Inverse problems in structural biology and flexible needle steering

Abstract

This dissertation for uses on several inverse problems and mathematical problems in structural biology and flexible needle steering. Among the many methods for determining structural information about biological macromolecules, electron tomography and Nuclear Magnetic Resonance (NMR) are considered here. Since reconstruction using the electron microscope is based on image modeling and processing, we develop a mathematical method for representing two-dimensional images using Hermite expansions. The special properties of the expansion are derived and are used for two-dimensional image processing. Based on this method, two-dimensional image reconstruction from random projections is performed as an intermediate step on the way to full three-dimensional image reconstruction. Furthermore, we solve the image deblurring problem using the Hermite image expansion and deconvolution on the Euclidean motion group. This deblurring technique is expected to be useful in electron tomography, when averaging projections for noise reduction leads to blurring. We also develop a framework for solving the assignment problem in experimental NMR with unassigned Residual Dipolar Coupling (RDC) measurements. Based on the relationship between crystal structure information and RDC data of a target biological macromolecule, we assign each measurement to the biological macromolecule using methods from linear algebra and the optional assignment algorithm. Finally, we propose a path planning algorithm for a flexible needle with a bevel tip. A nonholonomic needle model and a diffusion process are used for this planning algorithm. We also propose a method for determining the noise parameters of the needle insertion system, which can be used for the planning.