Many of the point cloud processing techniques have their origin in image processing. But mathematical morphology, despite being one of the most used image processing techniques, has not yet been clearly adapted to point clouds. The aim of this work is to design the basic operations of mathematical morphology applicable to 3D point cloud data, without the need to transform point clouds to 2D or 3D images and avoiding the associated problems of resolution loss and orientation restrictions. The object shapes in images, based on pixel values, are assumed to be the existence or absence of points, therefore, morphological dilation and erosion operations are focused on the addition and removal of points according to the structuring element. The structuring element, in turn, is defined as a point cloud with characteristics of shape, size, orientation, point density, and one reference point. The designed method has been tested on point clouds artificially generated, acquired from real case studies, and the Stanford bunny model. The results show a robust behaviour against point density variations and consistent with image processing equivalent. The proposed method is easy and fast to implement, although the selection of a correct structuring element requires previous knowledge about the problem and the input point cloud. Besides, the proposed method solves well-known point cloud processing problems such as object detection, segmentation, and gap filling.

许多点云处理技术起源于图像处理。但是，数学形态学尽管是最常用的图像处理技术之一，但仍未明确适应于点云。这项工作的目的是设计适用于3D点云数据的数学形态学的基本操作，而无需将点云转换为2D或3D图像，并且避免了分辨率降低和方向限制的相关问题。基于像素值的图像中的对象形状被假定为存在点或不存在点，因此，根据结构元素，形态扩展和腐蚀操作着重于点的添加和去除。反过来，结构元素被定义为具有形状，大小，方向，点密度和一个参考点。该设计方法已在人工生成的点云上进行了测试，该点云是从实际案例研究中获得的，以及斯坦福兔子模型。结果显示出针对点密度变化的鲁棒行为，并且与图像处理等效物一致。尽管选择正确的结构元素需要有关问题和输入点云的先前知识，但所提出的方法易于实现且快速。此外，该方法解决了众所周知的点云处理问题，例如目标检测，分割和间隙填充。结果显示出针对点密度变化的鲁棒行为，并且与图像处理等效物一致。尽管选择正确的结构元素需要有关问题和输入点云的先前知识，但所提出的方法易于实现且快速。此外，该方法解决了众所周知的点云处理问题，例如目标检测，分割和间隙填充。结果显示出针对点密度变化的鲁棒行为，并且与图像处理等效物一致。尽管选择正确的结构元素需要有关问题和输入点云的先前知识，但所提出的方法易于实现且快速。此外，该方法解决了众所周知的点云处理问题，例如目标检测，分割和间隙填充。