  Spectrograms are fundamental tools in time-frequency analysis, being the
squared magnitude of the so-called short time Fourier transform (STFT). Signal
analysis via spectrograms has traditionally explored their peaks, i.e. their
maxima. This is complemented by a recent interest in their zeros or minima,
following seminal work by Flandrin and others, which exploits connections with
Gaussian analytic functions (GAFs). However, the zero sets (or extrema) of GAFs
have a complicated stochastic structure, complicating any direct theoretical
analysis. Standard techniques largely rely on statistical observables from the
analysis of spatial data, whose distributional properties for spectrograms are
mostly understood only at an empirical level. In this work, we investigate
spectrogram analysis via an examination of the stochastic geometric properties
of their level sets. We obtain rigorous theorems demonstrating the efficacy of
a spectrogram level sets based approach to the detection and estimation of
signals, framed in a concrete inferential set-up. Exploiting these ideas as
theoretical underpinnings, we propose a level sets based algorithm for signal
analysis that is intrinsic to given spectrogram data, and substantiate its
effectiveness via extensive empirical studies. Our results also have
theoretical implications for spectrogram zero based approaches to signal
analysis. To our knowledge, these results are arguably among the first to
provide a rigorous statistical understanding of signal detection and
reconstruction in this set up, complemented with provable guarantees on
detection thresholds and rates of convergence.
