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 (CD)

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 (EMD)

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

Which one to be used?

• 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”.