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Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames

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Abstract

Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames for improving the orientation selectivity of tensor product real-valued wavelet bases/frames. In this paper, we studied the general theory of discrete Gabor frames for finite signals, and constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation. Besides high orientation selectivity, the proposed multi-scale discrete 2D Gabor frames also allow us to simultaneously exploit sparsity prior of cartoon image regions in spatial domain and the sparsity prior of textural image regions in local frequency domain. Using a composite sparse image model, we showed the advantages of the proposed discrete Gabor frames over the existing wavelet frames in several image recovery experiments.

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Notes

  1. http://www3.math.tu-berlin.de/numerik/www.shearlab.org/software.

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Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful comments and suggestions. This work was partially supported by Singapore MOE AcRF Research Grant MOE2012-T3-1-008 and R-146-000-165-112.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore, 119076, Singapore

    Hui Ji, Zuowei Shen & Yufei Zhao

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  1. Hui Ji
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  2. Zuowei Shen
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  3. Yufei Zhao
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Correspondence to Hui Ji.

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Communicated by Akram Aldroubi.

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Ji, H., Shen, Z. & Zhao, Y. Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames. J Fourier Anal Appl 23, 729–757 (2017). https://doi.org/10.1007/s00041-016-9487-5

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  • DOI: https://doi.org/10.1007/s00041-016-9487-5

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