On the capabilities of multilayer perceptrons
Journal of Complexity - Special Issue on Neural Computation
Multilayer feedforward networks are universal approximators
Neural Networks
Approximation capabilities of multilayer feedforward networks
Neural Networks
Minimal representations for translation-invariant set mappings by mathematical morphology
SIAM Journal on Applied Mathematics
Optimal mean-square N-observation digital morphological filters: i. optimal binary filters
CVGIP: Image Understanding
Optimal mean-square N-observation digital morphological filters: ii. optimal gray-scale filters
CVGIP: Image Understanding
Lower bounds on the Vapnik-Chervonenkis dimension of multi-layer threshold networks
COLT '93 Proceedings of the sixth annual conference on Computational learning theory
Computational mathematical morphology
Signal Processing - Special issue on mathematical morphology and its applications to signal processing
Mean-absolute-error representation and optimization of computational-morphological filters
Graphical Models and Image Processing
Optimal stack filters under rank selection and structural constraints
Signal Processing
Secondarily constrained Boolean filters
Signal Processing
Signal Processing
Coefficient of determination in nonlinear signal processing
Signal Processing - Special section on signal processing technologies for short burst wireless communications
Multiresolution Analysis for Optimal Binary Filters
Journal of Mathematical Imaging and Vision
Enhancement and Restoration of Digital Documents: Statistical Design of Nonlinear Algorithms
Enhancement and Restoration of Digital Documents: Statistical Design of Nonlinear Algorithms
IEEE Transactions on Image Processing
Optimal Filters with Multiresolution Apertures
Journal of Mathematical Imaging and Vision
Journal of Mathematical Imaging and Vision
Nonlinear Filter Design Using Envelopes
Journal of Mathematical Imaging and Vision
Generating segmented meshes from textured color images
Journal of Visual Communication and Image Representation
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A body of research has developed within the context of nonlinear signal and image processing that deals with the automatic, statistical design of digital window-based filters. Based on pairs of ideal and observed signals, a filter is designed in an effort to minimize the error between the ideal and filtered signals. The goodness of an optimal filter depends on the relation between the ideal and observed signals, but the goodness of a designed filter also depends on the amount of sample data from which it is designed. In order to lessen the design cost, a filter is often chosen from a given class of filters, thereby constraining the optimization and increasing the error of the optimal filter. To a great extent, the problem of filter design concerns striking the correct balance between the degree of constraint and the design cost. From a different perspective and in a different context, the problem of constraint versus sample size has been a major focus of study within the theory of pattern recognition. This paper discusses the design problem for nonlinear signal processing, shows how the issue naturally transitions into pattern recognition, and then provides a review of salient related pattern-recognition theory. In particular, it discusses classification rules, constrained classification, the Vapnik-Chervonenkis theory, and implications of that theory for morphological classifiers and neural networks. The paper closes by discussing some design approaches developed for nonlinear signal processing, and how the nature of these naturally lead to a decomposition of the error of a designed filter into a sum of the following components: the Bayes error of the unconstrained optimal filter, the cost of constraint, the cost of reducing complexity by compressing the original signal distribution, the design cost, and the contribution of prior knowledge to a decrease in the error. The main purpose of the paper is to present fundamental principles of pattern recognition theory within the framework of active research in nonlinear signal processing.