# Neural-Singular-Hessian: Implicit Neural Representation of Unoriented Point Clouds by Enforcing Singular Hessian

arXiv: https://arxiv.org/abs/2309.01793 · HTML: https://mlsanity.com/paper/2309.01793/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 73 references that are in the map.
- First posted on arXiv 4 Sep 2023.

## Builds on (most cited first, 30 of 43)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [pi-GAN: Periodic Implicit Generative Adversarial Networks for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2012.00926/) (2020)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)
- [ABC: A Big CAD Model Dataset For Geometric Deep Learning](https://mlsanity.com/paper/1812.06216/) (2018)
- [Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstruction](https://mlsanity.com/paper/2003.10983/) (2020)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Neural Unsigned Distance Fields for Implicit Function Learning](https://mlsanity.com/paper/2010.13938/) (2020)
- [Shape As Points: A Differentiable Poisson Solver](https://mlsanity.com/paper/2106.03452/) (2021)
- [Thingi10K: A Dataset of 10,000 3D-Printing Models](https://mlsanity.com/paper/1605.04797/) (2016)
- [Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces](https://mlsanity.com/paper/2011.13495/) (2020)
- [Modulated Periodic Activations for Generalizable Local Functional Representations](https://mlsanity.com/paper/2104.03960/) (2021)
- [Points2Surf: Learning Implicit Surfaces from Point Cloud Patches](https://mlsanity.com/paper/2007.10453/) (2020)
- [Deep Geometric Prior for Surface Reconstruction](https://mlsanity.com/paper/1811.10943/) (2018)
- [POCO: Point Convolution for Surface Reconstruction](https://mlsanity.com/paper/2201.01831/) (2022)
- [DiGS : Divergence guided shape implicit neural representation for unoriented point clouds](https://mlsanity.com/paper/2106.10811/) (2021)
- [Dual Octree Graph Networks for Learning Adaptive Volumetric Shape Representations](https://mlsanity.com/paper/2205.02825/) (2022)
- [Geometry-Consistent Neural Shape Representation with Implicit Displacement Fields](https://mlsanity.com/paper/2106.05187/) (2021)
- [Reconstructing Surfaces for Sparse Point Clouds with On-Surface Priors](https://mlsanity.com/paper/2204.10603/) (2022)
- [SA-ConvONet: Sign-Agnostic Optimization of Convolutional Occupancy Networks](https://mlsanity.com/paper/2105.03582/) (2021)
- [Surface Reconstruction from Point Clouds by Learning Predictive Context Priors](https://mlsanity.com/paper/2204.11015/) (2022)

## Built on by (most cited first, 30 of 35)

- [RNG: Relightable Neural Gaussians](https://mlsanity.com/paper/2409.19702/) (2024)
- [Gaussian Splatting with Discretized SDF for Relightable Assets](https://mlsanity.com/paper/2507.15629/) (2025)
- [NeurCADRecon: Neural Representation for Reconstructing CAD Surfaces by Enforcing Zero Gaussian Curvature](https://mlsanity.com/paper/2404.13420/) (2024)
- [GS-ROR2: Bidirectional-guided 3DGS and SDF for Reflective Object Relighting and Reconstruction](https://mlsanity.com/paper/2406.18544/) (2024)
- [MultiPull: Detailing Signed Distance Functions by Pulling Multi-Level Queries at Multi-Step](https://mlsanity.com/paper/2411.01208/) (2024)
- [NeurCross: A Neural Approach to Computing Cross Fields for Quad Mesh Generation](https://mlsanity.com/paper/2405.13745/) (2024)
- [HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition](https://mlsanity.com/paper/2411.14628/) (2024)
- [MeshMosaic: Scaling Artist Mesh Generation via Local-to-Global Assembly](https://mlsanity.com/paper/2509.19995/) (2025)
- [Details Enhancement in Unsigned Distance Field Learning for High-fidelity 3D Surface Reconstruction](https://mlsanity.com/paper/2406.00346/) (2024)
- [GSurf: Learning Signed Distance Fields from Splatting Opaque Gaussians for High-quality 3D Reconstruction](https://mlsanity.com/paper/2411.15723/) (2024)
- [GS-Octree: Octree-based 3D Gaussian Splatting for Robust Object-level 3D Reconstruction Under Strong Lighting](https://mlsanity.com/paper/2406.18199/) (2024)
- [DiffCD: A Symmetric Differentiable Chamfer Distance for Neural Implicit Surface Fitting](https://mlsanity.com/paper/2407.17058/) (2024)
- [Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds](https://mlsanity.com/paper/2410.18477/) (2024)
- [CoFie: Learning Compact Neural Surface Representations with Coordinate Fields](https://mlsanity.com/paper/2406.03417/) (2024)
- [FlatCAD: Fast Curvature Regularization of Neural SDFs for CAD Models](https://mlsanity.com/paper/2506.16627/) (2025)
- [NumGrad-Pull: Numerical Gradient Guided Tri-plane Representation for Surface Reconstruction from Point Clouds](https://mlsanity.com/paper/2411.17392/) (2024)
- [SDFs from Unoriented Point Clouds using Neural Variational Heat Distances](https://mlsanity.com/paper/2504.11212/) (2025)
- [NeuVAS: Neural Implicit Surfaces for Variational Shape Modeling](https://mlsanity.com/paper/2506.13050/) (2025)
- [Joint Neural SDF Reconstruction and Semantic Segmentation for CAD Models](https://mlsanity.com/paper/2510.03837/) (2025)
- [Scheduling the Off-Diagonal Weingarten Loss of Neural SDFs for CAD Models](https://mlsanity.com/paper/2511.03147/) (2025)
- [Self-Supervised Implicit Attention Priors for Point Cloud Reconstruction](https://mlsanity.com/paper/2511.04864/) (2025)
- [Neural Octahedral Field: Octahedral prior for simultaneous smoothing and sharp edge regularization](https://mlsanity.com/paper/2408.00303/) (2024)
- [Approximating Signed Distance Fields With Sparse Ellipsoidal Radial Basis Function Networks: A Dynamic Multi-Objective Optimization Strategy](https://mlsanity.com/paper/2505.02350/) (2025)
- [Wavelet-based Global Orientation and Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/2506.16299/) (2025)
- [NoiseSDF2NoiseSDF: Learning Clean Neural Fields from Noisy Supervision](https://mlsanity.com/paper/2507.13595/) (2025)
- [Neural Visibility of Point Sets](https://mlsanity.com/paper/2509.24150/) (2025)
- [A Finite Difference Approximation of Second Order Regularization of Neural-SDFs](https://mlsanity.com/paper/2511.08980/) (2025)
- [Recovering 3D Shapes from Ultra-Fast Motion-Blurred Images](https://mlsanity.com/paper/2602.07860/) (2026)
- [Joint Orientation and Weight Optimization for Robust Watertight Surface Reconstruction via Dirichlet-Regularized Winding Fields](https://mlsanity.com/paper/2602.13801/) (2026)
- [Generative Shape Reconstruction with Geometry-Guided Langevin Dynamics](https://mlsanity.com/paper/2603.27016/) (2026)

## Abstract (arXiv)

Neural implicit representation is a promising approach for reconstructing surfaces from point clouds. Existing methods combine various regularization terms, such as the Eikonal and Laplacian energy terms, to enforce the learned neural function to possess the properties of a Signed Distance Function (SDF). However, inferring the actual topology and geometry of the underlying surface from poor-quality unoriented point clouds remains challenging. In accordance with Differential Geometry, the Hessian of the SDF is singular for points within the differential thin-shell space surrounding the surface. Our approach enforces the Hessian of the neural implicit function to have a zero determinant for points near the surface. This technique aligns the gradients for a near-surface point and its on-surface projection point, producing a rough but faithful shape within just a few iterations. By annealing the weight of the singular-Hessian term, our approach ultimately produces a high-fidelity reconstruction result. Extensive experimental results demonstrate that our approach effectively suppresses ghost geometry and recovers details from unoriented point clouds with better expressiveness than existing fitting-based methods.