Priors in geometrical processing

Abstract

Priors are often used to help provide a starting point to solve specific problems, or cut down the solution space which makes the problem solvable in geometrical processing. In this thesis, we present three different tasks in geometrical processing and show how the priors are used to solve these problems.

First, 3D mesh models created by human users and shared through online platforms and datasets flourish recently. While the creators generally have spent large efforts in modeling the visually appealing shapes with both large scale structures and intricate details, a majority of the meshes are unfortunately flawed in terms of having duplicate faces, mis-oriented regions, disconnected patches and so on, due to multiple factors involving both human errors and software inconsistencies. All these artifacts have severely limited the possible low-level and high-level processing tasks that can be applied to the rich datasets. In this part, we present a novel approach to fix these man-made meshes such that the outputs are guaranteed to be oriented manifold meshes that preserve the original structures, big and small, as much as possible. Our key observation is that the models all visually look meaningful, which leads to our strategy of repairing the flaws while always preserving the visual quality. We apply local refinements and removals only where necessary to achieve minimal intrusion of the original meshes, and global adjustments through robust optimization to ensure the outputs are valid manifold meshes with optimal connections. We test the approach on large-scale 3D datasets, and obtain quality meshes that are more readily usable for further geometry processing tasks.

In the second task, we investigate the optimal partition of a geometric domain with specified capacity constraints on the partitioned regions which is an important problem in many fields, ranging from engineering to economics. It is known that the power diagram provides the appropriate framework for computing these optimal partitions, assuming the squared L2 metric. We present a super-linear convergent method for this problem that outperforms the state-of-the-art in an order of magnitude. We show its effectiveness by three separate application occasions in computer graphics and geometric processing: displacement interpolation, blue-noise point sampling, and optimal convex decomposition of 2D domains. Furthermore, the proposed method can be extended to capacity-constrained optimal partition with respect to general metrics beyond the squared L2 metric.

In the last task, we present a novel approach for completing and reconstructing 3D shapes from incomplete scanned data by using deep neural networks. Rather than being trained on supervised completion tasks and applied on a testing shape, the network is optimized from scratch on the single testing shape, to fully adapt to the shape and complete the missing data using contextual guidance from the known regions of the shape. The ability to complete missing data by an untrained neural network is usually referred as the deep prior. In this part, we interpret the deep prior from a neural tangent kernel (NTK) perspective, and show that the completed shape patches by the trained CNN are naturally similar to existing patches, as they are proximate in the kernel feature space induced by NTK. The interpretation allows us to design more efficient network structures and learning mechanisms for the shape completion and reconstruction task. Being more aware of structural regularities than both traditional and other unsupervised learning based reconstruction methods, our approach completes large missing regions with plausible shapes, and complements supervised learning based methods that use database priors by requiring no extra training data sets and showing flexible adaptation to a particular shape instance.