We obtain generalizations of the main result in 10, and then provide geometric interpretations of linear combinations of the mean curvature integrals that appear in the Gauss–Bonnet formula for ...hypersurfaces in space forms Mλn. Then, we combine these results with classical Morse theory to obtain new rotational integral formulae for the k-th mean curvature integrals of a hypersurface in Mλn.
The structure model index (SMI) is a means of subsuming the topology of a homogeneous random closed set under just one number, similar to the isoperimetric shape factors used for compact sets. ...Originally, the SMI is defined as a function of volume fraction, specific surface area and first derivative of the specific surface area, where the derivative is defined and computed using a surface meshing. The generalised Steiner formula yields however a derivative of the specific surface area that is – up to a constant – the density of the integral of mean curvature. Consequently, an SMI can be defined without referring to a discretisation and it can be estimated from 3D image data without need to mesh the surface but using the number of occurrences of 2×2×2 pixel configurations, only. Obviously, it is impossible to completely describe a random closed set by one number. In this paper, Boolean models of balls and infinite straight cylinders serve as cautionary examples pointing out the limitations of the SMI. Nevertheless, shape factors like the SMI can be valuable tools for comparing similar structures. This is illustrated on real microstructures of ice, foams, and paper.
A new method is presented for estimating the specific fiber length from 3D images of macroscopically homogeneous fiber systems. The method is based on a discrete version of the Crofton formula, where ...local knowledge from 3x3x3-pixel configurations of the image data is exploited. It is shown that the relative error resulting from the discretization of the outer integral of the Crofton formula amonts at most 1.2%. An algorithmic implementation of the method is simple and the runtime as well as the amount of memory space are low. The estimation is significantly improved by considering 3x3x3-pixel configurations instead of 2x2x2, as already studied in literature.
This paper deals with the analysis of spatial images taken from microscopically heterogeneous but macroscopically homogeneous microstructures. A new method is presented, which is strictly based on ...integral‐geometric formulae such as Crofton's intersection formulae and Hadwiger's recursive definition of the Euler number. By means of this approach the quermassdensities can be expressed as the inner products of two vectors where the first vector carries the ‘integrated local knowledge’ about the microstructure and the second vector depends on the lateral resolution of the image as well as the quadrature rules used in the discretization of the integral‐geometric formulae. As an example of application we consider the analysis of spatial microtomographic images obtained from natural sandstones.
Under the assumptions that
E
λ
n
is an
n
-dimensional, simply connected Riemannian manifold of constant sectional curvature
λ
and
L
λ
r
is an
r
-dimensional, totally geodesic submanifold of
E
λ
n
, ...the paper investigates the
q
-th integral of the mean curvature
M
q
n
of a convex body
K
r
in
E
λ
n
and gives the expression of
M
q
n
in the terms of
M
p
r
, where
M
p
r
is the
p
-th integral of the mean curvature of
K
r
> in
L
λ
r
. A result of L. A. Santaló 2 holds in particular.
In this paper, we investigate some properties of a compact domain in
R
n
and develop a new lower bound for the integral of the (
n − 2)nd mean curvature over the boundary of a compact domain in
R
n
.
PDF
