Anne Marie Svane

Local Digital Algorithms for Estimating the Integrated Mean Curvature of r-Regular Sets

Suppose an r-regular set is sampled on a random lattice. A fast algorithm for estimating the integrated mean curvature is to use a weighted sum of

Estimation of Intrinsic Volumes from Digital Grey-Scale Images

2\times \cdots \times 2

COBORDISM OBSTRUCTIONS TO INDEPENDENT VECTOR FIELDS

2×⋯×2 configuration counts. We show that for a randomly translated lattice, no asymptotically unbiased estimator of this type exists in dimensions larger than two, while for stationary isotropic lattices, asymptotically unbiased estimators are plenty. The basis for this is a formula for the asymptotic behavior of hit-or-miss transforms of r-regular sets.

A Koksma-Hlawka inequality for general discrepancy systems

Local digital algorithms based on n×⋯×n configuration counts are commonly used within science for estimating intrinsic volumes from binary images. This paper investigates multigrid convergence of such algorithms. It is shown that local algorithms for intrinsic volumes other than volume are not multigrid convergent on the class of convex polytopes. In fact, counter examples are plenty. On the other hand, for convex particles in 2D with a lower bound on the interior angles, a multigrid convergent local algorithm for the Euler characteristic is constructed. Also on the class of r-regular sets, counter examples to multigrid convergence are constructed for the surface area and the integrated mean curvature.

A geometric interpretation of the homotopy groups of the cobordism category

Local algorithms are common tools for estimating intrinsic volumes from black-and-white digital images. However, these algorithms are typically biased in the design based setting, even when the resolution tends to infinity. Moreover, images recorded in practice are most often blurred grey-scale images rather than black-and-white. In this paper, an extended definition of local algorithms, applying directly to grey-scale images without thresholding, is suggested. We investigate the asymptotics of these new algorithms when the resolution tends to infinity and apply this to construct estimators for surface area and integrated mean curvature that are asymptotically unbiased in certain natural settings.

On functions of bounded variation

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.

Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry

We define an invariant for the existence of r pointwise linearly independent sections in the tangent bundle of a closed manifold. For low values of r, explicit computations of the homotopy groups of certain Thom spectra combined with classical obstruction theory identifies this invariant as the top obstruction to the existence of the desired sections. In particular, this shows that the top obstruction is an invariant of the underlying manifold in these cases, which is not true in general. The invariant is related to cobordism theory and this gives rise to an identification of the invariant in terms of well-known invariants. As a corollary to the computations, we can also compute low-dimensional homotopy groups of the Thom spectra studied by Galatius, Tillmann, Madsen, and Weiss.

Rotational Crofton formulae for Minkowski tensors and some affine counterparts

Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D . As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D -variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.