Bilaplacian reconstruction of point clouds

A key process of the geometry processing pipeline is the reconstruction of surfaces from point clouds. The traditional problem addressed by surface reconstruction is to recover the digital representation of the shape that has been inputted, where the data could potentially contain a wide variety of drawbacks. The goal of this thesis would be to test the Bilaplacian Smoothness in order to enforce the smooth prior to the surface reconstruction. By considering our thesis goal we will build an application that not only will solve different sparse linear systems of equations using different possible methods for position, normal, and smoothness equation's constraints but also will make use of more complex and effective surface reconstruction solving techniques such as the multigrid or quadree reconstruction

Greedy low-rank algorithm for spatial connectome regression

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regression problem (Harris et al., 2016), but existing techniques do not scale to the whole-brain setting. The corresponding matrix equation is challenging to solve due to large scale, ill-conditioning, and a general form that lacks a convergent splitting. We propose a greedy low-rank algorithm for connectome reconstruction problem in very high dimensions. The algorithm approximates the solution by a sequence of rank-one updates which exploit the sparse and positive definite problem structure. This algorithm was described previously (Kressner and Sirkovi\'c, 2015) but never implemented for this connectome problem, leading to a number of challenges. We have had to design judicious stopping criteria and employ efficient solvers for the three main sub-problems of the algorithm, including an efficient GPU implementation that alleviates the main bottleneck for large datasets. The performance of the method is evaluated on three examples: an artificial "toy" dataset and two whole-cortex instances using data from the Allen Mouse Brain Connectivity Atlas. We find that the method is significantly faster than previous methods and that moderate ranks offer good approximation. This speedup allows for the estimation of increasingly large-scale connectomes across taxa as these data become available from tracing experiments. The data and code are available online