We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians, which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model. Notably, our method has the following features: (1) It accepts multivariate signals, i.e., sampled multivariate functions, histograms, time series, images, etc., as input. (2) The method can handle general (i.e., ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for Gaussian mixture model decomposition simultaneously enjoys all these features. We also prove an upper bound, which cannot be improved by a global constant, for the distance from any mode of a Gaussian mixture model to the set of corresponding means. For mixtures of spherical Gaussians with common variance σ2, the bound takes the simple form nσ. We evaluate our method on one- and two-dimensional signals. Finally, we discuss the relation between clustering and signal decomposition, and compare our method to the baseline expectation maximization algorithm.

This paper studies ill-posed tomographic imaging problems where the image is sparsely represented by a non-negative linear combination of Gaussians. Our main contribution is to develop a scheme for directly recovering the Gaussian mixture representation of an image from tomographic data, which here is modeled as noisy samples of the parallel-beam ray transform. An important aspect of this non-convex reconstruction problem is the choice of initial guess. We propose an initialization procedure that is based on a filtered back projection type of operator tailored for the Gaussian dictionary. This operator can be evaluated efficiently using an approximation of the Riesz-potential of an anisotropic Gaussian which is based on an exact closed form expression for the Riesz-potential of an isotropic Gaussian. The proposed method is evaluated on simulated data.

Three-dimensional reconstruction of the electron-scattering potential of biological macromolecules from electron cryo-microscopy (cryo-EM) projection images is an ill-posed problem. The most popular cryo-EM software solutions to date rely on a regularization approach that is based on the prior assumption that the scattering potential varies smoothly over three-dimensional space. Although this approach has been hugely successful in recent years, the amount of prior knowledge that it exploits compares unfavorably with the knowledge about biological structures that has been accumulated over decades of research in structural biology. Here, a regularization framework for cryo-EM structure determination is presented that exploits prior knowledge about biological structures through a convolutional neural network that is trained on known macromolecular structures. This neural network is inserted into the iterative cryo-EM structure-determination process through an approach that is inspired by regularization by denoising. It is shown that the new regularization approach yields better reconstructions than the current state of the art for simulated data, and options to extend this work for application to experimental cryo-EM data are discussed.

The central theme of this thesis is theoretical and algorithmic aspects of 3D electron microscopy (3D-EM). In particular, the thesis explores three parts of this theme. The first part concerns analysis of forward operators that, compared to those traditionally used, better account for the wave properties of the imaging electron. The second part concerns the adoption of data-driven methods in 3D-EM. The third part concerns the use of Gaussian dictionaries in image decomposition and image reconstruction.

The thesis consists primarily of five papers, which are preceded by two introductory chapters. The first chapter provides a background for the thesis and the second one constitutes a summary of the papers.

In paper A we propose a fast non-linear reconstruction method for joint phase-retrieval and image reconstruction in cryo electron tomography. We evaluate the method on simulated and real data. 

In paper B we train a deep convolutional neural network on a database of previously determined molecular structures. This network is used to model a prior distribution in single particle analysis (SPA) within a maximum-a-posteriori framework. We show in a simulation study that the proposed method is able to significantly improve on one of the current state-of-the-art methods.

In paper C we propose a greedy method for decomposing a signal as a mixture of Gaussians. We also derive an upper bound for the distance from any local maximum of a Gaussian mixture to the set of mean vectors.

In paper D we generalise the method in paper C and introduce an algorithm for reconstructing a mixture of Gaussians from its ray-transform projection images. We also derive exact and approximate expressions for the Riesz potential of isotropic and anisotropic Gaussians, respectively.

In paper E we prove a uniqueness theorem for an Ewald sphere corrected model for SPA. The theorem shows that accounting for a non-zero curvature of the Ewald sphere renders the noise-free SPA problem uniquely solvable, including the hand of the structure.

Det centrala temat för denna avhandling är teoretiska och algoritmiskaaspekter av 3D elektronmikroskopi (3D-EM). I synnerhet utforskar avhandlingentre delar av detta tema. Den första delen handlar om analys av framåtoperatorersom bättre tar hänsyn till elektronens vågengenskaper, jämförtmed tradionella modeller. Den andra delen handlar om data-drivna metoderi 3D-EM. Den tredje delen handlar om Gaussiska basfunktioner inom bildrepresentationoch bildrekonstruktion.Avhandlingen består i huvudsak av fem artiklar, som föregås av två inledandekapitel. Det första kaptilet ger en bakgrund till avhandlingen och detandra kapitlet utgör en sammanfattning av artiklarna.I artikel A föreslår vi en snabb icke-linjär rekonstruktionsmetod för gemensamfas- och bildrekonstruktion inom kryo-elektrontomografi. Vi utvärderarmetoden på simulerat och verkligt data.I artikel B tränar vi ett djupt neuralt faltningsnätverk på en databas avstrukturbestämda molekyler. Nätverket används för att modellera en a-priorifördelning i single particle analysis (SPA) inom ett maximum-a-posterioriramverk. Vi visar i en simuleringsstudie att den föreslagna metoden presteraravsevärt bättre än en av de ledande metoderna i fältet.I artikel C föreslår vi en girig metod för signalrepresentation med Gaussiskafunktioner. Dessutom härleder vi en övre begränsning för avståndet frånett godtyckligt lokalt maximum av en Gaussisk mixtur till mängden av väntevärdesvektorer.I artikel D generaliserar vi metoden i artikel C och introducerar därigenomen algoritm för att rekonstruera en Gaussisk mixtur från dess ray-transformprojektionsbilder. Vi härleder även exakta och approximativa uttryck för Rieszpotentialen av isotropa respektive anisotropa Gaussiska funktioner.I artikel E bevisar vi en entydighetssats för en Ewaldsfär-korrigerad modellför SPA. Denna sats visar att nollskild krökning av Ewaldsfären leder till ettunikt lösbart SPA problem, inklusive strukturens hänthet.

We perform three-dimensional (3D) reconstructions from simulated and real cryogenic electron tomography (cryo-ET) data. Our reconstructions are based on a nonlinear and phaseless forward model very reminiscent of a commonly used model for phase contrast x-ray tomography.