To interpolate between data points \( \fvec{x}_i \) which are not somehow ordered in a grid structure the scattered data interpolation (Shepard-Interpolation) algorithm can be used

\begin{equation*} f(\fvec{x}) = \sum_{i=1}^n \frac{w_i(\fvec{x}) f(\fvec{x}_i)}{\sum_{i=1}^n w_i(\fvec{x})}. \end{equation*}

The algorithm creates a smooth interpolation between the points based on their distances. At each interpolating position \( \fvec{x} \) all data points are taken into account but weighted accordingly to their distances (nominator, higher weight for nearer distances) with the weighting function

\begin{equation*} w_i(\fvec{x}) = \frac{1}{\left\| \fvec{x} - \fvec{x}_i \right\|^{\alpha}}. \end{equation*}

In the end, the total result is normalized by all weights (denominator). The weighting is controlled by the \(\alpha\) parameter. If this parameter is increased up to infinity, the result is a Voronoi decomposition.


Figure 1: Increasing the \(\alpha\) parameter results in the Voronoi decomposition.

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