Mixed-integer quadrangulation

We present a novel method for quadrangulating a given triangle mesh. After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose iso-parameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixed-integer problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.