The Texture Gradient Equation for Recovering Shape from Texture

ÐThis paper studies the recovery of shape from texture under perspective projection. We regard Shape from Texture as a statistical estimation problem, the texture being the realization of a stochastic process. We introduce warplets, which generalize wavelets over the 2D affine group. At fine scales, the warpogram of the image obeys a transport equation, called Texture Gradient Equation. In order to recover the 3D shape of the surface, one must estimate the deformation gradient, which measures metric changes in the image. This is made possible by imposing a notion of homogeneity for the original texture, according to which the deformation gradient is equal to the velocity of the Texture Gradient Equation. By measuring the warplet transform of the image at different scales, we obtain a deformation gradient estimator.