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3D Reconstruction of 2D Crystals in Real Space
"... Abstract—A new algorithm for threedimensional reconstruction of twodimensional crystals from projections is presented, and its applicability to biological macromolecules imaged using transmission electron microscopy (TEM) is investigated. Its main departures from the traditional approach is that i ..."
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Abstract—A new algorithm for threedimensional reconstruction of twodimensional crystals from projections is presented, and its applicability to biological macromolecules imaged using transmission electron microscopy (TEM) is investigated. Its main departures from the traditional approach is that it works in real space, rather than in Fourier space, and it is iterative. This has the advantage of making it convenient to introduce additional constraints (such as the support of the function to be reconstructed, which may be known from alternative measurements) and has the potential of more accurately modeling the TEM image formation process. Phantom experiments indicate the superiority of the new approach even without the introduction of constraints in addition to the projection data. Index Terms—3D reconstruction, crystals, electron microscopy, image reconstruction, projections. I.
Iterative Methods for Image Reconstruction
, 2008
"... These annotated slides were prepared by Jeff Fessler for attendees of the ISBI tutorial on statistical image reconstruction methods. The purpose of the annotation is to provide supplemental details, and particularly to provide extensive literature references for further study. For a fascinating hist ..."
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These annotated slides were prepared by Jeff Fessler for attendees of the ISBI tutorial on statistical image reconstruction methods. The purpose of the annotation is to provide supplemental details, and particularly to provide extensive literature references for further study. For a fascinating history of tomography, see [1]. For broad coverage of image science, see [2]. For further references on image reconstruction, see review papers and chapters, e.g., [3–9].
On the Effectiveness of Projection Methods for Convex Feasibility Problems with Linear Inequality Constraints
"... The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many realworld applications. ..."
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The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many realworld applications. This is supported by experimental evidence provided in this paper on problems of various sizes (up to tens of thousands of unknowns satisfying up to hundreds of thousands of constraints) and by a discussion of the demonstrated efficacy of projection methods in numerous scientific publications and commercial patents (dealing with problems that can have over a billion unknowns and a similar number of constraints).
Image Reconstruction Techniques for PET
"... this report lies on the reconstruction of PET images. Therefore we start, in x2, with a statistical description of the PET measurement process. The algorithms used to reconstruct these images depend on the medical scanner and on the noise in the data. They are subdivided into three major groups. Ana ..."
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this report lies on the reconstruction of PET images. Therefore we start, in x2, with a statistical description of the PET measurement process. The algorithms used to reconstruct these images depend on the medical scanner and on the noise in the data. They are subdivided into three major groups. Analytical algorithms [2] are based on a continuous description of the image and the data. They formulate a continuous solution which is discretized before being implemented as a computer program. These algorithms assume that the measurement space has been uniformly sampled by the scanner and that the noise in the data can be neglected. Sometimes the available data do not satisfy these constraints, or sometimes the measurement space has been sampled too sparsely to obtain an adequate discretisation of the continuous solution. In these cases one needs to use iterative algorithms [3]. Iterative algorithms start from a discretized description of the image as a linear combination of a limited set of basis functions. They try to find the most appropriate weights according to the available data. Iterative algorithms are further subdivided into two groups, depending on whether or not the reconstruction is based on a statistical description of the measurement process. In x3 we are interested in the discretisation of images for the use of iterative reconstruction algorithms. We define constraints on the basis functions in the spatial and in the frequency domain. We find that for PET and CT spatially limited and for MRI frequency limited basis functions result in the most efficient implementations. For PET and CT we also find that the basis functions should decay as fast as possible in the frequency domain, and that for MRI the basis functions should decay as fast as possible in the spati...
Lower Bounds on the Mean Square Error in Emission Tomography for Different Image Approximations
, 1996
"... A lower bound on the mean square error (MSE) of image reconstruction in emission tomography is obtained and used in the evaluation of different image approximations. One component of the lower bound on the MSE is the Cram'erRao (CR) bound on the error variance. The CR bound for different image appr ..."
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A lower bound on the mean square error (MSE) of image reconstruction in emission tomography is obtained and used in the evaluation of different image approximations. One component of the lower bound on the MSE is the Cram'erRao (CR) bound on the error variance. The CR bound for different image approximations is derived by knowing how the CR bound is modified when one is interested in some functions of the model parameters. The other component of the lower bound on the MSE is the square of the bias. The study of the lower bound on the MSE for different image approximations is the study of the tradeoff between bias and error variance. It is shown that the relative contribution of the two components changes for different image approximations, total number of counts and emission densities. I. Introduction Accuracy and precision are important quality criteria in many engineering problems. They are also vital in emission tomography, where image quality is governed in part by the statistic...
ART for helical conebeam CT reconstruction
, 2001
"... We report on our first results on the use of Algebraic Reconstruction Techniques (ART) on helical conebeam Computerized Tomography (CT) data. Two variants of ART have been implemented: a standard one which considers a single ray in an iterative step and a block version which groups several conebeam ..."
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We report on our first results on the use of Algebraic Reconstruction Techniques (ART) on helical conebeam Computerized Tomography (CT) data. Two variants of ART have been implemented: a standard one which considers a single ray in an iterative step and a block version which groups several conebeam projections in calculating an iterative update. Both seem to produce highquality reconstructions, although the number of cycles through the data to achieve those (between 15 and 20), while not huge, is larger than the number of cycles through the data needed for reconstructing volumes from data acquired from different modalities (1 iteration for PET data and 1 to 4 iterations for EM data). The reason for that maybe due to the uneven coverage of points by the data collection geometry, resulting in a slower rate of convergence. I.
Parallel ART for image reconstruction in CT using processor arrays
, 2004
"... Algebraic Reconstruction Technique (ART) is a widelyused iterative method for solving sparse systems of linear equations. This method (originally due to Kaczmarz) is inherently sequential according to its mathematical definition since, at each step, the current iterate is projected toward one of th ..."
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Algebraic Reconstruction Technique (ART) is a widelyused iterative method for solving sparse systems of linear equations. This method (originally due to Kaczmarz) is inherently sequential according to its mathematical definition since, at each step, the current iterate is projected toward one of the hyperplanes defined by the equations. The main advantages of ART are its robustness, its cyclic convergence on inconsistent systems, and its relatively good initial convergence. ART is widely used as an iterative solution to the problem of image reconstruction from projections in computerized tomography (CT), where its implementation with a small relaxation parameter produces excellent results. It is shown that for this particular problem, ART can be implemented in parallel on a linear processor array. pffiffiffi Reconstructing an image of n pixels from Q(n) equations can be done on a linear array of p Oð nÞ processors with optimal efficiency (linear speedup) and O(n/p) memory for each processor. The parallel technique can be applied to various geometric models of image reconstruction, as well as to 3D reconstruction with spherically symmetric volume elements, using a 2D rectangular meshconnected array of processors.
VolumetoImage Registration
"... et de nationalités serbe acceptée sur proposition du jury: Prof. M. Unser, directeur de thèse Dr M. Bierlaire, rapporteur ..."
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et de nationalités serbe acceptée sur proposition du jury: Prof. M. Unser, directeur de thèse Dr M. Bierlaire, rapporteur
A new representation and projection model for tomography, based on separable Bsplines
, 2012
"... Abstract—Data modelization in tomography is a key point for iterative reconstruction. The design of the projector, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a discrete basis of functions. Standard projector models are ..."
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Abstract—Data modelization in tomography is a key point for iterative reconstruction. The design of the projector, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a discrete basis of functions. Standard projector models are voxel or ray driven; more advanced models such as distance driven, use simple staircase voxels, giving rise to modelization errors due to their anisotropic behaviour. Moreover approximations made at the projection step amplify these errors. Though a more accurate projection could reduce approximation errors, characteristic functions of staircase voxels constitute a too coarse basis for representing a continuous function. As a result, pure modelization errors still hold. Spherically symmetric volume elements (blobs) have already been studied to eradicate such errors, but at the cost of increased