ℱ‐homogeneous functions and a generalization of directional monotonicity

Abstract

A function that takes n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor λ yields the original output multiplied by that same factor. This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function g and the effect of the factor λ by an automorphism. However, the effect of parameter λ remains unchanged for all the input values. In this study, we generalize further the condition of abstract homogeneity by introducing MJX-tex-caligraphicnormalℱ‐homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs. Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions. To achieve that, we generalize directional monotonicity by MJX-tex-caligraphicnormalℱ directional monotonicity, which is defined with respect to a family of functions MJX-tex-caligraphicnormalℱ and a family of vectors MJX-tex-caligraphicscriptV. Finally, we show how the introduced concepts could be applied in two different problems of computer vision: a snow detection problem and image thresholding improvement.