Noise characterization of combined Bellini-type attenuation correction and frequency-distance principle restoration filtering [SPECT]

We investigate how successive processing of the projection data with Bellini-type (BT) attenuation correction, frequency distance principle (FDP) restoration filtering, and filtered-backprojection (FBP) reconstruction propagates noise into the reconstructed image. The BT methods, which are Fourier-domain correction techniques, are all exact solutions to the attenuated Radon transform under the constraints of uniform attenuation and a convex object contour. The FDP states that points in the object at a specified distance from the center-of-rotation will contribute predominantly to particular regions of the Fourier transform-series expansion of the sinogram. Using this frequency-distance information, the nonstationary response of the collimator can ideally be deconvolved using an inverse filter. In the presence of noise, however, the FDP filter needs to be regularized in order to control noise amplification caused by the deconvolution. The noise is characterized by calculating the population covariance matrix of the projection data after successive processing with BT attenuation correction and FDP filtering and of the reconstructed image after FBP reconstruction. This is done for a simulated point-source located near the edge of an elliptical, uniformly attenuating medium. In our study, we consider the two implementational combinations of BT correction and FDP filtering that allow for sequential processing of the projection data, which we call mirroring and no mirroring. The results show that the mirroring strategy introduces local positive angular noise correlations in the processed sinogram that are caused by smoothing of the data in the projection angular direction. Furthermore, mirroring highly correlates the noise in opposing projection views. The impact on the image covariance is the presence of positive angular (arc-like) noise correlations. In contrast, the no mirroring strategy yields local positive and negative noise correlations with negative side-lobes in the processed sinogram. Also, no mirroring does not correlate noise in opposing projection views. The result on the image covariance structure is the presence of positive and negative angular correlations with negative side-lobes. It was also noted that the no mirroring scenario produces a much higher noise variance relative to the mirroring implementation of BT attenuation correction and FDP filtering.