A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.

Practical cone-beam algorithm

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Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density ? = ?(x, y, z) generates, and is to be reconstructed from, the number of counts n*(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate ? of ? which maximizes the probability p(n*|?) of observing the actual detector count data n* over all possible densities ?. Let independent Poisson variables n(b) with unknown means ?(b), b = 1, ···, B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ···, D with p(b, d) a one-step transition matrix, assumed known. We observe the total number n* = n*(d) of emissions in each detector unit d and want to estimate the unknown ? = ?(b), b = 1, ···, B. For each ?, the observed data n* has probability or likelihood p(n*|?). The EM algorithm of mathematical statistics starts with an initial estimate ?0 and gives the following simple iterative procedure for obtaining a new estimate ?new, from an old estimate ?old, to obtain ?k, k = 1, 2, ···, ?new(b)= ?old(b) ?Dd=1 n*(d)p(b,d)/??old(b?)p(b?,d),b=1,···B.

Maximum Likelihood Reconstruction for Emission Tomography

J+=j

Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm.

The Fourier reconstruction may be viewed simply in the spatial domain as the sum of each line integral times a weighting function of the distance from the line to the point of reconstruction A modified weighting function simultaneously achieves accuracy, simplicity, low computation time, as well as low sensitivity to noise Using a simulated phantom, the authors compare the Fourier algorithm and a search algorithm The search algorithm required 12 iterations to obtain a reconstruction of accuracy and resolution comparable to that of the Fourier reconstruction, and was more sensitive to noise To speed the search algorithm by using fewer interactions leaves decreased resolution in the region just inside the skull which could mask a subdural hematoma

The Fourier reconstruction of a head section

Several algorithms have been proposed for reconstructing the variable attenuation coefficients of tissues inside the skull from x-ray transmission measurements. We show that a simulated head section is accurately reconstructed by a modification, which is both fast and simple, of a previously studied algorithm. The possibility of using the algorithm to perform absolute measurements of the attenuation coefficient at points interior to the skull is high.

Reconstructing Interior Head Tissue from X-Ray Transmissions

Computerized tomographic imaging of the heart has great potential diagnostic value. The difficulty is that heart motion limits the data acquisition time to below 0.1 sec., but standard X-ray sources are not of sufficient intensity or mobility. A simple solution is proposed by using a rotating cathode filament with a fixed large stationary circular anode. In this design, X-ray detectors are placed on a concentric circle interior to the anode. To allow true planar X-ray projections, the detectors near the focal spot which would block the X-ray beam are moved synchronously out of the slice plane. An alternative method is also discussed where the detectors remain stationary and the nearer blocking detectors act as a filter for the X-ray beam.

L. A. Shepp

Papers

The Fourier reconstruction of a head section

Reconstructing Interior Head Tissue from X-Ray Transmissions

A New Design for High Speed Computerized Tomography