Subsampling approaches for generation of single lower-count image estimates derived from higher-count data.

Objectives Many researchers [1-7] have used bootstrap techniques for generating multiple samples of data from one or a few acquired list-mode data sets; this approach can be useful for characterizing the statistical properties of nuclear medicine images. Other investigators [8,9] have used binomial subsampling of image data from one scan of each of many subjects in a clinical study to generate a single noise realization of the subject's image that may be statistically consistent with a shorter acquisition time or lower-dose scan of that subject. This approach is fundamentally different than that of bootstrapping, so the properties of various resampled image estimators may also differ between these two cases. We have compared the performance of one parametric method, binomial subsampling, and two nonparametric methods, list-mode subsampling with and without replacement of selected events, by evaluating their ability to generate single estimates of lower-count images from each of many subjects that are consistent with the expected statistical distribution.

Methods We used a numerical phantom similar to that described in [1]; we simulated single-photon planar projections acquired end-on from a 20-cm-diameter cylinder containing three, 5-cm-diameter cylindrical compartments that yield, respectively, an average of 200, 150, and 100 counts per pixel, while the surrounding background yields 10 counts/pixel. We generated 10,000 spatially uncorrelated, independent Poisson-noise realizations of these images, each of which was also converted into a pseudo-list-mode data set. Images corresponding to 10%, 50%, and 90% of the initial total counts were produced by binomial subsampling of each pixel value using the Numerical Recipes BNLDEV routine with p=0.1, 0.5, and 0.9, respectively. The list-mode data were also randomly subsampled to the same three count levels both with and without replacement of selected events. Statistical moments (mean, variance, and skewness) of each pixel value was computed over the ensemble of images that were subsampled (one from each of the 10,000 initial data sets), and the moments were then averaged over each region and compared with expected values for uncorrelated Poisson noise. The spatial noise covariance was also computed at several different locations in the phantom. Results for some conditions were also compared to those obtained by calculating moments over image estimates that were all bootstrapped from a single noisy data set.

Results For all three levels of count reduction, the parametric binomial subsampling method and the nonparametric list-mode subsampling method without replacement of selected events yielded ensemble moments that were all extremely close to those expected for Poisson-distributed noise realizations of the lower-count images, whereas the variance and skewness of the image data produced by list-mode subsampling with replacement were, for all levels of count reduction, significantly larger than expected. The table lists the ensemble moments averaged over all four regions for each subsampling method for p = 0.5 (50% count reduction). The noise remained 'white' (spatially uncorrelated) for all three subsampling methods. On the other hand, when statistical moments were calculated, instead, over multiple image estimates that were all obtained from the same initial image, list-mode subsampling with replacement yielded more accurate values of variance and skewness than either of the other two subsampling methods.

Conclusions Binomial subsampling and list-mode subsampling without replacement both outperform list-mode subsampling with replacement for generating single lower-count images from an initial higher-count noisy image or list-mode file. The values of statistical moments calculated over ensembles of single lower-count images, each subsampled from its own higher-count data set, are different than those computed over multiple samples drawn from a single data set.