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 appearance manifold" (IAM) in a high-dimensional ambient space. In this talk, we will expand on the observation that typical IAMs are not differentiable, in particular if the images contain sharp edges. However, all is not lost, since IAMs have an intrinsic multiscale geometric structure. In fact, each IAM has a family of approximate tangent spaces, each one good at a certain resolution. In the first part of the talk, we will focus on the particular inverse problem of estimating, from a given image on or near an IAM, the underlying parameters that produced it. Putting the multiscale 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. This algorithm is reminiscent of recently proposed algorithms for multiscale image registration and super-resolution. In the second part of the talk, we will explore IAMs in the context of "Compressive Imaging" (CI), where we attempt to recover an image from a small number of (potentially random) projections. To date, CI has focused on sparsity-based image models; we will discuss how IAM models could offer better performance for geometry-rich images. This is joint work with Michael Wakin, Hyeokho Choi, and David Donoho.