Isotropic Regularization

Regularization of ill-posed problems has been applied in various ways to surface reconstruction problems. Typically the smoothness term has been found in an ad hoc manner. In this work the a priori knowledge of the surface is used to construct the smoothness terms. If all surface normals are considered equally probable, different smoothness terms can be derived for different projections using Bayesian estimation. These smoothness terms implement discontinuous regularization by the Lorentzian estimator under orthographic and perspective projection, and imply a convex solution space, if the standard deviation of the noise is smaller than some quantity. Under stereo projection, occluded areas are punished dependent on the distance from the cameras, as they are more probable to exist on nearby objects than on distant objects. A general scheme of developing smoothness terms under assumptions of isotropy or anisotropy is outlined.