  We present a framework for robust electric network frequency (ENF) extraction
from real-world audio recordings, featuring multi-tone ENF harmonic enhancement
and graph-based optimal harmonic selection. Specifically, We first extend the
recently developed single-tone ENF signal enhancement method to the multi-tone
scenario and propose a harmonic robust filtering algorithm (HRFA). It can
respectively enhance each harmonic component without cross-component
interference, thus further alleviating the effects of unwanted noise and audio
content on the much weaker ENF signal. In addition, considering the fact that
some harmonic components could be severely corrupted even after enhancement,
disturbing rather than facilitating ENF estimation, we propose a graph-based
harmonic selection algorithm (GHSA), which finds the optimal combination of
harmonic components for more accurate ENF estimation. Noticeably, the harmonic
selection problem is equivalently formulated as a maximum weight clique (MWC)
problem in graph theory, and the Bron-Kerbosch algorithm (BKA) is adopted in
the GHSA. With the enhanced and optimally selected harmonic components, both
the existing maximum likelihood estimator (MLE) and weighted MLE (WMLE) are
incorporated to yield the final ENF estimation results. The proposed framework
is extensively evaluated using both synthetic signals and our ENF-WHU dataset
consisting of $130$ real-world audio recordings, demonstrating substantially
improved capability of extracting the ENF from realistically noisy observations
over the existing single- and multi-tone competitors. This work further
improves the applicability of the ENF as a forensic criterion in real-world
situations.
