Gabor T. Herman

Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography.

Abstract We give a new method for direct reconstruction of three-dimensional objects from a few electron micrographs taken at angles which need not exceed a range of 60 degrees. The method works for totally asymmetric objects, and requires little computer time or storage. It is also applicable to X-ray photography, and may greatly reduce the exposure compared to current methods of body-section radiography.

Three-dimensional display of human organs from computed tomograms

Methods of hidden surface removal and shading for computer-displayed surfaces are proposed. If the surface to be displayed is approximated by a large number of square faces of restricted orientation, the methods proposed in this paper work at least an order of magnitude faster than previously published methods. A situation in which such approximation is natural is the display of organs based on data obtained by the reconstruction of the internal structure of a body from multiple shadowgraphs such as X-ray photographs (computed tomography). The methods can also be used for rapidly achieving some but not all of the aims of surface modeling in more general situations. The hidden surface removal method is based on the Z-buffer algorithm (no sorting is required), with modifications which make use of the nature of the assumed surface approximation. The artifact caused by approximating the curved surface with squares is reduced by low-pass filtering of the picture to be displayed. The results are demonstrated by displays of actual human organs based on data provided by computed tomography.

Algebraic reconstruction techniques can be made computationally efficient (positron emission tomography application)

Algebraic reconstruction techniques (ART) are iterative procedures for recovering objects from their projections. It is claimed that by a careful adjustment of the order in which the collected data are accessed during the reconstruction procedure and of the so-called relaxation parameters that are to be chosen in an algebraic reconstruction technique, ART can produce high-quality reconstructions with excellent computational efficiency. This is demonstrated by an example based on a particular (but realistic) medical imaging task, showing that ART can match the performance of the standard expectation-maximization approach for maximizing likelihood (from the point of view of that particular medical task), but at an order of magnitude less computational cost.

Iterative reconstruction algorithms

Abstract In this paper a survey of recent results on iterative reconstruction algorithms is given. These results, many of which have not yet appeared elsewhere, are applicable to a very general formulation of the reconstruction problem based on the series expansion approach. A set of optimization criteria and a number of iterative reconstruction algorithms are stated, together with theorems on the convergence of the algorithms to optimum images. The efficacy of the algorithms is compared to that of the convolution method. In particular, the falseness of the claim that ART and the backprojection method are the same is demonstrated.

Fundamentals of Computerized Tomography

This revised and updated text presents the computational and mathematical procedures underlying data collection, image reconstruction, and image display in computerized tomography. New topics: the fast calculation of a ray sum for a digitized picture, the task-oriented comparison of reconstruction algorithm performance, blob basis functions and the linogram method for image reconstruction. Features: Describes how projection data are obtained and the resulting reconstructions are used; Presents a comparative evaluation of reconstruction methods; Investigates reconstruction algorithms; Explores basis functions, functions to be optimized, norms, generalized inverses, least squares solutions, maximum entropy solutions, and most likely estimates; Discusses SNARK09, a large programming system for image reconstruction; Concludes each chapter with helpful Notes and References sections. An excellent guide for practitioners, it can also serve as a textbook for an introductory graduate course.

Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization

Although three-dimensional electron microscopy (3D-EM) permits structural characterization of macromolecular assemblies in distinct functional states, the inability to classify projections from structurally heterogeneous samples has severely limited its application. We present a maximum likelihood–based classification method that does not depend on prior knowledge about the structural variability, and demonstrate its effectiveness for two macromolecular assemblies with different types of conformational variability: the Escherichia coli ribosome and Simian virus 40 (SV40) large T-antigen.

The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm

In many three-dimensional imaging applications the three-dimensional scene is represented by a three-dimensional array of volume elements, or voxels for short. A subset Q of the voxels is specified by some property. The objects in the scene are then defined as subsets of Q formed by voxels which are “connected” in some appropriate sense. It is often of interest to detect and display the surface of an object in the scene, specified, say, by one of the voxels in it. In this paper, the problem of surface detection is translated into a problem of traversal of a directed graph, G. The nodes of G correspond to faces separating voxels in Q from voxels not in Q. It is proven that connected subgraphs of G correspond to surfaces of connected components of Q (i.e., of objects in the scene). Further properties of the directed graph are established which allow us to keep the number of marked nodes (needed to avoid loops in the graph traversal) to a small fraction of the total number of visited nodes. This boundary detection algorithm has been implemented. We discuss the interaction between the underlying mathematical theory and the design of the working software. We illustrate the software by some clinical studies in which the input is computed tomographic (CT) data and the output is dynamically rotating three-dimensional displays of isolated organs. Even though the medical application leads to very large-scale problems, our theory and design allows us to use our method routinely on the minicomputer of a CT scanner.

Three-Dimensional Reconstruction from Projections: A Review of Algorithms

Publisher Summary This chapter provides an overview of three-dimensional reconstruction from projections. The chapter reviews the algorithms that have been proposed to solve the reconstruction problem. The known reconstruction algorithms are classified into four categories—summation, the use of Fourier transform, analytic solution of the integral equations, and series expansion approaches. For each class of algorithms several points need to be considered—a general intuitive description, a precise mathematical description of a typical reconstruction method of the class, and a brief description of other methods in the class. All algorithms for reconstruction take as input the projection data, and all produce as output an estimate of the original structure based on the available data. The estimate varies from method to method. The relative performance of the various methods depends on the object and how the data are collected. The simplest algorithm for reconstruction is to estimate the density at a point by adding all the ray sums of the rays through that point. The Fourier method depends on transforming the projections into the Fourier space, where they define part of the Fourier transform of the whole object. Each projection may be shown to yield values on a central section of the Fourier space, which is a line or plane through the origin at an angle corresponding to the direction of the projection in real space.

Shape-based interpolation

Extensions to a shape-based interpolation method in which pixels that share a boundary edge (one inside and the other outside the object) are considered to be at a distance between adjacent pixel centers are proposed. Using such an initialization for distance calculations, a generalization of the chamfer distance calculation is developed. The generalization allows the simultaneous calculation of distances within the object and its background by two consecutive chamfering processes. The performances of a number of variants of the methods are evaluated. It is shown that the shape-based interpolation using a near-optimal 3*3 distance and modified cubic spline between-slice interpolation has superior properties to previously proposed methods for estimating object locations in missing slices in tomographic radiology.<<ETX>>

Correction for beam hardening in computed tomography

We investigate how one can estimate from the total attenuation, p, of a polyenergetic X-ray beam what the total attenuation, m, of a monoenergetic beam would have been along the same ray. We find that for beams with typical diagnostic X-ray spectra passing through the human body one can find a simple function f such that f(p) is a sufficiently close estimate of m to allow good reconstructions. We also find that m cannot be accurately estimated from p based on the assumption that the human body consists of water alone. Our results are demonstrated by reconstructions of a mathematical model of a cross-section of the human thorax. This article is self-contained and includes in its Appendices a detailed discussion of the mathematical nature of the problem of bean hardening in computed tomography.