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

Geometrical approach to mutually unbiased bases

We propose a unifying phase-space approach to the construction of mutually unbiased bases for a two-qubit system. It is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the Bloch-sphere principal axes. We suggest how to generalize the method to systems in dimensions that are powers of a prime.Comment: 10 pages. Some typos in the journal version have been correcte

Discrete phase space based on finite fields

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a 2N x 2N discrete phase space for a system with N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having N elements. There exists such a field if and only if N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our N x N phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.Comment: 60 pages; minor corrections and additional references in v2 and v3; improved historical introduction in v4; references to quantum error correction in v5; v6 corrects the value quoted for the number of similarity classes for N=

Automated detection of breast cancer using SAXS data and wavelet features

The overarching goal of this project was to improve breast cancer screening protocols first by collecting small angle x-ray scattering (SAXS) images from breast biopsy tissue, and second, by applying pattern recognition techniques as a semi-automatic screen. Wavelet based features were generated from the SAXS image data. The features were supplied to a classifier, which sorted the images into distinct groups, such as “normal” and “tumor”. The main problem in the project was to find a set of features that provided sufficient separation for classification into groups of “normal” and “tumor.” In the original SAXS patterns, information useful for classification was obscured. The wavelet maps allowed new scale-based information to be uncovered from each SAXS pattern. The new information was subsequently used to define features that allowed for classification. Several calculations were tested to extract useful features from the wavelet decomposition maps. The wavelet map average intensity feature was selected as the most promising feature. The wavelet map intensity feature was improved by using pre-processing to remove the high central intensities from the SAXS patterns, and by using different wavelet bases for the wavelet decomposition. The investigation undertaken for this project showed very promising results. A classification rate of 100% was achieved for distinguishing between normal samples and tumor samples. The system also showed promising results when tested on unrelated MRI data. In the future, the semi-automatic pattern recognition tool developed for this project could be automated. With a larger set of data for training and testing, the tool could be improved upon and used to assist radiologists in the detection and classification of breast lesions