  The aim of two-dimensional line spectral estimation is to super-resolve the
spectral point sources of the signal from time samples. In many associated
applications such as radar and sonar, due to cut-off and saturation regions in
electronic devices, some of the numbers of samples are corrupted by spiky
noise. To overcome this problem, we present a new convex program to
simultaneously estimate spectral point sources and spiky noise in two
dimensions. To prove uniqueness of the solution, it is sufficient to show that
a dual certificate exists. Construction of the dual certificate imposes a mild
condition on the separation of the spectral point sources. Also, the number of
spikes and detectable sparse sources are shown to be a logarithmic function of
the number of time samples. Simulation results confirm the conclusions of our
general theory.
