Rotational integral geometry of tensor valuations

We derive a new rotational Crofton formula for Minkowski tensors. In special cases, this formula gives (1) the rotational average of Minkowski tensors defined on linear subspaces and (2) the functional defined on linear subspaces with rotational average equal to a Minkowski tensor. Earlier results obtained for intrinsic volumes appear now as special cases.

[1]  Daniel Hug,et al.  Integral geometry of tensor valuations , 2008, Adv. Appl. Math..

[2]  S. Alesker.  Description of Continuous Isometry Covariant Valuations on Convex Sets , 1999 .

[3]  R. Ambartzumian.  Stochastic and integral geometry , 1987 .

[4]  C K Poh,et al.  Magnetic scattering effects in two-band superconductor: the ferromagnetic dopants in MgB2 , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[5]  Claus Beisbart,et al.  Vector- und Tensor-Valued Descriptors for Spatial Patterns , 2002 .

[6]  E. B. Vedel Jensen,et al.  Closed form of the rotational Crofton formula , 2012 .

[7]  R. Schneider,et al.  The space of isometry covariant tensor valuations , 2007 .

[8]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[9]  K. Mecke,et al.  Tensorial density functional theory for non-spherical hard-body fluids , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[10]  Richard J. Prokop,et al.  A survey of moment-based techniques for unoccluded object representation and recognition , 1992, CVGIP Graph. Model. Image Process..

[11]  Eva B. Vedel Jensen,et al.  A rotational integral formula for intrinsic volumes , 2008, Adv. Appl. Math..

[12]  D. Hug,et al.  Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures , 2011, Advanced materials.

[13]  Continuous rotation invariant valuations on convex sets , 1999, math/9905204.

[14]  K. Mecke,et al.  Tensorial Minkowski functionals and anisotropy measures for planar patterns , 2010, Journal of microscopy.

[15]  H. Wagner,et al.  Extended morphometric analysis of neuronal cells with Minkowski valuations , 2005, cond-mat/0507648.

[16]  Eva B. Vedel Jensen,et al.  Expressing intrinsic volumes as rotational integrals , 2010, Adv. Appl. Math..