Multiresolution analysis on compact Riemannian manifolds

The problem of representation and analysis of manifold defined functions (signals, images, and data in general) is ubiquities in neuroscience, medical and biological applications. In the context of modeling the computations of the cortex, some twenty years ago, Mumford noted: “... the set of higher level concepts will automatically have geometric structure”. Indeed, in Vision input images can be thought of as points in a high-dimensional measurement space (with each input dimension corresponding to the activity of retinal ganglion cells whose axons project from the eye to the brain), however, perceptually meaningful structures lay on a manifold embedded in this space [53]. In a general context, structural and functional connectivity of the brain are usually described within network theory [Chapters XX in this volume by D. Bassett et al. [4]; by M. Pesenson [49]; and by P. Ninez et al. [36]]. However, when a network of perceptual neurons can be considered as a discrete approximation of a manifold, multiresolution analysis of manifold defined functions becomes a powerful tool. In the last decade, the importance of these and other applications triggered the development of various generalized wavelet bases suitable for the unit spheres S and S and the rotation group of R. The goal of the present study is to describe a generalization of those approaches by constructing bandlimited and localized frames in a space L2(M), whereM is a compact Riemannian manifold. The following classes of manifolds will be considered: compact manifolds without boundary, compact homogeneous manifolds, bounded domains with smooth boundaries in Euclidean spaces. One can think of a manifold as of a surface in a Euclidean space. A homogeneous manifold is a surface with ”many” symmetries like the sphere x1 + ... + x 2 d = 1 in Euclidean space R. An important example of a bounded domain is a ball x1 + ...+ x 2 d ≤ 1 in R. As it will be demonstrated below, our construction of frames in a function space L2(M) heavily depends on a proper sampling of a manifold M itself. However, it is worth stressing that our main objective is the sampling of functions on manifolds. In sections 26 it is shown how to construct on a compact manifold (with or without boundary) a ”nearly” tight bandlimited and strongly localized frame. In other words, on a very fine scale members of our frame look almost like Dirac measures. In section 7 we consider the case of compact homogeneous manifolds, i.e. which have the form M = G/H , where G is a compact Lie group and H is its closed subgroup. For such manifolds we are able to construct tight, bandlimited and