# Analytic and data-driven methods for 3D electron microscopy

**Time: **
Fri 2020-10-23 14.00

**Location: **
via Zoom https://kth-se.zoom.us/j/68027481949, Stockholm (English)

**Doctoral student: **
Gustav Zickert
, Matematik (Avd.)

**Opponent: **
Assistant Professor Roy Lederman, Yale University, New Haven, USA

**Supervisor: **
Professor Pär Kurlberg, Matematik (Avd.); Ozan Öktem, Strategiskt centrum för industriell och tillämpad matematik, CIAM, Matematik (Avd.); Timo Koski, Matematisk statistik, Matematik

## Abstract

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.