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WP1 - Valuation theory

The main focus is on tensor valuations which are tensor-valued additive functionals on the families of convex or more general sets. Special cases are the classical intrinsic volumes (volume, surface area, length, Euler number) which are Minkowski tensors of rank zero.

In stochastic geometry, the focus has in recent years turned to Minkowski tensors of rank one or higher which provide information about position, orientation and shape of spatial structures.

WP1 - Subprojects

WP1.1: Tensor valuations and integral geometry
WP1.2: Isoperimetric inequalities with tensor constraints
WP1.3: Uniqueness of measurement functions in integral formulae
WP1.4: Stereology of tensors

Research questions

In the second funding period of CSGB, we take advantage of very recent theoretical advances. These concern the algebraic structure (product, convolution, Fourier transform) of tensor valuations and the locally defined Minkowski tensors.

  • Further and stronger results for integral geometry of Minkowski tensors are expected, using the new algebraic results. One of the key ideas is to use trace-free tensors which are closely related to spherical harmonics. Indeed, the mathematical theory suggests that trace-free tensors are in several respects more natural than the ordinary tensors.
  • The new locally defined Minkowski tensors will be used in an attempt to establish a principal rotational formula for Minkowski tensors. Such a formula will give a class of measurement techniques in optical microscopy a common formulation and special cases of the formula are expected to provide new microscopy techniques.

In the second funding period, it is also planned to transfer local stereological estimators of Minkowski tensors to particle populations. Important statistical issues are here

  • the development of confidence regions for shape and orientation, based on finite samples of particles
  • statistical comparison of shape and orientation in two particle populations
  • asymptotics in an expanding window regime

Selected references

  • Hasselholt, S., Hahn, U., Jensen, E.B.V. & Nyengaard, J.R. (2019): Practical implementation of the planar and spatial rotator in a complex tissue – the brain. J. Microsc. 273, 26–35.
  • Kiderlen, M. & Jensen, E.B.V. (2017, eds.): Tensor Valuations and their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics 2177. Springer.
  • Kousholt, A., Ziegel, J.F., Kiderlen, M. & Jensen, E.B.V. (2017): Stereological estimation of mean particle volume in R^3 from vertical sections. In Tensor Valuations and their Applications in Stochastic Geometry and Imaging (eds. M. Kiderlen and E.B.V. Jensen), Lecture Notes in Mathematics 2177, Springer, pp. 423-434.
  • Larsen, N.Y., Ziegel, J.F., Nyengaard, J.R. & Jensen, E.B.V. (2019): Stereological estimation of particle shape from vertical sections. J. Microsc. 275, 183–194.
  • Svane, A.M. & Jensen, E.B.V. (2017): Rotational Crofton formulae for Minkowski tensors and some affine counterparts. Adv. Appl. Math. 91, 44-75.

Project leaders

Eva B.V. Jensen, AU-math
Markus Kiderlen, AU-math

Participating scientists

AU-math:
Rikke Krog Eriksen
Andrew du Plessis
Mads Stehr

AU-bio:
Johnny B. Andersen
Karl-Anton Dorph-Petersen
Nick Yin Larsen
Jens R. Nyengaard

Focus points

  • the focus has in recent years turned to Minkowski tensors of rank one or higher
  • these tensors are important descriptors of position, orientation and shape of random particles
  • further and stronger results for integral geometry of Minkowski tensors are expected

Applications

Minkowski tensors have been used extensively as shape descriptors in physical applications; for instance as anisotropy indices for galaxies and ice grains in arctic drill cores. They have also been used as stress and shape descriptors of cells in sheared random foams. The potential of Minkowski tensors in biological applications is currently under investigation.

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Revised 20.09.2019