Karhunen–Loève Expansion Using a Parametric Model of Oscillating Covariance Function

The Karhunen–Loève (KL) expansion decomposes a stochastic process into a set of orthogonal functions with random coefficients. The basic idea of the decomposition is to solve the Fredholm integral equation associated with the covariance kernel of the process. The KL expansion is a powerful mathematical tool used to represent stochastic processes in a compact and efficient way. It has wide applications in various fields including signal processing, image compression, and complex systems modeling. The KL expansion has high computational complexity, especially for large data sets, since there is no single unique transformation for all stochastic processes and there are no fast algorithms for computing eigenvalues and eigenfunctions. One way to solve this problem is to use parametric models for the covariance function. In this paper, an explicit analytical solution of the KL expansion for a parametric model of oscillating covariance function with a small number of parameters is proposed. This model approximates the covariance functions of real images very well. Computer simulation results using a real image are presented and discussed.