New sequential exact Euclidean distance transform algorithms based on convex analysis

We present several sequential exact Euclidean distance transform algorithms. The algorithms are based on fundamental transforms of convex analysis: The Legendre Conjugate or Legendre–Fenchel transform, and the Moreau envelope or Moreau-Yosida approximate. They combine the separability of the Euclidean distance with convex properties to achieve an optimal linear-time complexity.We compare them with a Parabolic Envelope distance transform, and provide several extensions. All the algorithms presented perform equally well in higher dimensions. They can naturally handle grayscale images, and their principles are generic enough to apply to other transforms.