IEEE Trans Med Imaging. 2014 Mar;33(3):618-35. doi: 10.1109/TMI.2013.2292805.
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).
单光子发射断层扫描（SPECT）中的系统设计可以基于在发射断层图像重建中可以实现的偏差和方差之间的基本权衡来评估。这种权衡可以通过使用克拉美-罗（Cramer-Rao）型界来分析得出，这意味着需要计算和反转 Fisher 信息矩阵（FIM）。FIM 的逆表示与断层图像相关的不确定性，从而能够比较系统设计。然而，对于 3D 成像系统，计算、存储和反转 FIM 是不切实际的。为了解决计算 FIM 的逆时的计算负担问题，之前已经提出了一种基于在 FIM 的单行上的单个点计算局部脉冲响应和方差的方法，以便进行系统设计。然而，这种逼近（循环逼近）无法捕获 SPECT 等移位变化系统中变量之间的全局相互依赖性，也无法例如，无法考虑数据截断或丢失数据。我们的新公式依赖于对 FIM 的子采样。FIM 是在覆盖整个体积的网格上排列的一组体素上计算的。网格点处 FIM 的每个元素都是精确计算的，考虑了采集几何形状和物体。这种新公式减少了估计不确定性的计算复杂性，但仍然考虑了变量之间的全局相互依赖性，从而能够探索受到循环逼近阻碍的设计空间。图形处理单元（GPU）加速算法的实现进一步减少了计算时间，使该算法成为实时优化自适应成像系统的良好候选。本文介绍了子采样 FIM 公式和实现细节。在并行孔准直器 SPECT 系统和自适应成像系统（类似于市售的 D-SPECT）的设计优化背景下，探讨了新逼近与循环逼近的优缺点。
