References

The Minkowski tensor algorithms implemented in karambola are described in the following publications:

• Schröder-Turk, G. E. et al. Minkowski Tensors of Anisotropic Spatial Structure, New Journal of Physics 15 (8), 083028, (2013)
• Schröder-Turk, G. E. et al. Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures, Advanced Materials 23(22-23), 2535–2553 (2011)

Some examples of shape analyses based on Minkowski functionals or tensors can be found in the following publications (the list of examples is still under construction, further suggestions are very welcome).

In statistical physics and condensed matter:

• Klatt, M. A. Morphometry of Random Spatial Structures in Physics. (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2016)

• Schaller, F. M. et al. Non-universal Voronoi cell shapes in amorphous ellipsoid packs, EPL (Europhysics Letters) 111(2), 24002 (2015)

• Scholz, C. et al. Direct relations between morphology and transport in Boolean models. Phys. Rev. E 92, 43023 (2015)

• Klatt, M. A. & Torquato, S. Characterization of maximally random jammed sphere packings: Voronoi correlation functions. Phys. Rev. E 90, 52120 (2014)

• Kapfer, S.C. Morphometry and Physics of Particulate and Porous Media. (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2012)

• Schröder-Turk, G. E. et al. Disordered spherical bead packs are anisotropic, EPL 90, 34001 (2010)

• Kapfer, S. C. et al. Local anisotropy of fluids using Minkowski tensors, Journal of Statistical Mechanics: Theory and Experiment, P11010:1–20 (2010)

• Mecke, K. & Stoyan, D. Statistical Physics and Spatial Statistics – The Art of Analyzing and Modeling Spatial Structures and Pattern Formation. 554, (Springer, 2000)

• Mecke, K. R. & Wagner, H. Euler characteristic and related measures for random geometric sets. J. Stat. Phys. 64, 843 (1991)

In mathematical models of physical systems:

• Klatt, M. A. et al. Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors. In Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (eds. Vedel Jensen, E. B. & Kiderlen, M.) 2177, 385–421 (Springer International Publishing, 2017)
• Klatt, M. A. et al. Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals. J. Stat. Mech. Theor. Exp. 2017, 23302 (2017)
• Hug, D., Last, G. & Schulte, M. Second-order properties and central limit theorems for geometric functionals of Boolean models. The Annals of Applied Probability 26, 73–135 (2016)
• Hörrmann, J., Hug, D., Klatt, M. A. & Mecke, K. Minkowski tensor density formulas for Boolean models. Adv. Appl. Math. 55, 48–85 (2014)

In nuclear physics:

• Schuetrumpf, B. et al. Appearance of the single gyroid network phase in ‘nuclear pasta’ matter. Physical Review C 91, (2015)

• Schuetrumpf, B. et al. Time-dependent Hartree-Fock approach to nuclear ‘pasta’ at finite temperature. Physical Review C 87, (2013)

• Watanabe, G. et al. Structure of cold nuclear matter at subnuclear densities by quantum molecular dynamics. Physical Review C 68, (2003)
• Watanabe, G. et al. Phases of hot nuclear matter at subnuclear densities. Physical Review C 69, (2004)
• Sonoda, H. et al. Phase diagram of nuclear ‘pasta’ and its uncertainties in supernova cores. Physical Review C 77, (2008)

In astronomy and cosmology:

• Göring, D., Klatt, M. A., Stegmann, C. & Mecke, K. Morphometric analysis in gamma-ray astronomy using Minkowski functionals: Source detection via structure quantification. Astron. Astrophys. 555, A38 (2013)
• Kerscher, M. et al. Morphological fluctuations of large-scale structure: The PSCz survey. Astron. Astrophys. 373, 1 (2001)
• Kerscher, M. et al. Non-Gaussian morphology on large scales: Minkowski functionals of the REFLEX cluster catalogue. Astronomy & Astrophysics 377, 1–16 (2001)

In biology and medicine:

• Wilts, B. D. et al. Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development. Science Advances 3, e1603119 (2017)
• Klatt, M. A., Schröder-Turk, G. E. & Mecke, K. Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative. Med. Phys. 44, 3663–3675 (2017)
• Barbosa, M., Natoli, R., Valter, K., Provis, J. & Maddess, T. Integral-geometry characterization of photobiomodulation effects on retinal vessel morphology. Biomed. Opt. Express 5, 2317 (2014)
• Beisbart, C., Barbosa, M. S., Wagner, H. & Costa, L. da F. Extended morphometric analysis of neuronal cells with Minkowski valuations. Eur. Phys. J. B 52, 531 (2006)

In pattern analysis:

• Scholz, C., Schröder-Turk, G. E. & Mecke, K. Pattern-fluid interpretation of chemical turbulence. Phys. Rev. E 91, 42907 (2015)
• Mantz, H., Jacobs, K. & Mecke, K. Utilising Minkowski Functionals for Image Analysis. J. Stat. Mech. 12, P12015 (2008)
• Becker, J. et al. Complex dewetting scenarios captured by thin-film models. Nat. Mater. 2, 59 (2003)
• Mecke, K. Morphological characterization of patterns in reaction-diffusion systems. Phys. Rev. E 53, 4794 (1996)

In statistics and stereology:

• Baddeley, A. & Vedel Jensen, E. B. Stereology for statisticians. 103, (Chapman & Hall/CRC, 2004).
• Ebner, B. et al. Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images. arXiv:1710.02333 (2017)