In image recognition applications, complex decision regions in the image space are needed. Linear filtering forms the decision regions by hyperplanes in the image space. We determine the decision region formed by Fourier-plane nonlinear filtering. In the case in which power law nonlinearity is applied in the Fourier plane, the decision region turns out to be approximately an n-dimensional parabola that opens toward the direction of the reference vector. That is, the intersection of the decision region with any plane (two-dimensional vector space) not containing any vector parallel to the reference vector is a bounded convex region enclosed by a closed curve. The size of the convex region depends on the filter nonlinearity, which determines the distortion robustness and discrimination capability of the filter. It can be adjusted by choosing different Fourier-plane nonlinearities and/or different threshold values at the output plane. These types of regions are desirable and well suited in image recognition. Analytical and numerical solutions are provided.
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