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Two-stage Geometric Information Guided Image Reconstruction

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Advances in Data Science

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 26))

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Abstract

In compressive sensing, it is challenging to reconstruct image of high quality from very few noisy linear projections. Existing methods mostly work well on piecewise constant images but not so well on piecewise smooth images such as natural images, medical images that contain a lot of details. We propose a two-stage method called GeoCS to recover images with rich geometric information from very limited amount of noisy measurements. The method adopts the shearlet transform that is mathematically proven to be optimal in sparsely representing images containing anisotropic features such as edges, corners, spikes etc. It also uses the weighted total variation (TV) sparsity with spatially variant weights to preserve sharp edges but to reduce the staircase effects of TV. Geometric information extracted from the results of stage I serves as an initial prior for stage II which alternates image reconstruction and geometric information update in a mutually beneficial way. GeoCS has been tested on incomplete spectral Fourier samples. It is applicable to other types of measurements as well. Experimental results on various complicated images show that GeoCS is efficient and generates high-quality images.

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Acknowledgements

The authors would like to thank the Research Collaboration Workshop for Women in Data Science and Mathematics held at ICERM during July 29–August 2, 2019. Qin is supported by the NSF grant DMS-1941197, and Guo is supported by the NSF grant DMS-1521582.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, USA

    Jing Qin

  2. Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH, USA

    Weihong Guo

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  1. Jing Qin
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  2. Weihong Guo
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Correspondence to Jing Qin .

Editor information

Editors and Affiliations

  1. Intel (United States), Hermosa Beach, CA, USA

    Ilke Demir

  2. School of Natural Sciences & Mathematics, The University of Texas at Dallas, Richardson, TX, USA

    Yifei Lou

  3. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    Xu Wang

  4. Fakultät für Maschinenbau, Helmut Schmidt University, Hamburg, Germany

    Kathrin Welker

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Qin, J., Guo, W. (2021). Two-stage Geometric Information Guided Image Reconstruction. In: Demir, I., Lou, Y., Wang, X., Welker, K. (eds) Advances in Data Science. Association for Women in Mathematics Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-79891-8_1

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