# Multiscale Representations for Manifold-Valued Data

@article{Rahman2005MultiscaleRF, title={Multiscale Representations for Manifold-Valued Data}, author={Inam Ur Rahman and Iddo Drori and Victoria Stodden and David L. Donoho and Peter Schr{\"o}der}, journal={Multiscale Model. Simul.}, year={2005}, volume={4}, pages={1201-1232} }

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… Expand

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#### References

SHOWING 1-10 OF 63 REFERENCES

Smoothness Analysis of Nonlinear Subdivision Schemes of Homogeneous and Affine Invariant Type

- Mathematics
- 2005

Abstract
Nonlinear subdivision schemes
arise from, among other applications, nonlinear multiscale signal processing
and shape preserving interpolation. For the univariate homogeneous
subdivision… Expand

Convergence and C1 analysis of subdivision schemes on manifolds by proximity

- Mathematics, Computer Science
- Comput. Aided Geom. Des.
- 2005

This work verifies that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition, and shows that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property. Expand

Smoothness of Nonlinear Median-Interpolation Subdivision

- Mathematics, Computer Science
- Adv. Comput. Math.
- 2004

We give a refined analysis of the Hölder regularity for the limit functions arising from a nonlinear pyramid algorithm for robust removal of non-Gaussian noise proposed by Donoho and Yu [6,7,17]. The… Expand

Nonlinear Pyramid Transforms Based on Median-Interpolation

- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 2000

Analytic and computational results are presented to show that in the presence of highly non-Gaussian noise, the coefficients of the nonlinear transform have much better properties than traditional wavelet coeffi- cients. Expand

On a Linearization Principle for Nonlinear p-mean Subdivision Schemes

- Mathematics
- 2003

In a recent preprint [6], the authors consider, motivated by nonlinear signal processing, a family of nonlinear subdivision operators based on interpolation-imputation of p-means. A linearization… Expand

Continuous M-Estimators and Their Interpolation by Polynomials

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2004

A host of variants of the robust nonlinear pyramid transforms proposed by Donoho and Yu are constructed and the inverse problem of interpolating a univariate polynomial of degree n with n + 1 prescribed values for any given continuous M-estimator on n +1 nonoverlapping intervals is a well-posed procedure. Expand

Smooth Wavelet Decompositions with Blocky Coefficient Kernels

- Mathematics
- 1993

We describe bases of smooth wavelets where the coe cients are obtained by integration against ( nite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this… Expand

A comprehensive introduction to differential geometry

- Physics
- 1975

Spivak's Comprehensive introduction takes as its theme the classical roots of contemporary differential geometry. Spivak explains his Main Premise (my term) as follows: "in order for an introduction… Expand

The Rotation of Eigenvectors by a Perturbation. III

- Mathematics
- 1970

When a Hermitian linear operator is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding… Expand

Spline functions and the theory of wavelets

- Mathematics
- 1999

Spline Functions: Introduction and summary by H. Brunner Radial extensions of vertex data by L. P. Bos and D. Holland The use of splines in the numerical solutions of differential and Volterra… Expand