Chamfer Distance and Earth Mover’s Distance

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To evaluate distance between point cloud to pointcloud, Chamfer Distance and Earth Mover’s distance are frequently used. What is the difference?

Chamfer Distance measures the distance between two point clouds $S_1$ and $S_2$ , defined as:

\text{CD}(S_1, S_2) = \sum_{x \in S_1} \min_{y \in S_2} \|x - y\|_2^2 + \sum_{y \in S_2} \min_{x \in S_1} \|x - y\|_2^2

Earth Mover’s Distance calculates the minimal cost to transform one point cloud $S_1$ into another $S_2$ , defined as:

\text{EMD}(S_1, S_2) = \min_{\phi : S_1 \to S_2} \sum_{x \in S_1} \|x - \phi(x)\|_2

Chamfer Distance:
- For each point in each point cloud, find the closest point in the other point cloud.
- Does not enforce global one-to-one correspondence.
- Because the matching is local, some points may be over-matched or ignored.
Earth Mover’s Distance:
- Match points between point clouds with optimal global one-to-one correspondence (or “transportation plan”).
- Each point in $S_1$ should be matched to exactly one point in $S_2$ , minimizing the overall “cost”.

Feature	CD	EMD
Matching	Local (Nearest Neighbors)	Global
Sensitivity	High	Low
Symmetry	Not Symmetric	Symmetric
Computational Speed	Fast	Slow
Application	Completion, Surface Reconstruction, Approximation	Stochastic Distribution, Global Matching