### Abstract

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.

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 |

### Fingerprint

### 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 Modeling and Simulation*,

*4*(4), 1201-1232. https://doi.org/10.1137/050622729

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

Research output: Contribution to journal › Article

*Multiscale Modeling and Simulation*, vol. 4, no. 4, pp. 1201-1232. https://doi.org/10.1137/050622729

}

TY - JOUR

T1 - Multiscale representations for manifold-valued data

AU - Rahman, Inam Ur

AU - Drori, Iddo

AU - Stodden, Victoria

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

VL - 4

SP - 1201

EP - 1232

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 4

ER -