Subsampled Fisher Information Matrix for efficient estimation of the uncertainty in emission tomography

Optimization of the system design in Emission Tomography is a difficult problem due to the computational complexity and to the challenges in the mathematical formulation of the problem. In order to compare different system designs, a method to compare the uncertainty in the reconstruction is needed. The Fisher Information Matrix (FIM) formalism can be employed to characterize such uncertainty. Unfortunately, computing, storing and inverting the FIM is not feasible with 3D imaging systems. In order to tackle the problem of the computational load in calculating the inverse of the FIM an approximation has been previously proposed. A method based on the Local Impulse Response and the variance obtained from a single row of the FIM has been applied for system design in Single Positron Emission Computer Tomography (SPECT) [5]. However this approximation (circulant approximation) does not capture the global interaction between the variables and it cannot account e.g. for the depth-dependency of the Point Spread Function. Our new formulation relies on a sub-sampling of the FIM. The FIM is calculated over a subset of voxels arranged in a grid that covers the whole volume. This formulation reduces the computational complexity in inverting the FIM but nevertheless accounts for the global interdependence between the variables. We show how, our new methodology applies to the optimization of a parallel-hole collimator for SPECT. In order to prove the reliability of our approximation, we show also that for three different subsamples of the FIM we get the same optimal collimator aperture. In order to emphasize the benefits of our new approximation of the FIM with respect to the aforementioned circulant approximation, we show how it can be employed to calculate the reconstructed image quality in the case of truncated projection data. We demonstrate influence on covariance not demonstrated by the circulant approximation.