  Classical cluster inference is hampered by the spatial specificity paradox.
Given the null-hypothesis of no active voxels, the alternative hypothesis
states that there is at least one active voxel in a cluster. Hence, the larger
the cluster the less we know about where activation in the cluster is.
Rosenblatt et al. (2018) proposed a post-hoc inference method, All-resolutions
Inference (ARI), that addresses this paradox by estimating the number of active
voxels of any brain region. ARI allows users to choose arbitrary brain regions
and returns a simultaneous lower confidence bound of the true discovery
proportion (TDP) for each of them, retaining control of the family-wise error
rate. ARI does not, however, guide users to regions with high enough TDP. In
this paper, we propose an efficient algorithm that outputs all maximal
supra-threshold clusters, for which ARI gives a TDP lower confidence bound that
is at least a chosen threshold, for any number of thresholds that need not be
chosen a priori nor all at once. After a preprocessing step in linearithmic
time, the algorithm only takes linear time in the size of its output. We
demonstrate the algorithm with an application to two fMRI datasets. For both
datasets, we found several clusters whose TDP confidently meets or exceeds a
given threshold in less than a second.
