Mathematical Challenges in Electron Microscopy

Development of electron microscopes first started nearly 100 years ago and they are now a mature imaging modality with many applications and vast potential for the future. The principal feature of electron microscopes is their resolution; they can be up to 1000 times more powerful than a visible light microscope and resolve even the smallest atoms. Furthermore, electron microscopes are also sensitive to many material properties due to the very rich interactions between electrons and other matter. Because of these capabilities, electron microscopy is used in applications as diverse as drug discovery, computer chip manufacture, and the development of solar cells. In parallel to this, the mathematical field of inverse problems has also evolved dramatically. Many new methods have been introduced to improve the recovery of unknown structures from indirect data, typically an ill-posed problem. In particular, sparsity promoting functionals such as the total variation and its extensions have been shown to be very powerful for recovering accurate physical quantities from very little and/or poor quality data. While sparsity-promoting reconstruction methods are powerful, they can also be slow, especially in a big-data setting. This trade-off forms an eternal cycle as new numerical tools are found and more powerful models are developed. The work presented in this thesis aims to marry the tools of inverse problems with the problems of electron microscopy: bringing state-of-the-art image processing techniques to bear on challenges specific to electron microscopy, developing new optimisation methods for these problems, and modelling new inverse problems to extend the capabilities of existing microscopes. One focus is the application of a directional total variation to overcome the limited angle problem in electron tomography, another is the proposal of a new inverse problem for the reconstruction of 3D strain tensor fields from electron microscopy diffraction data. The remaining contributions target numerical aspects of inverse problems, from new algorithms for non-convex problems to convex optimisation with adaptive meshes.Cantab Capital Institute for Mathematics of Informatio

Tomographic Reconstruction Methods for Decomposing Directional Components

Decomposition of tomographic reconstructions has many different practical application. We propose two new reconstruction methods that combines the task of tomographic reconstruction with object decomposition. We demonstrate these reconstruction methods in the context of decomposing directional objects into various directional components. Furthermore we propose a method for estimating the main direction in a directional object, directly from the measured computed tomography data. We demonstrate all the proposed methods on simulated and real samples to show their practical applicability. The numerical tests show that decomposition and reconstruction can combined to achieve a highly useful fibre-crack decomposition

Two Dimensional Clipping Based Segmentation Algorithm for Grayscale Fingerprint Images

One of the huge methods in Automated Fingerprint Identification System (AFIS) is the segment or separation of the fingerprint. The process of decomposing an image into exclusive components is referred as segmentation. Fingerprint segmentation is the one of the predominant process involved in fingerprint pre-processing and it refers to the method of dividing or separating the image into disjoint areas as the foreground and the background region. The foreground also called as Region of Interest (ROI) due to the fact only the region which contains ridge and valley structure is used for processing, whilst the background carries noisy and irrelevant content material and so that it will be discarded in later enhancement or orientation or classification method. The challenge proper right here is to decide which a part of the image belongs to the foreground, retrieved as an input from the fingerprint sensor device or from benchmark datasets and which part belongs to the background. A 100% correct segmentation is continually very tough, specifically inside the very poor quality image or partial image together with the presence of latent. In this paper, we discuss a modified clipped based segmentation algorithm by adopting threshold value and canny edge detection techniques. We segment the background image is x and y dimensions or in other words left the edge, right edge, top edge and bottom edge of the image. For the purpose of analyzing the algorithm FVC ongoing 2002 benchmark dataset is considered. The entire algorithm is implemented using MATLAB 2015a. The algorithm is able to find affectively ROI of the fingerprint image or separates the foreground region from the background area of the fingerprint image very effectively. In high configuration system proposed algorithm achieves execution time of 1.75 seconds

Enhancing the spatial resolution of hyperpolarized carbon‐13 MRI of human brain metabolism using structure guidance

Funder: Mark Foundation Institute for Cancer ResearchFunder: Cambridge Experimental Cancer Medicine CentreFunder: Alan Turing Institute; Id: http://dx.doi.org/10.13039/100012338Funder: Cantab Capital Institute for the Mathematics of InformationPurpose: Dynamic nuclear polarization is an emerging imaging method that allows noninvasive investigation of tissue metabolism. However, the relatively low metabolic spatial resolution that can be achieved limits some applications, and improving this resolution could have important implications for the technique. Methods: We propose to enhance the 3D resolution of carbon‐13 magnetic resonance imaging (13C‐MRI) using the structural information provided by hydrogen‐1 MRI (1H‐MRI). The proposed approach relies on variational regularization in 3D with a directional total variation regularizer, resulting in a convex optimization problem which is robust with respect to the parameters and can efficiently be solved by many standard optimization algorithms. Validation was carried out using an in silico phantom, an in vitro phantom and in vivo data from four human volunteers. Results: The clinical data used in this study were upsampled by a factor of 4 in‐plane and by a factor of 15 out‐of‐plane, thereby revealing occult information. A key finding is that 3D super‐resolution shows superior performance compared to several 2D super‐resolution approaches: for example, for the in silico data, the mean‐squared‐error was reduced by around 40% and for all data produced increased anatomical definition of the metabolic imaging. Conclusion: The proposed approach generates images with enhanced anatomical resolution while largely preserving the quantitative measurements of metabolism. Although the work requires clinical validation against tissue measures of metabolism, it offers great potential in the field of 13C‐MRI and could significantly improve image quality in the future

Multicontrast MRI reconstruction with structure-guided total variation

Magnetic resonance imaging (MRI) is a versatile imaging technique that allows different contrasts depending on the acquisition parameters. Many clinical imaging studies acquire MRI data for more than one of these contrasts---such as for instance T1 and T2 weighted images---which makes the overall scanning procedure very time consuming. As all of these images show the same underlying anatomy one can try to omit unnecessary measurements by taking the similarity into account during reconstruction. We will discuss two modifications of total variation---based on i) location and ii) direction---that take structural a priori knowledge into account and reduce to total variation in the degenerate case when no structural knowledge is available. We solve the resulting convex minimization problem with the alternating direction method of multipliers that separates the forward operator from the prior. For both priors the corresponding proximal operator can be implemented as an extension of the fast gradient projection method on the dual problem for total variation. We tested the priors on six data sets that are based on phantoms and real MRI images. In all test cases exploiting the structural information from the other contrast yields better results than separate reconstruction with total variation in terms of standard metrics like peak signal-to-noise ratio and structural similarity index. Furthermore, we found that exploiting the two dimensional directional information results in images with well defined edges, superior to those reconstructed solely using a priori information about the edge location.Engineering and Physical Sciences Research Council (Grant ID: EP/H046410/1)This is the final version of the article. It first appeared from Society for Industrial and Applied Mathematics via http://dx.doi.org/10.1137/15M1047325