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Constrained Total Variation Minimization and Application in Computerized Tomography

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Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2005)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 3757))

Abstract

We present a simple framework for solving different ill-posed inverse problems in image processing by means of constrained total variation minimizations. We argue that drawbacks commonly attributed to total variation algorithms (slowness and incomplete fit to the image model) can be easily bypassed by performing only a few number of iterations in our optimization process. We illustrate this approach in the context of computerized tomography, that comes down to inverse a Radon transform obtained by illuminating an object by straight and parallel beams of x-rays. This problem is ill-posed because only a finite number of line integrals can be measured, resulting in an incomplete coverage of the frequency plane and requiring, for a direct Fourier reconstruction, frequencies interpolation from a polar to a Cartesian grid. We introduce a new method of interpolation based on a total variation minimization constrained by the knowledge of frequency coefficients in the polar grid, subject to a Lipschitz regularity assumption. The experiments show that our algorithm is able to avoid Gibbs and noise oscillations associated to the direct Fourier method, and that it outperforms classical reconstruction methods such as filtered backprojection and Rudin-Osher-Fatemi total variation restoration, in terms of both PSNR and visual quality.

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Authors and Affiliations

  1. LMAM, Université de Bretagne Sud, Campus de Tohannic – Y. Coppens, BP573, 56017, Vannes, France

    Xiao-Qun Zhang & Jacques Froment

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  1. Xiao-Qun Zhang
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  2. Jacques Froment
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Editors and Affiliations

  1. Department of CISE, University of Florida, Gainesville, FL, USA

    Anand Rangarajan

  2. University of Florida, 32611, Gainesville, FL, USA

    Baba Vemuri

  3. Department of Statistics, University of California, Los Angeles

    Alan L. Yuille

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© 2005 Springer-Verlag Berlin Heidelberg

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Zhang, XQ., Froment, J. (2005). Constrained Total Variation Minimization and Application in Computerized Tomography. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2005. Lecture Notes in Computer Science, vol 3757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11585978_30

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  • DOI: https://doi.org/10.1007/11585978_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30287-2

  • Online ISBN: 978-3-540-32098-2

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