For continuous tomography Helgason and Ludwig developed consistency conditions. They were used by others to overcome defects in the measurements. In this paper we introduce a consistency criterion for discrete tomography. We indicate how the consistency criterion can be used to overcome defects in measurements.

In a recent work, we have proposed a novel way to approximate point sets with grids using the LLL algorithm, which operates in polynomial time. Now, we show how this approach can be applied to pattern recognition purposes with interpreting the rate of approximation as a new feature for regularity measurement. Our practical problem is the characterization of pigment networks in skin lesions. For this task we also introduce a novel image processing method for the extraction of the pigment network. Then, we show how our grid approximation framework can be applied with specializing it for the recognition of hexagonal patterns. The classification performance of our approach for the pigment network characterization problem is measured on a database annotated by a clinical expert. Throughout the paper we address several practical issues that may help to apply our general framework to other practical tasks, as well.

Discrete tomography deals with tomographic reconstruction of greyscale images for which the set of possible grey levels is discrete and small. Here, we develop a discrete approximate reconstruction algorithm. Our algorithm computes an image that has only grey values belonging to a given finite set. It also guarantees that the difference between the given projections and the projections of the reconstructed discrete image is bounded. The bound, which is computable, is independent of the image size. We present reconstruction experiments for a range of phantom images and a varying number of grey values.

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and $f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed. Here $A$ was a block in $\mathbb{Z}^n$ with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that $A$ is an arbitrary or convex finite set in $\mathbb{Z}^n$. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of the following type: if the line sums in $k$ directions of a function $h: A \to [0,1]$ are known, then there exists a function $f: A \to \{0,1\}$ such that its line sums differ by at most $k$ from the corresponding line sums of $h$.