Symmetrization of 2D Polygonal Shapes Using Mixed-Integer Programming

Symmetry widely exists in nature and man-made shapes, but it is unavoidably distorted during the process of growth, design, digitalization, and reconstruction steps. To enhance symmetry, traditional methods follow the detect-then-symmetrize paradigm, which is sensitive to noise in the detection phase, resulting in ambiguities for the subsequent symmetrization step. In this work, we propose a novel optimization-based framework that jointly detects and optimizes symmetry for 2D shapes represented as polygons. Our method can detect and optimize symmetry using a single objective function. Specifically, we formulate symmetry detection and optimization as a mixed-integer program. Our method first generates a set of candidate symmetric edge pairs, which are then encoded as binary variables in our optimization. The geometry of the shape is expressed as continuous variables, which are then optimized together with the binary variables. The symmetry of the shape is enforced by the designed hard constraints. After the optimization, both the optimal symmetric edge correspondences and the geometry are obtained. Our method simultaneously detects all the symmetric primitive pairs and enhances the symmetry of a model while minimally altering its geometry. We have tested our method on a variety of shapes from designs and vectorizations, and the results have demonstrated its effectiveness.