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Multiscale Representations for Manifold-Valued Data

Ur Rahman, Inam and Drori, Iddo and Stodden, Victoria C. and Donoho, David L. and Schröder, Peter (2005) Multiscale Representations for Manifold-Valued Data. Multiscale Modeling and Simulation, 4 (4). pp. 1201-1232. ISSN 1540-3459. doi:10.1137/050622729.

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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.

Item Type:Article
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Donoho, David L.0000-0003-1830-710X
Schröder, Peter0000-0002-0323-7674
Additional Information:©2005 Society for Industrial and Applied Mathematics. Reprinted with permission. Received by the editors January 16, 2005; accepted for publication (in revised form) May 18, 2005; published electronically December 7, 2005. This work was partially supported by NSF DMS 00-77261 and 01-40698 (FRG), NIH, and ONR-MURI. It was also supported in part by NSF (DMS-0220905, DMS-0138458, ACI-0219979), DOE (W-7405-ENG-48/B341492), the Packard Foundation, the Center for Integrated Multiscale Modeling and Simulation, nVidia, Alias, and Pixar. We thank Roberto Altschul and Sabyasachi Basu from Boeing for data on aircraft orientations. We thank Robert Dougherty and Brian Wandell from Stanford for the magnetic resonance data (R01 EY015000). DLD would like to thank Leo Guibas and Claire Tomlin of Stanford University for helpful discussions, as well as Nira Dyn of Tel-Aviv University, John F. Hughes of Brown University, and Jonathan Kaplan of Harvard University.
Subject Keywords:symmetric space; Lie group; two-scale refinement scheme; wavelets; compression; denoising; nonlinear refinement scheme
Issue or Number:4
Record Number:CaltechAUTHORS:RAHmms05
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9402
Deposited By: Archive Administrator
Deposited On:18 Dec 2007
Last Modified:08 Nov 2021 20:59

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