Abstract
Measurements in nanoscopic imaging suffer from blurring effects concerning 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 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 an extension to EM-TV, based on Bregman iterations and inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. We illustrate our techniques by synthetic and experimental biological data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bertero, M., Lantéri, H., Zanni, L.: Iterative image reconstruction: a point of view. In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT). CRM series, vol. 8 (2008)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Le, T., Chartrand, R., Asaki, T.J.: A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imaging Vision 27, 257–263 (2007)
Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Transactions on Medical Imaging 1(2), 113–122 (1982)
Richardson, W.H.: Bayesian-based iterative method of image restoration. J. Opt. Soc. Am. 62, 55–59 (1972)
Lucy, L.B.: An iterative technique for the rectification of observed distributions. The Astronomical Journal 79, 745–754 (1974)
Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10, 1217–1229 (1994)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. Multiscale Modelling and Simulation 4, 460–489 (2005)
Burger, M., Gilboa, G., Osher, S., Xu, J.: Nonlinear inverse scale space methods. Commun. Math. Sci. 4(1), 179–212 (2006)
Burger, M., Frick, K., Osher, S., Scherzer, O.: Inverse total variation flow. SIAM Multiscale Modelling and Simulation 6(2), 366–395 (2007)
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum Likelihood from Incomplete Data via the EM Algorithm. J. of the Royal Statistical Society, B 39, 1–38 (1977)
Natterer, F., Wübbeling, F.: Mathematical methods in image reconstruction. SIAM Monographs on Mathematical Modeling and Computation (2001)
Resmerita, E., et al.: The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis. Inverse Problems 23, 2575–2588 (2007)
Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for positron emission tomography. J. of the American Statistical Association 80(389), 8–20 (1985)
Iusem, A.N.: Convergence analysis for a multiplicatively relaxed EM algorithm. Mathematical Methods in the Applied Sciences 14, 573–593 (1991)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Giusti, E.: Minimal surfaces and functions of bounded variation. Birkhäuser, Basel (1984)
Chambolle, A.: An algorithm for total variation minimization and applications. J. of Mathematical Imaging and Vision 20, 89–97 (2004)
Resmerita, E., Anderssen, S.: Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems. Math. Meth. Appl. Sci. 30, 1527–1544 (2007)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305. Springer, Heidelberg (1993)
Brune, C., Sawatzky, A., Wübbeling, F., Kösters, T., Burger, M.: EM-TV methods for inverse problems with poisson noise (in preparation) (2009)
Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comp. Math. and Math. Phys. 7, 200–217 (1967)
Klar, T.A., et al.: Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission. PNAS 97, 8206–8210 (2000)
Hell, S., Schönle, A.: Nanoscale resolution in far-field fluorescence microscopy. In: Hawkes, P.W., Spence, J.C.H. (eds.) Science of Microscopy. Springer, Heidelberg (2006)
Kittel, J., et al.: Bruchpilot promotes active zone assembly, Ca2+ channel clustering, and vesicle release. Science 312, 1051–1054 (2006)
Willig, K.I., Harke, B., Medda, R., Hell, S.W.: STED microscopy with continuous wave beams. Nature Meth. 4(11), 915–918 (2007)
Sawatzky, A., Brune, C., Wübbeling, F., Kösters, T., Schäfers, K.: Accurate EM-TV algorithm in PET with low SNR. In: IEEE Nucl. Sci. Symp. (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brune, C., Sawatzky, A., Burger, M. (2009). Bregman-EM-TV Methods with Application to Optical Nanoscopy. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-02256-2_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02255-5
Online ISBN: 978-3-642-02256-2
eBook Packages: Computer ScienceComputer Science (R0)