Comparison Between Parallel Hole and Rotating Slat Collimation: Analytical Noise Propagation Models

Abstract

We have previously proposed a method to compare tomographic systems. It is assumed that each system acquires a tomographic scan of a certain tracer distribution in the same acquisition time. From this scan, each system is forced to reconstruct an image with a predefined spatial resolution. The system that can perform this task with the "most favorable" noise propagation is considered as the best system. The variance on pixel values or region-of-interest (ROI) values is used to assess the noise in the reconstructed image. In this paper, we extend this idea to compare the performance of parallel hole (PH) and rotating slat (RS) collimations. Two different analytical approaches were used to analyze the variance of the reconstructed pixel/ROI values. The first method is based on the filtered-backprojection (FBP) theory, and was applied to the central point of a uniform symmetrical phantom. It yields analytical expressions for the optimal collimator aperture and the corresponding variance of the reconstructed pixel values, but it can only be applied to highly symmetrical configurations. The second method is based on approximations for the Fisher information matrix. It provides numerical results, and it is more general and can be applied to nonsymmetrical objects and shift-variant tomographic systems. The collimations were compared for both planar imaging and volume imaging. The main results are as follows. 1) For cases where both methods are valid, they are in excellent agreement. 2a) The optimal collimator aperture varies linearly with the target resolution. 2b) For a fixed target resolution, the optimal collimator aperture depends on the collimator type and the imaging mode (planar or volume). 2c) The optimal aperture of PH is a factor of √2 larger than that of RS. 3a) The relative performance of the two collimators is determined by both the object size and the object-to-detector distance. 3b) Pixel variance and variances of ROIs with varying sizes yield very similar relative performance for RS versus PH.

Efficient Determination of the Uncertainty for the Optimization of SPECT System Design: A Subsampled Fisher Information Matrix

“…We can see also how the optimal aperture varies almost linearly in relation with the target resolution imposed in the reconstruction. Similar results were presented in another study [4] which claims that the FWHM of the parallel hole collimator aperture that yields the minimum variance, equals the spatial resolution divided by . 3) Truncated Projection Data: Fig.…”

“…For this specific experiment we observe a decrease in CNR, for a voxel in the center of the sphere, of 8% compared to the nontruncated case, using the reference method based on multiple noise realizations for the calculation of the variance. An important observation is that we see no effect due to truncation with the circulant approximation of the FIM (as stated in [4]) whereas with the sumbsampled FIM, since we ac-count for the interdependence between the voxels, we see a decreased CNR (increased variance) with increased level of truncation. This is an important feature of the method that we have introduced, because it enables the optimization of systems for interior imaging, which is not possible with existing methods.…”

“…In order to tackle the problem of the computational load in calculating and inverting the FIM, it has been proposed to approximate it with a circulant matrix (see Section II-C). The use of the circulant approximation has been explored for the purpose of measuring the image quality in [9], [10], [5], [15] and for the purpose of system design optimization in [20], [4]. The computational complexity of the reference method, involving the reconstruction to convergence of thousands of noise realizations, has precluded a systematic evaluation of the effect and the limitations of the circulant approximation.…”

“…In the case of emission tomography, where the conditional probability distribution associated with photon counting is Poisson distributed with expectation (3) the maximum a posteriori (MAP) estimate consists in selecting the value of the unknown variable that maximizes the following cost function: (4) where the log likelihood function , is given by (5) In this paper, we restrict the discussion to the quadratic penalty function…”

“…SPECT systems may also be evaluated based on the fundamental trade-off between bias and variance that can be achieved in the reconstruction of emission tomograms [20], [1], [2], [4]. Such tradeoffs may be derived analytically using the Cramer-Rao type bounds [9], [18], [19] which imply the calculation and the inversion of the Fisher information matrix (FIM).…”