Primal and Dual Bregman Methods with Application to Optical Nanoscopy

  • Authors:

  • Christoph Brune;Alex Sawatzky;Martin Burger

  • Affiliations:

  • Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Münster, Germany 48149;Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Münster, Germany 48149;Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Münster, Germany 48149

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 2011

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Abstract

Measurements in nanoscopic imaging suffer from blurring effects modeled with different point spread functions (PSF). Some apparatus even have PSFs that are locally dependent on phase shifts. Additionally, raw data are affected by Poisson noise resulting from laser sampling and "photon counts" in fluorescence microscopy. In these applications standard reconstruction methods (EM, filtered backprojection) deliver unsatisfactory and noisy results. Starting from a statistical modeling in terms of a MAP likelihood estimation we combine the iterative EM algorithm with total variation (TV) regularization techniques to make an efficient use of a-priori information. Typically, TV-based methods deliver reconstructed cartoon images suffering from contrast reduction. We propose extensions to EM-TV, based on Bregman iterations and primal and dual inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. Besides further generalizations of the primal and dual scale space methods in terms of general, convex variational regularization methods, we provide error estimates and convergence rates for exact and noisy data. We illustrate the performance of our techniques on synthetic and experimental biological data.