Fuzzy histograms, weak fuzzification and accumulation of periodic quantities

The influence of the scale of a fuzzy membership function used to fuzzify a histogram is analysed. It is shown that for a class of fuzzifying functions it is possible to indicate the limit for fuzzification, at which the mode of the histogram equals the mean of the data accumulated in it. The fuzzification functions for which this appears are: the quadratic function for aperiodic histograms and the cosine square function for periodic ones. The scaled and clipped versions of these functions can be used to control the degree of fuzzification belonging to the interval [0,1]. While the quadratic function is related to the widely known Huber-type clipped mean or the kernel function derived from the Epanechnikov kernel, the clipped cosine square seems to be less known. The indications for using strong or weak fuzzification, according to the value of the fuzzification degree, are justified by examples in two applications: classic Hough transform-based image registration and novel accumulation-based line detection. Typically, the weak fuzzification is recommended. The images used are related to simulation images from teleradiotherapy and to mammographic images.