Boundary conditions and fast algorithms for surface reconstructions from synthetic aperture radar data

Most attempts to determine surface height from noiseless synthetic aperture radar (SAR) data involve approximating the surface by solving a related standard shape from shading (SFS) problem. Through analysis of the underlying partial differential equations for both the original SAR problem and the approximating standard SFS problem, the authors demonstrate significant differences between them. For example, if it is known that the surface is smooth, the standard SPS problem can generally be uniquely solved from knowledge of the height and concavity at one surface point, whereas for SAR, multiple valid solutions will generally exist unless height information is specified along entire curves on the surface (i.e., boundary conditions). Unlike the standard SFS approximation, the underlying SAR equation can be reexpressed as a time-dependent Hamilton-Jacobi equation. This transformation allows the authors to compute the correct surface topography from noiseless SAR data with boundary conditions extremely quickly. Finally, they consider the effect of radar noise on the computed surface reconstruction and discuss the ability of the presented PDE method to quickly compute an initial surface that will significantly cut the computational time needed by cost minimization algorithms to approximate surfaces from noisy radar data