Analysis and optimization of an algorithm for discrete tomography

Binary tomography is concerned with recovering binary images from a finite number of discretely sampled projections. Hajdu and Tijdeman outlined an algorithm for this type of problem. In this paper we analyze the algorithm and present several ways of improving the time complexity. We also give the results of experiments with an optimized version which is much faster than the original implementation, up to a factor of 50 or more (depending on the problem instance)

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

Algebraic filter approach for fast approximation of nonlinear tomographic reconstruction methods

We present a computational approach for fast approximation of nonlinear tomographic reconstruction methods by filtered backprojection (FBP) methods. Algebraic reconstruction algorithms are the methods of choice in a wide range of tomographic applications, yet they require significant computation time, restricting their usefulness. We build upon recent work on the approximation of linear algebraic reconstruction methods and extend the approach to the approximation of nonlinear reconstruction methods which are common in practice. We demonstrate that if a blueprint image is available that is sufficiently similar to the scanned object, our approach can compute reconstructions that approximate iterative nonlinear methods, yet have the same speed as FBP

Quality bounds for binary tomography with arbitrary projection matrices

Binary tomography deals with the problem of reconstructing a binary image from a set of its projections. The problem of finding binary solutions of underdetermined linear systems is, in general, very difficult and many such solutions may exist. In a previous paper we developed error bounds on differences between solutions of binary tomography problems restricted to projection models where the corresponding matrix has constant column sums. In this paper, we present a series of computable bounds that can be used with any projection model. In fact, th

Accurately approximating algebraic tomographic reconstruction by filtered backprojection

In computed tomography, algebraic reconstruction methods tend to produce reconstructions with higher quality than analytical methods when presented with limited and noisy projection data. The high computational requirements of algebraic methods, however, limit their usefulness in practice. In this paper, we propose a method to approximate the algebraic SIRT method by the computationally efficient filtered backprojection method. The method is based on an efficient way of computing a special angle-dependent convolution filter for filtered backprojection. Using this method, a reconstruction quality that is similar to SIRT can be achieved by existing efficient implementations of the filtered backprojection method. Results for a phantom image show that the method is indeed able to produce reconstructions with a quality similar to algebraic methods when presented with limited and noisy projection data