Abstract: Mapping landforms on the surface of the Moon and the planets is important for resources exploration and future human settlements, and it presents interesting and difficult mathematical challenges. Laser Altimeters provide low-resolution elevation maps with pixel sizes of up to hundreds of meters, but the topographic maps can be upscaled to a higher resolution using images from orbiters with high-quality cameras. The method exploits the relation between topographic gradients and degrees of shading of incoming sunlight. 

Earlier published methods are iterative, computationally demanding, and do not provide the reliable uncertainty quantification needed for risk evaluation in connection with spacecraft landing. We propose a 2-step linear method with high computational efficiency and built-in Bayesian uncertainty estimation. Its efficiency is derived from formulating the problem through the Sylvester Equation, which is linear, but not solvable through well-known, classical techniques.

Our the method operates on multiple images and incorporates albedo variations. It avoids/reduces the use of arbitrary tuning parameters, since all weighting of data and model parameters is based data uncertainties and reasonable bounds on the model. Applied to Lunar data, our results increase the resolution of the topography from ∼60 ​m per pixel to 0.9 ​m per pixel, bringing it to the same pixel resolution as optical images, allowing in some cases detection of craters as small as ∼3 ​m of diameter. We estimate uncertainties of the topographic model due to noise in the images, and in the low-resolution model.