Reconstruction of polygonal shapes from sparse Fourier samples

In this paper, we want to reconstruct polygonal shapes in the real plane from as few Fourier samples as possible, that is, we want to recover an original polygonal domain $D$ with $N$ vertices by using sparse sampling values of the Fourier transform of the characteristic function of the polygonal domain. We consider only simply‐connected polygons, i.e. polygons with non‐intersecting edges. For this purpose, we need to reconstruct the vertices of the polygon. In the case of non‐convex polygons, we also need to reconstruct the order of the vertices to determine the correct boundary line segments. The method presented here is based on the Prony method.