B.3 Quality measure for a potential field

In this section a method is introduced that determines the quality of the match between a potential field and a data distribution {$\bf x_{i}^{}$}. The overlap is computed between the data distribution and a region of same volume enclosed by an iso-potential curve (figure B.1). The method relies on the data points being uniformly distributed over a closed region $\mathcal {G}$ with volume A (as it is the case for the ring-line-square and vortex distributions).

Figure B.1: Illustration of an iso-potential curve surrounding a region of same volume as the data distribution.

Let B_c be the volume of the closed region defined by {$\bf x$ | p($\bf x$)$\le$c}, which is the set of points surrounded by an iso-potential curve with value c. The volume B_c was calculated using Monte-Carlo integration.

The computation of the quality measure has two steps. First, choose c, such that B_c = A. Second, count the number of data points $\bf x_{i}^{}$ fulfilling p($\bf x_{i}^{}$)$\le$c. The quality measure is the percentage of this number on the total number of data points.