A mathematical morphology approach to Euclidean distance transformation

This paper presents a distance transformation technique for a binary digital image using a gray scale mathematical morphology approach. A distance transformation converts a binary image which consists of object (foreground) and nonobject (background) pixels into an image where every object pixel has a value corresponding to the minimum distance from the background. The distance computation is, in fact, a global operation. Morphological erosion is an operation which selects the minimum value from the combination of an image and the predefined weighted structuring element within a window. Hence, mathematical morphology is the most appropriate approach to distance transformation. Applying well-developed decomposition properties of mathematical morphology, we can significantly reduce the tremendous cost of global operations to that of small neighborhood operations suitable for parallel pipelined computers. In the first part of this paper, the distance transformation using mathematical morphology is developed. In the second part, several approximations of the Euclidean distance are discussed. In the third part, the decomposition of the Euclidean distance structuring element is presented. The decomposition technique employs a set of 3 by 3 gray scale morphological erosions with suitable weighted structuring elements and combines the outputs using the minimum operator. Real-valued distance transformations are considered during the processes and the result is approximated to the closest integer in the final output image.