Optical EngineeringSize distributions for multivariate morphological granulometries: texture classification and statistical properties
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As introduced by Matheron (1975), granulometries depend on a single sizing parameter for each structuring element forming the filter. Size distributions resulting from these granulometries have been used to classify texture by using as features the moments of the resulting pattern spectra. The concept of granulometry is extended in such a way that each structuring element has its own sizing parameter and the size distribution is multivariate. Whereas with univariate granulometries the normalized size distribution (pattern spectrum) is easily shown to be a probability distribution function, this proposition is more difficult to show for multivariate granulometries. Its demonstration is the main theoretical result. The classical single-structuring-element granulometries appear as marginal size distributions and the single-parameter multiple-structuringelement granulometries result from setting all parameters equal in a multivariate granulometry. Because of the greatly expanded freedom in choosing parameters, multivariate granulometries can discriminate textures that are indistinguishable using single-parameter granulometries. Texture classification proceeds by taking either the Walsh or moment transform of the multivariate pattern spectrum, obtaining a reduced feature set by applying the Karhunen-Loe`ve transform to the Walsh or moment features, and classifying textures via a Gaussian maximumlikelihood classifier. For the disjoint multiprimitive random set model, multivariate granulometric moments are represented in terms of sizingdistribution moments and shown to be asymptotically normal. Formulas are given for their asymptotic mean and variance.