  The authors discuss how general regularization schemes, in particular linear
regularization schemes and projection schemes, can be used to design tests for
signal detection in statistical inverse problems. It is shown that such tests
can attain the minimax separation rates when the regularization parameter is
chosen appropriately. It is also shown how to modify these tests in order to
obtain (up to a $\log\log$ factor) a test which adapts to the unknown
smoothness in the alternative. Moreover, the authors discuss how the so-called
\emph{direct} and \emph{indirect} tests are related via interpolation
properties.
