Bias in ROI estimators and an unbiased solution

Signal activity is typically estimated by summing voxels from a reconstructed image. We introduce an alternative estimation scheme that operates on the raw projection data and offers a substantial improvement, as measured by the ensemble-mean squared error (EMSE), when compared to using voxel values from a maximum-likelihood expectation-maximization (MLEM) reconstructed ROI. The scanning-linear estimator is derived as a special case of maximum-likelihood (ML) techniques with a series of approximations to make the calculation tractable. The approximated likelihood accounts for background randomness, measurement noise, and variability in the signal’s activity. The resulting estimate of the signal activity is an unbiased estimator: the average estimate equals the true value. By contrast, algorithms that operate on reconstructed data are subject to unpredictable bias arising from the null functions of the imaging system and the object. Using visual inspection of reconstructed data to select an ROI is tantamount to estimating a location and size of the signal. In general, this procedure would be less than ideal, but we remove this source of error by estimating the activity of a spherical signal whose radius and centroid are known. The signal shape and location fully specify a binary ROI template in object space. Although the scanning-linear method can be generalized to more complicated estimation tasks, we will demonstrate its use for estimating only signal amplitude. Noisy projection data are realistically emulated using measured calibration data from the multi-module multi-resolution (M3R) small-animal SPECT imaging system. The scanning-linear estimate of signal activity is computed for 800 image samples. The same set of images are reconstructed using the MLEM algorithm (80 iterations), and the mean as well as the maximum value within the ROI is calculated.