EM-TV Methods for Inverse Problems with Poisson Noise

  • Alex Sawatzky
  • Christoph Brune
  • Thomas Kösters
  • Frank Wübbeling
  • Martin Burger
Part of the Lecture Notes in Mathematics book series (LNM, volume 2090)


We address the task of reconstructing images corrupted by Poisson noise, which is important in various applications such as fluorescence microscopy (Dey et al., 3D microscopy deconvolution using Richardson-Lucy algorithm with total variation regularization, 2004), positron emission tomography (PET; Vardi et al., J Am Stat Assoc 80:8–20, 1985), or astronomical imaging (Lantéri and Theys, EURASIP J Appl Signal Processing 15:2500–2513, 2005). Here we focus on reconstruction strategies combining the expectation-maximization (EM) algorithm and total variation (TV) based regularization, and present a detailed analysis as well as numerical results. Recently extensions of the well known EM/Richardson-Lucy algorithm received increasing attention for inverse problems with Poisson data (Dey et al., 3D microscopy deconvolution using Richardson-Lucy algorithm with total variation regularization, 2004; Jonsson et al., Total variation regularization in positron emission tomography, 1998; Panin et al., IEEE Trans Nucl Sci 46(6):2202–2210, 1999). However, most of these algorithms for regularizations like TV lead to convergence problems for large regularization parameters, cannot guarantee positivity, and rely on additional approximations (like smoothed TV). The goal of this lecture is to provide accurate, robust and fast EM-TV based methods for computing cartoon reconstructions facilitating post-segmentation and providing a basis for quantification techniques. We illustrate also the performance of the proposed algorithms and confirm the analytical concepts by 2D and 3D synthetic and real-world results in optical nanoscopy and PET.


Positron Emission Tomography Nonnegativity Constraint Order Optimality Condition Total Variation Regularization Bregman Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work has been supported by the German Ministry of Education and Research (BMBF) through the project INVERS: Deconvolution problems with sparsity constraints in nanoscopy and mass spectrometry. This research was performed when the second author was with the Mathematical Imaging Group at University of Münster. C. Brune acknowledges further support by the Deutsche Telekom Foundation. The work has been further supported by the German Science Foundation DFG through the SFB 656 Molecular Cardiovascular Imaging and the project Regularization with singular energies. The authors thank Florian Büther and Klaus Schäfers (both European Institute for Molecular Imaging, University of Münster) for providing real data in PET and useful discussions. The authors thank Katrin Willig and Andreas Schönle (both Max Planck Institute for Biophysical Chemistry, Göttingen) for providing real data in optical nanoscopy.


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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Alex Sawatzky
    • 1
  • Christoph Brune
    • 2
  • Thomas Kösters
    • 3
  • Frank Wübbeling
    • 1
  • Martin Burger
    • 1
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  3. 3.European Institute for Molecular ImagingUniversity of MünsterMünsterGermany

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