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Directional Multiscale Processing of Images Using Wavelets with Composite Dilations

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Abstract

It is widely recognized that the performance of many image processing algorithms can be significantly improved by applying multiscale image representations with the ability to handle very efficiently directional and other geometric features. Wavelets with composite dilations offer a flexible and especially effective framework for the construction of such representations. Unlike traditional wavelets, this approach enables the construction of waveforms ranging not only over various scales and locations but also over various orientations and other orthogonal transformations. Several useful constructions are derived from this approach, including the well-known shearlet representation and new ones, introduced in this paper. In this work, we introduce and apply a novel multiscale image decomposition algorithm for the efficient digital implementation of wavelets with composite dilations. Due to its ability to handle geometric features efficiently, our new image processing algorithms provide consistent improvements upon competing state-of-the-art methods, as illustrated on a number of image denoising and image enhancement demonstrations.

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Notes

  1. Even though the shearlet system is not exactly an example of wavelets with composite dilations, it is closely related to this framework being defined as a union of two such systems [26, 34].

  2. Recall that an orthonormal basis is a special case of a Parseval frame; however, the elements of a Parseval frame need not be orthogonal.

  3. Some ideas of this construction were introduced by one of the authors and collaborators in [27].

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Acknowledgements

D. Labate acknowledges partial support from NSF grants DMS 1008900 and DMS 1005799 and NHARP grant 003652-0136-2009.

Author information

Authors and Affiliations

  1. System Planning Corporation, Arlington, VA, USA

    Glenn R. Easley

  2. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Demetrio Labate

  3. University of Maryland, College Park, MD, 20742, USA

    Vishal M. Patel

Authors
  1. Glenn R. Easley
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  2. Demetrio Labate
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  3. Vishal M. Patel
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Corresponding author

Correspondence to Glenn R. Easley.

Appendix

Appendix

Below is a Matlab-based pseudo-code to generate the hyperbolic composite wavelet filters restricted to the fourth quadrant. The complete filters are found by adding the appropriate flip with zero padding. N 0 is the quadrant size of the image and k≥2 is a multiplier to determine the number of sequence elements. Hr and Ht denote the window functions for the radial and time parameters.

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Easley, G.R., Labate, D. & Patel, V.M. Directional Multiscale Processing of Images Using Wavelets with Composite Dilations. J Math Imaging Vis 48, 13–34 (2014). https://doi.org/10.1007/s10851-012-0385-4

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