Multiscale Representations for Manifold-Valued Data

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Abstract
ODER† We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as: the sphere S2, the special orthogonal group SO(3), the positive definite matrices SP D(n), and the Grassmann manifolds G(n, k). The representations are based on the deployment of Deslauriers-Dubuc and Average-Interpolating pyramids 'in the tangent plane' of such manifolds, using the Exp and Log maps of those manifolds. The representations provide 'wavelet coefficients' which can be thresholded, quantized, and scaled much as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds, but is particularly suited to the manifolds we consider, i.e. Riemannian symmetric spaces, such as Sn−1, SO(n), G(n, k), where the Exp and Log maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.

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