Teun Schilperoort - Structural Similarity in inverse problems: How working together is better!
Inverse problems are often difficult to solve as they can be ill-posed, ill-conditioned and there is often noise present.
If multiple measurements are performed of the same physical system, it is possible to couple the respective inverse problems. Even when completely different methods for data-gathering are used, there is still structural similarity hidden in the data. A large value change in the data at one location often implies a large change at the same location for the other measurement as well.
This so-called joint inversion has been used in practice for more than a decade in MRI-PET scans and geophysical surveys with good results.
I have developed a novel theoretical understanding of structural joint inversion and found general necessary criteria for any specific coupling to work. This has been put to the test in an image deblurring numerical application.
Liang Xu - Probabilistic Methods for Point Cloud Registration Problem
Over the past few decades, probabilistic methods have been one of the popular approaches for registering point clouds generated by LiDAR systems. Such methods consist of two major steps, i.e., using probabilistic models to represent point clouds and finding optimal transformation to align point clouds with the help of some statistical distances. In this thesis, a theoretical framework of probabilistic methods is studied, which provides a foundation for understanding and implementing point cloud registration with some specific probabilistic models and distance measures. Specifically, the concepts of Gaussian mixture models and kernel density estimation are explored, with a detailed discussion on their practical implementation. Furthermore, Kullback–Leibler divergence and Wasserstein distance, including the computation of Wasserstein distance through Kantorovich-Rubinstein duality, are also studied. An approximation of Wasserstein distance that preserves differentiability through linear programming techniques is proposed, which enables the use of gradient-based method on Wasserstein distance to find optimal transformation that aligns two point clouds. The algorithms that contribute to a complete procedure for solving point cloud registration problem with our proposed methods are also discussed and demonstrated.
