Markus Kiderlen

On Infinitesimal Increase of Volumes of Morphological Transforms

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝ d , and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝ d . If the union B ∪ W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W , provided that A is regular enough ( e.g. , polyconvex). An analogous formula is obtained for the case when the conditions B ⊂ A and W ⊂ A C are replaced by prescribed threshold volumes of B in A and W in A C . Applications in stochastic geometry are discussed. First, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B . An analogous result holds for the hit-or-miss function. Second, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.

Estimating the Euler Characteristic of a planar set from a digital image

A new estimator (approximation) for the Euler-Poincare characteristic of a planar set K in the extended convex ring is suggested. As input, it uses only the digital image of K, which is modeled as the set of all points of a regular lattice falling in K. The key idea is to estimate the two planar Betti numbers of K (number of connected components and number of holes) by approximating K and its complement by polygonal sets derived from the digitization. In contrast to earlier methods, only certain connected components of these approximations are counted. The estimator of the Euler characteristic is then defined as the difference of the estimators for the two Betti numbers. Under rather weak regularity assumptions on K, it is shown that all three estimators yield the correct result, whenever the resolution of the image is sufficiently high.

Algorithms to estimate the rose of directions of a spatial fibre system

The directional measure (which is up to normalization the rose of directions) is used to quantify anisotropy of stationary fibre processes in three‐dimensional space. There exist a large number of approaches to estimate this measure from the rose of intersections (which is the mean number of intersections of fibres with lower dimensional test sets). Three recently suggested nonparametric algorithms to solve this problem are reviewed and compared. They were obtained from solutions of a least squares problem, a more general convex optimization problem and a linear program, respectively. Application to two different carbon fibre architectures and to simulated data allow an empirical comparison of these approaches. In addition, estimators for the associated zonoid (or Steiner compact) are suggested. This set turns out to be an intuitive tool for visualization.

Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows approximation of these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.

Estimating the surface area of nonconvex particles from central planar sections.

In this paper, we present a new surface area estimator in local stereology. This new estimator is called the ‘Morse‐type surface area estimator’ and is obtained using a two‐stage sampling procedure. First a plane section through a fixed reference point of a three‐dimensional structure is taken. In this section plane, a modification of the area tangent count method is used. The Morse‐type estimator generalizes Cruz‐Orives pivotal estimator for convex objects to nonconvex objects. The advantages of the Morse‐type estimator over existing local surface area estimators are illustrated in a simulation study. The Morse‐type estimator is well suited for computer‐assisted confocal microscopy and we demonstrate its practicability in a biological application: the surface area estimation of the nuclei of giant‐cell glioblastoma from microscopy images. We also present an interactive software that allows the user to efficiently obtain the estimator.

PHASE RETRIEVAL FOR CHARACTERISTIC FUNCTIONS OF CONVEX BODIES AND RECONSTRUCTION FROM COVARIOGRAMS

We propose strongly consistent algorithms for reconstructing the characteristic function 1K of an unknown convex body K in R n from possibly noisy measurements of the modulus of its Fourier transform c 1K. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k n ) noisy covariogram measurements to construct a convex polytope Pk that approximates K or its reflection −K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three- dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between Pk and ±K tends to zero as k tends to infinity.

Efficient calculation of local dose distributions for response modeling in proton and heavier ion beams

We present an algorithm for fast and accurate computation of the local dose distribution in MeV beams of protons, carbon ions or other heavy charged particles. It uses compound Poisson modeling of track interaction and successive convolutions for fast computation. It can handle arbitrary complex mixed particle fields over a wide range of fluences. Since the local dose distribution is the essential part of several approaches to model detector efficiency and cellular response it has potential use in ion-beam dosimetry, radiotherapy, and radiobiology.

Introduction into integral geometry and stereology

This chapter is a self-contained introduction into integral geometry and its applications in stereology. The most important integral geometric tools for stereological applications are kinematic formulae and results of Blaschke–Petkantschin type. Therefore, Crofton’s formula and the principal kinematic formula for polyconvex sets are stated and shown using Hadwiger’s characterization of the intrinsic volumes. Then, the linear Blaschke–Petkantschin formula is proved together with certain variants for flats containing a given direction (vertical flats) or contained in an isotropic subspace. The proofs are exclusively based on invariance arguments and an axiomatic description of the intrinsic volumes. These tools are then applied in model-based stereology leading to unbiased estimators of specific intrinsic volumes of stationary random sets from observations in a compact window or a lower dimensional flat. Also, Miles-formulae for stationary and isotropic Boolean models with convex particles are derived. In design-based stereology, Crofton’s formula leads to unbiased estimators of intrinsic volumes from isotropic uniform random flats. To estimate the Euler characteristic, which cannot be estimated using Crofton’s formula, the disector design is presented. Finally we discuss design-unbiased estimation of intrinsic volumes from vertical and from isotropic sections.

Determination of the mean normal measure from isotropic means of flat sections

Let be the mean normal measure of a stationary random set Z in the extended convex ring in ℝ d . For k ∈ {1,…,d-1}, connections are shown between and the mean of . Here, the mean is understood to be with respect to the random isotropic k-dimensional linear subspace ξ k and the mean normal measure of the intersection is computed in ξ k . This mean to be well defined, a suitable spherical lifting must be applied to before averaging. A large class of liftings and their resulting means are discussed. In particular, a geometrically motivated lifting is presented, for which the mean of liftings of determines uniquely for any fixed k ∈ {2,…,d-1}.

Rotation Invariant Valuations

In this chapter, we focus on rotation invariant valuations. We give an overview of the results available in the literature, concerning characterization of such valuations. In particular, we discuss the characterization theorem, derived in Alesker (Ann Math 149:977–1005, 1999), for continuous rotation invariant polynomial valuations on \(\mathcal{K}^{n}\). Next, rotational Crofton formulae are presented. Using the new kinematic formulae for trace-free tensor valuations presented in Chap. 4, it is possible to extend the rotational Crofton formulae for tensor valuations, available in the literature. Principal rotational formulae for tensor valuations are also discussed. These formulae can be derived using locally defined tensor valuations, as introduced in Chap. 2 A number of open questions in rotational integral geometry are presented.