The Fourier reconstruction of a head section
TL;DR: The authors compare the Fourier algorithm and a search algorithm using a simulated phantom 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.
Abstract: 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
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TL;DR: In this article, a convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections, which has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation.
Abstract: 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.
5,356 citations
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TL;DR: In this article, a convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections, which has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation.
Abstract: 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.
5,329 citations
TL;DR: In this paper, the authors proposed a more accurate general mathematical model for ET where an unknown emission density generates, and is to be reconstructed from, the number of counts n*(d) in each of D detector units d. Within the model, they gave 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?.
Abstract: 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.
4,288 citations
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2,428 citations
TL;DR: This implementation of the Algebraic Reconstruction Technique appears to have a computational advantage over the more traditional implementation of ART and potential applications include image reconstruction in conjunction with ray tracing for ultrasound and microwave tomography.
1,539 citations