The images generated by varying the underlying articulation parameters of an object (pose, attitude, light source position, and so on) can be viewed as points on a low-dimensional image parameter articulation manifold (IPAM) in a high-dimensional ambient space. In this paper, we develop theory and methods for the inverse problem of estimating, from a given image on or near an IPAM, the underlying parameters that produced it. Our approach is centered on the observation that, while typical image manifolds are not differentiable, they have an intrinsic multiscale geometric structure. In fact, each IPAM has a family of approximate tangent spaces, each one good at a certain resolution. Putting this structural aspect to work, we develop a new algorithm for high-accuracy parameter estimation based on a coarse-to-fine Newton iteration through the family of approximate tangent spaces. We test the algorithm in several idealized registration and pose estimation problems.