Why Triangular Membership Functions Work Well in F-Transform: A Theoretical Explanation

In many practical applications, it is useful to represent a signal or an image by its average values on several fuzzy sets. The corresponding F-transform technique has many useful applications in signal and image processing. In principle, we can use different membership functions. Somewhat surprisingly, in many applications, the best results occur when we use triangular membership functions. In this paper, we provide a possible theoretical explanation for this empirical phenomenon. 1 Triangular Membership Functions Work Well in F-Transform: An Empirical Fact Need for approximating signals and images. In many practical situations, it is important to process signals and images. From the mathematical viewpoint, a signal is a function x(t) that describe the recorded value of the corresponding quantity at different moments of time t. Similarly, an image is a function I(x, y) that describe the intensity at a spatial point with coordinates (x, y). In principle, we can represent each signal by describing, for each moment t, the corresponding value x(t); however, this representation often requires too much computer memory, more than the current device has – and/or too much computer time to process all these values, more than the time we need to make a decision. In such situations, it is desirable to come up with a smaller number of values representing the original signal – the values from which the signal can be reproduced with a good approximation accuracy.