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 τ=1000 (i.e., σ≈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 τ=500 (i.e., σ≈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 τ=30 (i.e., σ≈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 τ=3000,30,0.3 (i.e., σ≈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
Figure 9: Add/remove proposal mechanism used in Update B* . The left and right panels illustrate the cases Bprop=Bold+1 (add) and Bprop=Bold−1 (remove), respectively. The complementary proposals G and F applied in the two cases determine the positioning of vertex hB , 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 τ=500 (i.e., σ≈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.
Atlas-based approaches allow high-quality, patient-specific shape reconstructions of cardiac anatomy from sparse and/or noisy data such as point clouds. However, these methods are mainly prior-driven, so the impact of uncertainty can be large, limiting their clinical reliability. We propose a probabilistic framework for uncertainty-aware cardiac shape reconstruction that combines Deep Signed Distance Functions (DeepSDFs) with Markov Chain Monte Carlo (MCMC) sampling. Cardiac geometries are modeled implicitly as zero-level sets of a neural network conditioned on learned latent codes, enabling multi-surface reconstruction of the left and right ventricles. By interpreting the reconstruction loss as a log-likelihood, we perform Bayesian inference in the latent space to obtain both maximum a posteriori (MAP) and posterior-sampled reconstructions. Experiments on a public cardiac dataset show that our approach produces accurate reconstructions and well-calibrated uncertainty estimates.
Jan Verhülsdonk, Thomas Grandits, Francisco Sahli Costabal +3
Institute for Applied Mathematics, University of Bonn, Germany · Department of Mathematics and Scientific Computing, University of Graz, Austria and with NAWI Graz · Institute for Biological and Medical Engineering, Pontificia Universidad Católica de Chile, Chile +3
We develop a framework for task-specific active next-best-view selection in 3D reconstruction from point clouds, by casting the problem in the language of Bayesian decision theory. Our framework works by (a) placing a prior distribution over the space of implicit surfaces, (b) using recently-developed stochastic surface reconstruction methods to calculate the resulting posterior distribution, then (c) using the posterior distribution to carefully reason about which view to scan next. This enables us to perform camera selection in a manner that is directly optimized for the intended use of the reconstructed data - meaning, we reduce uncertainty only in those regions that make a difference in the task at hand, as opposed to prior approaches that reduce it uniformly across space. We evaluate our method across three distinct downstream tasks: semantic classification, segmentation, and PDE-guided physics simulation. Experimental results demonstrate that our framework achieves superior task performance with fewer views compared to commonly used baselines and prior general uncertainty-reduction techniques.
Jingsen Zhu, Silvia Sellán, Alexander Terenin
Cornell University, USA · Columbia University, USA
We propose a novel framework for joint magnetic resonance image reconstruction and uncertainty quantification using under-sampled k-space measurements. The problem is formulated as a Bayesian linear inverse problem, where prior distributions are assigned to the unknown model parameters. Specifically, we assume the target image is sparse in its spatial gradient and impose a total variation prior model. A Markov chain Monte Carlo (MCMC) method, based on a split-and-augmented Gibbs sampler, is then used to sample from the resulting joint posterior distribution of the unknown parameters. Experiments conducted using single- and multi-coil datasets demonstrate the superior performance of the proposed framework over optimisation-based compressed sensing algorithms. Additionally, our framework effectively quantifies uncertainty, showing strong correlation with error maps computed from reconstructed and ground-truth images.
Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander
Department of Computer Science, University of Exeter, UK · UCL Hawkes Institute, Department of Computer Science, University College London, UK