Symmetric PH and EP distributions and their applications to the probability Hough transform (Q1767797)

The aim of this paper is to investigate the properties of two novel distributions defined by the authors: a symmetric phase type (PH) distribution, and a symmetric exponential-polynomial type (EP) distribution based on the PH distribution. The contents of the paper are organized as follows: Section 2 introduces a univariate symmetric PH distribution and then proposes a univariate symmetric EP distribution based on the univariate symmetric PH distribution. Some graphical characteristics of the univariate PH and EP distributions are discussed. Section 3 illustrates the fact that the class of univariate symmetric EP distributions is large enough such that an arbitrary symmetric probability density function in \(L_2(-\infty,+\infty)\) (the space of square integrable functions on the real line) can be approximated in \((-\infty,+\infty)\) by a sequence of symmetric EP probability density functions. This result is proved using Laguerre and Hermite spectrum orthogonal decompositions. Section 4 provides a moment-based approach for simply constructing a sequence of symmetric EP density functions. Sections 5 and 6 propose two-dimensional, respectively multi-dimensional, symmetric EP and PH distributions and investigate their properties. Section 7 applies the symmetric PH and EP distributions to the study of probability Hough transform. Numerical algorithms and statistical behaviour of the class of symmetric PH and EP distributions have a large range of possible applications in pattern recognition, image analysis, computer vision etc.