Christoph Florian Schaller - FU Berlin, FB MI

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Christoph Schaller - STORMicroscopy 26 Here M rot is an arbitrary rotation matrix and as already mentioned −→ x n does not denote the localization of a spot in frame n, but a mean localization of the surrounding frames. Now every assigned pair ( −→ x n , x −−→ n+1 ) provides two equations to estimate the six parameters −→ v = ( ) v 1 v 2 and M rot = ( ) m 1 m 2 . m 3 m 4 of the drift −−→ x n+1 − −→ x n . Hence we need to locate at least three beads in the surrounding frames. In case of more localizations we calculate the linear least squares t since the drift is no longer parameterdependant in a non-linear way. At this point we only have to describe how to obtain such a t. Assume i ∈ N equations a i,1 λ 1 + a i,2 λ 2 + ... + a i,k λ k = b i for known right hand sides b i and factors a i,k are given and we want to estimate the k ∈ N unknowns λ k ∈ R. This can be accomplished by minimizing the error in the Euclidean norm, i.e. ||b − Aλ|| 2 ≤ ||b − Av|| 2 ∀v ∈ R k . Now according to [18], Theorem 2.14, λ ∈ R k is a minimizer, if and only if A T Aλ = A T b (5.2) Furthermore it is unique if A is injective. In our case the right hand sides are the occuring drifts, the unknowns are the free parameters and the respective factors follow from (5.1): ( ) ( b 2i := b 2i+1 ) ( x n+1,i − y n+1,i λ := (v 1 , v 2 , m 1 , m 2 , m 3 , m 4 ), a 2i := (1, 0, x n,i , y n,i , 0, 0) and a 2i+1 := (0, 1, 0, 0, x n,i , y n,i ). ) x n,i , y n,i As we will not locate two spots in the exact same place A is obviously injective and thus a unique minimizer is given by (5.2), which can be evaluated using basic linear algebra.
Christoph Schaller - STORMicroscopy 27 6 xStorm In the context of this thesis an independent software package for spot localization was created. 6.1 Necessity of the development Even if RapidSTORM is open source code and the possibility of including other tting kernels was originally build in, there are several reasons against that. Above all, the code was once well-structured, but soon became confusing due to several bug- and hotxes and only sparse annotations. Furthermore the documentation is mainly meant for users, while the code contains many parts that are not relevant for the present thesis. Consequently we decided to develop our own tting environment for experimental data called xStorm, where x is an abbreviation of exact. In particular this allows us to include several algorithms for the analysis of external inuences such as noise or drift. Nonetheless Steve Wolter's diploma thesis [14] and the article published by his group on their experiences and results while designing and working with RapidSTORM [19] were very helpful. In terms of the programming language we decided in favor of C++, which combines computational eciency with availability of diverse functions, extensions and packages, e.g. for user interfaces and multithreading. In addition the G++ compiler guarantees the portability of the C++ code to the most common operating systems. Fortran would have been a reasonable decision as well, however making computational power accessible for the user as well as programming itself is more complex here. MATLAB R○ , which we used for our simulations, is very intuitive as it has many mathematical functions already built-in, but it is simply too ineecient (i.e. slow) when it comes to working with huge amounts of data. Similarly, these computational disadvantages apply to Java and Python. 6.2 Input and output As an output from a STORM experiment, one obtains a .tif- or .ti-le containing several frames. We will have a closer look at this format in the following Chapter 6.3, for now we just assume that we can extract framewise matrices of intensity values from it. Furthermore we need to know the pixel size in nanometers to provide meaningful nal results. Optionally we would like to know the emission wavelength (λ [nm]) and the numerical aperture (NA) of the used microscope, because this gives us the spot size (cf. (1.1)). One t nally consists of only four numbers. Those are x- and y-coordinates of the spot center, the number of tted photons and the frame the spot was found in. Consequently we use a simple text le with four columns (and a space as separator) to print our results and condense a complete STORM image into a list of found spots, alike the output format of RapidSTORM. 6.3 The tagged image le format (TIFF) Nowadays the TIFF TM is controlled by Adobe Systems and known as a exible le format that uses so called header tags to structure data and multiple images. The latest version (6.0) of its specication [20] can be found on Adobe's homepage.
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Christoph Florian Schaller - FU Berlin, FB MI

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