
DISCRETE TOMOGRAPHY
In the following part we outline the stateoftheart relevant to this project. In particular the focus is on the three main topics related to the reconstruction, uniqueness and stability questions, and on the different approaches for dealing with these problems.
The main problems of Discrete Tomography are the reconstruction and the uniqueness problems. First one consists in the retrieval of an unknown discrete set from the knowledge of its Xrays taken along a given set of directions, while the second problem consists in deciding when a lattice set is uniquely determined by the Xrays corresponding to a given set of directions. An overview of the topics treated in Discrete Tomography and of relevant results in the field are collected in [HK99] and [HK07]. The main application of Discrete Tomography is the reconstruction of threedimensional crystals from Xrays taken by an electron microscope [SK93,KS95]. The high energy electron beam in the microscope necessary to produce the discrete radiographies of a given crystal can damage the specimen if too many X rays are employed and for this reason only a few Xrays of the structure can be physically taken. This is the case, for instance in industrial nondestructive testing, in order to contain the costs, or in electron microscopy, because of the damage by radiation. Further examples include quality control in semiconductor industry, image processing, data compression and data security [IJ94,JB04,SG82]. In all these applications the conventional techniques of the Computerized Tomography cannot be applied. The research developed since '90 in Discrete Tomography follows different approaches. A first approach is based on image processing, medical imaging techniques and origins by Computerized Tomography (see for an example [MV99]).
A second approach is based on computerized and convex geometry, and optimization for providing the relevant mathematical methods for tomographic reconstruction of crystalline structures. Results in this subject concern the computational complexity of the reconstruction problem. If the data is measured in only two directions, then reconstruction can be done in polynomial time [R57](however, in general reconstruction will not be unique). In contrast, reconstruction from Xrays taken in three or more nonparallel directions is NPhard [GG96]. The same complexity bound holds for uniqueness. In spite of the hardness in the complexitytheoretic sense, applications demands for algorithms for solving the problem. For this reason, approximation algorithms have been investigated [GdV00] and in case of special structures exact algorithms which exploit particular properties of the objects to be reconstructed [BD03]. There are results in this direction also concerning uniqueness. A main result is that convex subsets of Z2 are uniquely determined by suitable sets of four directions [GG97,GG99]. The same directions distinguish socalled Qconvex sets [D05]. We also mention that the authors of [BD01] obtained several related negative results, based on combinatory and discrete mathematics, in the more general case when the discrete set to be reconstructed is assumed to be a so called horizontally and vertically convex polyomino. This latter approach is geometric since uses geometric properties such as convexity to determine uniqueness results. In the more general case when no such properties are considered, this approach cannot easily be extended to work. Connections between the notion of uniqueness and additivity are studied in [FL96,V]. A different approach is that introduced by Hadju and Tijdeman based on generating functions and divisibility properties of polynomials [HT01]. This algebraic approach has been turned out useful for studying both the reconstruction and the uniqueness problems. They provided an algorithm for solving tomographycal problems and extended their results to the case of emission tomography with absorption [HT03]. Moreover they studied uniqueness when discrete sets have the constraint to belong to a discrete grid of fixed size. In this case there exists a set of four directions depending only on the size of the grid that permits to distinguish any two subsets [H05]. In case of ambiguous reconstruction, different discrete sets yield the same Xrays. The difference of two density functions whose corresponding configurations have equal Xrays is a function with zero line sums along the given directions. This means that ambiguous configurations often appear, and, in general, these are addressed as switching configurations. In the case of row and column sums such switching configurations were already studied by H. Ryser under the name of interchanges, and later extended for more than two directions by several authors.
An algebraic theory of their structure, based on switching components of minimal size, socalled switching atoms, has been developed since 2001 by L. Hajdu and R. Tijdeman. Differently, the study of ambiguous configurations under the convexity constraint has been considered in several papers dealing with the geometric structure of so called Upolygons, both from continuous and from discrete point of view [D08,DP07]. In particular, when a lattice Upolygon exists, it is easy to construct two different convex lattice sets with equal discrete parallel Xrays in the directions in U. In 1997 R. Gardner and P. Gritzmann proved that in fact the nonexistence of a lattice Upolygon is necessary and sufficient for the discrete parallel Xrays in the directions in U to determine convex lattice sets (provided U has at least two nonparallel directions). The study of the difference of two discrete sets tomographically equivalent allows to measure in some sense how far from determining uniqueness is the considered set of directions. If data are affected by errors, bounds on the difference of any two discrete sets having bounded difference of their Xrays gives a measure of stability or instability in the reconstruction [AB07,D09A,D09B]. It is worth remarking that questions of stability are relevant for the applications, since noise in the data cannot be avoided. It can be shown that the reconstruction of lattice sets from Xrays taken along more than two directions is highly unstable [AG00,AG06]. This instability persists even when the Xrays uniquely determine the object. In [BD05] the authors proved that, under some extra assumptions, a stability result holds when the error on the data is “small”. In particular, they obtained an upper bound for the symmetric difference of two lattice sets depending on the distance of their Xrays and the maximal size of the sets.
References
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[AG06] A.Alpers and P.Gritzmann, On stability, error correction, and noise compensation in discrete tomography, SIAM J. Discrete Math. 20 (2006), pp.22739.
[AG01] A.Alpers, P.Gritzmann, and L.Thorens, Stability and instability in discrete tomography, in Digital and Image Geometry 2000 , Lecture Notes in Computer Science 2243, Springer. Berlin, 2001, pp.17586.
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