Certain embodiments involve refining local parameterizations that apply two-dimensional (“2D”) images to three-dimensional (“3D”) models. For instance, a particular parameterization-initialization process is select based on one or more features of a target mesh region. An initial local parameterization for a 2D image is generated from this parameterization-initialization process. A quality metric for the initial local parameterization is computed, and the local parameterization is modified to improve the quality metric. The 3D model is modified by applying image points from the 2D image to the target mesh region in accordance with the modified local parameterization.