Minkowski functionals and tensors are versatile, sensitive and robust shape descriptors with a rigorous mathematical foundation. Here, we present an introduction to the rich mathematical background, and we provide free-to-use software for two and three dimensions.


23.10.2017: New 2D-online tool is available now, see here.
20.09.2017: This website is under construction, feedback is welcome.


Minkowski functionals have already been successfully applied to a lot of mathematical and physical problems. Exemplary applications as well as especially nice examples are provided in the examples section.


The following articles describe the Minkowski Tensor Algorithms implemented by karambola:

The concepts of Minkowski functionals have been used in the spatial analysis of physical systems, mathematical models, astronomy, cosmology, biology, …. see reference section for details.


We thank the interdisciplinary research unit “Geometry and Physics of Spatial Random Systems” set up by the German Research Foundation (DFG) in February 2011. This unit comprises mathematicians from the Karlsruhe Institute of Technology, working on stochastic and integral geometry and spatial stochastics, theoretical physicists from the University of Erlangen-Nuremberg and Murdoch University in Perth, working on statistical physics of spatially complex and disordered materials, and the stochastic geometry group from the University of Aarhus, working on spatial statistics and image analysis. The research unit aims at building bridges between mathematics and physics and at exploring the geometry and physics of random structures in space.

We also thank the Institute of Stochastics at the Karlsruhe Institute of Technology, the Institute of Theoretical Physics at the University of Erlangen-Nuremberg and Murdoch University in Perth.