Nonlinear image operators for the evaluation of local intrinsic dimensionality

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Abstract
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 cortex.

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