The first part of this thesis studies a modified version of mean curvature flow, the ``conformalized mean curvature flow" (cMCF), developed by Kazhdan, Solomon, and Ben-Chen. The cMCF is a conformal mapping algorithm but it runs into numerical issues when it is applied on meshes with protrusions. We improve the cMCF with an initialization step which first maps the initial mesh onto a sphere. This initialization step is shown to improve the performance of cMCF so that it can be applied on meshes with long protrusions. More importantly, we give the first algorithm named ``Sphericalized cMCF" to construct a homotopy from a degree one map to a homeomorphism from a unit sphere onto a unit sphere. We provide results from numerical experiments of applying this algorithm to closed surfaces of genus zero that are embedded in $\Rl^3$, on which we construct a homotopy from a degree one map to a conformal homeomorphism onto a unit sphere.

The second part of this thesis focuses on my work in x-ray emission tomography at the Lawrence Livermore National Laboratory. This project is conducted over eighteen months of a student internship and within the framework of the inertial confinement fusion (ICF) experiments performed at the National Ignition Facility (NIF). We present a novel approach to reconstruct the 3D electron temperature distribution of ICF hotspots. Using very limited number of 2D x-ray projection images, we reconstruct 3D x-ray emission distributions of an ICF hotspot from different x-ray energy channels ranging from 20 to 30 keV. The x-ray input images are processed using the algebraic reconstruction technique (ART) to reconstruct 3D x-ray emission distributions in different energy channels, which can characterize and compare the thermophysical states of the fusion plasma such as its electron temperature. We compute the 3D electron temperature using the energy channel ratios. We present both synthetic and experimental results showing high accuracy and applicability of our method on different complex hotspot geometries.