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
The algebraic basis of mathematical morphology. I. dilations and erosions
Computer Vision, Graphics, and Image Processing
The algebraic basis of mathematical morphology
CVGIP: Image Understanding
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
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Treating a binary image as a random process results in the granulometric pattern spectrum being a random function and its moments being random variables. Because these moments are used as image signatures and as local texture descriptors, their statistical distributions, and in particular their moments, are of importance. The present paper employs a theorem of Cramer to show for a certain class of image models that the pattern-spectrum-moment distributions are asymptotically normal and, using asymptotic representations of Cramer, derives asymptotic expressions for moments of the spectrum moments. Asymptotic normality permits the application of customary normal statistical methods in estimation, classification, and hypothesis testing. The analysis proceeds in two parts, first treating a random image generated by a single grain primitive of random size. The Euclidean granulometric theory of Matheron is then employed to generalize the analysis to a more complicated textural model in which each image is generated by a number of randomly sized primitives. To facilitate application of Cramer's theory the paper introduces the notion of an orthogonal granulometric generator. The resulting new class of generators is necessary because Matheron's original granulometric theory permits pattern spectra whose moments are not sufficiently well behaved to allow application of Cramer's asymptotic formulations.