(JD). The JDs are then warped to a common coordinate system for the entire `ADNI` dataset using existing non-linear deformations from [@GMF_ISBI_matching; @GMF_ISBI_optimization]. Each such spatially normalized JD is then averaged within a region where the rate of atrophy is significantly associated with Alzheimer’s disease (AD), i.e., within a *statistical region of interest (stat-ROI)* (see Fig. \[fig:ROI\]). Specifically, we quantify atrophy as $$\left(1 - \frac{1}{|\omega|}  \int_{\omega} \text{det}(D \phi (x))~dx\right) \times 100$$ where $\text{det}(\cdot)$ denotes the determinant and $|\cdot|$ the cardinality/size of a set; $\omega$ is an area in the temporal lobes which was determined in prior studies [@GMF_ISBI_matching; @GMF_ISBI_optimization] to be significantly associated with accelerated atrophy in Alzheimer’s disease. The resulting scalar value is an estimate of the relative volume change experienced by that region between the baseline and a follow-up image. Hence, its sign is positive when the region has lost volume over time and is negative if the region has gained volume over time.

We limited our experiments to the applications in [@Xue_ADNI1; @Xue_ADNI2], wherein nonlinear registration/regression is used to quantify atrophy within regions known to be associated to varying degrees with AD ($2$), mild cognitive impairment (MCI) ($1$) (including LMCI[^5]), and normal ageing (NC: normal control) ($0$) 