1,784 research outputs found
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
Euclidean Distance Matrices: Essential Theory, Algorithms and Applications
Euclidean distance matrices (EDM) are matrices of squared distances between
points. The definition is deceivingly simple: thanks to their many useful
properties they have found applications in psychometrics, crystallography,
machine learning, wireless sensor networks, acoustics, and more. Despite the
usefulness of EDMs, they seem to be insufficiently known in the signal
processing community. Our goal is to rectify this mishap in a concise tutorial.
We review the fundamental properties of EDMs, such as rank or
(non)definiteness. We show how various EDM properties can be used to design
algorithms for completing and denoising distance data. Along the way, we
demonstrate applications to microphone position calibration, ultrasound
tomography, room reconstruction from echoes and phase retrieval. By spelling
out the essential algorithms, we hope to fast-track the readers in applying
EDMs to their own problems. Matlab code for all the described algorithms, and
to generate the figures in the paper, is available online. Finally, we suggest
directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine
- change of title in the last revisio
Geometric reconstruction methods for electron tomography
Electron tomography is becoming an increasingly important tool in materials
science for studying the three-dimensional morphologies and chemical
compositions of nanostructures. The image quality obtained by many current
algorithms is seriously affected by the problems of missing wedge artefacts and
nonlinear projection intensities due to diffraction effects. The former refers
to the fact that data cannot be acquired over the full tilt range;
the latter implies that for some orientations, crystalline structures can show
strong contrast changes. To overcome these problems we introduce and discuss
several algorithms from the mathematical fields of geometric and discrete
tomography. The algorithms incorporate geometric prior knowledge (mainly
convexity and homogeneity), which also in principle considerably reduces the
number of tilt angles required. Results are discussed for the reconstruction of
an InAs nanowire
Coherent X-ray Diffractive Imaging; applications and limitations
The inversion of a diffraction pattern offers aberration-free
diffraction-limited 3D images without the resolution and depth-of-field
limitations of lens-based tomographic systems, the only limitation being
radiation damage. We review our experimental results, discuss the fundamental
limits of this technique and future plans.Comment: 7 pages, 8 figure
A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems
Classical iterative methods for tomographic reconstruction include the class
of Algebraic Reconstruction Techniques (ART). Convergence of these stationary
linear iterative methods is however notably slow. In this paper we propose the
use of Krylov solvers for tomographic linear inversion problems. These advanced
iterative methods feature fast convergence at the expense of a higher
computational cost per iteration, causing them to be generally uncompetitive
without the inclusion of a suitable preconditioner. Combining elements from
standard multigrid (MG) solvers and the theory of wavelets, a novel
wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to
significantly speed-up Krylov convergence. The performance of the
WMG-preconditioned Krylov method is analyzed through a spectral analysis, and
the approach is compared to existing methods like the classical Simultaneous
Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods
on a 2D tomographic benchmark problem. Numerical experiments are promising,
showing the method to be competitive with the classical Algebraic
Reconstruction Techniques in terms of convergence speed and overall performance
(CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13
figures, 3 table
Phase Retrieval with Application to Optical Imaging
This review article provides a contemporary overview of phase retrieval in
optical imaging, linking the relevant optical physics to the information
processing methods and algorithms. Its purpose is to describe the current state
of the art in this area, identify challenges, and suggest vision and areas
where signal processing methods can have a large impact on optical imaging and
on the world of imaging at large, with applications in a variety of fields
ranging from biology and chemistry to physics and engineering
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
- …