Localized Principal Components of Natural Images - an Analytic Solution

The structure of receptive elds in the visual cortex is believed to be shaped by unsupervised learning. A simple variant of unsupervised learning is the extraction of principal components. In this paper, we derived analytically the form of the principal components of natural images. An assumption is made that only small circular regions of the images are being used as training patterns. The derivation relies on results about the correlation function of natural images (Field, 1987). Our results predict both the shapes and the phases of the receptive elds. We also compared these resluts to previous simulation results (Hancock et al., 1992). Finally, the biological relevance of our results is discussed.