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In the field of Volume Rendering, the pre-integration of arbitrary transfer functions has certainly led to the most significant and convincing results both quality and performance wise, allowing high quality visualization 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. Furthermore, we adapt an existing technique to compute our pre-integration tables using the GPU, thus making our approach suitable for transfer function manipulations.