Samuel Matej

Practical considerations for 3-D image reconstruction using spherically symmetric volume elements

Spherically symmetric volume elements with smooth tapering of the values near their boundaries are alternatives to the more conventional voxels for the construction of volume images in the computer. Their use, instead of voxels, introduces additional parameters which enable the user to control the shape of the volume element (blob) and consequently to control the characteristics of the images produced by iterative methods for reconstruction from projection data. For images composed of blobs, efficient algorithms have been designed for the projection and discrete back-projection operations, which are the crucial parts of iterative reconstruction methods. The authors have investigated the relationship between the values of the blob parameters and the properties of images represented by the blobs. Experiments show that using blobs in iterative reconstruction methods leads to substantial improvement in the reconstruction performance, based on visual quality and on quantitative measures, in comparison with the voxel case. The images reconstructed using appropriately chosen blobs are characterized by less image noise for both noiseless data and noisy data, without loss of image resolution.

A methodology for testing for statistically significant differences between fully 3D PET reconstruction algorithms

We present a practical methodology for evaluating 3D PET reconstruction methods. It includes generation of random samples from a statistically described ensemble of 3D images resembling those to which PET would be applied in a medical situation, generation of corresponding projection data with noise and detector point spread function simulating those of a 3D PET scanner, assignment of figures of merit appropriate for the intended medical applications, optimization of the reconstruction algorithms on a training set of data, and statistical testing of the validity of hypotheses that say that two reconstruction algorithms perform equally well (from the point of view of a particular figure of merit) as compared to the alternative hypotheses that say that one of the algorithms outperforms the other. Although the methodology was developed with the 3D PET in mind, it can be used, with minor changes, for other 3D data collection methods, such as fully 3D cr or SPECT.

Iterative image reconstruction using geometrically ordered subsets with list-mode data

In positron emission tomography (PET), the format in which the data is stored has a major influence on the image reconstruction procedure. The use of the list-mode format preserves all of the measured attributes of the detected photon pairs but the events are stored in the order that they were measured, which allows only sequential access to the data. This fact limits the number of applicable algorithms and often computing speed or memory capacity constraints require the use of algorithms that do not make full use of the original precise information in the data. In this paper we show how through a change of the format in which the data is stored one can keep all the initial information about the individual events while providing random access to subsets of events belonging to given geometrical regions, thus making possible the use of maximum likelihood ordered subsets (OSEM) type algorithms with data provided as a collection of individual events (list-mode), and facilitating the adaptation of other types of algorithms. The structured data format also allows for more compact (compressed) storage of the information compared to the simple list-mode format.

Performance of the Fourier rebinning algorithm for PET with large acceptance angles

The recently proposed Fourier rebinning (FORE) technique of 3D PET reconstruction is investigated over a wide range of axial acceptance angles. In this study we evaluate the performance of the FORE technique using spatial resolution, contrast and noise figures of merit and compare reconstruction performance of the FORE (followed by multislice 2D reconstruction) to the 3D-RP technique for large-acceptance-angle data (+/-26.25 degrees). Our results show that the FORE technique does not affect the transverse resolution. On the other hand the axial resolution using FORE deteriorates faster, compared with the 3D-RP, at large radii as the acceptance angle increases. Concerning the noise behaviour, we have found that filtering has better ability to suppress the noise in the FORE reconstruction, compared with the 3D-RP reconstruction, especially in the slices near the edge of the axial field of view. Overall, the combination of good performance and fast reconstruction time makes the FORE technique a practical choice for 3D PET applications.

A high-speed reconstruction from projections using direct Fourier method with optimized parameters-an experimental analysis

Some issues of the direct Fourier method (DFM) implementation are discussed. A hybrid spline-linear interpolation for the DFM is proposed. The results of comprehensive simulation research are presented. The following reconstruction problems and parameters are emphasized: interpolation, increasing the radial density of the polar raster, filtering, the 2-D inverse Fourier transformation dimension, and considering the cases of noiseless and noisy input data. For the a priori prescribed resolution of the reconstructed image, values of reconstruction parameters have been determined which are optimal with regard to reconstruction quality and computation cost. The computational requirements of the DFM algorithm which correspond to distinct interpolation schemes are compared to one another for CT and MR tomography, respectively. The estimations obtained are compared to computational characteristics of the convolution backprojection method.

Efficient 3-D TOF PET Reconstruction Using View-Grouped Histo-Images: DIRECT—Direct Image Reconstruction for TOF

For modern time-of-flight (TOF) positron emission tomography (PET) systems, in which the number of possible lines of response and TOF bins is much larger than the number of acquired events, the most appropriate reconstruction approaches are considered to be list-mode methods. However, their shortcomings are relatively high computational costs for reconstruction and for sensitivity matrix calculation. Efficient treatment of TOF data within the proposed DIRECT approach is obtained by 1) angular (azimuthal and co-polar) grouping of TOF events to a set of views as given by the angular sampling requirements for the TOF resolution, and 2) deposition (weighted-histogramming) of these grouped events, and correction data, into a set of ldquohisto-images,rdquo one histo-image per view. The histo-images have the same geometry (voxel grid, size and orientation) as the reconstructed image. The concept is similar to the approach involving binning of the TOF data into angularly subsampled histo-projections - projections expanded in the TOF directions. However, unlike binning into histo-projections, the deposition of TOF events directly into the image voxels eliminates the need for tracing and/or interpolation operations during the reconstruction. Together with the performance of reconstruction operations directly in image space, this leads to a very efficient implementation of TOF reconstruction algorithms. Furthermore, the resolution properties are not compromised either, since events are placed into the image elements of the desired size from the beginning. Concepts and efficiency of the proposed data partitioning scheme are demonstrated in this work by using the DIRECT approach in conjunction with the row-action maximum-likelihood (RAMLA) algorithm.

Binary Tomography Using Gibbs Priors

The problem of reconstructing a binary image (usually an image in the plane and not necessarily on a Cartesian grid) from a few projections translates into the problem of solving a system of equations,which is very underdetermined and leads in general to a large class of solutions. It is desirable to limit the class of possible solutions, by using appropriate prior information, to only those which are reasonably typical of the class of images which contains the unknown image that we wish to reconstruct. One may indeed pose the following hypothesis: if the image is a typical member of a class of images having a certain distribution, then by using this information we can limit the class of possible solutions to only those which are close to the given unknown image. This hypothesis is experimentally validated for the specific case of a class of binary images defined on the hexagonal grid, where the probability of the occurrence of a particular image of the class is determined by a Gibbs distribution and reconstruction is to be done from the three natural projections. Another case for which the hypothesis is tested is reconstruction, from the three projections, of semiconductor surface phantoms defined on the square grid. The time-consuming nature of the stochastic reconstruction algorithm is ameliorated by a preprocessing step that discovers image locations at which the value is the same in all images having the given projections; this reduces the search space considerably. We discuss, in particular, a linear-programming approach to finding such “invariant” locations.

Binary Tomography for Triplane Cardiography

The problem of reconstructing a binary image (usually an image in the plane and not necessarily on a Cartesian grid) from a few projections translates into the problem of solving a system of equations which is very underdetermined and leads in general to a large class of solutions. It is desirable to limit the class of possible solutions, by using appropriate prior information, to only those which are reasonably typical of the class of images which contains the unknown image that we wish to reconstruct. One may indeed pose the following hypothesis: if the image is a typical member of a class of images having a certain distribution, then by using this information we can limit the class of possible solutions to only those which are close to the given unknown image. This hypothesis is experimentally validated for the Specific case of a class of binary images representing cardiac cross-sections, where the probability of the occurrence of a particular image of the class is determined by a Gibbs distribution and reconstruction is to be done from the three noisy projections.

Fisher Information-Based Evaluation of Image Quality for Time-of-Flight PET

The use of time-of-flight (TOF) information during reconstruction is generally considered to improve the image quality. In this work we quantified this improvement using two existing methods: (1) a very simple analytical expression only valid for a central point in a large uniform disk source, and (2) efficient analytical approximations for post-filtered maximum likelihood expectation maximization (MLEM) reconstruction with a fixed target resolution, predicting the image quality in a pixel or in a small region based on the Fisher information matrix. The image quality was investigated at different locations in various software phantoms. Simplified as well as realistic phantoms, measured both with TOF positron emission tomography (PET) systems and with a conventional PET system, were simulated. Since the time resolution of the system is not always accurately known, the effect on the image quality of using an inaccurate kernel during reconstruction was also examined with the Fisher information- based method. First, we confirmed with this method that the variance improvement in the center of a large uniform disk source is proportional to the disk diameter and inversely proportional to the time resolution. Next, image quality improvement was observed in all pixels, but in eccentric and high-count regions the contrast-to-noise ratio (CNR) increased slower than in central and low- or medium-count regions. Finally, the CNR was seen to decrease when the time resolution was inaccurately modeled (too narrow or too wide) during reconstruction. Although the optimum is rather flat, using an inaccurate TOF kernel might introduce artifacts in the reconstructed image.

Binary tomography on the hexagonal grid using Gibbs priors

The problem of reconstructing a binary image (usually an image in the plane and not necessarily on a Cartesian grid) from a few projections translates into the problem of solving a system of equations which is very underdetermined and leads in general to a large class of solutions. It is desirable to limit the class of possible solutions, by using appropriate prior information, to only those which are reasonably typical of the class of images which contains the unknown image that we wish to reconstruct. One may indeed pose the following hypothesis: If the image is a typical member of a class of images having a certain distribution, then by using this information we can limit the class of possible solutions to only those which are close to the given unknown image. This hypothesis is experimentally validated for the specific case of a class of binary images defined on the hexagonal grid, where the probability of the occurrence of a particular image of the class is determined by a Gibbs distribution and reconstruction is to be done from the three natural projections.