Reflectional symmetry is a potentially very useful feature which many real-world objects exhibit. It is instrumental in a variety of applications such as object alignment, compression, symmetrical editing or reconstruction of incomplete objects. In this paper, we propose a novel differentiable symmetry measure, which allows using gradient-based optimization to find symmetry in geometric objects. We further propose a new method for symmetry plane detection in 3D objects based on this idea. The method performs well on perfectly as well as approximately symmetrical objects, it is robust to noise and to missing parts. Furthermore, it works on discrete point sets and therefore puts virtually no constraints on the input data. Due to flexibility of the symmetry measure, the method is also easily extensible, e.g., by adding more information about the input object and using it to further improve its performance. The proposed method was tested with very good results on many objects, including incomplete objects and noisy objects, and was compared to other state-of-the-art methods which it outperformed in most aspects.

反射对称性是许多现实世界对象展现的潜在非常有用的功能。它在各种应用中都很有用，例如对象对齐，压缩，对称编辑或不完整对象的重建。在本文中，我们提出了一种新颖的可微对称性度量，它允许使用基于梯度的优化来找到几何对象中的对称性。基于这一思想，我们进一步提出了一种用于3D对象对称平面检测的新方法。该方法在完美以及近似对称的对象上均表现良好，对噪声和缺失零件具有鲁棒性。此外，它适用于离散点集，因此实际上对输入数据没有任何限制。由于对称度量的灵活性，该方法也易于扩展，例如，通过添加有关输入对象的更多信息并使用它来进一步提高其性能。所提出的方法在许多对象（包括不完整的对象和嘈杂的对象）上都进行了测试，结果非常好，并且与其他在大多数方面都表现出色的最新方法进行了比较。