grid $\OO$. However, even in the case of piecewise constant function whose (distribution) gradient is sparse in the continuous domain, its singularities may not necessarily agree with the discrete grid $\OO$, leading to the problem of *basis mismatch* [@Y.Chi2010; @G.Ongie2016; @J.Ying2017]. Such a basis mismatch between the true singularities and the discrete grid may result in the loss of sparsity of $\Ph\bu$, and thus degrades the performance of \[CSType\] [@J.Ying2017].

The direct restoration of continuous domain signal is emerging as a powerful alternative to the discrete domain CS approach [@B.N.Bhaskar2013; @E.J.Candes2014; @Y.Chen2014; @G.Ongie2018]. The ability of these “off-the-grid” schemes to allieviate the basis mismatch makes them especially attractive in signal restorations from partial Fourier samples [@G.Ongie2018]. One of the most successful examples is the finite-rate-of-innovation (FRI) framework [@T.Blu2008] which extends the Prony’s method [@Kay1988; @S.M.Kay1981; @Prony1795; @P.Stoica1997] to the so-called nonuniform “Dirac streams” with unknown knot locations. In addition, the recovery of Dirac streams from a few Fourier samples can be achieved by the atomic norm minimization [@E.J.Candes2013; @E.J.Candes2014; @G.Tang2015], which can be viewed as the continuous domain generalization of the CS framework. While these frameworks work well for one dimensional piecewise constant signals [@E.J.Candes2013] and two dimensional Dirac streams 