The boolean map distance: theory and efficient computation

We propose a novel distance function, the boolean map distance (BMD), that defines the distance between two elements in an image based on the probability that they belong to different components after thresholding the image by a randomly selected threshold value. This concept has been explored in a number of recent publications, and has been proposed as an approximation of another distance function, the minimum barrier distance (MBD). The purpose of this paper is to introduce the BMD as a useful distance function in its own right. As such it shares many of the favorable properties of the MBD, while offering some additional advantages such as more efficient distance transform computation and straightforward extension to multi-channel images

Harmonic mappings and distance function

We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with $C^{1,\alpha}$ ($\alpha<1$), respectively $C^{1,1}$ compact boundary is bi-Lipschitz. The distance function with respect to the boundary of the image domain is used. This in turn extends a similar result of the author in \cite{kalajan} for Jordan domains, where stronger boundary conditions for the image domain were needed.Comment: 10 pages, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienz