedral partitions by a Moving Least Squares approximation of the volume data.…”

Direct Isosurface Visualization of Hex-Based High-Order Geometry and Attribute Representations

“…When represents a uniform scalar grid, efficient and interactive methods exist to directly visualize isosurfaces, including a GPU approach to visualize trivariate splines with respect to tetrahedral partitions that transform each patch to its Bernstein-Bézier form [20]. Earlier, a direct rendering paradigm of trivariate B-spline functions for large data sets with interactive rates was presented in the work in [38], where the rendering is conducted from a fixed viewpoint in two phases suitable for sculpting operations.…”

“…If there is such a piece of the isosurface silhouette, then u is computed so that VðuÞ lies on the isosurface silhouette and u is added to S S S . As discussed in Section 3, an isosurface that intersects the frustum (type 3) must have an isosurface silhouette in the frustum, i.e., it must satisfy (20). Given ðP i;';k ðuÞ; i;';k ðuÞÞ with sign changes both in the coefficients defining i;';k ðuÞ and defining i;';k ðuÞ, a patch Q i;';k ðuÞ is constructed, where Note that this additional solution step to find points on an isosurface silhouette within a ray frustum is executed only at isosurface silhouettes, when there are sign changes in the coefficients defining i;';k ðuÞ and i;';k ðuÞ.…”