Abstract-We present a non-parametric regression method for denoising 3D image sequences acquired via fluorescence microscopy. The proposed method exploits the redundancy of the 3D+time information to improve the signal-to-noise ratio of images corrupted by PoissonGaussian noise. A variance stabilization transform is first applied to the image-data to remove the dependence between the mean and variance of intensity values. This preprocessing requires the knowledge of parameters related to the acquisition system, also estimated in our approach. In a second step, we propose an original statistical patchbased framework for noise reduction and preservation of space-time discontinuities. In our study, discontinuities are related to small moving spots with high velocity observed in fluorescence video-microscopy. The idea is to minimize an objective non-local energy functional involving spatiotemporal image patches. The minimizer has a simple form and is defined as the weighted average of input data taken in spatially-varying neighborhoods. The size of each neighborhood is optimized to improve the performance of the pointwise estimator. The performance of the algorithm (which requires no motion estimation) is then evaluated on both synthetic and real image sequences using qualitative and quantitative criteria.Index Terms-Video-microscopy, fluorescence, image sequence denoising, patch-based approach, Poisson noise, variance stabilization, adaptive estimation, energy minimization.

The relation between random normal matrices and conformal mappings discovered by Wiegmann and Zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set. It is shown that for a suitable class of potentials the asymptotic density of eigenvalues is uniform with support in the interior domain of a simple smooth curve.

In this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed, and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting (see Schuster et al. [13]). We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer [11], and generalize existing convergence rates results. The key tool in our analysis is a homogeneous source condition, which we put into relation to the other existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces (see Groetsch [7]).

In this chapter a general mathematical model of Optical Coherence Tomography (OCT) is presented on the basis of the electromagnetic theory. OCT produces high resolution images of the inner structure of biological tissues. Images are obtained by measuring the time delay and the intensity of the backscattered light from the sample considering also the coherence properties of light. The scattering problem is considered for a weakly scattering medium located far enough from the detector. The inverse problem is to reconstruct the susceptibility of the medium given the measurements for different positions of the mirror. Different approaches are addressed depending on the different assumptions made about the optical properties of the sample. This procedure is applied to a full field OCT system and an extension to standard (time and frequency domain) OCT is briefly presented.