Abstract
AbstractIn recent years, scale-discretized directional wavelets and second-generation curvelets have been introduced on the unit sphere, yielding directional and localized polynomial frames for band-limited signals. In this paper, we show that these functions are able to detect the positions and orientations of all higher order jump discontinuities which lie along circles on the 2-sphere. Specifically, we prove upper and lower estimates for the magnitude of the corresponding inner products when the analysis function is concentrated in the neighborhood of such a singularity. Although similar results already exist in certain two-dimensional settings, this paper is the first one to consider frames and signals that are given on the 2-sphere. As a side product of our investigations, we also develop a new localization bound as well as an explicit formula for the auto-correlation function for second-generation curvelets.
Publisher
Springer Science and Business Media LLC
Subject
General Earth and Planetary Sciences,Modeling and Simulation
Reference24 articles.
1. Chan, J.Y.H., Leistedt, B., Kitching, T.D., et al.: Second-generation curvelets on the sphere. IEEE Trans. Signal Process. 65(1), 5–14 (2017). https://doi.org/10.1109/TSP.2016.2600506
2. Conrad, M., Prestin, J.: Multiresolution on the sphere. In: Tutorials on multiresolution in geometric modelling. In: European Summer School Lecture Notes, Munich University of Technology, Germany, August 22–30, 2001, pp. 165–202. Springer, Berlin (2002)
3. Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monogr. Math., Springer, New York (2013). https://doi.org/10.1007/978-1-4614-6660-4
4. Daugavet, I.K.: Rafal’son SZ (1972) Einige Ungleichungen vom Markov-Nikol’skiischen Typ für algebraische Polynome. Vestn. Leningr. Univ. Mat. Mekh. Astron 1, 15–25 (1972)
5. Driscoll, J.R., Healy, D.M.J.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15(2), 202–250 (1994). https://doi.org/10.1006/aama.1994.1008