能否用向量中位数替代均值计算3D网格顶点法线？

Absolutely—this is a perfect solution to the problem you're facing with unevenly subdivided meshes! Let me break down why this works, how it helps, and what to keep in mind:

Why Vector Median Fixes Your Offset Issue #

The arithmetic mean of face normals minimizes the L2 (squared) distance to all input vectors, which makes it sensitive to large groups of similar vectors (like the dense subdivided faces in your corner example). These over-represented normals pull the average away from the ideal surface direction (your yellow line).

The vector median, on the other hand, minimizes the L1 (absolute) distance to all input vectors. Think of it as the "middle" direction that sits equidistant (in terms of absolute vector distance) from the majority of your face normals. It’s far more robust to skewed distributions—those dense subdivided faces won’t overpower the result, keeping your vertex normal aligned with the true surface trend.

Key Advantages Over the Mean #

• Robust to uneven subdivision: Doesn’t get pulled by clusters of similar normals from over-subdivided areas
• Better geometric fidelity: Aligns more closely with the intuitive "ideal" surface normal, especially at corners or where mesh density changes abruptly
• Can be weighted: You can easily extend it to a weighted vector median, where each face normal’s contribution is scaled by its area. This makes sense because larger faces should have more influence on the vertex normal than tiny subdivided ones.

Quick Implementation Notes #

Unlike the mean (which is just sum(normals) / count), the vector median requires an iterative approach. The most common method is the Weiszfeld algorithm:

1. Start with an initial guess (the arithmetic mean works great here)
2. Iteratively update the candidate median using this formula:

   median = sum( (normal_i / ||median_prev - normal_i||) ) / sum( 1 / ||median_prev - normal_i|| )
   

   Add a small epsilon to the denominator to avoid division by zero if two normals are identical
3. Stop when the change between iterations is below a small threshold (like 1e-6)

For simpler use cases, you can also approximate it by clustering normals by direction and picking the cluster center that’s closest to the majority of vectors—but Weiszfeld gives a more precise result.

Final Verdict #

Yes, swapping the mean for the vector median is an excellent choice for your scenario. It’ll fix those offset vertex normals in unevenly subdivided regions and give you more accurate, visually consistent surface shading.

内容的提问来源于stack exchange，提问作者Tyson