  In this paper we propose a multiscale scanning method to determine active
components of a quantity $f$ w.r.t. a dictionary $\mathcal{U}$ from
observations $Y$ in an inverse regression model $Y=Tf+\xi$ with linear operator
$T$ and general random error $\xi$. To this end, we provide uniform confidence
statements for the coefficients $\langle \varphi, f\rangle$, $\varphi \in
\mathcal U$, under the assumption that $(T^*)^{-1} \left(\mathcal U\right)$ is
of wavelet-type. Based on this we obtain a multiple test that allows to
identify the active components of $\mathcal{U}$, i.e. $\left\langle f,
\varphi\right\rangle \neq 0$, $\varphi \in \mathcal U$, at controlled,
family-wise error rate. Our results rely on a Gaussian approximation of the
underlying multiscale statistic with a novel scale penalty adapted to the
ill-posedness of the problem. The scale penalty furthermore ensures weak
convergence of the statistic's distribution towards a Gumbel limit under
reasonable assumptions. The important special cases of tomography and
deconvolution are discussed in detail. Further, the regression case, when $T =
\text{id}$ and the dictionary consists of moving windows of various sizes
(scales), is included, generalizing previous results for this setting. We show
that our method obeys an oracle optimality, i.e. it attains the same asymptotic
power as a single-scale testing procedure at the correct scale. Simulations
support our theory and we illustrate the potential of the method as an
inferential tool for imaging. As a particular application we discuss
super-resolution microscopy and analyze experimental STED data to locate single
DNA origami.
