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.

A local method for estimating surface area and surface area measure of three-dimensional objects from discrete binary images is presented. A weight is assigned to each 2 × 2 × 2 configuration of voxels and the total surface area of an object is given by summation of the local area contributions. The method is based on an exact asymptotic result that holds for increasing resolution of the digitization. It states that the number of occurrences of a 2 × 2 × 2 configuration is asymptotically proportional to an integral of its “h-function” with respect to the surface area measure of the object. We find explicit representations for these h-functions. Analyzing them in detail, we determine weights that lead to an asymptotic worst case error for surface area estimation of less than 4%. We show that this worst case error is the best possible. Exploiting the local nature of the asymptotic result, we also establish two parametric estimators for the surface area measure. The latter allow to quantify anisotropy of the object under consideration. Simulation studies illustrate the validity of the estimation procedure also for finite, but sufficiently high resolution.

We introduce a new algorithm for reconstructing an unknown shape from a finite number of noisy measurements of its support function. The algorithm, based on a least squares procedure, is very easy to program in standard software such as Matlab, and it works for both 2D and 3D reconstructions (in fact, in principle, in any dimension). Reconstructions may be obtained without any pre- or post-processing steps and with no restriction on the sets of measurement directions except their number, a limitation dictated only by computing time. An algorithm due to Prince and Willsky was implemented earlier for 2D reconstructions, and we compare the performance of their algorithm and ours. But our algorithm is the first that works for 3D reconstructions with the freedom stated in the previous paragraph. Moreover, under mild conditions, theory guarantees that outputs of the new algorithm will converge to the input shape as the number of measurements increases. In addition we offer a linear program version of the new algorithm that is much faster and better, or at least comparable, in performance at low levels of noise and reasonably small numbers of measurements. Another modification of the algorithm, suitable for use in a "focus of attention" scheme, is also described.

A new estimator (approximation) for the Euler–Poincaré 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.