Deconvolving the Wedge: Maximum-Likelihood Power Spectra via
  Spherical-Wave Visibility Modeling
  Direct detection of the Epoch of Reionization (EoR) via the red-shifted 21-cm
line will have unprecedented implications on the study of structure formation
in the infant Universe. To fulfill this promise, current and future 21-cm
experiments need to detect this weak EoR signal in the presence of foregrounds
that are several orders of magnitude larger. This requires extreme noise
control and improved wide-field high dynamic-range imaging techniques. We
propose a new imaging method based on a maximum likelihood framework which
solves for the interferometric equation directly on the sphere, or equivalently
in the $uvw$-domain. The method uses the one-to-one relation between spherical
waves and spherical harmonics (SpH). It consistently handles signals from the
entire sky, and does not require a $w$-term correction. The spherical-harmonics
coefficients represent the sky-brightness distribution and the visibilities in
the $uvw$-domain, and provide a direct estimate of the spatial power spectrum.
Using these spectrally-smooth SpH coefficients, bright foregrounds can be
removed from the signal, including their side-lobe noise, which is one of the
limiting factors in high dynamics range wide-field imaging. Chromatic effects
causing the so-called "wedge" are effectively eliminated (i.e. deconvolved) in
the cylindrical ($k_{\perp}, k_{\parallel}$) power spectrum, compared to a
power spectrum computed directly from the images of the foreground visibilities
where the wedge is clearly present. We illustrate our method using simulated
LOFAR observations, finding an excellent reconstruction of the input EoR signal
with minimal bias.
