  In brain oncology, it is routine to evaluate the progress or remission of the
disease based on the differences between a pre-treatment and a post-treatment
Positron Emission Tomography (PET) scan. Background adjustment is necessary to
reduce confounding by tissue-dependent changes not related to the disease. When
modeling the voxel intensities for the two scans as a bivariate Gaussian
mixture, background adjustment translates into standardizing the mixture at
each voxel, while tumor lesions present themselves as outliers to be detected.
In this paper, we address the question of how to standardize the mixture to a
standard multivariate normal distribution, so that the outliers (i.e., tumor
lesions) can be detected using a statistical test. We show theoretically and
numerically that the tail distribution of the standardized scores is favorably
close to standard normal in a wide range of scenarios while being conservative
at the tails, validating voxelwise hypothesis testing based on standardized
scores. To address standardization in spatially heterogeneous image data, we
propose a spatial and robust multivariate expectation-maximization (EM)
algorithm, where prior class membership probabilities are provided by
transformation of spatial probability template maps and the estimation of the
class mean and covariances are robust to outliers. Simulations in both
univariate and bivariate cases suggest that standardized scores with soft
assignment have tail probabilities that are either very close to or more
conservative than standard normal. The proposed methods are applied to a real
data set from a PET phantom experiment, yet they are generic and can be used in
other contexts.
