Projection space image reconstruction using strip functions to calculate pixels more "natural" for modeling the geometric response of the SPECT collimator

SVD for imaging systems with discrete rotational symmetry

The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors will augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

An efficient analytical calculation of probability matrix in 2D SPECT

Expectation Maximization iterative reconstruction algorithms are being widely used in PET and SPECT imaging. The system probability matrix is usually calculated by using Monte Carlo simulations, since the analytical calculation is a rather complicated problem, especially in 3D reconstruction; however, realistic Monte Carlo simulations in 3D are time consuming and simplifications are necessary. In this paper, the probability matrix in the case of 2D SPECT is analytically calculated, using only two basic parameters: (i) the total number of image pixels and (ii) the number of projection angles. In the more general phase three more parameters will be taken into account: (i) the distance between the detectors and the object to be imaged; (ii) the collimator parameters; (iii) scintillator cells parameters (in the case of pixilated scintillators) and relative position between scintillator cell and collimator hole. It is shown that the accuracy of the probability matrix affects the quality of reconstructed images, especially in the case of pixilated scintillators, which are used in many dedicated SPECT systems. The methods presented here can be extended to 3D SPECT and also 2D and 3D PET. In addition, this analytically calculated matrix can be a reference matrix in order to be compared with Monte Carlo generated matrices.

Iterative reconstruction techniques in emission computed tomography

In emission tomography statistically based iterative methods can improve image quality relative to analytic image reconstruction through more accurate physical and statistical modelling of high-energy photon production and detection processes. Continued exponential improvements in computing power, coupled with the development of fast algorithms, have made routine use of iterative techniques practical, resulting in their increasing popularity in both clinical and research environments. Here we review recent progress in developing statistically based iterative techniques for emission computed tomography. We describe the different formulations of the emission image reconstruction problem and their properties. We then describe the numerical algorithms that are used for optimizing these functions and illustrate their behaviour using small scale simulations.

Reconstruction of 2D PET data with Monte Carlo generated system matrix for generalized natural pixels

In discrete detector PET, natural pixels are image basis functions calculated from the response of detector pairs. By using reconstruction with natural pixel basis functions, the discretization of the object into a predefined grid can be avoided. Here, we propose to use generalized natural pixel reconstruction. Using this approach, the basis functions are not the detector sensitivity functions as in the natural pixel case but uniform parallel strips. The backprojection of the strip coefficients results in the reconstructed image. This paper proposes an easy and efficient way to generate the matrix M directly by Monte Carlo simulation. Elements of the generalized natural pixel system matrix are formed by calculating the intersection of a parallel strip with the detector sensitivity function. These generalized natural pixels are easier to use than conventional natural pixels because the final step from solution to a square pixel representation is done by simple backprojection. Due to rotational symmetry in the PET scanner, the matrix M is block circulant and only the first blockrow needs to be stored. Data were generated using a fast Monte Carlo simulator using ray tracing. The proposed method was compared to a listmode MLEM algorithm, which used ray tracing for doing forward and backprojection. Comparison of the algorithms with different phantoms showed that an improved resolution can be obtained using generalized natural pixel reconstruction with accurate system modelling. In addition, it was noted that for the same resolution a lower noise level is present in this reconstruction. A numerical observer study showed the proposed method exhibited increased performance as compared to a standard listmode EM algorithm. In another study, more realistic data were generated using the GATE Monte Carlo simulator. For these data, a more uniform contrast recovery and a better contrast-to-noise performance were observed. It was observed that major improvements in contrast recovery were obtained with MLEM when the correct system matrix was used instead of simple ray tracing. The correct modelling was the major cause of improved contrast for the same background noise. Less important factors were the choice of the algorithm (MLEM performed better than ART) and the basis functions (generalized natural pixels gave better results than pixels).

A three-dimensional theoretical model incorporating spatial detection uncertainty in continuous detector PET

In this paper, we will describe a theoretical model of the spatial uncertainty for a line of response, due to the imperfect localization of events on the detector heads of a positron emission tomography (PET) camera. The forward acquisition problem is modelled by a Gaussian distribution of the position of interaction on a detector head, centred at the measured position. The a posteriori probability that an event originates from a certain point in the field of view (FOV) is calculated by integrating all the possible lines of response (LORs) through this point, weighted with the Gaussian detection likelihood at the LOR's end points. We have calculated these a posteriori probabilities both for perpendicular and oblique coincidences. For the oblique coincidence case it was necessary to incorporate the effect of the crystal thickness in the calculations. We found in the perpendicular incidence case as well as in the oblique incidence case that the probability density function cannot be analytically expressed in a closed form, and it was thus calculated by means of numerical integration. A Gaussian was fit to the transversal profiles of this function for a given distance to the detectors. From these fits, we can conclude that the profiles can be accurately approximated by a Gaussian, both for perpendicular and oblique coincidences. The FWHM reaches a maximum at the detector heads, and decreases towards the centre of the FOV, as was expected. Afterwards we extended this two-dimensional model to three dimensions, thus incorporating the spatial uncertainty in both transversal directions. This theoretical model was then evaluated and a very good agreement was found with theoretical calculations and with geometric Monte Carlo simulations. Possible improvements for the above-described incorporation of crystal thickness are discussed. Therefore a detailed Monte Carlo study has been performed in order to investigate the interaction probability of photons of different energies along their path in several detector materials dedicated to PET. Finally two approaches for the incorporation of this theoretical model in reconstruction algorithms are outlined.

Differences due to collimator blurring in cardiac images with use of circular and elliptic camera orbits

BACKGROUND

In cardiac imaging systems, an elliptic acquisition orbit about the patient can be used to enhance resolution of single photon emission computed tomography (SPECT) images by minimizing the distance between the object imaged and the rotating detector system. In this study artifacts from images acquired with the standard circular acquisition are compared with those acquired with various elliptic acquisitions.

METHODS AND RESULTS

With the use of elliptic camera orbits of different eccentricities, simulated projection data were generated from a slice through the left ventricle (LV). The projection data included a simulation of the degradation due to the depth-dependent response of the collimator. As is common in many clinical systems, SPECT images were reconstructed with the standard filtered backprojection algorithm without correction for the collimator response. When the ratio of the major-to-minor axis of the acquisition arc is changed from 1 (circular) to 1.5 (elliptic), reconstructed SPECT images show an additional loss of counts (about 10%) in the apical region of the LV. The severity of the apical defect is also dependent on the starting angle of the acquisition arc. When the starting angle is changed from 0 degrees (detector parallel to the major axis of the LV) to 60 degrees, the ratio between the minimum count in the apical region and the maximum count in the left ventricular myocardial wall decreases by as much as 20%.

CONCLUSIONS

SPECT image artifacts from elliptic acquisitions are significantly more severe than those from circular acquisitions. Because of the significant difference in images reconstructed from circular and elliptic acquisitions, standardized normal files acquired from circular acquisitions should not be used for comparisons with patient data acquired from elliptic acquisitions.

A three-dimensional ray-driven attenuation, scatter and geometric response correction technique for SPECT in inhomogeneous media

The qualitative and quantitative accuracy of SPECT images is degraded by physical factors of attenuation, Compton scatter and spatially varying collimator geometric response. This paper presents a 3D ray-tracing technique for modelling attenuation, scatter and geometric response for SPECT imaging in an inhomogeneous attenuating medium. The model is incorporated into a three-dimensional projector-backprojector and used with the maximum-likelihood expectation-maximization algorithm for reconstruction of parallel-beam data. A transmission map is used to define the inhomogeneous attenuating and scattering object being imaged. The attenuation map defines the probability of photon attenuation between the source and the scattering site, the scattering angle at the scattering site and the probability of attenuation of the scattered photon between the scattering site and the detector. The probability of a photon being scattered through a given angle and being detected in the emission energy window is approximated using a Gaussian function. The parameters of this Gaussian function are determined using physical measurements of parallel-beam scatter line spread functions from a non-uniformly attenuating phantom. The 3D ray-tracing scatter projector-backprojector produces the scatter and primary components. Then, a 3D ray-tracing projector-backprojector is used to model the geometric response of the collimator. From Monte Carlo and physical phantom experiments, it is shown that the best results are obtained by simultaneously correcting attenuation, scatter and geometric response, compared with results obtained with only one or two of the three corrections. It is also shown that a 3D scatter model is more accurate than a 2D model. A transmission map is useful for obtaining measurements of attenuation and scatter in SPECT data, which can be used together with a model of the geometric response of the collimator to obtain corrected images with quantitative and diagnostically accurate information.