Order-configuration functions: Mathematical characterizations and applications to digital signal and image processing

We call a function f in n variables an order-configuration function if for any x₁,…, x_n such that x_i1 … x_in we have f(x₁,…, x_n) = x_t, where t is determined by the n-tuple (i₁,…, i_n) corresponding to that ordering. Equivalently, it is a function built as a minimum of maxima, or a maximum of minima. Well-known examples are the minimum, the maximum, the median, and more generally rank functions, or the composition of rank functions. Such types of functions are often used in nonlinear processing of digital signals or images (for example in the median or separable median filter, min-max filters, rank filters, etc.). In this paper we study the mathematical properties of order-configuration functions and of a wider class of functions that we call order-subconfiguration functions. We give several characterization theorems for them. We show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order-configuration functions.