Fast fully 3-D image reconstruction in PET using planograms

Radon-to-Helmholtz mappings and nonlinear diffraction tomography

This paper addresses a number of approximate, analytically invertible solutions of the scalar Helmholtz equation. Primary attention is devoted to the Glauber approximation (GA) derived for the far-field pattern. It is shown that the GA has the form of a nonlinear Radon-to-Helmholtz (RtH) mapping, which transforms a sinogram of the scattering potential into an approximate solution of the Helmholtz equation. A proposal of how to construct a position space counterpart of the GA is formulated. Also, it is established that a paraxial version of the Glauber model coincides, up to an inessential constant factor, with a momentum-space representation of the Mazar-Felsen propagator, which describes forward-scattered waves. For weakly scattering objects, these solutions are reduced to the conventional Born/Rytov approximations, which may, however, preserve the parametrization and sampling formats of the original nonlinear models. Since all RtH mappings are analytically invertible, they can be applied to the (nonlinear) diffraction tomography of penetrable objects. In particular, the Glauber model, which has been largely ignored for years, is shown to provide efficient inversion of synthetic data. The resulting tomograms clearly outperform the Born inversions, even for moderately scattering potentials.

Fourier rebinning and consistency equations for time-of-flight PET planograms

Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

Transmission tomography of forward-scattering structures

A method of multiple-shot scatter correction in transmission tomography systems is considered. It is assumed that the scattering is confined within a cone of relatively small angles, and the propagation is governed by a standard parabolic wave equation. An approximate directed-wave propagator may then be obtained, which relates a set of diffraction patterns measured for a multitude of point sources, to a linogram of the object. Introducing a specially designed coordinate transformation, we rewrite this propagator in a separable form as a paired Fresnel transform. Further, a corresponding inverse operator is applied to extract emulated projections (constituting jointly a sinogram) of the object from the measured data. Apart from the original algorithm, which suffers essentially from truncation errors, its improved, considerably more robust version is also constructed. The estimated sinograms may be inverted by any of the existing methods including a filtered back-projection or algebraic reconstruction techniques. The results of simulations show an essential reduction of diffraction/scattering artifacts and improved resolution of reconstructed images. At the same time it is found that scatter correction may cause object-dependent ring artifacts observed in the tomograms.

Full data consistency conditions for cone-beam projections with sources on a plane

Cone-beam consistency conditions (also known as range conditions) are mathematical relationships between different cone-beam projections, and they therefore describe the redundancy or overlap of information between projections. These redundancies have often been exploited for applications in image reconstruction. In this work we describe new consistency conditions for cone-beam projections whose source positions lie on a plane. A further restriction is that the target object must not intersect this plane. The conditions require that moments of the cone-beam projections be polynomial functions of the source positions, with some additional constraints on the coefficients of the polynomials. A precise description of the consistency conditions is that the four parameters of the cone-beam projections (two for the detector, two for the source position) can be expressed with just three variables, using a certain formulation involving homogeneous polynomials. The main contribution of this work is our demonstration that these conditions are not only necessary, but also sufficient. Thus the consistency conditions completely characterize all redundancies, so no other independent conditions are possible and in this sense the conditions are full. The idea of the proof is to use the known consistency conditions for 3D parallel projections, and to then apply a 1996 theorem of Edholm and Danielsson that links parallel to cone-beam projections. The consistency conditions are illustrated with a simulation example.

DOI-based reconstruction algorithms for a compact breast PET scanner

PURPOSE

The authors discuss the design and evaluate the performance of combined event estimation and image reconstruction algorithms designed for a proposed high-resolution rectangular breast PET scanner (PETX). The PETX scanner will be capable of measuring the depth of interaction by utilizing detector modules composed of depth-of-interaction microcrystal element (dMiCE) crystal pairs. This design allows a unique combination of event estimation and fast projection methods.

METHODS

The authors implemented a Monte Carlo simulator to model the PETX system using only true coincident events. The performance of the dMiCE crystal pairs was determined experimentally over a range of depths of interaction. This distribution was used to simulate the noisy dMiCE detector signals and to estimate the line of response for each decay. Three different statistical methods were implemented to determine photon event positioning. Images were reconstructed from these line of response estimators with the exact planogram frequency distance rebinning algorithm, which is an exact analytical reconstruction algorithm for planar systems. Reconstructed images were analyzed with contrast, noise, and spatial resolution metrics.

RESULTS

The authors' simulations demonstrate the ability for the PETX system to produce quantitatively accurate images from true coincident events with a contrast recovery coefficient of greater than 0.8 for 5 mm spheres at the axial center of the scanner and a spatial resolution (FWHM) of 3 mm throughout most of the imaging field of view.

CONCLUSIONS

The authors' proposed positioning and reconstruction algorithms for the PETX system show the potential for creating high-quality, high-resolution, and quantitatively accurate images within a clinically feasible reconstruction time.

Advancements to the planogram frequency-distance rebinning algorithm

In this paper we consider the task of image reconstruction in positron emission tomography (PET) with the planogram frequency-distance rebinning (PFDR) algorithm. The PFDR algorithm is a rebinning algorithm for PET systems with panel detectors. The algorithm is derived in the planogram coordinate system which is a native data format for PET systems with panel detectors. A rebinning algorithm averages over the redundant four-dimensional set of PET data to produce a three-dimensional set of data. Images can be reconstructed from this rebinned three-dimensional set of data. This process enables one to reconstruct PET images more quickly than reconstructing directly from the four-dimensional PET data. The PFDR algorithm is an approximate rebinning algorithm. We show that implementing the PFDR algorithm followed by the (ramp) filtered backprojection (FBP) algorithm in linogram coordinates from multiple views reconstructs a filtered version of our image. We develop an explicit formula for this filter which can be used to achieve exact reconstruction by means of a modified FBP algorithm applied to the stack of rebinned linograms and can also be used to quantify the errors introduced by the PFDR algorithm. This filter is similar to the filter in the planogram filtered backprojection algorithm derived by Brasse et al. The planogram filtered backprojection and exact reconstruction with the PFDR algorithm require complete projections which can be completed with a reprojection algorithm. The PFDR algorithm is similar to the rebinning algorithm developed by Kao et al. By expressing the PFDR algorithm in detector coordinates, we provide a comparative analysis between the two algorithms. Numerical experiments using both simulated data and measured data from a positron emission mammography/tomography (PEM/PET) system are performed. Images are reconstructed by PFDR+FBP (PFDR followed by 2D FBP reconstruction), PFDRX (PFDR followed by the modified FBP algorithm for exact reconstruction) and planogram filtered backprojection image reconstruction algorithms. We show that the PFDRX algorithm produces images that are nearly as accurate as images reconstructed with the planogram filtered backprojection algorithm and more accurate than images reconstructed with the PFDR+FBP algorithm. Both the PFDR+FBP and PFDRX algorithms provide a dramatic improvement in computation time over the planogram filtered backprojection algorithm.

Fast, accurate and shift-varying line projections for iterative reconstruction using the GPU

List-mode processing provides an efficient way to deal with sparse projections in iterative image reconstruction for emission tomography. An issue often reported is the tremendous amount of computation required by such algorithm. Each recorded event requires several back- and forward line projections. We investigated the use of the programmable graphics processing unit (GPU) to accelerate the line-projection operations and implement fully-3D list-mode ordered-subsets expectation-maximization for positron emission tomography (PET). We designed a reconstruction approach that incorporates resolution kernels, which model the spatially-varying physical processes associated with photon emission, transport and detection. Our development is particularly suitable for applications where the projection data is sparse, such as high-resolution, dynamic, and time-of-flight PET reconstruction. The GPU approach runs more than 50 times faster than an equivalent CPU implementation while image quality and accuracy are virtually identical. This paper describes in details how the GPU can be used to accelerate the line projection operations, even when the lines-of-response have arbitrary endpoint locations and shift-varying resolution kernels are used. A quantitative evaluation is included to validate the correctness of this new approach.

Rotate-and-slant projector for fast LOR-based fully-3-D iterative PET reconstruction

One of the greatest challenges facing iterative fully-3-D positron emission tomography (PET) reconstruction is the issue of long reconstruction times due to the large number of measurements for 3-D mode as compared to 2-D mode. A rotate-and-slant projector has been developed that takes advantage of symmetries in the geometry to compute volumetric projections to multiple oblique sinograms in a computationally efficient manner. It is based upon the 2-D rotation-based projector using the three-pass method of shears, and it conserves the 2-D rotator computations for multiple projections to each oblique sinogram set. The projector is equally applicable to both conventional evenly-spaced projections and unevenly-spaced line-of-response (LOR) data. The LOR-based version models the location and orientation of the individual LORs (i.e., the arc-correction), providing an ordinary Poisson reconstruction framework. The projector was implemented in C with several optimizations for speed, exploiting the vertical symmetry of the oblique projection process, depth compression, and array indexing schemes which maximize serial memory access. The new projector was evaluated and compared to ray-driven and distance-driven projectors using both analytical and experimental phantoms, and fully-3-D iterative reconstructions with each projector were also compared to Fourier rebinning with 2-D iterative reconstruction. In terms of spatial resolution, contrast, and background noise measures, 3-D LOR-based iterative reconstruction with the rotate-and-slant projector performed as well as or better than the other methods. Total processing times, measured on a single cpu Linux workstation, were approximately 10x faster for the rotate-and-slant projector than for the other 3-D projectors studied. The new projector provided four iterations fully-3-D ordered-subsets reconstruction in as little as 15 s--approximately the same time as FORE + 2-D reconstruction. We conclude that the rotate-and-slant projector is a viable option for fully-3-D PET, offering quality statistical reconstruction in times only marginally slower than 2-D or rebinning methods.

Planogram rebinning with the frequency-distance relationship

We present an efficient rebinning algorithm for positron emission tomography (PET) systems with panel detectors. The rebinning algorithm is derived in the planogram coordinate system which is the native data format for PET systems with panel detectors and is the 3-D extension of the 2-D linogram transform developed by Edholm. Theoretical error bounds and numerical results are included.

Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters

Fourier-based approaches for three-dimensional (3-D) reconstruction are based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The critical step in the Fourier-based methods is the estimation of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for forward-projection (reprojection). The Fourier-based approaches have the potential for very fast reconstruction, but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions. In our previous work, we have investigated optimal interpolation parameters for the Fourier-based forward and back-projectors for iterative image reconstruction. The optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator. This work presents an optimization of interpolation parameters of the 3-D direct Fourier method with Fourier reprojection (3D-FRP) for fully 3-D positron emission tomography (PET) data with incomplete oblique projections. The reprojection step is needed for the estimation (from an initial image) of the missing portions of the oblique data. In the 3D-FRP implementation, we use the gridding interpolation strategy, combined with proper weighting approaches in the transform and image domains. We have found that while the 3-D reprojection step requires similar optimal interpolation parameters as found in our previous studies on Fourier-based iterative approaches, the optimal interpolation parameters for the main 3D-FRP reconstruction stage are quite different. Our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling, which is a common practice to decrease errors caused by interpolation in Fourier reconstruction.

[Technical limitations of the positron emission tomography (PET) for small laboratory animals]

Visualization and quantification of organ function by PET in small laboratory animals is becoming an outstanding tool for characterization of phenotype of transgenic and knock-out animals, for the study of animal models of human diseases, and for the development of new therapeutic drugs and diagnostic biochemical probes. To be able to make use of the PET with small laboratory animals in the same way as it is operated with humans it is necessary to account for the volumetric scale factor as well as for the requirement of maintaining the counting statistics. This work sketches the problems that these requirements represent for the technical design of the scanners and for the execution of the experiments. Finally, some characteristics of commercially available scanners (microPET/ FOCUS, HiDAC, eXplore VISTA, MOSAIC, YAP-(S)PET and rPET) are briefly discussed.

Fast 3D-EM reconstruction using Planograms for stationary planar positron emission mammography camera

At the University of Pisa we are building a PEM prototype, the YAP-PEM camera, consisting of two opposite 6 x 6 x 3 cm3 detector heads of 30 x 30 YAP:Ce finger crystals, 2 x 2 x 30 mm3 each. The camera will be equipped with breast compressors. The acquisition will be stationary. Compared with a whole body PET scanner, a planar Positron Emission Mammography (PEM) camera allows a better, easier and more flexible positioning around the breast in the vicinity of the tumor: this increases the sensitivity and solid angle coverage, and reduces cost. To avoid software rejection of data during the reconstruction, resulting in a reduced sensitivity, we adopted a 3D-EM reconstruction which uses all of the collected Lines Of Response (LORs). This skips the PSF distortion given by data rebinning procedures and/or Fourier methods. The traditional 3D-EM reconstruction requires several times the computation of the LOR-voxel correlation matrix, or probability matrix {p(ij)}; therefore is highly time-consuming. We use the sparse and symmetry properties of the matrix {p(ij)} to perform fast 3D-EM reconstruction. Geometrically, a 3D grid of cubic voxels (FOV) is crossed by several divergent 3D line sets (LORs). The symmetries occur when tracing different LORs produces the same p(ij) value. Parallel LORs of different sets cross the FOV in the same way, and the repetition of p(ij) values depends on the ratio between the tube and voxel sizes. By optimizing this ratio, the occurrence of symmetries is increased. We identify a nucleus of symmetry of LORs: for each set of symmetrical LORs we choose just one LOR to be put in the nucleus, while the others lie outside. All of the possible p(ij) values are obtainable by tracking only the LORs of this nucleus. The coordinates of the voxels of all of the other LORs are given by means of simple translation rules. Before making the reconstruction, we trace the LORs of the nucleus to find the intersecting voxels, whose p(ij) values are computed and stored with their voxel coordinates on a hard disk. Only the non-zero p(ij) are considered and their computation is performed just once. During the reconstruction, the stored values are loaded and are available in the random access memory for all of the operations of normalization, backprojection and projection: these are now performed rapidly, because the application of the translation rules is much faster than the probability computations. We tested the algorithm on Monte Carlo data fully simulating the typical YAP-PEM clinical condition. The adopted algorithm gives an excellent positioning capability for hot spots in the camera FOV. To use all of the possible skew LORs in the FOV avoids the software rejection of collected data. Reconstructed images indicate that a 5mm diameter tumor of 37 kBq/cm3, in an active breast with a 10:1 Tissue to Background ratio (T/B), with a 10 min acquisition, for a head distance of 5 cm, can be detected by the YAP-PEM with a SNR of 8.7+/-1.0. The obtained SNR values depend linearly on the tumor volume. The algorithm allows one to discriminate between two hot sources of 5.0 mm diameter if they do not lie on the same axis. The YAP-PEM is now in the assembly stage.

Iterative tomographic image reconstruction using Fourier-based forward and back-projectors

Iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems, especially in emission tomography. With the fast increase of the sizes of the tomographic data, reduction of the computation demands of the reconstruction algorithms is of great importance. Fourier-based forward and back-projection methods have the potential to considerably reduce the computation time in iterative reconstruction. Additional substantial speed-up of those approaches can be obtained utilizing powerful and cheap off-the-shelf fast Fourier transform (FFT) processing hardware. The Fourier reconstruction approaches are based on the relationship between the Fourier transform of the image and Fourier transformation of the parallel-ray projections. The critical two steps are the estimations of the samples of the projection transform, on the central section through the origin of Fourier space, from the samples of the transform of the image, and vice versa for back-projection. Interpolation errors are a limitation of Fourier-based reconstruction methods. We have applied min-max optimized Kaiser-Bessel interpolation within the nonuniform FFT (NUFFT) framework and devised ways of incorporation of resolution models into the Fourier-based iterative approaches. Numerical and computer simulation results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic forward and back-projection than conventional interpolation methods. Our studies have further confirmed that Fourier-based projectors using the NUFFT approach provide accurate approximations to their space-based counterparts but with about ten times faster computation, and that they are viable candidates for fast iterative image reconstruction.