Feature-Aligned Parametrization in Penner Coordinates

Parametrization is a key element of many geometric modeling tasks. Seamless parametrization, in particular, is needed as a starting point for many algorithms for quadrangulation and conversion to high-order patches, as well as for the construction of seamless texture maps and displacement maps. Seamless parametrizations are difficult to compute robustly, in part because, in general, it is not known if one exists for a given mesh connectivity or for a particular configuration of singularities. Recently, Penner-coordinate-based methods that allow for connectivity changes have been shown to achieve a perfect success rate on a widely used dataset (Thingi10k). However, previously proposed Penner coordinate methods do not support sharp feature alignment or soft alignment with preferred directions on the surface, both of which are important for practical applications, especially those involving models with sharp features. In this paper, we extend Penner coordinates to surfaces with sharp features to which the parametrization needs to be aligned. Our algorithm extends the holonomy signature description of seamless parametrizations to surfaces with marked feature curves. We describe sufficient conditions for obtaining feasible solutions and describe a two-phase method to efficiently enforce feature constraints or minimize residual errors when solutions are unattainable. We demonstrate that the resulting algorithm works robustly on the Thingi10k dataset with automatic feature labeling, and the resulting seamless parametrizations can be optimized, quantized, and quadrangulated, completing the quad mesh generation pipeline.