Automated focusing and astigmatism correction in electron microscopy

Nowadays electron microscopy still requires an expert operator in order to manually obtain in-focus and astigmatism-free images. Both the defocus and the twofold astigmatism have to be adjusted regularly during the image recording process. Possible reasons are for instance the instabilities of the environment and the magnetic nature of some samples. For some applications the high level of repetition severely strains the required concentration. Therefore, a robust and reliable autofocus and twofold astigmatism correction algorithm is a necessary tool for electron microscopy automation. Most of the automatic focusing methods are based on a sharpness function, which delivers a real-valued estimate of an image quality. In this thesis we study sharpness functions based on image derivative, image Fourier transform, image variance, autocorrelation and histogram. A new method for rapid automated focusing is developed, based on a quadratic interpolation of the derivative-based sharpness function. This function has been already used before on heuristic grounds. In this thesis we give a more solid mathematical foundation for this function and get a better insight into its analytical properties. Further we consider a focus series method, which could act as an extension for an autofocus technique. The method is meant to obtain the astigmatism information from the focus series of images. The method is based on the moments of the image Fourier transforms. After all the method of simultaneous defocus and astigmatism correction is developed. The method is based on a three-parameter optimization (Nelder-Mead simplex method or interpolation-based trust region method) of a sharpness function. All the three methods are employed for the scanning transmission electron microscopy. To be more specific, we have implemented them in the FEI scanning transmission electron microscope and successfully tested their performance as a part of a real-world application

Derivative-based image quality measure for autofocus in electron microscopy

Automatic focusing methods are based on an image quality measure, which is a realvalued estimation of an image’s sharpness. In this paper we study L_1- or L_2-norm derivative-based image quality measures. For a bench mark case these measures turn out to be quadratic, which implies that after obtaining of at least three images one can find the position of the optimal defocus. The resulting autofocus method is demonstrated for a reference scanning transmission electron microscopy application. Keywords: Electron microscopy · Autofocus · Linear image formation · Image quality measure

Orientation identification of the power spectrum

The image Fourier transform is widely used for defocus and astigmatism correction in electron microscopy. The shape of a power spectrum (the square of a modulus of image Fourier transform) is directly related to the three microscope’s controls, namely defocus and two-fold (two-parameter) astigmatism. In this paper the new method for power spectrum orientation identification is proposed. The method is based on the three measures which are related to the microscope’s controls. The measures are derived from the mathematical moments of the power spectrum. The method is tested with the help of a Gaussian benchmark, as well as with the scanning electron microscopy experimental images. The method can be used as an assisting tool for increasing the capabilities of defocus and astigmatism correction a of non-experienced scanning electron microscopy user, as well as a basis for automated application

Analysis, numerics, and optimization of algae growth

We extend the mathematical model for algae growth as described in [11] to include new effects. The roles of light, nutrients and acidity of the water body are taken into account. Important properties of the model such as existence and uniqueness of solution, as well as boundedness and positivity are investigated. We also discuss the numerical integration of the resulting system of ordinary differential equations and derive a condition which guarantees positivity of the numerical solution. An optimization problem is formulated which demonstrates an application of the model

A derivative-based fast autofocus method

Most automatic focusing methods are based on a sharpness function, which delivers a real-valued estimate of an image quality. In this paper, we study an L2-norm derivative-based sharpness function, which has been used before based on heuristic consideration. We give a more solid mathematical foundation for this function and get a better insight into its analytical properties. Moreover an efficient autofocus method is presented, in which an artificila blur variable plays an important role. We show that for a specific choice of the artificial blur control variable, the function is approximately a quadratic polynomial, which implies that after obtaining of at least three images one can find the approximate position of the optimal defocus. This provides the speed improvement in comparison with existing approaches, which usually require recording of more than ten images for autofocussing. The new autofocus method is employed for the scanning transmission electron microscopy. To be more specific, it has been implemented in the FEI scanning transmission electron microscope and its performance has been tested as a part of a particle analysis application

Autofocus and two-fold astigmatism correction in HAADF-STEM

A new simultaneous autofocus and two-fold astigmatism correction method is proposed for High Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF-STEM). The method makes use of a modification of an image variance, which has already been used before as an image quality measure for different types of microscopy. In this paper we describe numerical simulations based on a classical HAADF-STEM linear image formation model showing that the modified variance reaches it's maximum for Scherzer focus and zero astigmatism. In order to find this maximum in a three-parameter space we employ the well-known Nelder-Mead simplex optimization algorithm. The method is implemented and tested on a FEI Tecnai F20.It successfully finds the optimal defocus and zero astigmatism with the time and accuracy, compared with the human operator. The method is iterative, and finding the optimal defocus and zero astigmatism requires obtaining typically 20-50 images

Design parameters for a siphon system

DHI are interested in understanding a rather unusual water extraction system that is operated by a water supply company. Typically when water is extracted from the ground a well is dug and a pump is installed in the well to push the water to the surface where it enters a distribution system of pipes. Such a system may consist of a dozen or so wells each connected to a single collection pipe. The system that DHI wish to more fully understand consists of a series of ten wells connected to a single collection pipe. The difference in the mode of operation is that the system contains no pumps in the wells. The force to collect the water comes from placing the end of the collection pipe in a tank that is continuously pumped to keep it at approximately 0.5 bar below atmospheric pressure. In this way the water is drawn out of the wells by a siphon mechanism. Such a system appears cheaper o install with fewer pumps and water supplied in this manner costs roughly half the price of water from a standard pump system. How this multiple siphon system works and how it might be controlled were the general problems of interest to the study group

Modeling and optimization of algae growth

The wastewater from greenhouses has a high amount of mineral contamination\ud and an environmentally-friendly method of removal is to use algae\ud to clean this runoff water. The algae consume the minerals as part of their\ud growth process. In addition to cleaning the water, the created algal bio-mass\ud has a variety of applications including production of bio-diesel, animal feed,\ud products for pharmaceutical and cosmetic purposes, or it can even be used as\ud a source of heating or electricity .\ud The aim of this paper is to develop a model of algae production and use\ud this model to investigate how best to optimize algae farms to satisfy the dual\ud goals of maximizing growth and removing mineral contaminants.\ud With this aim in mind the paper is split into five main sections. In the\ud first a review of the biological literature is undertaken with the aim of determining\ud what factors effect the growth of algae. The second section contains\ud a review of exciting mathematical models from the literature, and for\ud each model a steady-state analysis is performed. Moreover, for each model\ud the strengths and weaknesses are discussed in detail. In the third section,a new two-stage model for algae production is proposed, careful estimation\ud of parameters is undertaken and numerical solutions are presented. In the\ud next section, a new one-dimensional spatial-temporal model is presented,\ud numerically solved and optimization strategies are discussed. Finally, these\ud elements are brought together and recommendations of how to continue are\ud drawn

Automated focusing and astigmatism correction in electron microscopy

Nowadays electron microscopy still requires an expert operator in order to manually obtain in-focus and astigmatism-free images. Both the defocus and the twofold astigmatism have to be adjusted regularly during the image recording process. Possible reasons are for instance the instabilities of the environment and the magnetic nature of some samples. For some applications the high level of repetition severely strains the required concentration. Therefore, a robust and reliable autofocus and twofold astigmatism correction algorithm is a necessary tool for electron microscopy automation. Most of the automatic focusing methods are based on a sharpness function, which delivers a real-valued estimate of an image quality. In this thesis we study sharpness functions based on image derivative, image Fourier transform, image variance, autocorrelation and histogram. A new method for rapid automated focusing is developed, based on a quadratic interpolation of the derivative-based sharpness function. This function has been already used before on heuristic grounds. In this thesis we give a more solid mathematical foundation for this function and get a better insight into its analytical properties. Further we consider a focus series method, which could act as an extension for an autofocus technique. The method is meant to obtain the astigmatism information from the focus series of images. The method is based on the moments of the image Fourier transforms. After all the method of simultaneous defocus and astigmatism correction is developed. The method is based on a three-parameter optimization (Nelder-Mead simplex method or interpolation-based trust region method) of a sharpness function. All the three methods are employed for the scanning transmission electron microscopy. To be more specific, we have implemented them in the FEI scanning transmission electron microscope and successfully tested their performance as a part of a real-world application