What's Best for My Mesh? Convex or Non-Convex Regularisation for Mesh Optimisation

Jason Pilbrough, Paul Amayo

Abstract: A 3D mesh offers a rich yet lightweight representation of geometry and topology for the metric and semantic understanding of a robot’s scene. Noisy features are often used to generate the mesh which furthers the need for accurate regularisation. Current approaches tightly couple front-end optimisation with regularisation making it difficult to evaluate the choice of discretisation and regularisation on mesh accuracy. In this work, we aim to explicitly query the performance of a set of well-known convex and non-convex regularisers on the mesh optimisation problem. We then apply these norms to dense depth estimation from a mesh representation and evaluate their performance in indoor and outdoor environments.While we show that the use of exotic, non-convex regularisers such as logTV and logTGV can result in more faithful structural reconstruction under noise, this comes at the cost of stronger outlier persistence that limits their performance when compared to their convex equivalents. This represents a significant departure from results achieved when the same regularisers are applied in denser "every-pixel" scenarios and suggests that current discretisation techniques adopted for this problem are more sensitive to triangulation. This sensitivity is often obscured in many practical robotic applications by a rigorous front-end that removes artefacts from the mesh to be optimised. By decoupling the front and back-ends we therefore show that further consideration must be taken to align current mesh discretisations with the classical definitions of variational regularisers to allow the full benefit of their well-documented properties to be realised.