Statistical image reconstruction from correlated data with applications to PET

Abstract

Most statistical reconstruction methods for emission tomography are designed for data modeled as conditionally independent Poisson variates. In reality, due to scanner detectors, electronics and data processing, correlations are introduced into the data resulting in dependent variates. In general, these correlations are ignored because they are difficult to measure and lead to computationally challenging statistical reconstruction algorithms. This work addresses the second concern, seeking to simplify the reconstruction of correlated data and provide a more precise image estimate than the conventional independent methods. In general, correlated variates have a large non-diagonal covariance matrix that is computationally challenging to use as a weighting term in a reconstruction algorithm. This work proposes two methods to simplify the use of a non-diagonal covariance matrix as the weighting term by (a) limiting the number of dimensions in which the correlations are modeled and (b) adopting flexible, yet computationally tractable, models for correlation structure. We apply and test these methods with simple simulated PET data and data processed with the Fourier rebinning algorithm which include the one-dimensional correlations in the axial direction and the two-dimensional correlations in the transaxial directions. The methods are incorporated into a penalized weighted least-squares 2D reconstruction and compared with a conventional maximum a posteriori approach.

Generalized least-squares CT reconstruction with detector blur and correlated noise models

“…Thus, there is a fundamental mismatch between the usual presumption of independent measurements and the nature of the acquisition data. There are a few reconstruction algorithms with the potential to model noise correlations with examples in nuclear imaging [9] and in sinogram deblurring [10]. In this work, we focus on a generalized model-based approach that accommodates both detector blur and a correlated noise model as part of the reconstruction process.…”

MAP reconstruction for Fourier rebinned TOF-PET data

“…An alternative approach to reconstruction is to use a weighted least squares (WLS) method as described for rebinned nonTOF data in (Alessio et al 2007). The appropriate weighting is the inverse of the covariance matrix.…”

“…This is because the Poisson model ignores correlation in a similar manner to the diagonal WLS method. Approximations to the covariance matrix that do take local correlations into account have been derived for FORE rebinned nonTOF data by Alessio et al (2007). However, the results obtained show only marginal improvements in image quality.…”

“…However, the results obtained show only marginal improvements in image quality. Based on the simulation above and the results from Alessio et al (2007), it appears that there are advantages of reweighting the data to match mean and variance and furthermore, that incorporating simplified models of the noise correlation structure into the estimator will lead to only minor additional improvements. However, we note with reference to figure 2 that there could be cases where reweighting produces inferior results relative to the unweighted case.…”

Simultaneous deblurring and iterative reconstruction of CBCT for image guided brain radiosurgery

“…The weights of the reconstruction are calculated from the covariance matrix of the colored noise estimated from the deblurred projection model. A similar method is proposed in (Alessio et al 2007) for reconstructing positron emission tomography (PET) images from correlated data. To decrease the computational burden of the noise correlation in weighted least square, a block diagonal covariance matrix is used.…”