Labelled Reconstruction of Binary Objects: A Vector Propagation Algorithm

Abstract

The quench function of a binary image is the distance transform of the image sampled on its skeleton. In principle the original image can be reconstructed from the quench function by drawing a disk at each point on the skeleton with radius given by the corresponding quench function value. This reconstruction process is of more than theoretical interest. One possible use is in coding of binary images, but our interest is in an applied image analysis context where the skeleton has been (1) reduced by, for example, deletion of barbs or other segments, and/or (2) labelled so that segments, or indeed individual pixels, have identifying labels. A useful reconstruction, or partial reconstruction, in such a case would be a labelled image, with labels propagated from the skeleton in some intuitive fashion, and the support of this labelled output would be the theoretical union of disks.

An algorithm which directly draws disks would, in many situations, be very inefficient. Moreover the label value for each pixel in the reconstruction is highly ambiguous in most cases where disks are highly overlapping. We propose a vector propagation algorithm based on Ragnelmalm’s Euclidean distance transform algorithm which is both efficient and provides a natural label value for each pixel in the reconstruction. The algorithm is based on near-exact Euclidean distances in the sense that the reconstruction from a single-pixel skeleton is, to a very good approximation, a Euclidean digital disk. The method is illustrated using a biological example of neurite masks originating from images of neurons in culture.