Wave propagation in digital image
Here's a trick that video game designers have used for years. A computationally cheaper approximation to Euclidean distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is to use
$$
d=\max(|x_1-x_2|, |y_1-y_2|)+\min(|x_1-x_2|, |y_1-y_2|)/2
$$
The relative error between this and the Euclidean distance is no more than 11.8%, which might be good enough for your purposes.