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.

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We present a fully automatic approach for reconstructing compact 3D building models from large-scale airborne point clouds. A major challenge of urban reconstruction from airborne LiDAR point clouds lies in that the vertical walls are typically missing. Based on the observation that urban buildings typically consist of planar roofs connected with vertical walls to the ground, we propose an approach to infer the vertical walls directly from the data. With the planar segments of both roofs and walls, we hypothesize the faces of the building surface, and the final model is obtained by using an extended hypothesis-and-selection-based polygonal surface reconstruction framework. Specifically, we introduce a new energy term to encourage roof preferences and two additional hard constraints into the optimization step to ensure correct topology and enhance detail recovery. Experiments on various large-scale airborne LiDAR point clouds have demonstrated that the method is superior to the state-of-the-art methods in terms of reconstruction accuracy and robustness. In addition, we have generated a new dataset with our method consisting of the point clouds and 3D models of 20k real-world buildings. We believe this dataset can stimulate research in urban reconstruction from airborne LiDAR point clouds and the use of 3D city models in urban applications@en