SummaryQuantitative measures for anisotropic characteristics of spatial structure are needed when relating the morphology of microstructured heterogeneous materials to tensorial physical properties such as elasticity, permeability and conductance. Tensor-valued Minkowski functionals, defined in the framework of integral geometry, provide a concise set of descriptors of anisotropic morphology. In this article, we describe the robust computation of these measures for microscopy images and polygonal shapes. We demonstrate their relevance for shape description, their versatility and their robustness by applying them to experimental data sets, specifically microscopy data sets of non-equilibrium stationary Turing patterns and the shapes of ice grains from Antarctic cores.

How does the clustering of galaxies depend on their inner properties like morphological type and luminosity ? We address this question in the mathematical framework of marked point processes and clarify the notion of luminosity and morphological segregation. A number of test quantities such as conditional mark-weighted two-point correlation functions are introduced. These descriptors allow for a scale-dependent analysis of luminosity and morphology segregation. Moreover, they break the degeneracy between an inhomogeneous fractal point set and actual present luminosity segregation. Using the Southern Sky Redshift Survey 2, we Ðnd both luminosity and morphological segregation at a high level of signiÐcance, conÐrming claims by Benoist and colleagues in 1996 and Willmer and colleagues in 1998 using these data. SpeciÐcally, the average luminosity and the Ñuctuations in the luminosity of pairs of galaxies are enhanced out to separations of 15 h~1 Mpc. On scales smaller than 3 h~1 Mpc the luminosities on galaxy pairs show a tight correlation. A comparison with the random Ðeld model indicates that galaxy luminosities depend on the spatial distribution and galaxy-galaxy interactions. Early-type galaxies are also more strongly correlated, indicating morphological segregation. The galaxies in the PSCz catalog do not show signiÐcant luminosity segregation. This again illustrates that mainly earlytype galaxies contribute to luminosity segregation. However, based on several independent investigations we show that the observed luminosity segregation cannot be explained by the morphology-density relation alone.

Abstract. In a follow-up study to a previous analysis of the IRAS 1.2 Jy catalogue, we quantify the morphological fluctuations in the PSCz survey. We use a variety of measures, among them the family of scalar Minkowski functionals. We confirm the existence of significant fluctuations that are discernible in volume-limited samples out to 200h −1 Mpc. In contrast to earlier findings, comparisons with cosmological N-body simulations reveal that the observed fluctuations roughly agree with the cosmic variance found in corresponding mock samples. While two-point measures, e.g. the variance of count-in-cells, fluctuate only mildly, the fluctuations in the morphology on large scales indicate the presence of coherent structures that are at least as large as the sample.

Abstract. Higher-rank Minkowski valuations are efficient means for describing the geometry and connectivity of spatial patterns. We show how to extend the framework of the scalar Minkowski valuations to vector-and tensor-valued measures. The versatility of these measures is demonstrated by using simple toy models as well as real data.