Approximations of the Euclidean distance by chamfer distances

TL;DR

This paper analyzes how well chamfer distances approximate Euclidean distances, providing optimal error bounds and explicit sequences for small neighborhoods, which are useful for efficient distance transforms.

Contribution

It determines the maximum relative error of chamfer distances under various boundary conditions and explicitly provides optimal sequences, especially for small neighborhoods.

Findings

Optimal maximum relative error bounds are established.

Explicit best approximating sequences are provided for small neighborhoods.

The results improve understanding of chamfer distance accuracy in practical applications.

Abstract

Chamfer distances play an important role in the theory of distance transforms. Though the determination of the exact Euclidean distance transform is also a well investigated area, the classical chamfering method based upon "small" neighborhoods still outperforms it e.g. in terms of computation time. In this paper we determine the best possible maximum relative error of chamfer distances under various boundary conditions. In each case some best approximating sequences are explicitly given. Further, because of possible practical interest, we give all best approximating sequences in case of small (i.e. 5 by 5 and 7 by 7) neighborhoods.