On estimating the variance of smoothed MLEM images

The maximum-likelihood expectation-maximization (MLEM) algorithm is being used increasingly as a routine reconstruction algorithm in positron emission tomography (PET). The response of the MLEM reconstruction algorithm is nonlinear and not shift-invariant. As a result, two problems occur when MLEM is used for tracer kinetic modeling. First, resolution and recovery may be different for different positions and for different time frames, adversely affecting the modeling results. Second, it is not trivial to determine appropriate weights for weighted least squares fitting to image derived time-activity curves. The first problem can be remedied by applying a "relatively high" number of iterations, followed by smoothing with a shift-invariant kernel. The smoothing kernel reduces the noise and imposes an (approximately) constant resolution. For the second problem, different approaches exist. Some authors have shown that the variance of pixel values in MLEM images is proportional to the mean of the pixel value. Others have suggested that the uncertainty about a reconstructed pixel value can be estimated from the Fisher information matrix. These two approaches produce different results. Our simulation study confirms that the pixel variances of unsmoothed MLEM images are approximately proportional to the pixel values. However, smoothing reduces the variance more in high count regions than in low count regions. As a result, the variances of smoothed MLEM pixel values correlate better with the reciprocal of the diagonal elements of the Fisher information matrix. Smoothing strongly affects the variance because neighboring pixel values are highly correlated. Because computing the mean of a region-of-interest is a smoothing operation, the Fisher information can be used to compute appropriate weight values for model based analysis of time-activity curves.