We propose a multiscale scanning method to determine active components of a quantity of interest w.r.t. a dictionary from observations coming from an inverse regression model. Based on a Gaussian approximation of the underlying multiscale statistic we provide uniform confidence statements for the coefficients under suitable smoothness assumptions on the dictionary. This gives then rise to a multiple test that allows to identify the active components at controlled, family-wise error rate. We furthermore show that this test has certain optimality properties.
In this talk we focus on the important special case of deconvolution and support our theory by simulations. As one particular application we illustrate the potential of the method as an inferential tool for imaging and discuss applicability to super-resolution microscopy. In this setting, we are able to apply our method to localize fluorescent markers in a real data example where we are able to discern objects below the resolution level of the corresponding microscope.
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