Neural Processing Letters 8: 117–129, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 117 Collinearity and Parallelism are Statistically Significant Second-Order Relations of Complex Cell Responses ⋆ NORBERT KRÜGER Institute für Neuroinformatik, Ruhr-Universität, Universitätsstraße 150, ND 03/72, 44081 Bochum, Germany. E-mail: nkrueger@neuroinformatik.ruhr-uni-bochum.de Abstract. By investigating the second-order statistics of Gabor wavelet responses derived from natural images, we show that collinearity and parallelism are conspicuous relations. We give a pre- cise mathematical characterization of these Gestalt principles by the conditional probability of two responses. Essential for our investigations is a non-linear transformation, initially utilized within the object recognition system [5], which transforms continuous Gabor wavelet responses into a binary code indicating the presence or absence of local oriented line segments. Key words: natural images, Gabor wavelets, Gestalt principles, learning, contextual information, Banana wavelets 1. Introduction A great deal of evidence exists (see, e.g., Daugman [1]) supporting the assumption that Gabor wavelets play an important role in the first stages of human visual processing. Jones et al. [4] recorded neurons with Gabor wavelet-like sensitivity in V1, and Gabor wavelet-like filters can be learned from the statistics of natural images by applying the restrictions ‘information preservation’ and ‘sparse coding’ (Ohlshausen et al. [6]). There is also strong evidence that the human object recogni- tion system processes the visual input through a couple of stages in which features of increasing complexity are extracted and in which Gabor wavelets represent only an early stage of processing (see, e.g., Hummel et al. [3] and Oram et al. [7]). Eval- uating the second-order statistics of Gabor wavelet responses, we give statistical evidence for important second-order relations for the class of natural images and therefore we support the understanding of the stages of the visual system beyond the extraction of Gabor wavelet responses. On the one hand there is, as of yet, little knowledge about the statistical proper- ties of the class of natural images. As the most precise quality Field [2] showed that ⋆ Supported by grants from the German Ministry for Science and Technology 01IN504E9 (NEUROS) and 01M3021A4 (Electronic Eye).