Single molecule localization microscopy (SMLM) has the potential to resolve structural details of biological samples at the nanometer length scale. Compared to room temperature experiments, SMLM performed under cryogenic temperature achieves higher photon yields and, hence, higher localization precision. However, to fully exploit the resolution it is crucial to account for the anisotropic emission characteristics of fluorescence dipole emitters with fixed orientation. In case of slight residual defocus, localization estimates may well be biased by tens of nanometers. We show here that astigmatic imaging in combination with information about the dipole orientation allows to extract the position of the dipole emitters without localization bias and down to a precision of 1 nm, thereby reaching the corresponding Cramér Rao bound. The approach is showcased with simulated data for various dipole orientations, and parameter settings realistic for real life experiments.
In this paper, we consider the minimization of a Tikhonov functional with an ℓ
1 penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an ℓ
2 penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton’s method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach.
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