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.

Figure 1

Sample covariance matrices of a simple one-dimensional example. K̃_e includes correlations amongst measurements and K̂_e assumes independent data.

Figure 2

Objective functions for WLS estimators using full covariance matrix (dashed) and diagonal matrix (dotted).

Figure 3

Block diagonal portion of the covariance matrix, K_ŷs for the covariance known simulation study.

Figure 4

The digital phantom used for contrast and noise comparison of methods applied to FORE data.

Figure 5

Axial relationship, H, for the 18 ring (35 plane) scanner.

Figure 6

Portion of the transaxial 2D covariance matrix for a 70 × 90 FORE rebinned sinogram. Neighboring radial bins occur directly off the diagonal, while neighboring angular bins occur 70 points away from the diagonal.

Figure 7

Normalized value of objective function at each iteration of the proposed algorithms.

Figure 8

Average contrast versus noise of all features in the simulated phantoms for the axial-modified reconstructions using Φ_RD. The curve is formed from varying the smoothing parameter.

Figure 9

Average contrast versus noise of all features in the simulated phantoms for the transaxial-modified reconstructions using Φ_MRF.

Figure 10

A transaxial slice of reconstructions of the Derenzo phantom.

Figure 11

Profile of the horizontal line through reconstructions comparing proposed methods with conventional FBP and MAP. FBP (dashed); MAP (dotted); Φ_RD Axial (solid); Φ_MRF Trans (dash-dot).