Let us consider the ‘th ROI which has been wrongly aligned to a phase-specific template at time ; in other words, the motion parameter has been wrongly estimated so that the ’th volume should be rejected as a volume outlier by setting the missingness as . Let be a voxel with fixed coordinates inside the ‘th ROI of the template (‘0’ denotes a template volume), and let be the actual location at time corresponding to the voxel . The volume misalignment leads to the spatial mismatch between and . For the sake of simplicity, the rotational components of volume misalignment are considered negligible. The geometric distance between two positions is then equivalent to the measurement error of object motion; that is, . As a result, the actual area of the ROI at time is updated as , and consequently the background at time would be given as where is a set of whole voxels. Finally, the volume misalignment leads to the spatial mismatch between and as illustrated in Fig. 4.