Particle-Wise Higher-Order SPH Field Approximation for DVR

TL;DR

This paper introduces a novel higher-order field approximation for direct volume rendering of SPH data, where each particle’s contribution along a viewing ray is approximated by a piecewise polynomial and then summed efficiently via a knot-based, localized-difference encoding. The approach yields an adaptive, data-resolution-aware representation that improves accuracy over traditional piecewise-constant or piecewise-linear DVR. A quantization strategy is developed to prevent higher-order floating-point error propagation, enabling exact arithmetic with two quantum values, and a combined error framework guides the selection of approximation order, number of pieces, and precision. The work provides theoretical guarantees of optimality for fixed knot positions, a method to optimize knot locations, and practical guidance for balancing accuracy and performance, with potential applicability to other DVR tasks involving scattered data.

Abstract

When employing Direct Volume Rendering (DVR) for visualizing volumetric scalar fields, classification is generally performed on a piecewise constant or piecewise linear approximation of the field on a viewing ray. Smoothed Particle Hydrodynamics (SPH) data sets define volumetric scalar fields as the sum of individual particle contributions, at highly varying spatial resolution. We present an approach for approximating SPH scalar fields along viewing rays with piece-wise polynomial functions of higher order. This is done by approximating each particle contribution individually and then efficiently summing the results, thus generating a higher-order representation of the field with a resolution adapting to the data resolution in the volume.

Figures (4)

• Figure 1: Sketch of measures involved in the positional relationship between viewing ray and particle. The volume of influence of a particle with smoothing length $\zeta$ is intersected by a viewing ray, defined by base point $\boldsymbol{b}$ and unit direction vector $\boldsymbol{v}$. The particle's contribution on the ray at a point $\mathop{\boldsymbol{x}}\mathopen{}\left(t\right)$ is determined by its distance to the particle position $\boldsymbol{\chi}$. We denote by $q$ the upper bound of the kernel function's support, such that $q\zeta$ is the radius of the particle's volume of influence.
• Figure 2: Extracts of sample renderings of the temperature field of an SPH data set using our higher-order SPH field approximation scheme and a transfer function emphasizing three rather low temperature value regions by mapping them to an emission of blue, yellow, and red light. All computations are performed in single precision on the GPU. (a) clearly shows "sprinkling" artifacts caused by higher-order rounding error propagation, which "randomly" cause the field approximation on the ray to stay within one of the highlighted temperature regions far "behind" the particle cluster. (b) shows the result of employing 10 slices of reinitialization to mitigate the problem.
• Figure 3: Plot of error values in the example case of the cubic B-Spline SPH kernel (\ref{['eq:cubic-kernel']}), for $D \leq 6$ and $K\leq 4$. The horizontal colored marks show the polynomial approximation error component $E_{K,D}$ on the left and the combined error $\sqrt{E_{K,D}^2+Q_{D}^2}$ on the right. The grey bars on the right-hand side depict the quantization error component $Q_{D}$, which has been computed assuming an integer bit length of 64 and a data variance factor $a_{\max}/\phi_\text{repr}=10^5$.
• Figure 4: Plots of the basis functions $\tilde{A}_{kd}$ for $k=1,\dots,3$, $d=1,\dots,4$, and an arbitrary setting of positive knot positions $\theta_{1}$, $\theta_{2}$, and $\theta_{3}$. These 12 functions are the elements of the basis $\tilde{\mathcal{A}}$ for $K=6$ and $D=4$. For $K=5$, there are still three positive knot positions but the elements $\tilde{A}_{11}$ and $\tilde{A}_{13}$ drawn in paler colors are excluded because they feature a polynomial change at $t=0$.

Theorems & Definitions (2)

• proof
• proof