A new formulation for the Karhunen–Loeve expansion
Introduction
In several domains of signal processing, it may be interesting to provide a second-moment characterization of a random signal in terms of uncorrelated random variables. We can find in the literature such an expansion referred to as the Karhunen–Loeve expansion. The basis functions of this expansion are the eigenfunctions, solutions of the Fredholm integral equation where the kernel is the autocorrelation function of the studied random process. Generally, the problem of finding the solutions of this integral equation is reduced to the question of finding the eigenvectors and the eigenvalues of a covariance matrix. Unfortunately, when we consider the discrete form of this integral equation, the solutions often constitute an insufficient approximation of the real solutions. Furthermore, applying the Karhunen–Loeve expansion in its discrete form can take a very long time. So, the Karhunen–Loeve expansion is rarely used in practice because we do not know how to estimate it quickly and efficiently [11], [12], [16]. We answer this question by proposing a new formulation and an algorithm that is cheaper and more accurate than classical approaches.
First of all, we recall in Section 1 the Karhunen–Loeve expansion in the case of one-dimensional signal. Then, we describe, in Section 2, a new method to approximate solutions of the Fredholm integral equation and we illustrate this method with several experiments. In order to limit memory problems and to quickly obtain a signal expansion, we propose, in Section 3, a new formulation of the Karhunen–Loeve expansion by using the discrete cosine transform. Next, in Section 4, we propose an extension of this new formulation to the case of a two-dimensional signal, in order to apply it in image processing. We finish this article with an example of an application for image interpolation. We propose several experiments to describe the quality of our results and we compare them with the results obtained using the classical formulation.
Section snippets
The Karhunen–Loeve expansion
In this section, we propose the Karhunen–Loeve development in its classical formulation. This development is based on the signal expansion into series of functions with uncorrelated random variables for decomposition coefficients. For this reason, first of all, we explain how to expand a random signal.
Fredholm integral equation, numerical method
It is possible to find in the literature several methods for solving the Fredholm integral equation. Some of them give an analytical expression for the solutions by substituting a differential equation for the integral equation [1], [7], [8], [9], [17]. So, these methods imply the solving of a differential equation. This may be difficult according to the kernel studied. For other methods the principle is to modify the formulation of the integral equation in order to find a matrix system [3],
New formulation of the Karhunen–Loeve expansion
We propose in this section a new formulation for the Karhunen–Loeve expansion by using the discrete cosine transform. In these conditions, we are looking for the expression of the coefficients of the random signal expanded into a series of cosines; we shall reconstruct the approximation of the random signal in the final phase of the treatment [4], [5], [6].
Let S(t) be the signal to be expanded and let be the reconstructed signal. We havewhere xn are the random
Extension to a two-dimensional random signal
We propose in this section an extension of this new formulation of the Karhunen–Loeve expansion to the case of a two-dimensional signal, in order to apply it in image processing.
An application: interpolation of a two-dimensional random signal
To evaluate the improvement due to our formulation, we propose in this section an example of an application of the Karhunen–Loeve expansion for image interpolation.
Conclusion
We have presented in this paper a new method for finding an analytical approximation for the solutions of the Fredholm integral equation and a new formulation of the Karhunen–Loeve expansion. We have proposed an application for image interpolation. Several experiments have shown that the interpolated images obtained with our formulation present better results than with the classical processing. Indeed, the restoration of the low frequency and the preservation of the contrasts are better and the
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