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1. Nonparametric regression for 3D point cloud learning

   Li, Xinyi; Yu, Shan; Wang, Yueying; Wang, Guannan; Wang, Li; Lai, Ming-Jun (January 2024, Journal of machine learning research)

   Shen, Xiaotong (Ed.)

   In recent years, there has been an exponentially increased amount of point clouds collected with irregular shapes in various areas. Motivated by the importance of solid modeling for point clouds, we develop a novel and efficient smoothing tool based on multivariate splines over the triangulation to extract the underlying signal and build up a 3D solid model from the point cloud. The proposed method can denoise or deblur the point cloud effectively, provide a multi-resolution reconstruction of the actual signal, and handle sparse and irregularly distributed point clouds to recover the underlying trajectory. In addition, our method provides a natural way of numerosity data reduction. We establish the theoretical guarantees of the proposed method, including the convergence rate and asymptotic normality of the estimator, and show that the convergence rate achieves optimal nonparametric convergence. We also introduce a bootstrap method to quantify the uncertainty of the estimators. Through extensive simulation studies and a real data example, we demonstrate the superiority of the proposed method over traditional smoothing methods in terms of estimation accuracy and efficiency of data reduction. 

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2. Visualizing Point Cloud Classifiers by Curvature Smoothing

   Ziwen, Chen; Wu, Wenxuan; Qi, Zhongang; Fuxin, Li (January 2020, The 31st British Machine Vision Virtual Conference)

   null (Ed.)

   Recently, several networks that operate directly on point clouds have been proposed. There is significant utility in understanding their mechanisms to classify point clouds, which can potentially help diagnosing these networks and designing better architectures. In this paper, we propose a novel approach to visualize features important to the point cloud classifiers. Our approach is based on smoothing curved areas on a point cloud. After prominent features were smoothed, the resulting point cloud can be evaluated on the network to assess whether the feature is important to the classifier. A technical contribution of the paper is an approximated curvature smoothing algorithm, which can smoothly transition from the original point cloud to one of constant curvature, such as a uniform sphere. Based on the smoothing algorithm, we propose PCI-GOS (Point Cloud Integrated-Gradients Optimized Saliency), a visualization technique that can automatically find the minimal saliency map that covers the most important features on a shape. Experiment results revealed insights into different point cloud classifiers. The code is available at https://github.com/arthurhero/PC-IGOS 

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3. An asymptotic and empirical smoothing parameters selection method for smoothing spline ANOVA models in large samples

   https://doi.org/10.1093/biomet/asaa047  

   Sun, Xiaoxiao; Zhong, Wenxuan; Ma, Ping (August 2020, Biometrika)

   null (Ed.)

   Summary Large samples are generated routinely from various sources. Classic statistical models, such as smoothing spline ANOVA models, are not well equipped to analyse such large samples because of high computational costs. In particular, the daunting computational cost of selecting smoothing parameters renders smoothing spline ANOVA models impractical. In this article, we develop an asympirical, i.e., asymptotic and empirical, smoothing parameters selection method for smoothing spline ANOVA models in large samples. The idea of our approach is to use asymptotic analysis to show that the optimal smoothing parameter is a polynomial function of the sample size and an unknown constant. The unknown constant is then estimated through empirical subsample extrapolation. The proposed method significantly reduces the computational burden of selecting smoothing parameters in high-dimensional and large samples. We show that smoothing parameters chosen by the proposed method tend to the optimal smoothing parameters that minimize a specific risk function. In addition, the estimator based on the proposed smoothing parameters achieves the optimal convergence rate. Extensive simulation studies demonstrate the numerical advantage of the proposed method over competing methods in terms of relative efficacy and running time. In an application to molecular dynamics data containing nearly one million observations, the proposed method has the best prediction performance. 

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4. Point Cloud Segmentation based on Hypergraph Spectral Clustering

   https://doi.org/10.1109/ITA50056.2020.9244954  

   Zhang, Songyang; Cui, Shuguang; Ding, Zhi (February 2020, 2020 Information Theory and Applications Workshop (ITA))

   null (Ed.)

   Hypergraph spectral analysis has emerged as an effective tool processing complex data structures in data analysis. The surface of a three-dimensional (3D) point cloud and the multilateral relationship among their points can be naturally captured by the high-dimensional hyperedges. This work investigates the power of hypergraph spectral analysis in unsupervised segmentation of 3D point clouds. We estimate and order the hypergraph spectrum from observed point cloud coordinates. By trimming the redundancy from the estimated hypergraph spectral space based on spectral component strengths, we develop a clustering-based segmentation method. We apply the proposed method to various point clouds, and analyze their respective spectral properties. Our experimental results demonstrate the effectiveness and efficiency of the proposed segmentation method. 

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5. A PARALLEL ALGORITHM FOR LOCAL POINT DENSITY INDEX COMPUTATION OF LARGE POINT CLOUDS

   https://doi.org/10.5194/isprs-annals-VIII-4-W2-2021-75-2021  

   Vo, A. V.; Lokugam Hewage, C. N.; Le Khac, N. A.; Bertolotto, M.; Laefer, D. (January 2021, ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences)

   Abstract. Point density is an important property that dictates the usability of a point cloud data set. This paper introduces an efficient, scalable, parallel algorithm for computing the local point density index, a sophisticated point cloud density metric. Computing the local point density index is non-trivial, because this computation involves a neighbour search that is required for each, individual point in the potentially large, input point cloud. Most existing algorithms and software are incapable of computing point density at scale. Therefore, the algorithm introduced in this paper aims to address both the needed computational efficiency and scalability for considering this factor in large, modern point clouds such as those collected in national or regional scans. The proposed algorithm is composed of two stages. In stage 1, a point-level, parallel processing step is performed to partition an unstructured input point cloud into partially overlapping, buffered tiles. A buffer is provided around each tile so that the data partitioning does not introduce spatial discontinuity into the final results. In stage 2, the buffered tiles are distributed to different processors for computing the local point density index in parallel. That tile-level parallel processing step is performed using a conventional algorithm with an R-tree data structure. While straight-forward, the proposed algorithm is efficient and particularly suitable for processing large point clouds. Experiments conducted using a 1.4 billion point data set acquired over part of Dublin, Ireland demonstrated an efficiency factor of up to 14.8/16. More specifically, the computational time was reduced by 14.8 times when the number of processes (i.e. executors) increased by 16 times. Computing the local point density index for the 1.4 billion point data set took just over 5 minutes with 16 executors and 8 cores per executor. The reduction in computational time was nearly 70 times compared to the 6 hours required without parallelism. 

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