Characterizing the grain structure of polycrystalline material is an important task in material science. The present paper introduces the concept of generalized balanced power diagrams as a concise alternative to voxelated mappings. Here, each grain is represented by (measured approximations of) its center-of-mass position, its volume and, if available, by its second-order moments (in the non-equiaxed case). Such parameters may be obtained from 3D x-ray diffraction. As the exact global optimum of our model results from the solution of a suitable linear program it can be computed quite efficiently. Based on verified real-world measurements we show that from the few parameters per grain (3, respectively 6 in 2D and 4, respectively 10 in 3D) we obtain excellent representations of both equiaxed and non-equiaxed structures. Hence our approach seems to capture the physical principles governing the forming of such polycrystals in the underlying process quite well.

A discrete tomography algorithm is presented for the reconstruction of grain maps based on X-ray diffraction data. This is the first algorithm for this task, inherently exploiting the discrete structure of grain maps. Gibbs potentials serve to characterize the statistics of the local morphology of the grain boundaries. A Monte Carlo based algorithm is applied as a restoration method for improving the quality of grain maps produced by a classical (non-discrete) tomography algorithm (ART). The quality of the restored maps is demonstrated and quantified by simulation studies. The robustness of the algorithm with respect to the choice of Gibbs potentials is investigated.

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A non-thermal gliding arc discharge was generated at atmospheric pressure in an air flow. The dynamics of the plasma column and tracer particles were recorded using two synchronized high-speed cameras. Whereas the data analysis for such systems has previously been performed in 2D (analyzing the single camera image), we provide here a 3D data analysis that includes 3D reconstructions of the plasma column and 3D particle tracking velocimetry based on discrete tomography methods. The 3D analysis, in particular, the determination of the 3D slip velocity between the plasma column and the gas flow, gives more realistic insight into the convection cooling process. Additionally, with the determination of the 3D slip velocity and the 3D length of the plasma column, we give more accurate estimates for the drag force, the electric field strength, the power per unit length, and the radius of the conducting zone of the plasma column.

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a b s t r a c tElectron tomography is becoming an increasingly important tool in materials science for studying the three-dimensional morphologies and chemical compositions of nanostructures. The image quality obtained by many current algorithms is seriously affected by the problems of missing wedge artefacts and non-linear projection intensities due to diffraction effects. The former refers to the fact that data cannot be acquired over the full 1801 tilt range; the latter implies that for some orientations, crystalline structures can show strong contrast changes. To overcome these problems we introduce and discuss several algorithms from the mathematical fields of geometric and discrete tomography. The algorithms incorporate geometric prior knowledge (mainly convexity and homogeneity), which also in principle considerably reduces the number of tilt angles required. Results are discussed for the reconstruction of an InAs nanowire.

The task of reconstructing binary images from the knowledge of their line sums (discrete X-rays) in a given finite number m of directions is ill-posed. Even some small noise in the physical measurements can lead to dramatically different yet still unique solutions.The present paper addresses in particular the following problems. Does discrete tomography have the power of error correction? Can noise be compensated by taking more X-ray images, and, if so, what is the quantitative effect of taking one more X-ray? Our main theorem gives the first nontrivial unconditioned (and best possible) stability result. In particular, we show that the Hamming distance between any two different sets of m X-ray images of the same cardinality is at least 2(m − 1), and this is best possible. As a consequence, this result implies a Rényi-type theorem for denoising and shows that the noise compensating effect of X-rays is linear in their number.Our theoretical results are complemented by determining the computational complexity of some underlying algorithmic tasks. In particular, we show that while there always is a certain inherent stability, the possibility of making (worst-case) efficient use of it is rather limited. Introduction.Discrete tomography deals with the reconstruction of finite sets from knowledge about their interaction with certain query sets. The most prominent example is that of the reconstruction of a finite subset F of Z d from its X-rays (i.e., line sums) in a small positive integer number m of directions. Applications of discrete tomography include quality control in semiconductor industry, image processing, graph theory, scheduling, statistical data security, game theory, etc. (see, e.g., [6], [8], [9], [13], [14], [17], [19]). The reconstruction task is an ill-posed discrete inverse problem, depicting (suitable variants of) all three Hadamard criteria [12]for ill-posedness. In fact, for general data there need not exist a solution, if the data is consistent, the solutions need not be uniquely determined, and even in the case of uniqueness, the solution may change dramatically with small changes of the data.The papers [1] and [2] show just how unstable the reconstruction task really is: For arbitrarily large lattice sets even of the same cardinality, a total error of only 2(m − 1) in the measurements can lead to unique but disjoint solutions. Clearly, this is an important issue for all practical applications where noise in the data cannot be avoided, particularly if the data stems from physical measurements.The main theorem of the present paper shows that this number 2(m − 1) is best possible in an ultimate sense. In Theorem 2.1 we prove that two finite sets of the same cardinality whose X-rays in a given set of m directions differ by a total of less than 2(m − 1) are "tomographically equivalent." This means that either the X-rays differ by at least 2(m − 1), or they do not differ at all. Note that the situation becomes trivial if the assumption on the equal cardinality of the lattice sets is omitted. Indeed, if...

Super-resolution imaging aims at improving the resolution of an image by enhancing it with other images or data that might have been acquired using different imaging techniques or modalities. In this paper we consider the task of doubling, in each dimension, the resolution of grayscale images of binary objects by fusion with double-resolution tomographic data that have been acquired from two viewing angles. We show that this task is polynomial-time solvable if the gray levels have been reliably determined. The problem becomes NP-hard if the gray levels of some pixels come with an error of ±1 or larger. The NP-hardness persists for any larger resolution enhancement factor. This means that noise does not only affect the quality of a reconstructed image but, less expectedly, also the algorithmic tractability of the inverse problem itself.