Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated to $X$. If $X$ has regions of variable density, which is more typical than not in practice, then this variability affects the resulting invariants. While this may be a feature and not a bug, **I am curious to know if (and especially where) alternative filtrations similar to the following construction may have been studied.**

Take the Voronoi regions corresponding to $X$ and dilate each of them (about their respective "anchor points") by a fixed scale, then build a simplicial complex using the intersections of these dilated regions.

For what it's worth, a casual search of Google Scholar yielded nothing of obvious relevance.