  We develop a multiscale scanning method to find anomalies in a
$d$-dimensional random field in the presence of nuisance parameters. This
covers the common situation that either the baseline-level or additional
parameters such as the variance are unknown and have to be estimated from the
data. We argue that state of the art approaches to determine asymptotically
correct critical values for multiscale scanning statistics will in general fail
when such parameters are naively replaced by plug-in estimators. Opposed to
this, we suggest to estimate the nuisance parameters on the largest scale and
to use (only) smaller scales for multiscale scanning. We prove a uniform
invariance principle for the resulting adjusted multiscale statistic (AMS),
which is widely applicable and provides a computationally feasible way to
simulate asymptotically correct critical values. We illustrate the implications
of our theoretical results in a simulation study and in a real data example
from super-resolution STED microscopy. This allows us to identify interesting
regions inside a specimen in a pre-scan with controlled family-wise error rate.
