关于三维有界空间中点集的均匀或相对均匀子集选取方法的咨询

Hey there! No worries at all about fumbling with math terms—research is all about asking questions, and we’ve all been there when jargon feels out of reach. Let’s dive into your problem since it’s tied directly to your research work.

First, let’s restate what you’re looking for clearly:
Given a set of points in a bounded 3D space, you want to select a subset where either:

• The points are strictly uniformly spread out across the entire space, or
• They’re not perfectly uniform, but relatively spread out so no single area is crowded with points.

You also shared hand-drawn diagrams to clarify:

• Original set: The initial collection of points (likely with some crowded regions)
• Ideal subset: Points evenly/relatively evenly distributed throughout the space
• Poor subset: Points clustered tightly in one small area

Now, here are practical methods tailored to both your needs:

Strictly Uniform Subset Selection #

If you need true uniformity, these approaches focus on enforcing consistent spacing across the space:

• Grid-based matching: Divide your 3D space into evenly sized cubic grid cells. Then, pick at most one point from each cell. The size of the grid will control how many points you end up with—smaller grids mean more points, larger grids mean fewer. This guarantees no two selected points are too close (relative to grid size) and covers the whole space.
• Poisson Disk Sampling (adapted for existing points): Poisson sampling is a classic method that ensures all points are at least a minimum distance apart. To apply it to your existing point set: start by randomly picking one point, then iteratively select the next point that’s farthest from all already chosen points (and meets the minimum distance threshold). Keep going until you have your desired subset size.

Relatively Spread Out Subset Selection #

If perfect uniformity isn’t necessary, just avoiding crowded clusters, these flexible methods work great:

• Density-based filtering: First calculate the local density for each point (e.g., count how many other points are within a small 3D radius around it). Then, prioritize selecting points with lower local density—this automatically skips the crowded areas. You can sort points by density and pick the top N least dense ones.
• Greedy farthest-point selection: Start with a random point from your set. Then, each time, select the point that’s farthest away from all the points you’ve already picked. Repeat this until you hit your target subset size. This is super intuitive and quickly spreads points out across the space.
• Cluster-based representative selection: Use a clustering algorithm like K-means to split your original points into K clusters (one for each point you want in your subset). Then, pick one representative point from each cluster (e.g., the point closest to the cluster’s center). This ensures every region of the space has at least one point in your subset.

And don’t stress about the math behind these—many of these methods have pre-built implementations in libraries like scipy, numpy, or open3d (for 3D point clouds). You can tweak parameters like grid size, minimum distance, or number of clusters to get the exact spread you need for your research.

备注：内容来源于stack exchange，提问作者Felis