Multiscale representations for manifold-valued data

Inam Ur Rahman, Iddo Drori, Victoria C. Stodden, David L. Donoho, Peter Schröder

Research output: Contribution to journalArticle

Abstract

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 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 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.

Original languageEnglish (US)
Pages (from-to)1201-1232
Number of pages32
JournalMultiscale Modeling and Simulation
Volume4
Issue number4
DOIs
StatePublished - Dec 1 2005
Externally publishedYes

Fingerprint

wavelet
compression
Wavelet Coefficients
software
matrix
simulation
Tangent plane
Grassmann Manifold
Contrast Enhancement
Riemannian Symmetric Space
Noise Removal
Positive definite matrix
Orthogonal Group
Stochastic Simulation
Pyramid
coefficients
pyramids
tangents
Compression
Grid

Keywords

  • Compression
  • Denoising
  • Lie group
  • Nonlinear refinement scheme
  • Symmetric space
  • Two-scale refinement scheme
  • Wavelets

ASJC Scopus subject areas

  • Chemistry(all)
  • Modeling and Simulation
  • Ecological Modeling
  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this

Multiscale representations for manifold-valued data. / Rahman, Inam Ur; Drori, Iddo; Stodden, Victoria C.; Donoho, David L.; Schröder, Peter.

In: Multiscale Modeling and Simulation, Vol. 4, No. 4, 01.12.2005, p. 1201-1232.

Research output: Contribution to journalArticle

Rahman, IU, Drori, I, Stodden, VC, Donoho, DL & Schröder, P 2005, 'Multiscale representations for manifold-valued data', Multiscale Modeling and Simulation, vol. 4, no. 4, pp. 1201-1232. https://doi.org/10.1137/050622729
Rahman, Inam Ur ; Drori, Iddo ; Stodden, Victoria C. ; Donoho, David L. ; Schröder, Peter. / Multiscale representations for manifold-valued data. In: Multiscale Modeling and Simulation. 2005 ; Vol. 4, No. 4. pp. 1201-1232.
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