Point Cloud Processing Based On Computational Topology And Deep Learning

In recent years,topological data analysis(TDA)has become a popular tool for studying high-dimensional data.It can extract the topological structure information of highdimensional data,record the birth and death times of connected components,loops,trapped volumes and other topological structures of three-dimensional point clouds,which can be used as a topological representation of point cloud data.This topological representation can be expressed as a multi-set on a two-dimensional plane.We call this point set the persistence diagram(PD).Computing PD from high-dimensional point cloud is a highly time-complex algorithm.Time-consuming Computation limits the application of topological data analysis.At the same time,PD,as a multi-set,is difficult to be applied to common deep learning algorithms.On the basis of PD,researchers have proposed persistence image(PI)to convert the multi-set into an 2D image,which is suitable for various classifiers and neural networks.However,the time-consuming calculation of PD severely limits its application.This article mainly makes the following contributions to this problem:1.A novel point cloud neural network is proposed,which we called TopologyNet.It directly computes the corresponding persistence image(PI)from the input high-dimensional data.There are two main contributions:1.While keeping the error small,the calculation speed is greatly accelerated.2.The discrete calculation process is replaced with the neural network framework.It is possible to allow the back propagation of the mean square error loss function between PIs.This loss is called the differentiable topological loss.2.At the same time,in the application of point cloud generation,a novel method is proposed to use the topological signal as additional input to improve the effect of the point cloud autoencoder,which we call topological autoencoder.The additional topological information reduces the error of generating the point cloud to a certain extent and reflects the effect of topological signals on point cloud generation.Using the topological autoencoder,applications such as topology editing,shape interpolation,and shape completion can be realized.Finally,the generative modeling of point cloud is realized by using the generative adversarial network.