On convergence in the methods of Strat and of Smith for Shape from Shading

摘要

We study the convergence of two iterative Shape from Shading methods: the methods of Strat and of Smith. We try to determine the spectral radius of the Jacobian matrix of each iteration at any possible fixed point. We show that the method of Strat diverges for any image containing at least four pixels forming a square, any reflectance map and any relative weight between the irradiance term and the integrability term. An example is provided, in which divergence occurs after a large number of iterations, even if the reconstructed surface approaches the real surface after only a few iterations. We show then by a similar way that the method of Smith diverges for any image containing at least nine pixels forming a square, any reflectance map and any relative weight between the irradiance term and the smoothing term.