Bijective Parameterization with Free Boundaries

We present a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces partitioned into charts. We do so by using a distortion metric that prevents local folds of triangles in the parameterization and a barrier function that prevents intersection of the chart boundaries. In addition, we show how to modify the line search of an interior point method to directly compute the singularities of the distortion metric and barrier functions to maintain a bijective map. By using an isometric metric that is efficient to compute and a spatial hash to accelerate the evaluation and gradient of the barrier function for the boundary, we achieve fast optimization times. Unlike previous methods, we do not require the boundary be constrained by the user to a non-intersecting shape to guarantee a bijection, and the boundary of the parameterization is free to change shape during the optimization to minimize distortion.