Triangulations

Hand Pose Estimation (HPE) is a fundamental technology for various applications such as AR/VR and robotics. In these applications, the visibility of each hand joint in the image is crucial for assessing the reliability of estimation results under occlusion. However, most existing HPE methods output joint positions without explicitly indicating their visibility. Although some methods account for occlusion or visibility, visibility estimation has mainly been used as an auxiliary signal for improving pose estimation. To our knowledge, per-joint hand visibility estimation has not been systematically studied as a standalone task. In this work, we propose Hand Visibility Detector, a model for estimating the visibility of individual hand joints, and present the first systematic investigation of visibility estimation as an independent task. We show that leveraging the prior knowledge of HPE models pretrained on large-scale data as a backbone yields high performance in this task. We further demonstrate the utility of Hand Visibility Detector on a downstream task of 3D hand pose annotation via multi-view triangulation of 2D keypoints, showing that visibility-weighted triangulation reduces reprojection error. Our method is released as a ready-to-use package, and the code and demo are available at https://github.com/ryhara/hand_visibility_detector .

We present an extension of Triangolazioni (a Cultural Heritage webapp) to enrich curated content with context-dependent, external information provided by Large Language Models (LLMs) within a loosely-coupled architecture agnostic to the LLM. The system supports context-dependent information search and presentation within an architecture agnostic to the exploited LLM.

This work proposes an adaptation of the attention mechanism for triangle meshes. The core observation is that endowing the attention mechanism with critical properties for learning over meshes -- intrinsicality and triangulation-agnosticism -- enables it to attain state-of-the-art results over several learning-based tasks in geometry-processing. The above is achieved by modifying the attention mechanism from the bottom up based on simple principles from geometry-processing. Namely, the quantities used within attention -- queries, keys and values -- are created by an intrinsic, triangulation-agnostic network, and treated as discretizations of continuous functions. From that, we devise an appropriate attention mechanism that operates over triangle meshes through standard FEM discretization of the resulting integrals of the above functions. Surprisingly, as far as we know, this straightforward approach has not been utilized for learning over meshes. Experiments show our method exceeds current state of the art, including both mesh-based architectures as well as point cloud transformers. Namely, we show significant improvements on several common benchmarks and tasks -- predicting canonical high-frequency signals; predicting deformations; computing dense correspondences, both between full shapes and partial ones; and predicting feature descriptors.

Recent radiance field methods represent scenes with 2D primitives that offer surface alignment and efficient rasterization, from Gaussian disks to triangles, yet all rely on convex boundaries: curved and concave structures demand excessive primitives. We introduce Deformable Triangle Splatting, which augments each triangle with $K$ control points per edge, each parameterized by a single learnable scalar displacement that shifts the boundary inward or outward, enabling non-convex shape representation while preserving the three base vertices that define the 3D plane. To render these non-convex primitives differentiably, we design a rasterization pipeline in the triangle's barycentric coordinate space, ensuring view-consistent rendering. A winding number test determines whether each pixel lies inside the deformed primitive, and a window function controlled by two learnable parameters, sharpness and corner smoothness, together with a per-primitive scalar opacity, produces the smooth opacity transition from interior to boundary. Validation is done in a variety of real-world scenes, outperforming recent works based on non-volumetric primitives in terms of visual quality and versatility while still achieving competitive rendering efficiency.

Accurate forest inventory and large-scale mapping are essential for ecosystem monitoring and sustainable forest management. Multiple low-cost edge platforms enable efficient large-area data acquisition, but merging independently constructed local maps in GNSS-denied understory environments still requires initialization-free loop closure detection and global registration. This task is challenging because low-cost LiDAR point clouds are sparse and noisy, while repetitive trunk layouts and the lack of distinctive geometric landmarks lead to severe perceptual aliasing and false correspondences. To address these issues, we propose DTIF (Delaunay Triangulation in Forests), a lightweight trunk-topology-based framework for forest loop closure detection and global registration. Tree trunks are first extracted as stable landmarks and encoded using a Delaunay topology for compact scene representation. Candidate submaps are then screened using edge-length and radius statistics, followed by edge--radius consistency verification and strong/weak vertex support aggregation to construct weighted vertex correspondences. Finally, topology-derived reliability weights are incorporated into a decoupled robust pose estimator that separately estimates yaw, horizontal translation, and elevation translation under gravity alignment. Experiments on simulated and real-world forest datasets demonstrate that DTIF achieves accurate registration with low computational overhead, providing a favorable balance among robustness, efficiency, and deployability on resource-constrained edge platforms.

We describe a method for computing signed distance to point clouds that allows fast pointwise evaluation at arbitrary spatial resolution. As input, our method takes a point cloud with normals; as output, it provides an analytical parameterization that allows queries of signed distance to the approximate underlying surface at arbitrary points - simultaneously providing reconstruction and distance. Our key idea is to reconstruct shapes by locally fitting point clouds with tori, which have closed-form signed distance functions. Tori are fitted in a feed-forward manner, using a pre-trained network to output per-point curvature and shift parameters. Importantly, our method does not require costly global optimization or spatial discretization, and is easily parallelizable. Underlying our method is a new theory that unifies signed distance with the classic reconstruction methods of winding numbers and Poisson surface reconstruction. We use our method to compute signed distance to point clouds arising from photogrammetry, meshes, 3D Gaussians, and neural implicits. Our method allows point clouds to be used directly in applications, without explicit surface reconstruction: as examples, we take offsets of point clouds, apply morphological and Boolean operations, and directly visualize offset surfaces using sphere tracing.

Triangulations, i.e., well-structured decompositions of geometric objects into triangle-like pieces, are central objects in many domains of mathematics and physics. In particular, fine, regular, and star triangulations (FRSTs) of 4D reflexive polytopes give rise to smooth Calabi-Yau threefolds, which are of significant interest in string theory. However, the high dimensionality and combinatorial complexity of triangulations make them particularly challenging to model with classical numerical methods or machine learning. In this work, we show that transformers, equipped with an appropriate encoding scheme, can be effectively trained to representatively generate new FRSTs across a range of polytope sizes. Moreover, these models can also self-improve through retraining on their own output. This opens the door to both concrete applications to the classification of Calabi-Yau manifolds and further research in physics, combinatorics and algebraic geometry.

Automated floor plan generation is an important problem in computational architectural design. The goal is to construct a floor plan from user-defined room numbers and door requirements. The user specifies which rooms must share a door and which rooms must not be adjacent. However, these requirements do not determine the exact placement or shape of the rooms. The task is therefore to arrange the rooms in a single floor plan so that all required door connections are satisfied and no rooms overlap. To address this problem, we propose DPLAN (Door Connectivity to Floor Plan Generation), a graph-based prototype that generates floor plans from door and non-adjacency constraints. The framework operates in three stages. First, the user-defined graph is examined and, if disconnected, additional edges are added to connect its components. Second, a bi-connected plane triangulation is constructed to ensure the existence of a floor plan without overlapping rooms or empty spaces. Third, the triangulated graph is transformed into floor plans. For rectangular floor plans (RFPs), separating triangles are removed by modifying edges without adding new vertices, thereby avoiding the creation of extra rooms. For orthogonal floor plans (OFPs), separating triangles are removed by introducing additional vertices, allowing rectilinear room shapes. By enforcing both door and non-adjacency requirements, the framework generates floor plans that satisfy the given constraints. The method is implemented in Python and includes a prototype for interactive constraint specification and floor plan visualization. Currently, the framework supports rectangular plot boundaries. Future work includes support for non-rectangular plots, dimension-based scaling, and circulation modeling.

While event cameras offer immense potential for underwater SLAM, existing Time Surface (TS)-based methods prove highly unreliable when deployed underwater. Fluctuating camera velocities severely degrade TS imaging quality, while wide stereo baselines and repetitive underwater textures induce critical matching failures, frequently triggering system failure. To overcome these challenges, we develop the first high-accuracy event-based underwater stereo SLAM system. A structure-aware metric for TS is designed based on structure tensor coherence and gradients to quantitatively evaluate TS structural information density. By decoupling the optimal TS generation into two distinct stages based on system initialization, Bayesian Optimization(BO) first predicts an optimal prior TS sequentially before initialization while we set an asynchronous online local searching method periodically to obtain appropriate TS in real-time during the tracking stage. We use the prior disparity to guarantee precise data association and "latest-observation-first'' triangulation mechanism to realize stable triangulation. As a benchmark for these solutions and a resource for the community, we also contribute UWE, the first high-quality real-world underwater event dataset containing variable camera motions, complex textures and different trajectory features. Extensive evaluations on public datasets and UWE show the competitive accuracy performance of the proposed SLAM system compared to the state-of-the-art event-based method. The code and data will be open-sourced.

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, $$(κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}.$$ of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $α, κ$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(α, κ)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

Various language domains have undergone remarkable changes in recent years; these shifts are largely attributed to the advent of Large Language Models and their misalignment with natural language usage. These misalignments are thought to partly originate in the preference-learning stage, e.g. Reinforcement Learning from Human Feedback, which generally makes models more useful but simultaneously may introduce systematic lexical bias. In terms of lexical behavior, this is visible in a model's preference for certain formats or the overuse of words (delve, furthermore), even when such patterns are not present in base model outputs. Research on lexical misalignment induced during preference training is constrained by reliance on manual curation. We address this, by introducing the Triangulated Preference Shift score, a metric that triangulates between human gold standards, base models, and instruct variants to isolate shifts induced specifically by preference learning, without manual curation. We provide data across six model families, anchor the results in the literature, and illustrate the general approach's utility by analyzing whether preference learning shifts models toward what could be interpreted as a "language of prestige". The metric provides an initial automated method to quantify behavioral shifts attributable to preference tuning, and thus, may help inform model alignment and development of trustworthy AI.

We present a dense RGB-D SLAM system using differentiable triangles as the 3D map representation. While 3D Gaussian Splatting has emerged as the leading method for novel-view synthesis, triangles remain the standard primitive for traditional rendering hardware, game engines, and downstream tasks requiring explicit geometry such as simulation, collision, and editing. Recent offline methods have demonstrated that an unstructured 'triangle soup' can be optimised into a photorealistic mesh via Delaunay triangulation across a set of posed images. Building upon this insight, we present the first dense SLAM system to employ Triangle Splatting to perform both tracking and mapping through online differentiable rendering of a triangle soup. The map can be converted into a connected mesh on-the-fly via restricted Delaunay triangulation, enabling new online capabilities such as mesh deformation and collision checking. On Replica and TUM-RGBD, our system outperforms baselines on 3D geometry, matches the camera-tracking accuracy, and enables online mesh-based scene editing.

We introduce TriSearch, a reinforcement learning framework for optimizing objectives over triangulations of a polytope via bistellar flips. The key idea is a circuit-supported subtriangulation action representation: feasible flips are encoded by their supporting circuit and realized local subtriangulation, enabling a learned policy to rank them using local geometric and combinatorial features. This yields a dimension-agnostic interface and enables efficient traversal of the flip graph without explicit enumeration of the full triangulation space. Instantiated in 3D and 4D, TriSearch generalizes zero-shot from small training instances to larger polytopes with exponentially larger search spaces. It achieves top performance on metric objectives in 3D and, in 4D, discovers more distinct Fine, Regular, Star triangulations of reflexive polytopes, corresponding to Calabi-Yau threefolds, than existing samplers under a fixed budget.

We introduce dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations of lattice polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by signed circuits' -- combinatorial invariants from the theory of oriented matroids. We show that these circuits are necessary and sufficient to determine a triangulation's regularity from the graph, provided certain magnitude information is retained. The model is independent of the polytope's point count and invariant under its orientation-preserving symmetries ($\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with $N_\mathrm{pts} \leq 40$, dualGNN is the only sampler we tested that is consistent with uniform sampling across all our diagnostics (KL divergence from uniformity, collision counts, and sample autocorrelation). The model is small ($\sim92$k parameters) and trains in $\sim7.5$ hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at $h^{1,1}=86$; we also sample CYs at $h^{1,1}=128$, observing no deviations from uniformity, but our diagnostics are weaker here. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN (pip install dualgnn), and dualGNN is integrated into CYTools.

Sparse-view 3D reconstruction is increasingly addressed with feed-forward splatting networks that predict explicit primitives directly from images. Yet most existing methods remain centered on Gaussian primitives and expose surfaces only indirectly: extracting a usable mesh for downstream simulation, physics reasoning, or embodied interaction still requires expensive post-hoc steps that break the feed-forward promise. This limitation is especially pronounced in pose-free settings, where scene structure and camera parameters must be estimated jointly from sparse observations. We present TriSplat, a feed-forward reconstruction network that represents scenes with oriented triangle primitives and directly exports simulation-ready mesh scenes from a single forward pass. Given input images, the network predicts local 3D point maps, triangle attributes, camera poses, and optional intrinsics. Rather than regressing triangle orientation as an unconstrained latent variable, our approach constructs geometry normals from the predicted point maps, refines them with an image-conditioned normal head, and converts them into stable local frames for triangle parameterization. A mono-normal bootstrap schedule further stabilizes early training, while opacity and blur scheduling progressively sharpens the learned surface representation for direct mesh extraction. Experiments on RealEstate10K and DL3DV show that this representation produces more geometry-faithful reconstructions than Gaussian feed-forward baselines while maintaining competitive novel-view rendering quality. Because the rendering primitives are themselves surface triangles, the output can be directly ingested by physics engines, collision detectors, and standard rendering pipelines without any conversion, making it a practical simulation-ready solution for feed-forward 3D scene reconstruction.

Advancements in 3D curve skeletonization are accelerating progress across a wide range of applications. However, developing robust skeletonization algorithms that capture intricate object details remains challenging. Skeletonization via Local Separators (LS) offers an efficient graph-based approach but suffers from representation inaccuracies due to its discrete nature. To address this, we introduce CSCD, a novel framework for Curve Skeletonization in the Continuous Domain, generalizing LS to manifolds. Specifically, we present two realizations: CSCD-M for meshes and CSCD-PC for point clouds. CSCD-M leverages the intrinsic triangulation of a mesh for resilience to noise and improved topological preservation, while CSCD-PC employs tufted Laplacians for enhanced robustness. To our knowledge, CSCD-M is the first intrinsic method for curve skeletonization. Our results show CSCD-M matches LS performance across diverse meshes and outperforms LS (TOG'21) on benchmarks like Thingi10k dataset. CSCD-PC qualitatively outperforms CoverageAxis++ (Eurographics'24) and EPCS (CAG'23). Finally, we demonstrate the efficacy of CSCD in a few downstream tasks: object classification, shape segmentation, identifying handles, tunnels, and constrictions in objects. Project Website: https://cscd-skel.pages.dev

This paper tackles the task of learning to generate signals over triangle meshes in a triangulation-agnostic manner, meaning the trained model can be applied to different meshes and triangulations effectively. Practically, the paper adapts the flow matching (FM) paradigm to a mesh-based, triangulation-agnostic setting. Theoretically, it proposes a specific noise distribution which is triangulation agnostic, to be used inside the FM model's denoising process. While noise distributions are usually trivial to devise for, e.g., images, devising a triangulation-agnostic distribution proves to be a much more difficult task. We formulate a mathematical definition of triangulation agnosticism of distributions, via their spectrum. We then show that a discretization of a specific Gaussian random field called a Matérn process holds these desired properties, and provides a simple and efficient sampling algorithm. We use it as our noise model, and adapt FM to the triangulation-agnostic setting by using a state-of-the-art approach for learning signals on meshes in the gradient domain -- PoissonNet -- as the denoiser. We conduct experiments on elaborate tasks such as sampling elastic rest states, and generating poses of humanoids. Our method is shown to be capable of producing highly realistic results for meshes of over one million triangles, significantly exceeding the state-of-the-art in quality and diversity.

We study Relational Semantic Decomposition, abbreviated as RSD, as a moving local triangular chart audit for language-model hidden states. For repeated occurrences of one target word, RSD fits a shared three-anchor membership chart $S_t$ at layer or token-time $t$. The hidden-state channel uses $X_t\approx S_tC_t$; the invariant readout $M_t=S_tS_t^\top$ is the induced occurrence co-membership relation, and $R_t=X_t-S_tC_t$ records what the fitted root chart leaves outside the chart. The broader joint audit reuses the same membership chart for relation data, $A_t\approx S_tB_tS_t^\top$, such as an attention-derived occurrence relation. The current GPT-2 evidence is the $X$-channel hidden-state audit with Word-in-Context labels used as an external same-sense versus different-sense reference relation. On full WiC train, the root chart passes 16 of 53 eligible target words; this is audit coverage, not GPT-2 task accuracy. Token-time and pair-level diagnostics show the main regimes: \texttt{make} and \texttt{break} align at the target state, \texttt{drive} and \texttt{stay} improve after right context in small-count exploratory cases, and \texttt{play} remains a localized root-chart failure whose final same-sense pairs are not closer and have larger residual discrepancy. The resulting claim is diagnostic: RSD reports where a sense relation is visible in root co-membership and which failures become residual branch candidates or attention-channel obligations.

Real-time Light Detection And Ranging (LiDAR) simulation must find, per emitted ray, the closest intersecting triangle even in dynamic scenes containing large numbers of moving and deformable objects. Dominant acceleration-structure approaches require rebuilding each frame for dynamic geometry -- a cost that compounds directly with scene dynamics and cannot be amortized regardless of how little actually changed. This paper presents the Gajmer Ray-Casting Algorithm (GRCA), which inverts the question: instead of asking what does each ray hit? it asks which rays can each triangle possibly hit? GRCA geometrically models spinning LiDAR emitters as rotation-traced cones or planes and uses each triangle's emitter-centric apparent area to cull, per triangle, which channels and the rays within those channels can possibly reach it -- without any acceleration structure. GRCA is compute-based and vendor-agnostic by design, targeting highly dynamic, high-resolution simultaneous multi-sensor simulation. At its core, GRCA is a general-purpose ray-casting algorithm: the emitter-centric inversion applies to any setting where rays originate from a known position, not only LiDAR. Benchmarks evaluate 2-8 simultaneous 128x4096-ray LiDARs (360deg/180deg) over complex dynamic scenes -- with just two sensors casting ~1M rays per frame. With range culling inactive, GRCA reaches up to 7.97x over hardware-accelerated OptiX (GPU) and 14.55x over Embree (CPU). Two independent extensions further boost performance even in the most complex scene (~22M triangles, ~9M of which are dynamic, 8 LiDARs): range culling at realistic deployment ranges (10-100m) reaches up to 7.02x GPU and 9.33x CPU; a hybrid pipeline -- GRCA for dynamic geometry, OptiX/Embree for static -- reaches up to 10.5x GPU and 19.2x CPU.

Translation averaging aims to recover camera locations from pairwise relative translation directions and is a fundamental component of global Structure-from-Motion pipelines. The problem is challenging because direction measurements contain no distance information, making the estimation problem highly ill-conditioned and highly sensitive to corrupted observations. In this paper, we propose TriP, a triangle-based framework for robust translation averaging. TriP first infers local relative edge scales from triangle geometry, and then synchronizes the scales of overlapping triangles in the logarithmic domain to recover globally consistent edge lengths and camera locations. By leveraging higher-order consistency across triangles, the proposed method is robust to adversarial, cycle-consistent, and other structured corruptions. In addition, TriP avoids the collapse issue without requiring any extra anti-collapse constraints, since log-scale synchronization excludes the degenerate zero-scale solution by construction. These structural advantages enable a particularly strong theory for exact location recovery. On the practical side, TriP is fully parallelizable, computationally efficient, and naturally scalable to graphs with millions of cameras. Moreover, it outperforms all previous translation averaging methods by a large margin on both synthetic and real datasets.

Pairwise translation directions are a key input to camera location estimation in global structure-from-motion. Existing estimators usually process each image pair independently, producing directions that may be locally plausible but inconsistent with the other relative directions in the viewing graph. To jointly estimate the direction, we propose TriDE, which exploits camera-triangle consistency as an efficient higher-order verification signal. Instead of solving a costly global nonlinear optimization problem that is sensitive to initialization, TriDE refines unreliable pairwise directions through message passing between directions and their incident weighted triangles. This information propagation strategy enables us to establish a strong phase-transition bound for exact recovery under a realistic random corruption model. Experiments on real image graphs show that TriDE improves direction accuracy by a large margin and yields better downstream camera locations, providing a practical link between local pairwise estimation and global camera pose geometry.

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, unlike prior claims, both graph neural networks (GNNs) and higher-order message passing (HOMP) methods can saturate the benchmark. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. We thus provide a novel evaluation protocol based on representational diversity and triangulation refinement. Surprisingly, we find no indication that existing models are capable of generalizing beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.

Recovering 3D human pose from multi-view imagery typically relies on precise camera calibration, which is often unavailable in real-world scenarios, thereby severely limiting the applicability of existing methods. To overcome this challenge, we propose an unconstrained framework that synergizes deep neural networks, algebraic priors, and temporal dynamics for uncalibrated multi-view human pose estimation. First, we introduce the Triangulation with Transformer Regressor (TTR), which reformulates classical triangulation into a data-driven token fusion process to bypass the dependency on explicit camera parameters. Second, to explicitly embed the inherent algebraic relations of the multi-view variety into the learning process, we propose the Gröbner basis Corrector (GC). This pioneering loss formulation enforces constraints derived from the multi-view variety to ensure the neural predictions strictly adhere to the laws of projective geometry. Finally, we devise the Temporal Equivariant Rectifier (TER), which exploits the equivariance property of human motion to impose temporal coherence and structural consistency, effectively mitigating scale ambiguity in uncalibrated settings. Extensive evaluations on standard benchmarks demonstrate that our framework establishes a new state-of-the-art for uncalibrated multi-view human pose estimation. Notably, our approach significantly closes the performance gap between calibration-free methods and fully calibrated oracles.

A fundamental challenge in causal inference with observational data is correct specification of a causal model. When there is model uncertainty, analysts may seek to use estimates from multiple candidate models that rely on distinct, and possibly partially overlapping, sets of identifying assumptions to infer the causal effect, a process known as triangulation. Principled methods for triangulation, however, remain underdeveloped. Here, we develop a framework for causal effect triangulation that combines model testability methods from causal discovery with statistical inference methods from semiparametric theory, while avoiding explicit model selection and post-selection inference problems. We propose a triangulation functional that combines identified functionals from each model with data-driven measures of model validity. We provide a bound on the distance of the functional from the true causal effect along with conditions under which this distance can be taken to zero. Finally, we derive valid statistical inference for this functional. Our framework formalizes robustness under causal pluralism without requiring agreement across models or commitment to a single specification. We demonstrate its performance through simulations and an empirical application.

Reconstructing continuous surfaces from unoriented and unordered 3D points is a fundamental challenge in computer vision and graphics. Recent advancements address this problem by training neural signed distance functions to pull 3D location queries to their closest points on a surface, following the predicted signed distances and the analytical gradients computed by the network. In this paper, we introduce NumGrad-Pull, leveraging the representation capability of tri-plane structures to accelerate the learning of signed distance functions and enhance the fidelity of local details in surface reconstruction. To further improve the training stability of grid-based tri-planes, we propose to exploit numerical gradients, replacing conventional analytical computations. Additionally, we present a progressive plane expansion strategy to facilitate faster signed distance function convergence and design a data sampling strategy to mitigate reconstruction artifacts. These components are synergistically integrated into a unified tri-plane-based pulling framework, in which numerical gradients, progressive expansion, and complementary sampling jointly address the locality and sparsity challenges of learning SDFs from unoriented point clouds. Our extensive experiments across a variety of benchmarks demonstrate the effectiveness and robustness of our approach. Codes are available at: https://github.com/cuiruikai/numgrad-pull.