From Transparent to Opaque: Rethinking Neural Implicit Surfaces with $α$-NeuS

Traditional 3D shape reconstruction techniques from multi-view images, such as structure from motion and multi-view stereo, primarily focus on opaque surfaces. Similarly, recent advances in neural radiance fields and its variants also primarily address opaque objects, encountering difficulties with the complex lighting effects caused by transparent materials. This paper introduces $\alpha$-NeuS, a new method for simultaneously reconstructing thin transparent objects and opaque objects based on neural implicit surfaces (NeuS). Our method leverages the observation that transparent surfaces induce local extreme values in the learned distance fields during neural volumetric rendering, contrasting with opaque surfaces that align with zero level sets. Traditional iso-surfacing algorithms such as marching cubes, which rely on fixed iso-values, are ill-suited for this data. We address this by taking the absolute value of the distance field and developing an optimization method that extracts level sets corresponding to both non-negative local minima and zero iso-values. We prove that the reconstructed surfaces are unbiased for both transparent and opaque objects. To validate our approach, we construct a benchmark that includes both real-world and synthetic scenes, demonstrating its practical utility and effectiveness. Our data and code are publicly available at https://github.com/728388808/alpha-NeuS.

Quasi-Medial Distance Field (Q-MDF): A Robust Method for Approximating and Discretizing Neural Medial Axis

The medial axis, a lower-dimensional shape descriptor, plays an important role in the field of digital geometry processing. Despite its importance, robust computation of the medial axis transform from diverse inputs, especially point clouds with defects, remains a significant challenge. In this paper, we tackle the challenge by proposing a new implicit method that diverges from mainstream explicit medial axis computation techniques. Our key technical insight is the difference between the signed distance field (SDF) and the medial field (MF) of a solid shape is the unsigned distance field (UDF) of the shape's medial axis. This allows for formulating medial axis computation as an implicit reconstruction problem. Utilizing a modified double covering method, we extract the medial axis as the zero level-set of the UDF. Extensive experiments show that our method has enhanced accuracy and robustness in learning compact medial axis transform from thorny meshes and point clouds compared to existing methods.

DCUDF2: Improving Efficiency and Accuracy in Extracting Zero Level Sets from Unsigned Distance Fields

Unsigned distance fields (UDFs) allow for the representation of models with complex topologies, but extracting accurate zero level sets from these fields poses significant challenges, particularly in preserving topological accuracy and capturing fine geometric details. To overcome these issues, we introduce DCUDF2, an enhancement over DCUDF--the current state-of-the-art method--for extracting zero level sets from UDFs. Our approach utilizes an accuracy-aware loss function, enhanced with self-adaptive weights, to improve geometric quality significantly. We also propose a topology correction strategy that reduces the dependence on hyper-parameter, increasing the robustness of our method. Furthermore, we develop new operations leveraging self-adaptive weights to boost runtime efficiency. Extensive experiments on surface extraction across diverse datasets demonstrate that DCUDF2 outperforms DCUDF and existing methods in both geometric fidelity and topological accuracy. We will make the source code publicly available.

UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs

Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and exhibits sub-optimal efficiency for certain PDE formulations, while data-driven neural methods typically lack mathematical guarantee of convergence and correctness. This paper articulates a mathematically rigorous neural solver for linear PDEs. The proposed UGrid solver, built upon the principled integration of U-Net and MultiGrid, manifests a mathematically rigorous proof of both convergence and correctness, and showcases high numerical accuracy, as well as strong generalization power to various input geometry/values and multiple PDE formulations. In addition, we devise a new residual loss metric, which enables unsupervised training and affords more stability and a larger solution space over the legacy losses.

Learning Unsigned Distance Fields from Local Shape Functions for 3D Surface Reconstruction

Unsigned distance fields (UDFs) provide a versatile framework for representing a diverse array of 3D shapes, encompassing both watertight and non-watertight geometries. Traditional UDF learning methods typically require extensive training on large datasets of 3D shapes, which is costly and often necessitates hyperparameter adjustments for new datasets. This paper presents a novel neural framework, LoSF-UDF, for reconstructing surfaces from 3D point clouds by leveraging local shape functions to learn UDFs. We observe that 3D shapes manifest simple patterns within localized areas, prompting us to create a training dataset of point cloud patches characterized by mathematical functions that represent a continuum from smooth surfaces to sharp edges and corners. Our approach learns features within a specific radius around each query point and utilizes an attention mechanism to focus on the crucial features for UDF estimation. This method enables efficient and robust surface reconstruction from point clouds without the need for shape-specific training. Additionally, our method exhibits enhanced resilience to noise and outliers in point clouds compared to existing methods. We present comprehensive experiments and comparisons across various datasets, including synthetic and real-scanned point clouds, to validate our method's efficacy.

GS-Octree: Octree-based 3D Gaussian Splatting for Robust Object-level 3D Reconstruction Under Strong Lighting

The 3D Gaussian Splatting technique has significantly advanced the construction of radiance fields from multi-view images, enabling real-time rendering. While point-based rasterization effectively reduces computational demands for rendering, it often struggles to accurately reconstruct the geometry of the target object, especially under strong lighting. To address this challenge, we introduce a novel approach that combines octree-based implicit surface representations with Gaussian splatting. Our method consists of four stages. Initially, it reconstructs a signed distance field (SDF) and a radiance field through volume rendering, encoding them in a low-resolution octree. The initial SDF represents the coarse geometry of the target object. Subsequently, it introduces 3D Gaussians as additional degrees of freedom, which are guided by the SDF. In the third stage, the optimized Gaussians further improve the accuracy of the SDF, allowing it to recover finer geometric details compared to the initial SDF obtained in the first stage. Finally, it adopts the refined SDF to further optimize the 3D Gaussians via splatting, eliminating those that contribute little to visual appearance. Experimental results show that our method, which leverages the distribution of 3D Gaussians with SDFs, reconstructs more accurate geometry, particularly in images with specular highlights caused by strong lighting.

Details Enhancement in Unsigned Distance Field Learning for High-fidelity 3D Surface Reconstruction

While Signed Distance Fields (SDF) are well-established for modeling watertight surfaces, Unsigned Distance Fields (UDF) broaden the scope to include open surfaces and models with complex inner structures. Despite their flexibility, UDFs encounter significant challenges in high-fidelity 3D reconstruction, such as non-differentiability at the zero level set, difficulty in achieving the exact zero value, numerous local minima, vanishing gradients, and oscillating gradient directions near the zero level set. To address these challenges, we propose Details Enhanced UDF (DEUDF) learning that integrates normal alignment and the SIREN network for capturing fine geometric details, adaptively weighted Eikonal constraints to address vanishing gradients near the target surface, unconditioned MLP-based UDF representation to relax non-negativity constraints, and a UDF-tailored method for extracting iso-surface with non-constant iso-values. These strategies collectively stabilize the learning process from unoriented point clouds and enhance the accuracy of UDFs. Our computational results demonstrate that DEUDF outperforms existing UDF learning methods in both accuracy and the quality of reconstructed surfaces. We will make the source code publicly available.

Topology-Aware Latent Diffusion for 3D Shape Generation

We introduce a new generative model that combines latent diffusion with persistent homology to create 3D shapes with high diversity, with a special emphasis on their topological characteristics. Our method involves representing 3D shapes as implicit fields, then employing persistent homology to extract topological features, including Betti numbers and persistence diagrams. The shape generation process consists of two steps. Initially, we employ a transformer-based autoencoding module to embed the implicit representation of each 3D shape into a set of latent vectors. Subsequently, we navigate through the learned latent space via a diffusion model. By strategically incorporating topological features into the diffusion process, our generative module is able to produce a richer variety of 3D shapes with different topological structures. Furthermore, our framework is flexible, supporting generation tasks constrained by a variety of inputs, including sparse and partial point clouds, as well as sketches. By modifying the persistence diagrams, we can alter the topology of the shapes generated from these input modalities.

Robust Zero Level-Set Extraction from Unsigned Distance Fields Based on Double Covering

In this paper, we propose a new method, called DoubleCoverUDF, for extracting the zero level-set from unsigned distance fields (UDFs). DoubleCoverUDF takes a learned UDF and a user-specified parameter $r$ (a small positive real number) as input and extracts an iso-surface with an iso-value $r$ using the conventional marching cubes algorithm. We show that the computed iso-surface is the boundary of the $r$-offset volume of the target zero level-set $S$, which is an orientable manifold, regardless of the topology of $S$. Next, the algorithm computes a covering map to project the boundary mesh onto $S$, preserving the mesh's topology and avoiding folding. If $S$ is an orientable manifold surface, our algorithm separates the double-layered mesh into a single layer using a robust minimum-cut post-processing step. Otherwise, it keeps the double-layered mesh as the output. We validate our algorithm by reconstructing 3D surfaces of open models and demonstrate its efficacy and effectiveness on synthetic models and benchmark datasets. Our experimental results confirm that our method is robust and produces meshes with better quality in terms of both visual evaluation and quantitative measures than existing UDF-based methods. The source code is available at https://github.com/jjjkkyz/DCUDF.

Bi-directional Deformation for Parameterization of Neural Implicit Surfaces

The growing capabilities of neural rendering have increased the demand for new techniques that enable the intuitive editing of 3D objects, particularly when they are represented as neural implicit surfaces. In this paper, we present a novel neural algorithm to parameterize neural implicit surfaces to simple parametric domains, such as spheres, cubes or polycubes, where 3D radiance field can be represented as a 2D field, thereby facilitating visualization and various editing tasks. Technically, our method computes a bi-directional deformation between 3D objects and their chosen parametric domains, eliminating the need for any prior information. We adopt a forward mapping of points on the zero level set of the 3D object to a parametric domain, followed by a backward mapping through inverse deformation. To ensure the map is bijective, we employ a cycle loss while optimizing the smoothness of both deformations. Additionally, we leverage a Laplacian regularizer to effectively control angle distortion and offer the flexibility to choose from a range of parametric domains for managing area distortion. Designed for compatibility, our framework integrates seamlessly with existing neural rendering pipelines, taking multi-view images as input to reconstruct 3D geometry and compute the corresponding texture map. We also introduce a simple yet effective technique for intrinsic radiance decomposition, facilitating both view-independent material editing and view-dependent shading editing. Our method allows for the immediate rendering of edited textures through volume rendering, without the need for network re-training. Moreover, our approach supports the co-parameterization of multiple objects and enables texture transfer between them. We demonstrate the effectiveness of our method on images of human heads and man-made objects. We will make the source code publicly available.