Gabriel Samberg, YoonHaeng Hur, Yuehaw Khoo +1 more
Matching two point clouds usually means finding which point corresponds to which. But in many real settings the points are samples from distributions with genuine cluster structure, and within a coherent region individual points are effectively interchangeable.
When that is true, insisting on exact point-to-point correspondence solves a harder problem than the one you have, and solves it fragilely: resample the same underlying object and every individual pairing changes even though the regions clearly still align. The stable, meaningful answer is region to region.
Bringing the graph Laplacian into optimal transport is a natural way to encode that, since the Laplacian is how you describe which points belong together before deciding what maps where. The method is told about the structure rather than being asked to work around it.
In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian…
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