Reconstruction of convex bodies from X-rays: Old and new, including an application to nanowire reconstruction

The main purpose of the talk is to discuss an implementation and application of an algorithm developed by the speaker and Markus Kiderlen for reconstructing planar convex bodies from a finite set of their parallel X-rays (called projections in the non-mathematical literature).

From earlier theoretical work, it is known that there are certain sets of four directions in the plane, for example those specified by the four vectors $(0,1)$, $(1,0)$, $(2,1)$, and $(-1,2)$, such that the exact X-rays of a planar convex body in these directions determine it uniquely among all planar convex bodies. The algorithm takes $k$ values of each X-ray in such a set of four directions and returns a convex polygon which converges to the convex body as $k$ tends to infinity, even when the measurements are affected by Gaussian noise of fixed variance.

The talk will describe the necessary background and mention related open problems. An application to nanowire reconstruction will be described, and the implementation of the algorithm demonstrated and compared to other algorithms. This part of the talk is based on joint work with A. Alpers, K.J. Batenburg, C.B. Boothroyd, R.E. Dunin-Borkowski, L. Houben, S. König, and R.S. Pennington.

Some asymptotic and approximation properties of Poisson driven random tessellations
Based on joint work with Julia Hörrmann and Rolf Schneider

For Poisson driven stationary random tessellations, the asymptotic shape of cells having large size has been explored in a succession of contributions devoted to Kendall's problem. In the case of hyperplane tessellations, for instance, it turned out that often the asymptotic or limit shape is determined by the direction distribution of the underlying hyperplane process, via its Blaschke body. Recently, also asymptotic results for the volume of the typical or the zero cell of a certain class of random tessellations, as the dimension goes to infinity,  have been obtained. We recall some of these findings and then continue by introducing, for a given convex body $K$, the $K$-cell $Z_K$ associated with a stationary Poisson hyperplane process $X$. This is the intersection of all closed halfspaces containing $K$ and bounded by hyperplanes of $X$. We pursue the question how well $K$ can be approximated by $Z_K$, with respect to the Hausdorff metric, if the intensity increases, and how the quality/speed of approximation depends on the direction distribution of $X$.

Wicksell's corpuscle problem in local stereology
joint work with Ólöf Thórisdóttir

Wicksell's classical corpuscle problem is one of the earliest examples of a particle reconstruction problem from lower dimensional sections. It deals with the retrieval of the size distribution of spherical particles from sections with randomly translated planes.
We discuss the problem in the framework of local stereology: each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters, unfolding of the arising integral equations and uniqueness issues are discussed.
We illustrate the approach by reconstructing from simulated data using a numerical unfolding algorithm.

Mean Section Bodies and Surface Area Measures
joint work with Paul Goodey

For $1\le k\le d-1$, the $k$-th mean section body $M_k(K)$ of a convex body $K\subset {\mathbb R}^d$ is the Minkowski sum of all its sections by $k$-dimensional affine flats. In previous work (1992, 1998, 2012), we showed that $M_1(K)$ is a ball, whereas $M_2(K)$, for $k\ge 2$, uniquely determines $K$. Here, we express the (centered) support function of $M_k(K)$ (for $k\ge 2$) in terms of certain Fourier transforms of the $(d+1-k)$-th surface area measure $S_{d+1-k}(K,\cdot)$ of $K$. From this we obtain an inversion formula, but we also deduce a stability version of the uniqueness result. We further give explicit expressions for $h(M_k(K),\cdot)$, using the functions introduced by Berg (1969) in his
solution of the Christoffel problem. This result yields an interesting connection between mean section bodies and the general Minkowski problem of characterizing surface area measures.