Generalization of form in visual pattern classification.

Human observers were trained to criterion in classifying compound Gabor signals with sym- metry relationships, and were then tested with each of 18 blob-only versions of the learning set. General- ization to dark-only and light-only blob versions of the learning signals, as well as to dark-and-light blob versions was found to be excellent, thus implying virtually perfect generalization of the ability to classify mirror-image signals. The hypothesis that the learning signals are internally represented in terms of a 'blob code' with explicit labelling of contrast polarities was tested by predicting observed generalization behaviour in terms of various types of signal representations (pixelwise, Laplacian pyramid, curvature pyramid, ON/OFF, local maxima of Laplacian and curvature operators) and a minimum-distance rule. Most representations could explain generalization for dark-only and light-only blob patterns but not for the high-thresholded versions thereof. This led to the proposal of a structure-oriented blob-code. Whether such a code could be used in conjunction with simple classifiers or should be transformed into a propo- sitional scheme of representation operated upon by a rule-based classification process remains an open question

Nonlinear image operators, higher-order statistics,and the AND-like combinations of frequency components

The frequency domain plays a key role in the description of signals and systems. In the classical approaches, the individual frequency components are treated as independent: In linear systems, the superposition principle restricts the filtering to an OR-like processing of independent complex exponentials. Likewise, the classical second-order statistic (the powerspectrum) measures only the occurrence of each individual frequency component, independent of whether it occurs in a systematic combination with other components or not. This basic limitation can be overcome by the extension of the classical approaches to nonlinear systems and higher-order statistics, which makes it possible to selectively address AND-like combinations of frequency components. We measure which AND combinations are statistically most relevant in natural images, and investigate how this statistical structure can be exploited by nonlinear Volterra filters

Nonlinear image operators for the evaluation of local intrinsic dimensionality

Local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters. We derive a class of polynomial operators for the detection of intrinsically 2-D image features like curved edges and lines, junctions, line ends, etc. Although it is a deterministic concept, intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal. Furthermore, there is an intimate connection to multidimensional surface geometry and to the concept of `Gaussian curvature'. Nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are, by the superposition principle, restricted to OR-combinations of their intrinsically 1-D eigenfunctions. The essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components. Therefore, such operators can provide the AND-combination of complex exponentials, which is required for the exploitation of intrinsic dimensionality. Using frequency design methods, we obtain a generalized class of quadratic Volterra operators that are selective to intrinsically 2-D signals. These operators can be adapted to the requirements of the signal processing task. For example, one can control the “curvature tuning” by adjusting the width of the stopband for intrinsically 1-D signals, or the operators can be provided in isotropic and in orientation-selective versions. We first derive the quadratic Volterra kernel involved in the computation of Gaussian curvature and then present examples of operators with other arrangements of stop and passbands. Some of the resulting operators show a close relationship to the end-stopped and dot-responsive neurons of the mammalian visual corte

Nonlinear mechanisms and higher-order statistics in biological vision and electronic image processing: review and perspectives

The classical approach in vision research—the derivation of basically linear filter models from experiments with simple artificial test stimuli—is currently undergoing a major revision. Instead of trying to keep the dirty environment out of our clean labs we put it now right into the focus of scientific exploration. An increasing number of scientists are using natural images in their experimental work, and concepts from statistics and information theory are employed for the theoretical modeling of the results. The new approach has a close relation to basic engineering strategies for electronic image processing, since its major concept is that biological sensory systems exploit the statistical redundancies of the environment by appropriate neural transformations. The standard engineering methods are not sufficient, however. Even such a basic biological feature as orientation selectivity requires the consideration of higher-order statistics, like multivariate wavelet statistics, cumulants, or polyspectra. Furthermore, there exists an abundance of nonlinear phenomena in biological vision, for example the phase invariance of complex cells, cortical gain control, end-stopping, and a variety of extra-classical receptive field properties. These amount to nonlinear combinations of linear wavelet filter outputs, which are required to exploit higher-order statistical dependencies, and make it necessary to consider unconventional modeling approaches like differential geometry or Volterra–Wiener systems. By use of such methods we cannot only gain a deeper understanding of the adaptation of the visual system to the complex natural environment, but we can also make the biological system an inspiring source for the design of novel strategies in electronic image processing