# StEik: Stabilizing the Optimization of Neural Signed Distance Functions and Finer Shape Representation

arXiv: https://arxiv.org/abs/2305.18414 · HTML: https://mlsanity.com/paper/2305.18414/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 50 references that are in the map.
- First posted on arXiv 28 May 2023.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [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)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [ShuffleNet V2: Practical Guidelines for Efficient CNN Architecture Design](https://mlsanity.com/paper/1807.11164/) (2018)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [NeRF++: Analyzing and Improving Neural Radiance Fields](https://mlsanity.com/paper/2010.07492/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [SALD: Sign Agnostic Learning with Derivatives](https://mlsanity.com/paper/2006.05400/) (2020)
- [DiGS : Divergence guided shape implicit neural representation for unoriented point clouds](https://mlsanity.com/paper/2106.10811/) (2021)
- [Neural Splines: Fitting 3D Surfaces with Infinitely-Wide Neural Networks](https://mlsanity.com/paper/2006.13782/) (2020)
- [Non-linear Convolution Filters for CNN-based Learning](https://mlsanity.com/paper/1708.07038/) (2017)
- [Phase Transitions, Distance Functions, and Implicit Neural Representations](https://mlsanity.com/paper/2106.07689/) (2021)
- [Universal Approximation with Quadratic Deep Networks](https://mlsanity.com/paper/1808.00098/) (2018)
- [QuadraLib: A Performant Quadratic Neural Network Library for Architecture Optimization and Design Exploration](https://mlsanity.com/paper/2204.01701/) (2022)
- [Accelerated Optimization in the PDE Framework: Formulations for the Manifold of Diffeomorphisms](https://mlsanity.com/paper/1804.02307/) (2018)

## Built on by (most cited first, 22 of 22)

- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition](https://mlsanity.com/paper/2411.14628/) (2024)
- [GS-SDF: LiDAR-Augmented Gaussian Splatting and Neural SDF for Geometrically Consistent Rendering and Reconstruction](https://mlsanity.com/paper/2503.10170/) (2025)
- [Inferring Neural Signed Distance Functions by Overfitting on Single Noisy Point Clouds through Finetuning Data-Driven based Priors](https://mlsanity.com/paper/2410.19680/) (2024)
- [Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds](https://mlsanity.com/paper/2410.18477/) (2024)
- [Bubbleformer: Forecasting Boiling with Transformers](https://mlsanity.com/paper/2507.21244/) (2025)
- [PBIR-NIE: Glossy Object Capture under Non-Distant Lighting](https://mlsanity.com/paper/2408.06878/) (2024)
- [Neural Surface Reconstruction and Rendering for LiDAR-Visual Systems](https://mlsanity.com/paper/2409.05310/) (2024)
- [Prof. Robot: Differentiable Robot Rendering Without Static and Self-Collisions](https://mlsanity.com/paper/2503.11269/) (2025)
- [SDFs from Unoriented Point Clouds using Neural Variational Heat Distances](https://mlsanity.com/paper/2504.11212/) (2025)
- [Surface Reconstruction from Point Clouds via Grid-based Intersection Prediction](https://mlsanity.com/paper/2403.14085/) (2024)
- [Scheduling the Off-Diagonal Weingarten Loss of Neural SDFs for CAD Models](https://mlsanity.com/paper/2511.03147/) (2025)
- [Neural Octahedral Field: Octahedral prior for simultaneous smoothing and sharp edge regularization](https://mlsanity.com/paper/2408.00303/) (2024)
- [Neural Shortest Path for Surface Reconstruction from Point Clouds](https://mlsanity.com/paper/2502.06047/) (2025)
- [T-MLP: Tailed Multi-Layer Perceptron for Level-of-Detail Signal Representation](https://mlsanity.com/paper/2509.00066/) (2025)
- [Gauss-Newton Natural Gradient Descent for Shape Learning](https://mlsanity.com/paper/2602.00099/) (2026)
- [Generative Shape Reconstruction with Geometry-Guided Langevin Dynamics](https://mlsanity.com/paper/2603.27016/) (2026)
- [Medial Axis Aware Learning of Signed Distance Functions](https://mlsanity.com/paper/2604.16512/) (2026)
- [SAND: Spatially Adaptive Network Depth for Fast Sampling of Neural Implicit Surfaces](https://mlsanity.com/paper/2604.25936/) (2026)
- [SC-MFJ: A Simple Haptic Quality Metric for Medical Image Segmentation](https://mlsanity.com/paper/2606.06199/) (2026)
- [Points as Tori: Fast Pointwise Signed Distance for Point Clouds](https://mlsanity.com/paper/2607.16946/) (2026)
- [M-plicits: Neural Implicit Surfaces via Nested Multiscale Residuals](https://mlsanity.com/paper/2609.28684/) (2026)

## Abstract (arXiv)

We present new insights and a novel paradigm (StEik) for learning implicit neural representations (INR) of shapes. In particular, we shed light on the popular eikonal loss used for imposing a signed distance function constraint in INR. We show analytically that as the representation power of the network increases, the optimization approaches a partial differential equation (PDE) in the continuum limit that is unstable. We show that this instability can manifest in existing network optimization, leading to irregularities in the reconstructed surface and/or convergence to sub-optimal local minima, and thus fails to capture fine geometric and topological structure. We show analytically how other terms added to the loss, currently used in the literature for other purposes, can actually eliminate these instabilities. However, such terms can over-regularize the surface, preventing the representation of fine shape detail. Based on a similar PDE theory for the continuum limit, we introduce a new regularization term that still counteracts the eikonal instability but without over-regularizing. Furthermore, since stability is now guaranteed in the continuum limit, this stabilization also allows for considering new network structures that are able to represent finer shape detail. We introduce such a structure based on quadratic layers. Experiments on multiple benchmark data sets show that our new regularization and network are able to capture more precise shape details and more accurate topology than existing state-of-the-art.