Iterative reconstruction

Figure 3.3 The errors produced by an iterative reconstruction algorithm have patterns which appear, at each iteration, completely random (A), but if successive patterns are memorised together, it is possible to observe regular structures appearing in the memory matrix (B).

Dedicated PET systems are usually based on lull ring detector systems with an axial field of view exceeding 15 cm, and can be operated in septa extended (2D mode) or septa retracted mode (3D) for patient examinations. Some systems only provide 3D-data acquisition modes. Most systems allow the simultaneous acquisition of 36 transversal slices and more with a theoretical slice thickness of 2-5 mm. Transmission scans for a total of 10 min are obtained prior to the radionuclide application, for the attenuation correction of the acquired emission tomographic images. All PET images are attenuation corrected and iteratively reconstructed. 

Fig. 3 Transaxial slices reconstructed with filtered back projection (top row) and iterative reconstruction (bottom row). Negative reconstructed values have been set to zero for the filtered back projection reconstruction. The streak artifacts and count defects caused by focal activity deposition in the central airways are clearly evident on the reconstructions with filtered back projection. The iterative reconstruction provides artifact free images.

Thus iterative reconstruction is the clear method of choice for fast lung SPECT studies. ... 

Figure 4.8. A conceptual step in iterative reconstruction methods. (Reprinted with the permission of The Cleveland Clinic Center for Medical Art Photography 2009. All Rights Reserved).

Figure 4.10. Comparison of filtered backprojection (FBP) and iterative (OSEM) methods with attenuation correction, (a) Lungs, (b) Normal Liver, (c) Liver with tumor, (d) Breast. FBP images with attenuation corrections are noisier than OSEM images with attenuation correction. (Reprinted by permission of Society of Nuclear Medicine from Riddell C et al (2001) Noise reduction in oncology FDG PET images by iterative reconstructions a quantitative assessment. J Nucl Med 42 1316)...

Figure 6.17 TEM analysis of aggregates of PNOEG-PNGLF (1). (a) Conventional TEM using negative staining, (b) cryoTEM image of a vitrified film, (c) gallery of z slices showing different cross-sections of a 3D SIRT (simultaneous iterative reconstruction...

To search for the forms of potentials we are considering here simple mechanical models. Two of them, namely cluster support algorithm (CSA) and plane support algorithm (PSA), were described in details in [6]. Providing the experiments with simulated and experimental data, it was shown that the iteration procedure yields the sweeping of the structures which are not volumetric-like or surface-like, correspondingly. While the number of required projections for the reconstruction is reduced by 10 -100 times, the quality of reconstruction estimated quantitatively remained quite comparative (sometimes even with less artefacts) with that result obtained by classic Computer Tomography (CT). 

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

No a priori information about the unknown profile is used in this algorithm, and the initial profile to start the iterative process is chosen as (z) = 1. Moreover, the solution of the forward problem at each iteration can be obtained with the use of the scattering matrices concept [8] instead of a numerical solution of the Riccati equation (4). This allows to perform reconstruction in a few seconds of a microcomputer time. The whole algorithm can be summarized as follows ... 

The algebraic methods of reconstruction give result at incomplete and complete set of initial projection data. But the iterative imhlementation of these methods requires large computing resources. Algebraic method can be used in cases, when the required accuracy is not great. 

W.C. Chew and Y.M. Wang. Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method. IEEE Transaetions on Medical Imaging, 9, 1990. 

Image Space Reconstruction Algorithm. ISRA (Daube-Witherspoon and MuehUehner, 1986) is a multiplicative and iterative method which yields the constrained maximum likelihood in the case of Gaussian noise. The ISRA solution is obtained using the recursion ... 

Daube-Witherspoon, M.E., Muehllehner, G., 1986, An iterative image space reconstruction algorithm suitable for volume etc., IEEE Trans. Med. Imaging, 5, 61... 

The commercially available software (Maximum Entropy Data Consultant Ltd, Cambridge, UK) allows reconstruction of the distribution a.(z) (or f(z)) which has the maximal entropy S subject to the constraint of the chi-squared value. The quantified version of this software has a full Bayesian approach and includes a precise statement of the accuracy of quantities of interest, i.e. position, surface and broadness of peaks in the distribution. The distributions are recovered by using an automatic stopping criterion for successive iterates, which is based on a Gaussian approximation of the likelihood. 

The reliability of the exit wave reconstruction process was independently confirmed by making available to the scientific community a reconstructed focal series from the OAM. L. J. Allen et al. demonstrated striking agreement for the identical data set with an iterative method for exit wave function reconstruction. 

Figure 8. Schematic showing the iterative, nonlinear maximum-likehhood reconstruction...

In addition to reconstruction within the CSE, it is important to constrain the 2-RDM to remain approximately Al-representable. The process of correcting a 2-RDM to satisfy A-representability constraints is known as purification. In the context of an iterative solution of the CSE, early algorithms by Valdemoro checked that the 2-RDM satisfies a number of fundamental inequahties such as... 

The CSE allows us to recast A-representability as a reconstruction problem. If we knew how to build from the 2-RDM to the 4-RDM, the CSE in Eq. (12) furnishes us with enough equations to solve iteratively for the 2-RDM. Two approaches for reconstruction have been explored in previous work on the CSE (i) the explicit representation of the 3- and 4-RDMs as functionals of the 2-RDM [17, 18, 20, 21, 29], and (ii) the construction of a family of higher 4-RDMs from the 2-RDM by imposing ensemble representability conditions [20]. After justifying reconstmction from the 2-RDM by Rosina s theorem, we develop in Sections III.B and III.C the functional approach to the CSE from two different perspectives—the particle-hole duality and the theory of cumulants. 

The 2-RDM is automatically antisymmetric, but it may require an adjustment of the trace to correct the normalization. The functionals in Table I from cumulant theory allow us to approximate the 3- and the 4-RDMs from the 2-RDM and, hence, to iterate with the contracted power method. Because of the approximate reconstruction the contracted power method does not yield energies that are strictly above the exact energy. As in the full power method the updated 2-RDM in Eq. (116) moves toward the eigenstate whose eigenvalue has the largest magnitude. 

Here we synthesize the concepts of the last four sections, (i) CSE, (ii) reconstruction, (iii) purification, and (iv) a contracted power method, to obtain an iterative algorithm for the direct calculation of the 2-RDM. 

Figure 1. The energy for the molecule CO is given as a function of the number of contracted power iterations. With first-order (U) reconstruction the CSE obtains the correlation energy within 1%.

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