Approximate X-rays reconstruction of special lattice sets

Sometimes the inaccuracy of the measurements of the X-rays can give rise to an inconsistent reconstruction problem. In this paper we address the problem of reconstructing special lattice sets in Z2 from their approximate X-rays in a finite number of prescribed lattice directions. The class of "strongly Q-convex sets" is taken into consideration and a polynomial time algorithm for reconstructing members of that class with line sums having possibly some bounded differences with the given X-ray values is provided. In particular, when these differences are zero, the algorithm exactly reconstructs any set. As a result, this algorithm can also be used to reconstruct convex subsets of Z2 from their exact X-rays in a finite set of suitable prescribed lattice directions

Reconstructing binary images from discrete X-rays

We present a new algorithm for reconstructing binary images from their projections along a small number of directions. Our algorithm performs a sequence of related reconstructions, each using only two projections. The algorithm makes extensive use of network flow algorithms for solving the two-projection subproblems. Our experimental results demonstrate that the algorithm can compute reconstructions which resemble the original images very closely from a small number of projections, even in the presence of noise. Although the effectiveness of the algorithm is based on certain smoothness assumptions about the image, even tiny, non-smooth details are reconstructed exactly. The class of images for which the algorithm is most effective includes images of convex objects, but images of objects that contain holes or consist of multiple components can also be reconstructed with great accurac

Network Flow Algorithms for Discrete Tomography

Tomography is a powerful technique to obtain images of the interior of an object in a nondestructive way. First, a series of projection images (e.g., X-ray images) is acquired and subsequently a reconstruction of the interior is computed from the available project data. The algorithms that are used to compute such reconstructions are known as tomographic reconstruction algorithms. Discrete tomography is concerned with the tomographic reconstruction of images that are known to contain only a few different gray levels. By using this knowledge in the reconstruction algorithm it is often possible to reduce the number of projections required to compute an accurate reconstruction, compared to algorithms that do not use prior knowledge. This thesis deals with new reconstruction algorithms for discrete tomography. In particular, the first five chapters are about reconstruction algorithms based on network flow methods. These algorithms make use of an elegant correspondence between certain types of tomography problems and network flow problems from the field of Operations Research. Chapter 6 deals with a problem that occurs in the application of discrete tomography to the reconstruction of nanocrystals from projections obtained by electron microscopy.The research for this thesis has been financially supported by the Netherlands Organisation for Scientific Research (NWO), project 613.000.112.UBL - phd migration 201

Generic iterative subset algorithms for discrete tomography

AbstractDiscrete tomography deals with the reconstruction of images from their projections where the images are assumed to contain only a small number of grey values. In particular, there is a strong focus on the reconstruction of binary images (binary tomography). A variety of binary tomography problems have been considered in the literature, each using different projection models or additional constraints. In this paper, we propose a generic iterative reconstruction algorithm that can be used for many different binary reconstruction problems. In every iteration, a subproblem is solved based on at most two of the available projections. Each of the subproblems can be solved efficiently using network flow methods. We report experimental results for various reconstruction problems. Our results demonstrate that the algorithm is capable of reconstructing complex objects from a small number of projections

SPEDEN: Reconstructing single particles from their diffraction patterns

Speden is a computer program that reconstructs the electron density of single particles from their x-ray diffraction patterns, using a single-particle adaptation of the Holographic Method in crystallography. (Szoke, A., Szoke, H., and Somoza, J.R., 1997. Acta Cryst. A53, 291-313.) The method, like its parent, is unique that it does not rely on ``back'' transformation from the diffraction pattern into real space and on interpolation within measured data. It is designed to deal successfully with sparse, irregular, incomplete and noisy data. It is also designed to use prior information for ensuring sensible results and for reliable convergence. This article describes the theoretical basis for the reconstruction algorithm, its implementation and quantitative results of tests on synthetic and experimentally obtained data. The program could be used for determining the structure of radiation tolerant samples and, eventually, of large biological molecular structures without the need for crystallization.Comment: 12 pages, 10 figure

Volume MLS Ray Casting

The method of Moving Least Squares (MLS) is a popular framework for reconstructing continuous functions from scattered data due to its rich mathematical properties and well-understood theoretical foundations. This paper applies MLS to volume rendering, providing a unified mathematical framework for ray casting of scalar data stored over regular as well as irregular grids. We use the MLS reconstruction to render smooth isosurfaces and to compute accurate derivatives for high-quality shading effects. We also present a novel, adaptive preintegration scheme to improve the efficiency of the ray casting algorithm by reducing the overall number of function evaluations, and an efficient implementation of our framework exploiting modern graphics hardware. The resulting system enables high-quality volume integration and shaded isosurface rendering for regular and irregular volume data.Engineering and Applied Science

Muon Track Reconstruction and Data Selection Techniques in AMANDA

The Antarctic Muon And Neutrino Detector Array (AMANDA) is a high-energy neutrino telescope operating at the geographic South Pole. It is a lattice of photo-multiplier tubes buried deep in the polar ice between 1500m and 2000m. The primary goal of this detector is to discover astrophysical sources of high energy neutrinos. A high-energy muon neutrino coming through the earth from the Northern Hemisphere can be identified by the secondary muon moving upward through the detector. The muon tracks are reconstructed with a maximum likelihood method. It models the arrival times and amplitudes of Cherenkov photons registered by the photo-multipliers. This paper describes the different methods of reconstruction, which have been successfully implemented within AMANDA. Strategies for optimizing the reconstruction performance and rejecting background are presented. For a typical analysis procedure the direction of tracks are reconstructed with about 2 degree accuracy.Comment: 40 pages, 16 Postscript figures, uses elsart.st

On the Reconstruction of Static and Dynamic Discrete Structures

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in $\mathbb{R}^d$). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems