Shape from shading of SAR imagery in fourier space

A new shape from shading technique is presented in this study. This technique is a further development of Pentland's linear shape from shading technique (1990). The equations and geometric modeling are extracted for SAR imagery. However, the modeling can be easily adapted for optical images as the geometric modeling is much simpler than SAR images due to the fact that the geometric transformation from 3D object space to 2D image space in optical imagery is modeled by a well-estimated orthogonal projection or perspective projection itself. This solution for shape from shading problem employs linear estimation of the reflectance model using Taylor expansion. As there are one known image intensity and two unknown surface gradients for every pixel, there is no direct solution. By using a global approach and solving all the image pixels at the same time, it is possible to estimate an answer and overcome the ambiguity problem in the incidence angle estimation. For this purpose, the Fourier transform of the expanded reflectance model will be taken and as the Fourier transform of the surface gradients is a linear function of the surface height, the number of equations and unknowns will be equal and the system is soluble. In the proposed algorithm, an iterative approach is used to improve the accuracy of the estimated surface height. Iteration starts with a low resolution digital terrain model (DTM) of the area for the initial guess of the surface gradients and based on the equation system in Fourier space, the iteration continues until a threshold is reached or the number of iterations exceeds a tolerance. The technique is tested on a Radarsat image from Death Valley area and the results are shown and validated with the available DTM of the area. Based on the results, it seems that the proposed method is sensitive to the noise. The noise problem should be carefully considered either during the reflectance modeling or during solving the problem in the Fourier space as a filtering problem.