I'm currently trying to improve a meshing algorithm for signed distance fields (which are of the form $f(x,y,z) = w$, where $x,y,z$ is the location of my query, $w$ indicates the distance to the nearest surface, $w > 0$ is outside of the volume and $w < 0$ is inside the volume) by moving vertices to defining edges and corners of the described volume.
Existing techniques (like Dual Contouring and Dual Marching Cubes) make use of pretty heavy least square computations to estimate where edges and corners are (and to be honest, those are all quite over my head), so I'm trying to replace those with a less sophisticated approach that is simpler to compute and easier to implement.
It occurred to me through this article that the problem may be as simple as taking regular measurements in a grid, using three spheres in a small distance from each other (where the radius for sphere at position $p$ is resolved from $f(p)$) and testing for intersections, which is in effect trilateration.
I did some initial experiments and it appears the technique only works for very tuned situations, where the distance and orientation between the spheres is "just right". The question is this:
Is there a smart way to place the measurement points so that the yield of intersections at surface level $f(\operatorname{intersect}(a,b,c)) = 0$ is maximum?
Alternatively:
Are there other alternatives to finding signed distance field features?