A motivic local Cauchy-Crofton formula

In this note, we establish a version of the local Cauchy-Crofton formula for definable sets in Henselian discretely valued fields of characteristic zero. It allows to compute the motivic local density of a set from the densities of its projections integrated over the Grassmannian.

Absorption probabilities for Gaussian polytopes and regular spherical simplices

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

On the tangent model for the density of lines and a Monte Carlo method for computing hypersurface area

Methods to estimate surface areas of geometric objects in 3D are well known. A number of these methods are of Monte Carlo type, and some are based on the Cauchy–Crofton formula from integral geometry. Employing this formula requires the generation of sets of random lines that are uniformly distributed in 3D. One model to generate sets of random lines that are uniformly distributed in 3D is called the tangent model (see [

IMPROVED ESTIMATION OF FIBER LENGTH FROM 3-DIMENSIONAL IMAGES

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.

Efficient N-Dimensional surface estimation using Crofton formula and run-length encoding

Unlike the measure of the area in 2D or of the volume in 3D, the perimeter and the surface are not easily measurable in a discretized image. In this article we describe a method based on the Crofton formula to measure those two parameters in a discritized image. The accuracy of the method is discussed and tested on several known objects. An algorithm based on the run-length encoding of binary objects is presented and compared to other approaches. An implementation is provided and integrated in the LabelObject/LabelMap framework contributed earlier by the authors.