On the distributions of the relative phase of complex wavelet coefficients

In this paper, the probability distributions of relative phase are studied. We proposed von Mises and wrapped Cauchy for the probability density function (pdf) of the relative phase in complex wavelet domain. The maximum-likelihood method is used to estimate the two parameters of von Mises and wrapped Cauchy. We demonstrate that the von Mises and wrapped Cauchy fit well with real data obtained from various real images including texture images as well as natural images. The von Mises and wrapped Cauchy models are compared, and the simulation results show that the wrapped Cauchy fits well with the peaky and heavy-tailed pdf of the relative phase and the von Mises fits well with the pdf which is in Gaussian shape. For most of the test images, the wrapped Cauchy model is more accurate than the von Mises, when images are decomposed by different complex wavelet transforms including dual-tree complex wavelet (DTCWT), pyramidal dual-tree directional filter bank (PDTDFB) and a modified version of curvelet.