Sea surface wave spectrum measurements are necessary for a host of basic research questions as well as for engineering and societal needs. However, most measurement techniques require great investment in infrastructure and time-intensive deployment techniques. We propose a new approach of wave measurement from standard video footage recorded by low-cost Unmanned Aerial Vehicles (UAV). We address UAV nadir imagery, which are particularly simple to obtain operationally. The method relies on the fact that optical contrast of surface gravity waves is proportional to their steepness. We present a robust methodology of regularized inversion of the optical imagery spectra, resulting in retrieval of the three-dimensional wavenumber-frequency sea surface height spectrum. The system was tested in several sea trials and in different bathymetric depths and sea state conditions. The resulting wave bulk parameters and spectral characteristics are in good agreement with collocated measurements from wave buoys and bottom-mounted acoustic sensors. Simple deployment, mobility, and flexibility in spatial coverage show a great potential of UAVs to significantly enhance the availability of wave measurements.