Joint asymptotic normality of granulometric moments under multiple structuring elements

For a binary image composed of randomly sized disjoint grains, morphological granulometries produce a pattern spectrum containing textural information that can be used for image classification, image segmentation and model estimation. It has been shown that as the number of grains per image increases, any finite collection of granulometric moments based on the same structuring element are jointly asymptotically normal with a known analytic form for the moments. However, many applications require the use of multiple structuring elements, for instance horizontal and vertical linear generators, to draw out important textural characteristics. In this work, we prove the joint asymptotic normality of granulometric moments from multiple structuring elements. We also derive analytic expressions for the asymptotic mean vector and covariance matrix of the granulometric moments, including the previously unknown off-diagonal elements of the asymptotic covariance matrix corresponding to granulometric moments from distinct structuring elements.