[@W.Xu2014], the extension to the (piecewise smooth) image restoration is not straightforward [@G.Ongie2016]. This is because the image singularities such as the edges and ridges in general form a continuous curve in a two dimensional domain, which violates the separation condition [@E.J.Candes2014] for the recovery guarantee.

Recently, there are several extensions of the FRI framework to the two dimensional continuous domain image restoration. Such extensions include the piecewise holomorphic complex image restoration [@H.Pan2014] and the piecewise constant real image restorations [@G.Ongie2018; @G.Ongie2015a; @G.Ongie2015; @G.Ongie2016]. These approaches are commonly based on the assumption that the singularity curves, i.e. the supports of the (real/complex) derivatives of a target image, lie in the zero level set of a band-limited periodic function, called the *annihilating polynomial*. The annihilation relation of the Fourier transform of the derivatives by convolution with the Fourier coefficients of the annihilating polynomial is then derived to recover the image singularities and eventually the image [@G.Ongie2016]. In addition, the authors in [@G.Ongie2016] established the necessary and sufficient conditions for the restoration of the edge set and the edge set aware recovery of the piecewise constant images from a finite number of uniformly sampled low frequency k-space data.

Given that the edges 