Abstract
Wavelets can be interpreted as coherent states and this group theoretic approach is very successful. Unfortunately, this approach fails on spheres and other manifolds. In this paper we concentrate on two-dimensional manifolds and construct continuous and discrete wavelets transforms and frames in L 2-spaces on them. By exploring a fundamental fact that every two-dimensional manifold is locally conformally equivalent to \(\mathbb{R}^{2}\) we transfer wavelets and frames from \(L_{2}(\mathbb{R}^{2})\) by using conformal maps. We are able to transfer continuous and discrete wavelet transforms from \(\mathbb{R}^{2}\) to a manifold in the case when a corresponding conformal map has no singular points. Otherwise, only frames can be transferred.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
By our definition the conformal factor e 2u > 0 is strictly positive. If a conformal factor is allowed to have zeros, then the map is called “weakly conformal.”
- 2.
An immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.
- 3.
In the differential geometry of surfaces, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvature in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction. The name “umbilic” comes from the Latin umbilicus - navel [29].
References
S.T. Ali, J.-P. Antoine, J.P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer Science + Business Media, New York, 2014)
J.-P. Antoine, P. Vandergheynst, Wavelets on the n-sphere and other manifolds. J. Math. Phys. 39, 3987–4008 (1998)
J.-P. Antoine, P. Vandergheynst, Wavelets on the 2-sphere: a group-theoretical approach. Appl. Comput. Harmon. Anal. 7, 262–291 (1999)
J.-P. Antoine, L. Demanet, L. Jacques, P. Vandergheynst, Wavelets on the sphere: implementation and approximations. Appl. Comput. Harmon. Anal. 13 (3), 177–200 (2002)
J.-P. Antoine, R. Murenzi, P. Vandergheynst, S.T. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge University Press, Cambridge, 2004)
J.-P. Antoine, D. Roşca, P. Vandergheynst, Wavelet transform on manifolds: old and new approaches. Appl. Comput. Harmon. Anal. 28, 189–202 (2010)
C. Bär, Elementare Differentialgeometrie (De Gruyter, Berlin, 2010)
S. Bernstein, A Lie group approach to diffusive wavelets, SampTA 2013, Available via Eurasip. http://www.eurasip.org/Proceedings/Ext/SampTA2013/proceedings.html
J. Bremer, R. Coifman, M. Maggioni, A. Szlam, Diffusion wavelet packets. Appl. Comput. Harmon. Anal. 21 (1), 95–112 (2006)
P.G. Casazza, G. Kutyniok, Frames of subspaces, in Wavelets, Frames and Operator Theory. Contemporary Mathematics, vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113
O. Christensen, Frames and Bases: An Introductory Course (Birkhäuser, Boston, 2008)
R.R. Coifman, M. Maggioni, Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53–94 (2006)
I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)
S. Ebert, J. Wirth, Diffusive wavelets on groups and homogeneous spaces. Proc. R. Soc. Edinb. A Math. 141 (3), 497–520, (2011)
M. Ferreira, Continuous wavelet transforms on the unit sphere, Ph.D. thesis, University of Aveiro, 2008
M. Ferreira, Spherical continuous wavelet transforms arising from sections of the Lorentz group. Appl. Comput. Harmon. Anal. 26 (2), 212–229 (2009)
W. Freeden, U. Windheuser, Combined spherical harmonic and wavelet expansion – a future concept on Earth’s gravitational determination. Appl. Comput. Harmon. Anal. 4, 1–37 (1997)
W. Freeden, T. Gervens, M. Schreiner, Constructive Approximation on the Spheres with Applications to Geomathematics. Numerical Mathematics and Scientific Computation (The Claredon Press/Oxford University Press, New York, 1998)
D. Geller, I. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21 (2), 334–371 (2011)
U. Hertich-Jeromin, Introduction to Möbius Differential Geometry. London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 2003)
M. Holschneider, Wavelets: An Analysis Tool (Oxford University Press, Oxford, 1995)
M. Holschneider, Continuous wavelet transforms on the sphere. J. Math. Phys. 37, 4156–4165 (1996)
A.K. Louis, P. Maass, A. Rieder, B.G. Wavelets, Teubner Studienbücher (1994)
R. Murenzi, Wavelet transforms associated to the n-dimensional Euclidean group with dilations: signal in more than one dimension, in Wavelets ed. By J.M. Combes, A. Grossmann, P. Tchamitchian (Springer, Berlin, 1990)
I. Pesenson, An approach to spectral problems on Riemannian manifolds. Pac. J. Math. 215 (1), 183–199 (2004)
I. Pesenson, Variational splines on Riemannian manifolds with applications to integral geometry. Adv. Appl. Math. 33 (3), 548–572 (2004)
D. Roşca, Wavelets on two-dimensional manifolds, Habilitation Thesis, University of Cluj-Napoca (Romania), 2012
D. Roşca, J.-P. Antoine, Locally supported orthogonal wavelet bases on the sphere via stereographic projection. Math. Probl. Eng. (2009). http://dx.doi.org/10.1155/2009/124904
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Bernstein, S., Keydel, P. (2017). Orthogonal Wavelet Frames on Manifolds Based on Conformal Mappings. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55550-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-55550-8_13
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-55549-2
Online ISBN: 978-3-319-55550-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)