Bivariate Quantile Interpolation For Ensemble Derived Probability Density Estimates

Probability distribution functions (PDFs) may be estimated from members in an ensemble. For an ensemble of 2D vector fields, this results in a bivariate PDF at each location in the field. Vector field analysis and visualization, e.g., stream line calculation, require an interpolation to be defined over these 2D density estimates. Thus, a non parametric PDF interpolation must advect features as opposed to cross-fading them, where arbitrary modalities in the distribution can be introduced. This is already achieved for 1D PDF interpolation via inverse cumulative distribution functions (CDFs). However, there is no closed-form extension to bivariate PDF. This paper presents one such direct extension of the 1D closed-form solution for bivariates. We show an example of physically coupled components (velocity) and correlated random variables. Our method does not require a complex implementation or expensive computation as does displacement interpolation Bonneel et al., ACM Trans. Graphics (TOG), 30(6):158, 2011. Additionally, our method does not suffer from ambiguous pair-wise linear interpolants, as does Gaussian Mixture Model Interpolation.