Distance transforms for real-valued functions

A set in a metric space gives rise to its distance function that associates with every point its distance to the nearest point in the set. This function is called the distance transform of the original set. In the same vein, given a real-valued function f we consider the expected distances from any point to alevelset of f taken at a random height. This produces another function called a distance transform of f. Such transforms are called grey-scale distance transforms to signpost their differences from the binary case when sets (or their indicators) give rise to conventional distance functions. Basic properties of the introduced grey-scale distance transform are discussed. The most important issue is the uniqueness problem whether two different functions may share the same distance transform. We answer this problem in a generality completely sufficient for all practical applications in imaging sciences, the full-scale problem remains open