Bayesian methods and Markov chain Monte Carlo algorithms for curve reconstruction and point cloud data analysis

Asir Intesar Tushar; Ioannis Sgouralis

cs.LGAug 27, 2026

Bayesian methods and Markov chain Monte Carlo algorithms for curve reconstruction and point cloud data analysis

Authors: Asir Intesar Tushar, Ioannis Sgouralis

Organizations: Department of Mathematics, University of Tennessee, Knoxville, TN, USA

Abstract

Point-cloud data routinely captured by modern imaging and sensor technologies provide detailed geometric descriptions of objects and environments, but their analysis is hindered by large data volumes, localization noise, and missing information. In addition, existing point-cloud reconstruction pipelines typically return a single best-fit structure without uncertainty quantification. We introduce a fully Bayesian framework for representing point-cloud data and reconstructing closed curves, in which observed points are modeled as noisy perturbations of latent locations constrained to lie on the underlying curve that is regularized by a non-parametric prior. Posterior inference in our framework is carried out using a series of Markov chain Monte Carlo samplers tailored to point-cloud characteristics. Numerical experiments, including synthetic examples and real-world LiDAR datasets, show accurate reconstructions and quantified uncertainty over the recovered curves.

Figures & tables

Figure 1: Graphical representation of the statistical model in this study. Following the common convention [ 23 , 25 ] , random quantities are demonstrated with circles, deterministic quantities are shown with diamonds, and quantities with known values are shaded. In addition, arrows specify dependencies between the various quantities and plates indicate repetition over the specified index.

Figure 2: Synthetic data analysis of a Lissajous curve. Reconstruction is applied on a noisy point-cloud of size $N=1500$ and noise $\tau=1000$ (i.e., $\sigma\approx 0.032$ ). A total of 3,000 MCMC iterations are performed using Sampler 3, with the first 50% discarded as burn-in and the remaining samples thinned at a ratio of 1:3 for visualization purposes. The panel on the left shows the point-cloud and reconstructed curves. The panels to the right summarize key posterior results.

Figure 3: Synthetic data analysis of an octagon curve. Reconstruction is performed on a noisy point-cloud of size $N=12000$ and noise $\tau=500$ (i.e., $\sigma\approx 0.045$ ). A total of 12,000 MCMC iterations are performed using Sampler 3, with the first 25% discarded as burn-in and the remaining samples thinned at a ratio of 1:3 for illustration purposes. The panel on the left shows the point-cloud and reconstructed curves. The panels to the right summarize key posterior results.

Figure 4: Effects of point-cloud size. Reconstruction is performed on three point-clouds with sizes $N=150,100,50$ , left to right, and fixed noise $\tau=30$ (i.e., $\sigma\approx 0.18$ ). For each experiment, $3{,}000$ MCMC iterations are performed using Sampler 3, with the first 30% discarded as burn-in and the remaining samples thinned at a ratio of 1:2 for illustration purposes. The top panels show the point-clouds used and reconstructed curves. The square marks the same neighborhood of the lower-right vertex in each panel, and the bottom panels show a magnified view of this region using identical axis limits. The enlarged panels highlight the increase in posterior uncertainty as point-cloud size decreases.

Figure 5: Effects of localization noise. Reconstruction is performed on point-clouds of size $N=1000$ and noise levels $\tau=3000,30,0.3$ (i.e., $\sigma\approx 0.018,0.18,1.8$ ), left to right. For each experiment, $3{,}000$ MCMC iterations are performed using Sampler 3, with the first 30% discarded as burn-in and the remaining samples thinned at a ratio of 1:3 for visualization purposes. The top panels show the point-clouds used and reconstructed curves. The square marks the same neighborhood of the lower-right vertex in each panel, and the bottom panels show a magnified view of this region using identical axis limits. The enlarged panels highlight the increase in posterior uncertainty as noise increases.

Figure 6: Comparison of MCMC samplers on the same synthetic point-cloud. The top panels show the point-clouds used and reconstructed curves. The bottom panels show traces of the corresponding log-posterior. From the three samplers, only Sampler 3 recovers the ground-truth; while, Samplers 1 and 2 remains trapped in local maxima corresponding to only partially matching configurations.

Figure 7: Reconstruction of the Butterfly example . A total of 5,000 MCMC iterations are performed using Sampler 3, with the first 50% discarded as burn-in and the remaining samples thinned at a ratio of 1:2 to improve visual clarity. The panel on the left shows the point-cloud and reconstructed curves. The panels to the right summarize key posterior results.

Figure 8: Reconstruction of field LiDAR point-clouds imaging rock glaciers in the San Juan Mountains, CO (top panels) and McBee Island in the Holston River, TN (bottom panels). In each case, a total of 3,000 MCMC iterations are performed using Sampler 3, with the first 50% discarded as burn-in and the remaining samples thinned at a ratio of 1:8 for visualization purposes. The left panels show reference outlines of the imaged structures and the right panels show the point-cloud used and reconstructed curves.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure 9: Add/remove proposal mechanism used in Update B* . The left and right panels illustrate the cases $B_{\rm prop}=B_{\rm old}+1$ (add) and $B_{\rm prop}=B_{\rm old}-1$ (remove), respectively. The complementary proposals $\mathbb{G}$ and $\mathbb{F}$ applied in the two cases determine the positioning of vertex $h^{B}$ , while the remaining inducing points remain unchanged.

Figure 10: Synthetic data analysis of a Lissajous curve. Reconstruction is applied on a noisy point-cloud of size $N=1000$ and noise $\tau=500$ (i.e., $\sigma\approx 0.045$ ). A total of 3,000 MCMC iterations are performed using Sampler 3, with the first 30% discarded as burn-in and the remaining samples thinned at a ratio of 1:3 for visualization purposes. The panel on the left shows the point-cloud and reconstructed curves. The panels to the right summarize key posterior results.

Figure 11: Reconstruction of the Sawtooth example . A total of 3,000 MCMC iterations are performed using Sampler 3, with the first 20% discarded as burn-in and the remaining samples thinned at a ratio of 1:3 for visualization purposes. The panel on the left shows the point-cloud and reconstructed curves. The panels to the right summarize key posterior results.

Figure 12: Reconstruction of the Bottle example . A total of 3,000 MCMC iterations are performed using Sampler 3, with the first 50% discarded as burn-in and the remaining samples thinned at a ratio of 1:5 for visual purposes. The panel on the left shows the point-cloud and reconstructed curves. The panels to the right summarize key posterior results.

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