# Kernel Methods for Higher Order Image Statistics

The conditions under which natural vision systems evolved show statistical regularities determined both by the environment and by the actions of the organism. Many aspects of biological vision can be understood as evolutionary adaptations to these regularities. This is demonstrated by the recent sucess in explaining properties of retinal and cortical neurons from the statistics of natural images. At the same time, we observe an increasing interest in statistical modeling techniques in the computer vision community. Here, the motivation comes from the need for powerful image models in image processing tasks such as super-resolution or denoising. In the literature, the statistical analysis of natural images has mainly been done with linear techniques such as Principal Component Analysis (PCA) or Fourier analysis. These techniques capture only the second-order statistics of an image ensemble. A large part of the interesting image structure, however, is contained in the higher-order statistics. Unfortunately, the estimation of these statistics involves a huge number of terms which makes their explicit computation for images infeasible in practice. Kernel methods provide an implicit access to higher-order statistics that avoids this combinatorial explosion. In the course, we start with an overview of existing approaches to image statistics. The need to go beyond the usual linear, second-order techniques will lead us to the classical higher-order statistics such as Wiener series, higher-order cumulants and spectra. We will see that the exponential number of terms involved in these statistics prevents them from being applied to images. This motivates the introduction of kernel techniques. Here, we will discuss two approaches: 1. The Wiener series can be estimated implicitly via polynomial kernel regression. We will use this technique to decompose an image into components that are characterized by pixel interactions of a given order. 2. Kernel PCA of image patches provides a powerful image model that takes higher-order statistics into account. We will show applications of this model to various image processing tasks.

*Author: Matthias O. Franz, Max Planck Institute for Biological Cybernetics*