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 language | English (US) |
|---|---|
| Pages (from-to) | 1201-1232 |
| Number of pages | 32 |
| Journal | Multiscale Modeling and Simulation |
| Volume | 4 |
| Issue number | 4 |
| DOIs | |
| State | Published - Dec 1 2005 |
| Externally published | Yes |
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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 journal › Article
}
TY - JOUR
T1 - Multiscale representations for manifold-valued data
AU - Rahman, Inam Ur
AU - Drori, Iddo
AU - Stodden, Victoria C.
AU - Donoho, David L.
AU - Schröder, Peter
PY - 2005/12/1
Y1 - 2005/12/1
N2 - 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.
AB - 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.
KW - Compression
KW - Denoising
KW - Lie group
KW - Nonlinear refinement scheme
KW - Symmetric space
KW - Two-scale refinement scheme
KW - Wavelets
UR - http://www.scopus.com/inward/record.url?scp=33745778314&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=33745778314&partnerID=8YFLogxK
U2 - 10.1137/050622729
DO - 10.1137/050622729
M3 - Article
AN - SCOPUS:33745778314
VL - 4
SP - 1201
EP - 1232
JO - Multiscale Modeling and Simulation
JF - Multiscale Modeling and Simulation
SN - 1540-3459
IS - 4
ER -