3d stochastic geometry modeling

This study addresses a key limitation of traditional marked point process models, which commonly assume independence between marks and locations—an assumption often violated in real-world applications such as forestry. To overcome this constraint, the authors propose a unified framework that, for the first time, enables comprehensive modeling, parameter estimation, simulation, and visualization of location-dependent marked point processes within the R programming environment. Grounded in spatial point process theory, the approach integrates statistical modeling with computational tools to support fitting to empirical data, model diagnostics, and generation of realistic spatial patterns. By relaxing the restrictive independence assumption, this work provides a practical and extensible analytical toolkit for researchers in ecology and related fields.

In finite three-dimensional wireless networks, the coverage probability lacks accurate closed-form analytical solutions due to enhanced spatial dependence among nodes within bounded domains and strong coupling between link distances and interference. Method: This paper proposes the first rigorous analytical framework based on a cylindrical-domain binomial point process (BPP), innovatively decoupling the inter-node distance distribution from the interference term—overcoming inherent limitations of Poisson point process (PPP) modeling in bounded spaces. Leveraging stochastic geometry, along with convolution and derivative properties of Laplace transforms, we derive a computationally efficient and mathematically rigorous closed-form expression for the coverage probability. Results: Monte Carlo simulations validate that the proposed model achieves significantly higher accuracy than conventional PPP-based approaches in constrained 3D scenarios—including UAV, underwater, and robotic networks—with error reductions of 30%–50%. The framework thus bridges theoretical rigor and practical engineering applicability.

Modern telecommunication data traffic exhibits spatial burstiness—characterized by clustering and multiscale scale-invariance—that cannot be adequately captured by Poisson processes. To address this, we propose a novel modeling framework based on sparse stable point processes. We introduce thinning stability—a concept previously unexplored in spatial point processes—to construct a theoretically interpretable, empirically adaptive non-Poisson dependence model capable of identifying bursts across multiple scales and quantifying their intensities. Integrating stochastic geometry, Bayesian inference, and empirical likelihood, we develop a computationally tractable joint parametric and nonparametric inference procedure. Evaluated on real-world network traffic data, our method achieves significantly improved accuracy in anomaly localization and intensity prediction: the AUC improves by 12.6% over classical models, with strong generalization performance across diverse traffic regimes.

Poisson surface reconstruction from partial or sequential point clouds using Gaussian processes (GPs) traditionally involves a two-stage pipeline—first interpolating point-wise normals via GP regression, then solving the volumetric Poisson PDE globally—entailing high computational cost and reliance on diagonal approximations of the kernel matrix inverse. Method: We propose a unified, single-stage framework that embeds geometric Gaussian processes directly into the Poisson reconstruction formulation, jointly modeling surface geometry and uncertainty. This integrates normal interpolation and PDE solving into one sparse linear system, enabling mesh-free function evaluation and local spatial reasoning—including collision detection, on-demand ray casting, and slice-level view planning—without explicit kernel matrix operations. Results: Experiments demonstrate superior reconstruction accuracy and real-time performance over conventional two-stage approaches, validating its efficacy for task-driven, online 3D perception.

This work addresses the challenge of jointly modeling variable point counts and spatial configurations in spatial point process generation by proposing the Existence Field Diffusion Model (EFDM). EFDM introduces, for the first time, an existence field into a diffusion framework, assigning each latent point a continuous existence variable to unify the modeling of point cardinality and location without requiring explicit discrete dimensional jumps. By constructing a joint continuous diffusion mechanism over both existence variables and spatial coordinates, EFDM enables symmetric, flexible, and unified generation of variable-cardinality point processes. Experimental results demonstrate that the proposed model significantly improves generation quality and modeling capability across multiple variable-cardinality datasets.

This study addresses the challenge of disentangling background spatial inhomogeneity—driven by site-specific attractiveness—from inter-individual repulsive interactions in pedestrian waiting behavior, using repeated observations of spatial point patterns. To this end, the authors propose a novel semi-parametric spatial point process model that integrates a determinantal point process with a Gibbs point process. For the first time, repeated spatial point patterns are incorporated into the inference framework of such models, enabling parameter estimation and model assessment based on multiple independent and identically distributed spatial realizations. Applied to real-world pedestrian waiting scenarios, the method successfully reproduces key empirical spatial characteristics, demonstrating its effectiveness in capturing complex crowd distributions and achieving a tight integration of methodological innovation with practical application.

This study addresses the challenge that observed spatial point patterns are often degraded by missed detections (sparsity) and positional errors (displacement), which can severely bias inference. Focusing on Poisson point processes, the authors propose a unified framework that jointly estimates the underlying intensity function nonparametrically while simultaneously learning a parametric model for both the sparsity mechanism and the displacement scale. Under mild conditions, they rigorously establish model identifiability—a first for enabling joint nonparametric inference of the degradation mechanisms and the true intensity function. Simulations demonstrate that the proposed method substantially outperforms misspecified models that ignore such observational degradations. The approach is successfully applied to real-world data on North Atlantic right whale calls in Cape Cod Bay, showcasing its practical utility.

This study addresses the joint optimization of coverage performance and energy efficiency for high-altitude platforms (HAPs) performing circular patrol missions. For the first time, a stochastic geometry framework based on small-circle Cox processes is developed under a spherical Earth model, yielding Poisson and binomial patrol trajectory models. By integrating communication interference analysis with a circular flight energy consumption model, closed-form expressions are derived for the nearest-neighbor distance distribution, coverage probability, and coverage energy efficiency. The work reveals fundamental differences between intensity-driven deployment and finite-formation strategies and provides a closed-form condition for the energy-optimal patrol radius, demonstrating that efficient operation requires joint optimization of patrol geometry, platform density, and cruising speed.