Abstract
The classical wavelet transform is a remarkably effective tool for the analysis of pointwise regularity of functions and distributions. During the last decade, the emergence of a new generation of multiscale representations has extended the classical wavelet approach leading to the introduction of a class of generalized wavelet transforms—most notably the shearlet transform—which offers a much more powerful framework for microlocal analysis. In this paper, we show that the shearlet transform enables a precise geometric characterization of the set of singularities of a large class of multidimensional functions and distributions, going far beyond the capabilities of the classical wavelet transform. This paper generalizes and extends several results that previously appeared in the literature and provides the theoretical underpinning for advanced applications from image processing and pattern recognition including edge detection, shape classification, and feature extraction.
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Notes
- 1.
Note that the distributional Fourier transform of \(f\) is \(\hat{f}(\xi ) = \frac{1}{2}\delta (\xi ) + \frac{1}{2\pi i} \text {p.v.} \frac{1}{\xi }\), but the term \(\frac{1}{2}\delta (\xi )\) gives no contribution in the computation for \(\langle f , \psi _{a,t} \rangle \) since \(\hat{\psi }(0)=0\).
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Acknowledgments
The authors are partially supported by NSF grant DMS 1008900/1008907. DL is also partially supported by NSF grant DMS 1005799.
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Guo, K., Houska, R., Labate, D. (2014). Microlocal Analysis of Singularities from Directional Multiscale Representations. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_10
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