Multiscale Representations for Manifold-Valued Data

• Published in: Multiscale modeling & simulation Vol. 4; no. 4; pp. 1201 - 1232
• Main Authors: Rahman, Inam Ur, Drori, Iddo, Stodden, Victoria C., Donoho, David L., Schröder, Peter
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2005
• Subjects: Algorithms; Arrays; Deformation; Lie groups; Noise; Signal processing; Software; Time series; Wavelet transforms
• ISSN: 1540-3459; 1540-3467
• DOI: 10.1137/050622729

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, 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 in much the same way 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 $S^{n-1}$, $SO(n)$, $G(n,k)$, 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.