Fast Convolutional Distance Transform

We propose “convolutional distance transform”—efficient implementations of distance transform. Specifically, we leverage approximate minimum functions to rewrite the distance transform in terms of convolution operators. Thanks to the fast Fourier transform, the proposed convolutional distance transforms have $\mathcal {O}(N\log N)$ complexity, where $N$ is the total number of pixels. The proposed acceleration technique is “distance metric agnostic.” In the special case that the distance function is a $p$ -norm, the distance transform can be further reduced to separable convolution filters; and for Euclidean norm, we achieve $\mathcal {O}(N)$ using constant-time Gaussian filtering.