Second Order Pre-Integrated Volume Rendering

Abstract

In the field of Volume Rendering, pre-integration techniques for arbitrary transfer functions has certainly led to the most significant and convincing results both quality and performance wise on standard PC consumer graphics. By showing that the ideal scalar signal along the cast rays is better approximated by a succession of polynomial curves as opposed to linear segments, we propose a new method for pre-integrated volume rendering. This method is based on a second order polynomial interpolation of the scalar values, allowing it to converge more rapidly towards the integration of a volume reconstructed by a trilinear filter. This approach manages to capture the smoothness of the volume's details without the need of further ray resampling, and consequently succeeds in reducing the visual artefacts in comparison to previous techniques. Futhermore, we adapt an existing technique to compute our pre-integration tables using the GPU, thus making our approach suitable for transfer function manipulations.

Verifying Volume Rendering Using Discretization Error Analysis

“…The solution with linearly interpolated attributes is presented by Williams and Max [44], with further discussions on its numerical stability by Williams et al [45]. Interpolant approximations and errors [5], [28], [29], [31], gradient computation [42] and opacity correction [23] are also the subject of analysis with regard to numerical accuracy. The idea of pre-integration enables high-quality, accurate and efficient algorithms using graphics hardware [7], [22], [37].…”

Volume Visualization: A Technical Overview with a Focus on Medical Applications

“…Furthermore, to improve the lookup table generation speed, Kye et al 58 presented an opacity approximation algorithm that used the arithmetic mean instead of the geometric mean in the integral computation. To improve the accuracy of approximating the pre-integrated TF integral, Hajjar and colleagues 59 used a secondorder function as a substitute for the piecewise linear segment. Recently, Kraus 60 proposed a space-covering volume sampling technique to extend the pre-integrated volume rendering to multidimensional TFs.…”

Pre-Integrated Volume Rendering with Non-Linear Gradient Interpolation

“…[11] restricts transfer functions to piecewise linear functions which allows one to precompute a pre-integration table independent of classification. Recently, El Hajjar et al [5] derived a method based on a second order polynomial for pre-integration at constant sampling steps. But they need to use a 3D table even though considering equidistant samples.…”