Some remarks on shape from shading (Q1902816)

Let \(\Sigma\) be a smooth surface in \(\mathbb{R}^3\), if \(\Sigma\) is made of a perfect material and is both illuminated and viewed from afar, its image will exhibit a shading from which one can attempt to reconstruct the surface itself. This is possible trying to solve the partial differential equation \[ f^2_x + f^2_y = W \tag{1} \] where \(\Sigma\) is locally the graph of \(f\) and \(W\) is a given function. In this paper, local and global uniqueness results are proved for the solutions of (1), under a nondegeneration assumption on the critical points of \(W\).