Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Morphological methods in image and signal processing
Morphological methods in image and signal processing
Pattern Spectrum and Multiscale Shape Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Optimal Morphological Pattern Restoration from Noisy Binary Images
IEEE Transactions on Pattern Analysis and Machine Intelligence
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
Journal of Mathematical Imaging and Vision
An Introduction to Nonlinear Image Processing
An Introduction to Nonlinear Image Processing
Extrapolation, Interpolation, and Smoothing of Stationary Time Series
Extrapolation, Interpolation, and Smoothing of Stationary Time Series
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Optimal Conjunctive Granulometric Bandpass Filters
Journal of Mathematical Imaging and Vision
Markovian Analysis of Adaptive Reconstructive Multiparameter τ-Openings
Journal of Mathematical Imaging and Vision
Granulometric Size Density for Segmented Random-Disk Models
Journal of Mathematical Imaging and Vision
Design of optimal binary filters under joint multiresolution-envelope constraint
Pattern Recognition Letters - Special issue: Sibgrapi 2001
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Euclidean granulometries are used to decompose a binary image into a disjoint union based on interaction between image shape and the structuring elements generating the granulometry. Each subset of the resulting granulometric spectral bands composing the union defines a filter by passing precisely the bands in the subset. Given an observed image and an ideal image to be estimated, an optimal filter must minimize the expected symmetric-difference error between the ideal image and filtered observed image. For the signal-union-noise model, and for both discrete and Euclidean images, given a granulometry, a procedure is developed for finding a filter that optimally passes bands of the observed noisy image. The key is characterization of an optimal filter in the Euclidean case. Optimization is achieved by decomposing the mean functions of the signal and noise size distributions into singular and differentiable parts, deriving an error representation based on the decomposition, and describingoptimality in terms of generalized derivatives for the singular parts and ordinary derivatives for the differentiable parts. Owing to the way in which spectral bands are optimally passed, there are strong analogies with the Wiener filter.