  Astronomy, biophysics, and material science often depend on the possibility
to extract information out of faint spatial signals. Here we present a
morphometric analysis technique to quantify the shape of structural deviations
in greyscale images. It identifies important features in noisy spatial data,
especially for short observation times and low statistics. Without assuming any
prior knowledge about potential sources, the additional shape information can
increase the sensitivity by 14 orders of magnitude compared to previous
methods. Rejection rates can increase by an order of magnitude. As a key
ingredient to such a dramatic increase, we accurately describe the distribution
of the homogeneous background noise in terms of the density of states
$\Omega(A,P,\chi)$ for the area $A$, perimeter $P$, and Euler characteristic
$\chi$ of random black-and-white images. The technique is successfully applied
to data of the H.E.S.S. experiment for the detection of faint extended sources.
