The original (left) and modified (right) Monte Carlo process for a 1 dimensional sample. The vertices on the blue line are the original sample points, the blue line itself the DTFE field. Black points are uniformly drawn in the area below the red line, those below the blue line are included in the new sample. In this 1D example, the lowering of the red line gives a factor 6 improvement in performance, in our MSG dataset this is more than a factor 10 000. The weights are given by the colored areas: the cyan cells have the same weight. The yellow cell has the same maximum density or volume as one of the cyan cells but a lower weight. (Densities are linear spatial densities, not probability densities).