  In this paper, we extend the $\beta$-CNMF to two dimensions and derive exact
multiplicative updates for its factors. The new updates generalize and correct
the nonnegative matrix factor deconvolution previously proposed by Schmidt and
M{\o}rup. We show by simulation that the updates lead to a monotonically
decreasing $\beta$-divergence in terms of the mean and the standard deviation
and that the corresponding convergence curves are consistent across the most
common values for $\beta$.
