We propose a deep autoencoder with graph topology inference and filtering to achieve compact representations of unorganized 3D point clouds in an unsupervised manner. The encoder of the proposed networks adopts similar architectures as in PointNet, which is a well-acknowledged method for supervised learning of 3D point clouds. The decoder of the proposed networks involves three novel modules: the folding module, the graph-topology-inference module, and the graph-filtering module. The folding module folds a canonical 2D lattice to the underlying surface of a 3D point cloud, achieving coarse reconstruction; the graph-topology-inference module learns a graph topology to represent pairwise relationships between 3D points; and the graph-filtering module designs graph filters based on the learnt graph topology to obtain the refined reconstruction. We further provide theoretical analyses of the proposed architecture. We provide an upper bound for the reconstruction loss and further show the superiority of graph smoothness over spatial smoothness as a prior to model 3D point clouds. In the experiments, we validate the proposed networks in three tasks, including reconstruction, visualization, and classification. The experimental results show that (1) the proposed networks outperform the state-of-the-art methods in various tasks, including reconstruction and transfer classification; (2) a graph topology can be inferred as auxiliary information without specific supervision on graph topology inference; and (3) graph filtering refines the reconstruction, leading to better performances.
Graph signal processing is a theoretical framework for the analysis of high-dimensional data with irregular structures @cite_72 @cite_9 @cite_23 . This framework extends classical discrete signal processing to signals with an underlying irregular structure. The framework models underlying structure by a graph and signals by graph signals, generalizing concepts and tools from classical discrete signal processing to the graph domain. Some techniques involve representations for graph signals @cite_8 @cite_39 , sampling for graph signals @cite_32 @cite_10 @cite_49 , recovery for graph signals @cite_35 @cite_53 , denoising @cite_30 @cite_24 , graph-based filter banks @cite_30 @cite_7 , graph-based transforms @cite_12 @cite_17 , graph topology inference @cite_64 , and graph neural networks @cite_47 @cite_73 . To process 3D point clouds, @cite_37 uses graph filters and graph-based resampling strategies to select most informative 3D nodes; @cite_41 uses graph convolutional neural networks to classify 3D point clouds. In this work, we use graph signal processing techniques to achieve filter design in the proposed graph-filtering module.
@cite_30 @cite_35 @cite_64 @cite_41 @cite_72 @cite_10 @cite_8 @cite_39 @cite_23 @cite_49 @cite_17 @cite_37 @cite_7 @cite_32 @cite_73 @cite_12 @cite_9 @cite_53 @cite_24 @cite_47
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Graph signal processing deals with the processing of signals defined on irregular domains and is an emerging area of research. Graph filter banks allow the wavelet transform to be extended for processing graph signals. Sakiyama and Tanaka (2015) 22 recently proposed a technique to convert linear-phase biorthogonal filter banks for regular domain signals to biorthogonal graph filter banks. Perfect reconstruction is preserved using the technique but the resulting spectral filter functions are transcendental and not polynomial. Polynomial function filters are desired for the localization property and implementation efficiency. In this work we present alternative techniques to perform the conversion. Perfect reconstruction is preserved with the proposed techniques and the resulting spectral filters are polynomial functions. HighlightsWe design filter banks for signal defined over bipartite graphs.Spectral filters are polynomials resulting in localization and efficient implementation.Technique is based on converting 1D regular domain filter banks.Perfect reconstruction is exactly preserved and spectral response shape is almost preserved. Signal processing on graph is attracting more and more attentions. For a graph signal in the low-frequency subspace, the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal. In this paper, the concept of local set is introduced and two local-set-based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods reweights the sampled residuals for different vertices, while the other propagates the sampled residuals in their respective local sets. These algorithms are built on frame theory and the concept of local sets, based on which several frames and contraction operators are proposed. We then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method. Furthermore, the correspondence between graph signal sampling and time-domain irregular sampling is analyzed comprehensively, which may be helpful to future works on graph signals. Computer simulations are conducted. The experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries, imprecise priori knowledge of cutoff frequency, and noisy scenarios. We propose a novel discrete signal processing framework for the representation and analysis of datasets with complex structure. Such datasets arise in many social, economic, biological, and physical networks. Our framework extends traditional discrete signal processing theory to structured datasets by viewing them as signals represented by graphs, so that signal coefficients are indexed by graph nodes and relations between them are represented by weighted graph edges. We discuss the notions of signals and filters on graphs, and define the concepts of the spectrum and Fourier transform for graph signals. We demonstrate their relation to the generalized eigenvector basis of the graph adjacency matrix and study their properties. As a potential application of the graph Fourier transform, we consider the efficient representation of structured data that utilizes the sparseness of graph signals in the frequency domain. Signal processing on graph is attracting more and more attention. For a graph signal in the low-frequency subspace, the missing data on the vertices of graph can be reconstructed through the sampled data by exploiting the smoothness of graph signal. In this paper, two iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods weights the sampled residual for different vertices, while the other conducts a limited propagation operation. Both the methods are proved to converge to the original signal under certain conditions. The proposed methods lead to a significantly faster convergence compared with the baseline method. Experiment results of synthetic graph signal and the real world data demonstrate the effectiveness of the reconstruction methods.