Fast Injective Mesh Parameterization via Beltrami Coefficient Prolongation

Abstract We present a highly efficient and robust method for free boundary injective parameterization of disk‐like triangle meshes with low isometric distortion. Harmonic function–based approaches, grounded in a strong mathematical framework, are widely employed. In particular, harmonic maps are valuable for guaranteeing injectivity under suitable boundary conditions. They are also computationally efficient, as they operate within a linear subspace [FW22]. However, this restriction to a limited subspace often introduces substantial isometric distortion, especially on highly curved surfaces. In contrast, methods that explore the full space of piecewise linear maps [SPSH*17] achieve much lower isometric distortion, but at the expense of increased computational cost. We propose a hybrid method that combines the speed and robustness of harmonic maps with the generality of full‐space methods to produce injective maps with low isometric distortion up to 50 times faster than state‐of‐the‐art methods. The core concept is simple but powerful. Instead of searching for the optimal parameterization over the original mesh via a full‐space method, we simplify the input fine mesh and parameterize the coarse mesh. We then prolong the Beltrami coefficients from the coarse mesh parameterization to the fine mesh, resulting in a customized Laplacian matrix that gives rise to a harmonic map in a modified metric [WGS23, FW25]. Our method ensures injectivity, offers great computational efficiency, and produces significantly lower isometric distortion compared to harmonic maps.