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The genus statistics of isodensity contours has become a well-establishedtool in cosmology. In this Letter we place the genus in the wider framework ofa complete family of morphological descriptors. These are known as theMinkowski functionals, and we here apply them for the first time to isodensitycontours of a continuous random field. By taking two equivalent approaches, onethrough differential geometry, the other through integral geometry, we derivetwo complementary formulae suitable for numerically calculating the Minkowskifunctionals. As an example we apply them to simulated Gaussian random fieldsand compare the outcome to the analytically known results, demonstrating thatboth are indeed well suited for numerical evaluation. The code used forcalculating all Minkowski functionals is available from the authors.Comment: 8 pages plus 1 figure; uses aaspp4.sty and flushrt.sty. Matches  version accepted for publication in Ap. J. Let

Beyond Genus Statistics: A Unifying Approach to the Morphology of Cosmic Structure