Optimization Methods for 3D Reconstruction: Depth Sensors, Distance Functions and Low-Rank Models

This thesis explores methods for estimating 3D models using depth sensors and
finding low-rank approximations of matrices. In the first part we focus on how to
estimate the movement of a depth camera and creating a 3D model of the scene.
Given an accurate estimation of the camera position, we can produce dense 3D
models using the images obtained from the camera. We present algorithms that
are both accurate, robust and in addition, fast enough for online 3D reconstruction
in real-time. The frame rate varies between about 5-20 Hz. It is shown in
experiments that these algorithms are viable for several different applications such
as autonomous quadrocopter navigation and object reconstruction.

In the second part we study the problem of finding a low-rank approximation
of a given matrix. This has several applications in computer vision such as rigid
and non-rigid Structure from Motion, denoising, photometric stereo and so on.
Two convex relaxations which take both the rank function and a data term into
account are introduced and analyzed together with a non-convex relaxation. It is
shown that these methods often avoid shrinkage bias and give better results than
the common heuristic of replacing the rank function with the nuclear norm.