Viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape-from-shading equations with discontinuous intensities

Abstract The Shape-From-Shading equation relating u(y, r), the unknown (angular) height of a surface, to I(y, r), the known Synthetic Aperture Radar intensity data from the surface, is where y and r are axial and radial cylindrical coordinates. Unlike the more common eikonal Shape-From-Shading equation needed to relate surface height in Cartesian coordinates to optical/photographic intensity data, the above radar equation can be transformed into Hamilton-Jacobi Cauchy form: u r + g(I, u y ) = 0. We explore the case where I is a discontinuous function, which occurs commonly in radar data. By considering sequences of continuous intensity functions that converge to I, we obtain corresponding sequences of viscosity solutions. We prove that these sequences must converge. We also establish conditions that guarantee that these sequences converge to a common limit, which we define as the solution to the radar equation. Finally, we establish and demonstrate that when this common limit exists, monotone numerical schemes must converge to this solution as the mesh size decreases

An approximation scheme for an Eikonal Equation with discontinuous coefficient

We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a-priori error estimate for the scheme in an integral norm. The last section contains some applications to control and image processing problems