Abstract
System designs in single photon emission tomography (SPECT) can be evaluated based on the fundamental trade-off between bias and variance that can be achieved in the reconstruction of emission tomograms. This trade off can be derived analytically using the Cramer-Rao type bounds, which imply the calculation and the inversion of the Fisher information matrix (FIM). The inverse of the FIM expresses the uncertainty associated to the tomogram, enabling the comparison of system designs. However, computing, storing and inverting the FIM is not practical with 3-D imaging systems. In order to tackle the problem of the computational load in calculating the inverse of the FIM, a method based on the calculation of the local impulse response and the variance, in a single point, from a single row of the FIM, has been previously proposed for system design. However this approximation (circulant approximation) does not capture the global interdependence between the variables in shift-variant systems such as SPECT, and cannot account e.g., for data truncation or missing data. Our new formulation relies on subsampling the FIM. The FIM is calculated over a subset of voxels arranged in a grid that covers the whole volume. Every element of the FIM at the grid points is calculated exactly, accounting for the acquisition geometry and for the object. This new formulation reduces the computational complexity in estimating the uncertainty, but nevertheless accounts for the global interdependence between the variables, enabling the exploration of design spaces hindered by the circulant approximation. The graphics processing unit accelerated implementation of the algorithm reduces further the computation times, making the algorithm a good candidate for real-time optimization of adaptive imaging systems. This paper describes the subsampled FIM formulation and implementation details. The advantages and limitations of the new approximation are explored, in comparison with the circulant approximation, in the context of design optimization of a parallel-hole collimator SPECT system and of an adaptive imaging system (similar to the commercially available D-SPECT).
Highlights
O PTIMIZATION of the system design in single photon emission tomography (SPECT) is a difficult problem due to the computational complexity and to the challenges in the mathematical formulation
In recent years there has been an increasing interest in optimizing system designs prospectively, by computer simulation, at low computational cost. Such optimization problems include the choice of a particular type of detector and collimator and tuning of their parameters, as well as the choice of the number of cameras and their position. While such class of design optimization problems may be referred to as hard optimization, the development of adaptive SPECT systems has introduced a second class of soft optimization problems, where the parameters of the imaging system may be modified during acquisition, in order to image certain desired properties of the underlying object and to adapt to the imaging conditions
From these images we can see how both the method based on the subsampled Fisher information matrix (FIM) and the method based on the circulant approximation of the FIM approximately predict the variance of the maximum a posteriori (MAP) estimator, presenting minor, but obvious, differences with respect to the variance obtained with the reference method
Summary
O PTIMIZATION of the system design in single photon emission tomography (SPECT) is a difficult problem due to the computational complexity and to the challenges in the mathematical formulation. In recent years there has been an increasing interest in optimizing system designs prospectively, by computer simulation, at low computational cost Such optimization problems include the choice of a particular type of detector and collimator and tuning of their parameters, as well as the choice of the number of cameras and their position. While such class of design optimization problems may be referred to as hard optimization, the development of adaptive SPECT systems has introduced a second class of soft optimization problems, where the parameters of the imaging system may be modified during acquisition, in order to image certain desired properties of the underlying object and to adapt to the imaging conditions. Characterization of the uncertainty associated with the measurement of activity enables the comparison of system designs
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