TetSphere Splatting: Representing High-Quality Geometry with Lagrangian Volumetric Meshes

## Abstract

Abstract We introduce TetSphere Splatting, a Lagrangian geometry representation
designed for high-quality 3D shape modeling.
TetSphere splatting leverages an underused yet powerful geometric primitive – volumetric tetrahedral meshes.
It represents 3D shapes by deforming a collection of tetrahedral spheres, with geometric regularizations and constraints that effectively resolve common mesh issues such as irregular triangles, non-manifoldness, and floating artifacts.
Experimental results on multi-view and single-view reconstruction highlight TetSphere splatting’s superior mesh quality while maintaining competitive reconstruction accuracy compared to state-of-the-art methods.
Additionally, TetSphere splatting demonstrates versatility by seamlessly integrating into generative modeling tasks, such as image-to-3D and text-to-3D generation.

## 1 Introduction

Accurate 3D shape modeling is critical for many real-world applications.
Recent advancements in reconstruction (Mildenhall et al., ; Wang et al., ; Kerbl et al., ), generative modeling (Poole et al., ; Liu et al., ; ; Long et al., ), and inverse rendering (Mehta et al., ; Nicolet et al., ) have significantly improved the geometric precision and visual quality of 3D shapes, pushing the boundaries of automatic digital asset generation.

Central to these advancements are geometry representations, which can be broadly categorized into two types: *Eulerian* and *Lagrangian* representations.
Eulerian representations describe a geometry on a set of pre-defined, fixed coordinates in 3D world space, where each coordinate position is associated with properties, such as occupancy within the volume or distance from the surface.
Widely used Eulerian representations include neural networks that take *continuous* spatial coordinates as input to model density fields (Mildenhall et al., ) or signed distance functions (Wang et al., ; Mehta et al., ), as well as deformable grids that use *discrete* coordinates, with signed distance values defined at grid vertices (Shen et al., ; Gao et al., ; Shen et al., ).
Despite their popularity, Eulerian representations face a trade-off between computational complexity and geometry quality:
Capturing intricate geometric details of the shape requires either a high-capacity neural network or a high-resolution grid, both of which are computationally expensive to optimize in terms of time and memory.
This trade-off often limits Eulerian representations when modeling thin, slender structures, as their pre-defined resolution is often insufficient to capture fine details.

Recently, there has been a growing shift in the community towards *Lagrangian* representations,
which are typically more computationally efficient than Eulerian methods (Kerbl et al., ; Guédon & Lepetit, ; Huang et al., ; Son et al., ; Chen et al., ).
Lagrangian representations describe a 3D shape by tracking the movement of a set of geometry primitives in 3D world space.
These geometry primitives can be adaptively positioned based on the local geometry of the shape, typically requiring fewer computational resources than Eulerian methods, especially when modeling shapes with fine geometric details.
An illustrative comparison between Eulerian and Lagrangian geometry representation is shown in Fig. 1 (a).
Two of the most commonly used Lagrangian primitives are 3D Gaussians (Kerbl et al., ), which represent geometry using point clouds, and surface triangles (Son et al., ; Chen et al., ), which are coupled with their connectivity to form surface meshes.
While these Lagrangian representations are favored for their computational efficiency, they often struggle with poor *mesh quality* due to their reliance on tracking individual points or triangles, which can lack overall structural coherence.
For instance, Gaussian point clouds can move freely in space, often resulting in noisy meshes, while surface triangles, when coupled with connectivity, can form non-manifold surfaces or irregular, and sometimes degenerated, triangles.
The resulting geometry, exhibiting these geometric issues,
is unsuitable for downstream tasks such as rendering and simulation, where high-quality meshes are crucial for high fidelity.

Figure: Figure 2: Visual comparison of mesh quality across widely used shape representations, including NeRF (Mildenhall et al., ), FlexiCubes (Shen et al., ) (Eulerian), DMesh (Son et al., ), and Gaussian Splatting (Huang et al., ) (Lagrangian). These methods exhibit mesh quality issues, such as irregular or degenerated triangles, non-manifoldness, and floating artifacts. Our method demonstrates uniform surface triangles, improved mesh quality, and structure integrity.
Refer to caption: x2.png

To address these challenges, we propose a novel Lagrangian geometry representation, *TetSphere Splatting*, designed to construct geometry with an emphasis on producing high-quality meshes.
Our key insights stem from the fact that existing Lagrangian primitives are too fine-grained to ensure high-quality meshes.
Mesh quality depends not only on individual primitives but also on their interactions:
For example, the absence of irregular or degenerated triangles relies on the proper alignment of primitives, while manifoldness depends on how well they are connected.
Our representation uses volumetric tetrahedral spheres, termed *TetSphere*, as geometric primitives.
Unlike existing primitives that are individual points or triangles, each TetSphere is a volumetric sphere composed of a set of points connected through tetrahedralization.
Initialized as a uniform sphere, each TetSphere can be deformed into complex shapes.
Together, a collection of these deformed TetSpheres represents a 3D shape, which is in line with the Lagrangian approach.
This more structured primitive allows geometric regularization and constraints to be imposed among points within each TetSphere, ensuring mesh quality is maintained throughout deformation.
The volumetric nature of TetSpheres also establishes a cohesive arrangement of points throughout the volume, ensuring structural integrity and effectively reducing common surface mesh issues such as irregular triangles or non-manifoldness.

We further present a computational framework for TetSphere splatting.
Similar to Gaussian splatting (Kerbl et al., ), our method “splats” TetSpheres to conform to the target shape.
We formulate the deformation of TetSphere as a geometric energy optimization problem, consisting of differentiable rendering loss, bi-harmonic energy of the deformation gradient field, and non-inversion constraints, all effectively solvable via gradient descent.
For evaluation, we conduct quantitative comparisons on two tasks to assess the geometry quality: multi-view and single-view reconstruction.
In addition to the commonly used metrics for evaluating reconstruction accuracy, we introduce three metrics to evaluate mesh quality, focusing on key aspects of 3D model usability: surface triangles uniformity, manifoldness, and structural integrity.
Compared to state-of-the-art methods, TetSphere splatting demonstrates superior mesh quality while maintaining competitive performance on other metrics.
Furthermore, as demonstrations of its versatility, we showcase TetSphere splatting’s utility in downstream applications of 3D shape generation from both single images and text.

## 2 Related Work

Eulerian and Lagrangian geometry representations.
The differentiation between Eulerian and Lagrangian representations originates from computational fluid dynamics (Chung, ) but extends more broadly into computational geometry and physics.
Using fluid simulation as an analogy, an Eulerian view would analyze fluid presence at fixed points in space, whereas a Lagrangian perspective follows specific fluid particles.
Neural implicit representations, such as DeepSDF (Park et al., ), NeRF (Mildenhall et al., ), and InstantNGP (Müller et al., ), are modern adaptations of Eulerian concepts, processing 3D positions as inputs to neural networks.
These methods theoretically allow for infinite resolution through NN-based parameterization but can result in slow optimization speeds due to NN training.
Beyond implicit ones, explicit or hybrid Eulerian representations, such as DMTet (Shen et al., ), DefTet (Gao et al., ), and TetGAN (Yang et al., ), incorporate explicit irregular grids but can still cause substantial memory usage for high-resolution shapes.
Gaussian splatting (Tang et al., ) exemplifies a Lagrangian approach by moving Gaussian point clouds in space.
Surface triangles (Son et al., ; Chen et al., ) are another common example of a Lagrangian approach, frequently used in mesh-based representations.
The surface of a 3D object is discretized into a collection of connected triangular elements, which are tracked individually.
Our proposed TetSphere is an explicit Lagrangian representation, which can be viewed as introducing explicit constraints among points due to tetrahedral meshing, with enhanced mesh quality.

3D object reconstruction.
3D reconstruction is an inherently ill-posed problem, and extensive research has been dedicated to addressing it (Fu et al., ; Fahim et al., ).
Early approaches utilized a combination of 2D image encoders and 3D decoders trained on 3D data
with both explicit representations, including voxels (Chen & Zhang, ; Xie et al., ; ), meshes (Wang et al., ; Gkioxari et al., ),
and point clouds (Mandikal et al., ; Groueix et al., ), and implicit
representations such as NeRF (Yu et al., ; Jang & Agapito, ; Müller et al., ), SDF(Park et al., ; Mittal et al., ; Xu et al., ), and occupancy networks(Mescheder et al., ; Bian et al., ).
Recently, an active research direction has been leveraging 2D generative models for 3D reconstruction,
including the use of SDS and supplementary losses (Lin et al., ; Sun et al., ; Liu et al., ; Melas-Kyriazi et al., ; Shen et al., ; Xu et al., ; Gu et al., ; Deng et al., ).
Alternative approaches train view-conditioned 2D diffusion models to directly produce multi-view images for 3D reconstruction (Liu et al., ; ; ; Chan et al., ; Tang et al., ; ; Long et al., ).
The recent introduction of large-scale 3D dataset propelled feed-forward large reconstruction models (Hong et al., ; Wang et al., ; Xu et al., ; Weng et al., ; Tang et al., ; He & Wang, ; Tochilkin et al., ; Wei et al., ; Zhang et al., ; Xu et al., ).
Their feed-forward inference accelerates the speed of 3D reconstruction, but often at a sacrifice of relatively low resolution and geometry quality.

Our method also relates to text-to-3D content generation as it is one of the applications of TetSphere splatting.
We leave a detailed discussion on these related works to Appendix F.

Figure: Figure 3: Overall pipeline: TetSphere splatting represents a 3D shape using a collection of TetSpheres. Each TetSphere is a tetrahedral sphere that can be deformed from its initial uniform state through deformation gradient. The deformation process is optimized by minimizing rendering loss and geometric energy terms.
Refer to caption: x3.png

## 3 TetSphere Splatting

We use tetrahedral spheres as our primitive of choice.
Unlike point clouds, tetrahedral meshes enforce structured local connectivity between points owing to tetrahedralization.
This preserves the geometric integrity of the 3D shape and also enhances the surface quality by imposing geometric regularization across the entire mesh interior.
We formulate the reconstruction of shapes through TetSphere splatting as a deformation of tetrahedron spheres.
Starting from a set of tetrahedral spheres, we adjust the positions of their vertices to align the rendered images of these meshes with the corresponding target multi-view images.
Vertex movement is constrained by two geometric regularizations on the tetrahedral meshes, derived from the field of geometry processing (Bærentzen et al., ).
These regularizations, which penalize the non-smooth deformation (via bi-harmonic energy) and prevent the inversion of mesh elements (via local injectivity), have proven effective in ensuring that the resulting tetrahedral meshes are of superior quality and maintain structural integrity.
Fig. 3 illustrates the overall pipeline.

### 3.1 Tetrahedral Sphere Primitive

The primitive of TetSphere splatting is a tetrahedralized sphere, called TetSphere, with $N$ vertices and $T$ tetrahedra.
By applying principles from the Finite Element Method (FEM) (Sifakis & Barbic, ),
the mesh of each sphere is composed of tetrahedral elements,
with each tetrahedron constituting a 3D discrete piecewise linear volumetric entity.
We denote the position vector of all vertices of the $i$-th deformed sphere mesh as ${x}_{i}\in\mathbb{R}^{3N}$.
The deformation gradient of the $j$-th tetrahedron in the $i$-th sphere is denoted as $\mathbf{F}^{(i,j)}_{\mathbf{x}}\in\mathbb{R}^{3\times 3}$,
which quantitatively describes how each tetrahedron’s shape transforms (Sifakis & Barbic, ).
Essentially, the deformation gradient $\mathbf{F}^{(i,j)}_{\mathbf{x}}$ serves as a measure of the spatial changes a tetrahedron undergoes from its original configuration to its deformed state.
Refer to Fig. 3 for a visual explanation and Appendix B for an in-depth derivation.

Rather than using a single sphere, our representation employs a collection of spheres to accurately represent arbitrary shapes. Consequently, the complete shape is the union of all spheres.
By adopting multiple spheres,
this approach ensures that each local region of a shape is detailed independently, enabling a highly accurate representation.
Moreover, it allows for the representation of shapes with arbitrary topologies.
Such a claim is theoretically guaranteed by the paracompactness property of manifold shapes (James, ).

Using tetrahedral spheres offers several technical benefits compared with prevalent representations for object reconstruction, as demonstrated in Fig. 2:

- •
Compared to neural representations (e.g., NeRF), our tetrahedral representation does not rely on neural networks, thus inherently accelerating the optimization process.
- •
Compared to Eulerian representations (such as DMTet), our approach entirely avoids the need for iso-surface extraction – an operation that often degrades mesh quality owing to the predetermined resolution of the grid space.
- •
Compared to other Lagrangian representations, such as Gaussian point clouds and triangle meshes, our method offers a volumetric representation through the use of tetrahedral meshes.
Each tetrahedron imposes constraints among vertices, leading to superior mesh quality.

### 3.2 TetSphere Splatting as Shape Deformation

To reconstruct the geometry of the target object, we deform the initial TetSpheres by changing their vertex positions.
Two primary goals govern this process: ensuring the deformed TetSpheres align with the input multi-view images and maintaining high mesh quality that adheres to necessary geometry constraints.
Fig. 4 illustrates the iterative process of TetSphere splatting.

To maintain the mesh quality, we leverage bi-harmonic energy – defined in the literature on geometry processing (Botsch & Sorkine, ) as an energy quantifying smoothness throughout a field – to the deformation gradient field.
This geometric regularization ensures the smoothness of the deformation gradient field across the deformation process, thus preventing irregular mesh or bumpy surfaces.
It is important to highlight that this bi-harmonic regularization does *not* lead to over-smoothness of the final result.
This is because the energy targets the deformation gradient field, which measures the *relative* changes in vertex positions, rather than the *absolute* positions themselves.
Such an approach allows for the preservation of sharp local geometric details, akin to techniques used in physical simulations (Wen et al., ).
Furthermore, we introduce a geometric constraint to guarantee local injectivity in all deformed elements (Schüller et al., ).
This ensures that the elements maintain their orientation during the deformation, avoiding inversions or inside-out configurations.
This constraint can be mathematically expressed as $\mathrm{det}(\mathbf{F}^{(i,j)}_{\mathbf{x}})>0.$
Importantly, these two terms – bi-harmonic energy for smoothness and local injectivity for element orientation – are universally applicable to any tetrahedral meshes, stemming from their fundamental basis in geometry processing (Bærentzen et al., ).

Figure: Figure 4: TetSphere splatting with deforming tetrahedral spheres. Color-coded regions represent the bi-harmonic energy values (red: high, blue: low) across tetrahedra, one of the geometric regularizations employed in our deformation optimization process.
Refer to caption: extracted/5896802/figures/optimization-seq-v2.5.png

Let $\mathbf{x}=[x_{1},...,x_{M}]\in\mathbb{R}^{3NM}$ denote the positions of vertices across all $M$ TetSpheres, and $\mathbf{F}_{\mathbf{x}}\in\mathbb{R}^{9MT}=[\mathrm{vec}(\mathbf{F}^{(1,1)}_{
\mathbf{x}}),...,\mathrm{vec}(\mathbf{F}^{(M,T)}_{\mathbf{x}})]$ denote the flattened deformation gradient fields of all TetSpheres.
In the bi-harmonic energy, the Laplacian matrix is defined based on the connectivity of the tetrahedron faces, denoted as $\mathbf{L}\in\mathbb{R}^{9MT\times 9MT}$.
This matrix is block symmetric, where each block $\mathbf{L}_{pq}\in\mathbb{R}^{9\times 9},p\neq q$ is set to a negative identity matrix $-I$ if the $p$-th and $q$-th tetrahedron shares a common triangle; or $kI$ for $\mathbf{L}_{pp}$, where $k$ is the number of neighbors of the $p$-th tetrahedron.
The deformation of the TetSpheres is formulated as an optimization problem:

$$ $\displaystyle\min_{\textbf{x}}\ \$ $\displaystyle\mathbf{\Phi}(R(\mathbf{x}))+||\mathbf{L}\mathbf{F}_{\mathbf{x}}| |_{2}^{2}$ $\displaystyle\mathrm{s.t.}\ \$ $\displaystyle\mathrm{det}(\mathbf{F}^{(i,j)}_{\mathbf{x}})>0,\ \forall i\in\{1 ,...,M\},\ j\in\{1,...,T\},$ (1) $$

where $R(\cdot)$ is the rendering function, $\mathbf{\Phi}(\cdot)$ is the rendering loss matching the union of deformed tetrahedral spheres with the input images.
The second term regulates the bi-harmonic energy across the deformation gradient field.
The non-inversion constraint ensures that tetrahedra maintain their orientation.
To manage this constrained optimization, we reformulate it by incorporating the non-inversion hard constraint as a soft penalty term into the objective,

$$ $\displaystyle\min_{\textbf{x}}\ \$ $\displaystyle\mathbf{\Phi}(R(\mathbf{x}))+w_{1}||\mathbf{L}\mathbf{F}_{\mathbf {x}}||_{2}^{2}+w_{2}\sum_{i,j}(\mathrm{min}\{0,\mathrm{det}(\mathbf{F}^{(i,j)} _{\mathbf{x}})\})^{2},$ (2) $$

allowing for optimization via standard gradient descent solvers.

In the proposed optimization framework, three considerations have been outlined.
1) The adaptive loss function $\mathbf{\Phi}(\cdot)$, designed to be flexible, supports a variety of metrics, including $l_{1}$ for color images, MSE for depth images, and cosine embedding loss for normal images. These losses can all be optimized using gradients provided by differentiable renderers, such as mesh rasterizers (Laine et al., ).. 2) Given that the tetrahedron is a linear element, the deformation gradient $\mathbf{F}^{(i,j)}_{\mathbf{x}}$ is a linear function of $\mathbf{x}$, making the bi-harmonic energy a quadratic term.
3) The weights $w_{1}$ and $w_{2}$ are dynamically adjusted using a cosine scheduler.
We provide details of the scheduler’s
hyperparameters in Appendix E.

## 4 TetSphere Initialization and Texture Optimization

TetSphere initialization.
Given multi-view images as inputs, we select feature points to initialize the 3D center positions of the TetSpheres.
We aim to achieve a uniform distribution of these TetSpheres within the object, ensuring comprehensive coverage of the silhouette depicted in the multi-view images.

We introduce an algorithm, silhouette coverage, inspired by Coverage Axis (Dou et al., ) to select initial centers of TetSpheres for an arbitrary shape automatically.
This process begins with constructing a coarse voxel grid and initially assigning a zero value to each voxel.
By projecting these voxels into the image spaces using the same camera poses as the input multi-view images, voxels within the foreground of all images are marked with a value of $1$.
These voxel positions are identified as candidate positions of TetSphere centers.
From these marked positions, the objective is to pick a minimal subset of candidates to ensure that all candidates are fully encapsulated by TetSpheres and centered on these points.
This involves placing uniform spheres with varying radius values at all candidate points and choosing a minimal subset that collectively covers all the candidate points.
We formulate a linear programming problem to perform the selection efficiently.
The detailed formulation is provided in Appendix C.
In our implementation, with a voxel grid resolution $300\times 300$ and $n=20$, the whole TetSphere initialization completes in ${\sim}1$ minute on average.

Texture optimization.
While the primary purpose of TetSphere splatting is to represent high-quality geometry, its explicit structure allows textures and materials to be directly applied to the surface vertices and faces of the TetSpheres.
A key advantage of TetSphere splatting is that the deformation of tetrahedral spheres preserves the surface topology, enabling the seamless integration of advanced material models, such as Disney’s principled BRDF (Burley & Studios, ), in physically-based rendering for applications like text-to-3D generation.
However, it is important to note that texture optimization is an optional feature of our representation and does not serve as the main focus of our comparisons.
The primary objective of TetSphere splatting remains the accurate representation of geometry, with textures and materials being secondary enhancements. Further details are provided in Appendix D.

## 5 Experiments and Results

We provide detailed quantitative evaluations on 3D reconstruction from both multi-view and single-view images to demonstrate the effectiveness of TetSphere.
Additionally, we showcase TetSphere’s versatility through case studies on image-to-3D and text-to-3D shape generation.
We further analyze the effect of energy coefficients and offer a comparative study on computational costs, particularly focusing on GPU requirements and memory usage for image-to-3D generation with SDS loss, where existing methods struggle with high computational demands.

### 5.1 Implementation Details

Our implementation of the TetSphere initialization algorithm is developed in C++ and uses the Gurobi linear programming solver.
The optimization of geometric energies is implemented using CUDA as a PyTorch extension to enhance computational efficiency.
1) For multi-view reconstruction, we render multi-view RGBA and depth images of the ground truth shape as the input of TetSphere splatting.
The optimization objective $\mathbf{\Phi}(\cdot)$
includes both rendering loss (MSE on the alpha mask) and depth loss, following (Son et al., ).
2) For single-view reconstruction, we use Wonder3d (Long et al., ) to generate six multi-view images
with predefined camera poses.
The optimization objective $\mathbf{\Phi}(\cdot)$
includes rendering loss $l_{1}$ norm on tone-mapped color and MSE on the alpha mask) and normal loss (cosine loss on normals), following (Munkberg et al., ).
The implementation details for the two applications are described in Appendix E.
To enhance robustness and efficiency, we apply a cosine scheduler to scale the coefficients of the geometry loss, formulated as $\eta=4^{\sin(\frac{t\pi}{2n})}$,
where $t$ denotes the current iteration and $T$ represents the total number of iterations.

**Table 1: Multi-View reconstruction results: Evaluating reconstruction accuracy with Chamfer Distance (Cham.) and Volume IoU, alongside mesh quality metrics: Area-Length Ratio (ALR), Manifoldness Rate (MR), and Connected Component Discrepancy (CC Diff.). For additional results on other metrics, please refer to Table 4 in Appendix A.1.**
| Method | Geo. Rep. | Cham. $\downarrow$ | Vol. IoU $\uparrow$ | ALR $\uparrow$ | MR(%) $\uparrow$ | CC Diff. $\downarrow$ |
| --- | --- | --- | --- | --- | --- | --- |
| NIE | Eulerian | 0.0254 | 0.1863 | 0.0273 | 72.3 | 7.0 |
| FlexiCubes | Eulerian | 0.0247 | 0.5887 | 0.0722 | 45.5 | 201.3 |
| 2DGS | Lagrangian | 0.0322 | 0.4923 | 0.0209 | 27.3 | 25.1 |
| DMesh | Lagrangian | 0.0136 | 0.5616 | 0.1193 | 9.09 | 3.75 |
| Ours | Lagrangian | 0.0184 | 0.6844 | 0.6602 | 100 | 0.0 |

### 5.2 Baselines and Evaluation Protocol

Baselines.
For multi-view reconstruction, we quantitatively compare our method with state-of-the-art geometry representations: Neural Implicit Evolution (NIE) (Mehta et al., ), FlexiCubes (Shen et al., ), 2DGS (Huang et al., ), and DMesh (Son et al., ).
The first two are Eulerian representations, whereas the latter two are Lagrangian representations.
For single-view reconstruction, we quantitatively compare with several state-of-the-art methods:
Magic123 (Qian et al., ),
One-2-3-45 (Liu et al., ),
SyncDreamer (Liu et al., ), Wonder3d (Long et al., ), Open-LRM (Hong et al., ; He & Wang, ), and DreamGaussian (Tang et al., ).
DreamGaussian is the only Lagrangian representation, as the field of single-view reconstruction is largely dominated by Eulerian methods.
For all baselines, we follow their original implementations and use their publicly available codebase to get the results in this paper.

Evaluation datasets.
For multi-view reconstruction, we follow Son et al. () and use four closed-surface models from the Thingi32 dataset (Zhou & Jacobson, ), four open-surface models from the DeepFashion3D dataset (Heming et al., ), and three additional models with both closed and open surfaces from the Objaverse dataset (Deitke et al., ) and Adobe Stock.
We also add the challenging sorter shape from Google Scanned Objects (GSO) dataset (Downs et al., ) as it features slender structures.
For single-view reconstruction,
following prior research (Liu et al., ; Long et al., ),
we use the GSO dataset for our evaluation, which covers a broad range of everyday objects.
The evaluation dataset aligns with those used by SyncDreamer and Wonder3D, featuring 30 diverse objects ranging from household items to animals.

Figure: Figure 5: Qualitative results on multi-view reconstruction, with surface mesh visualizations and rendered normal maps. Our method excels over baseline methods regarding mesh quality, less bumpy surface, correct surface orientation, and accurately capturing slender and thin structures.
Refer to caption: x4.png

**Table 2: Single-View reconstruction results on the GSO Dataset: Evaluating reconstruction accuracy with Chamfer Distance (Cham.) and Volume IoU, alongside mesh quality metrics: Area-Length Ratio (ALR), Manifoldness Rate (MR), and Connected Component Discrepancy (CC Diff.).**
| Method | Geo. Rep. | Cham. $\downarrow$ | Vol. IoU $\uparrow$ | ALR $\uparrow$ | MR(%) $\uparrow$ | CC Diff. $\downarrow$ |
| --- | --- | --- | --- | --- | --- | --- |
| Magic123 | Eulerian | 0.0516 | 0.4528 | 0.0383 | 100 | 13.7 |
| One-2-3-45 | Eulerian | 0.0629 | 0.4086 | 0.0574 | 96 | 0.83 |
| SyncDreamer | Eulerian | 0.0261 | 0.5421 | 0.0201 | 10 | 0.3 |
| Wonder3d | Eulerian | 0.0329 | 0.5768 | 0.0281 | 100 | 0.0 |
| Open-LRM | Eulerian | 0.0285 | 0.5945 | 0.0252 | 100 | 0.0 |
| DreamGaussian | Lagrangian | 0.0641 | 0.3476 | 0.0812 | 100 | 237.4 |
| Ours | Lagrangian | 0.0351 | 0.6317 | 0.3665 | 100 | 0.0 |

Metrics.
To assess the accuracy of the reconstruction, we use two commonly used metrics: Chamfer Distance (Cham.) and Volume IoU (Vol. IoU), comparing the ground-truth shapes with the reconstructed ones.
Following established practices, we use the rigid Iterative Closest Point (ICP) algorithm to align the generated shapes with their ground-truth counterparts before metric calculation.
For multi-view reconstruction, we also report F-Score, Normal Consistency, Edge Chamfer Distance, and Edge F-Score to the ground truth mesh following prior research (Shen et al., ; Son et al., ).

While these metrics effectively measure the reconstruction accuracy, they fail to evaluate mesh quality.
We use three additional metrics to assess the geometry quality of the reconstructed shapes: 1) Area-length Ratio (ALR): This metric computes the average ratio of a triangle’s area to its perimeter (scaled by a constant coefficient) within the surface mesh. Values range from 0 0 to $1$, where meshes with higher ALR values contain mostly equilateral triangles, thereby indicating superior triangle quality; 2) Manifoldness Rate (MR): Manifoldness verifies whether a mesh qualifies as a closed manifold. Non-manifold meshes can manifest anomalies, such as edges shared by more than two faces, vertices connected by edges but not by a surface, isolated vertices and edges, and vertices where more than one distinct surface meets, which can cause problems in downstream applications such as simulation and rendering. We report the percentage of manifold shapes within the evaluation dataset as MR; and 3) Connected Component Discrepancy (CC Diff.) from the ground-truth shape: This measure identifies the presence of floating artifacts or structural discontinuities within the mesh, highlighting the integrity and cohesion of the reconstructed shape.

**Table 3: Comparison of memory cost and run-time speed on image-to-3D generation with SDS loss. We report the maximal batch size of $256\times 256$ images that can occupy a 40GB A100 and the run-time speed for training with batch size $4$.**
| Method | \CenterstackMaximal |  |  |
| --- | --- | --- | --- |
|  | \CenterstackSpeed$\uparrow$ | Batch Size$\uparrow$ |  |
|  | (#iter./s) |  |  |
| Eulerian | Make-it-3D | 4 | 1.22 |
| Magic123 | 4 | 1.03 |  |
| SyncDreamer | 48 | 1.8 |  |
| DreamCraft3D | 8 | 1.23 |  |
| GeoDream | 8 | 1.30 |  |
| Lagrangian | DreamGaussian | 80 | 4.43 |
| Lagrangian | Ours | 120 | 6.59 |

### 5.3 Results

Multi-view reconstruction.
Table 1 and Table 4 in Appendix A.1 present the quantitative comparison results.
Fig. 5 shows the qualitative comparison on four different shapes.
Our method excels over baseline methods regarding mesh quality while achieving similar reconstruction accuracy.
Specifically, our approach features uniform meshing (as indicated by ALR) and eliminates floating artifacts (as shown by CC Diff.).
This improvement is attributed to the volumetric representation of TetSphere and its geometric energy regularization.
The qualitative results further support these findings: our method accurately captures slender and thin structures, such as the fingers of Lucy’s statue, the sorter, and the dress, which other methods struggle with.
Additionally, our method demonstrates correct surface orientation, whereas baseline methods, particularly FlexiCubes, and DMesh, often exhibit inverted triangles, as seen in the noisy rendered normal images.
However, our TetSphere approach is less effective when modeling thin shells, such as the leaves of the plant, where DMesh shows better accuracy.

Single-view reconstruction.
Table 2 shows the comparison results.
Fig. 8 in Appendix A.2 shows qualitative results of our method.
Our TetSphere splatting excels over baseline methods in terms of mesh quality while also achieving competitive levels of reconstruction accuracy.
This improvement is attributed to the regularizations embedded within the TetSphere splatting optimization, ensuring uniform meshing and eliminating common artifacts like floating components.
These findings are consistent with the results observed in multi-view reconstruction.

Figure: Figure 7: Results on text-to-3D shape generation. Our TetSphere splatting excels in handling slender, thin structures, such as the wine glass stem and the dragon head. More results are available in Appendix A.4.
Refer to caption: extracted/5896802/figures/exp-3.png

### 5.4 Analysis

Applications to generative modeling.
TetSphere splatting is a versatile approach that can be effectively applied to generative modeling tasks.
We showcase two applications: image-to-3D and text-to-3D generation.
For image-to-3D generation, we include a variety of internet-collected images in different styles in our evaluation.
For text-to-3D generation, we employ text prompts from the official implementation of RichDreamer.
Fig. 9 in Appendix A.3 illustrate the results on image-to-3D generation.
Our approach outperforms Magic123 and DreamCraft3D regarding mesh quality, achieving smoother surfaces for broad regions, such as animal bodies while retaining local sharp details in areas like eyes and noses.
Magic123 tends to produce overly smooth surfaces that sometimes deviate from the correct geometry.
DreamCraft3D suffers from noisy and bumpy surface meshes, indicating lower mesh quality.
Fig. 7 and Fig. 10 in Appendix A.4 shows comparison results on text-to-3D generation.
Our TetSphere splatting can produce slender structures such as the dragon’s head and the goblet.
Furthermore, the results demonstrate that the integration of TetSphere splatting with SDS enhances the geometric detail of the generated shapes, resulting in both diverse and high-quality textures.

Analysis on computational cost
As shown in Table 4 in Appendix A.1, our method significantly outperforms the Eulerian method NIE regarding running time while remaining competitive with other geometry representation methods.
To further assess the performance of TetSphere splatting, we conduct an extreme test by applying it to the task of image-to-3D generation with SDS loss. We evaluate several baseline methods and report the maximum batch size of $256\times 256$ images that can be processed on a 40GB A100 GPU, as well as the training run-time speed with a batch size of $4$, as shown in Table 3.
TetSphere splatting stands out for its minimal memory usage and achieves the fastest run-time speed.
This efficiency underscores the benefits of TetSphere’s explicit and Lagrangian properties.

Effects of energy coefficients.
Fig. 6 demonstrates how different energy coefficients influence the reconstruction outcome. Larger coefficients lead to a smoother surface, but excessively small coefficients may cause tetrahedron inversion. In our experiments, we choose $w_{1}=5\times 10^{-6},w_{2}=2\times 10^{-5}$ and apply a cosine increasing schedule.

## 6 Conclusion

We introduced TetSphere splatting, a Lagrangian geometry representation for high-quality 3D shape modeling.
Our computational framework leverages geometric energy optimization and differentiable rendering to deform TetSpheres into complex shapes, producing results that demonstrate superior mesh quality compared to state-of-the-art methods.
Future work could extend TetSphere splatting to leverage direct 3D supervision with volumetric data, eliminating the need for rendering to images.
The current limitation of TetSphere splatting lies in its inability to guarantee topology preservation due to the union of all tetrahedral spheres.
This highlights the need for future development in shape generation that can better adhere to topology constraints.

## Appendix A Additional Results

### A.1 More Quantitative Results on Multi-view Reconstruction

Table 4 shows comparisons of multi-view reconstruction on more evaluation metrics.

### A.2 Qualitative Results on Single-view Reconstruction

Fig. 8 shows the results of single-view reconstruction performed on the GSO dataset.
Our TetSphere splatting demonstrates superior mesh quality, effectively capturing sharp geometric features, including the boundaries of the shoes.

Figure: Figure 9: Qualitative comparison on image-to-3D generation with surface mesh visualizations and rendered normal maps. Our generated meshes exhibit less bumpiness and high mesh quality with more regular triangle meshing.
Refer to caption: extracted/5896802/figures/exp2-1.png

Figure: Figure 10: More results on text-to-3D shape generation.
Refer to caption: x5.png

### A.3 Qualitative Results on Image-to-3D Generation

Fig. 9 shows qualitative comparisons of image-to-3D generation with surface mesh visualizations and rendered normal maps. Our generated meshes exhibit minimal noise and high quality, featuring more regular triangle meshing.

### A.4 More Results on Text-to-3D Shape Generation

Fig. 10 shows additional results of text-to-3D shape generation.
These results highlight our method’s capability to construct complicated material, such as reflections, by leveraging the explicit geometry representation.

## Appendix B Tetrahedron and its Deformation Gradient

Following (Sifakis & Barbic, ), we treat a tetrahedron as a piecewise linear element.
The initial (undeformed) positions of the four vertices of a tetrahedron are denoted by $\mathbf{X}=[\mathbf{X}^{(1)},\mathbf{X}^{(2)},\mathbf{X}^{(3)},\mathbf{X}^{(4)}]$, with each $\mathbf{X}^{(i)}\in\mathbb{R}^{3}$. Similarly, the positions of the four vertices of a deformed tetrahedron are represented by $\mathbf{x}=[\mathbf{x}^{(1)},\mathbf{x}^{(2)},\mathbf{x}^{(3)},\mathbf{x}^{(4)}]$, where each $\mathbf{x}^{(i)}\in\mathbb{R}^{3}$. The deformation gradient $\mathbf{F}\in\mathbb{R}^{3\times 3}$, which quantifies the local deformation of the tetrahedron, is given by:

$$ $\displaystyle\mathbf{F}$ $\displaystyle=\mathbf{D}_{s}\mathbf{D}_{m}^{-1},$ (3) $\displaystyle\mathbf{D}_{s}$ $\displaystyle:=\left[\begin{array}[]{ccc}\mathbf{x}^{(1)}-\mathbf{x}^{(4)}&\ \mathbf{x}^{(2)}-\mathbf{x}^{(4)}&\ \mathbf{x}^{(3)}-\mathbf{x}^{(4)}\end{ array}\right],$ (5) $\displaystyle\mathbf{D}_{m}$ $\displaystyle:=\left[\begin{array}[]{ccc}\mathbf{X}^{(1)}-\mathbf{X}^{(4)}&\ \mathbf{X}^{(2)}-\mathbf{X}^{(4)}&\ \mathbf{X}^{(3)}-\mathbf{X}^{(4)}\end{ array}\right].$ (7) $$

The deformation gradient $\mathbf{F}$ essentially captures how a tetrahedron transforms from its initial state to its deformed state, encompassing both rotation and stretching effects.

## Appendix C Formulation of TetSphere Initialization

Assuming there are a total of $m$ candidate positions obtained from the coarse voxel grid (as shown in Fig. 5), our goal with TetSphere initialization is to select a subset of these candidate points such that the object’s shape is adequately covered by tetrahedral spheres centered at these positions.
We first initialize a sphere of fixed radius at each candidate position, where the radius is calculated as
$\alpha r+\beta$, where $r$ is the minimum distance from each candidate position to the voxel surface.
We use $\alpha=1.2,\beta=0.07$ in all our examples.
The objective is to select a subset of spheres that collectively cover all voxel positions.

We define a coverage matrix $\mathbf{D}\in\{0,1\}^{m\times m}$, where each element $d_{ji}\in\{0,1\}$ indicates whether voxel position $j$ is covered by a sphere centered on candidate position $i$.
A binary vector $\mathbf{v}\in\{0,1\}^{m}$ identifies selected candidate positions, with each element denoting the selection status of corresponding voxel positions.
The selection of feature points is formulated as a mixed-integer linear programming problem:

$$ $\displaystyle\min_{\mathbf{v}}|\mathbf{v}|\ \ \ \mathrm{s.t.}\ \ \mathbf{D} \mathbf{v}\geq\mathbbm{1},$ (8) $$

where $|\cdot|$ is the $l_{1}$ norm,
$\mathbbm{1}$ is a vector of ones.
This optimization is efficiently solved using standard linear programming solvers.

## Appendix D Texture and PBR Material Optimization

Material optimization is facilitated by using differentiable rasterizers (Laine et al., ), which adjust the textures or materials to closely match the input multi-view color images.
A significant advantage of TetSphere splatting is that the deformation of tetrahedral spheres does not alter the surface topology.
Unlike methods such as DMTet, which require isosurface extraction and subsequent texture parameterization at each step due to potential changes in the underlying shape, our method necessitates only a single texture parameterization at the beginning of optimization.
This parameterization remains consistent throughout the process, significantly enhancing the efficiency of texture optimization.

For scenarios with dense input views, we have found that using textured images as optimization variables is straightforward and yields high-quality results.
In cases with sparse input views, we adopt a two-layer MLP that takes the surface vertex positions as inputs and outputs the material parameters, a practice in line with existing methods (Sun et al., ; Qiu et al., ).

## Appendix E More Implementation Details

For image-to-3D shape generation, we obtain multi-view images from the initial stage of DreamCraft3D (coarse NeRF fitting only), generating $360$ views sampled on a Fibonacci sphere (González, ).
The texture optimization directly employs a $2048\times 2048$ 2D texture image,
circumventing the need for additional neural networks.
Here, the optimization objective $\mathbf{\Phi}(\cdot)$ focuses solely on the rendering loss.
For text-to-3D generation, we use the initial stage of RichDreamer (Qiu et al., ) to obtain multi-view images.
We optimize our TetSphere splatting using both the rendering loss and the SDS loss.
The SDS loss for geometry is calculated using the Normal-depth diffusion model as described in (Qiu et al., ).
The SDS loss for texture leverages the albedo diffusion model from the same work.

Regarding baselines:
For image-to-3D generation, we compare TetSphere splatting with Magic123 and Dreamcraft3D (Sun et al., ).
Both are multi-stage methods starting with NeRF optimization followed by DMTet to optimize mesh and texture.
For text-to-3D, our comparison focuses on RichDreamer, a state-of-the-art method known for incorporating PBR materials and generating shapes, showcasing notable results in 3D generation from text prompts.

## Appendix F Related work – Text-to-3D content generation

The recent success of text-to-2D image generation models has spurred a growing interest in generating 3D output from text input.
In light of the limited availablity of text-annotated 3D datasets, methods have been developed that leverage the pre-trained text-to-image models to reason between 2D renderings of 3D models and text descriptions.
Early works (Khalid et al., ; Jain et al., ) adopt the pre-trained CLIP model (Radford et al., ) to supervise the generation by aligning the clip text and image embeddings.
Recently, 2D diffusion-based generative models (Saharia et al., ; Rombach et al., ) have powered direct supervision in the image/latent space and achieved superior 3D quality (Poole et al., ; Tang et al., ; Chen et al., ; Lin et al., ; Wang et al., ).
Notably, DreamFusion (Poole et al., ) introduces the Score Distillation Sampling (SDS) for supervising the NeRF (Mildenhall et al., ) optimization using diffusion priors as a score function (i.e., by minimizing the added noise and the predicted noise under the text condition).
Follow-up works have since been proposed to improve score sampling formula with Perturb-and-Average and variational method (Wang et al., ; ), improved noise sampling schedules (Huang et al., ), various 3D representations (Lin et al., ; Tsalicoglou et al., ; Chen et al., ), text prompts (Armandpour et al., ), 3D consistency (Li et al., ; Hong et al., ), and prior quality with dedicated diffusion models (Qiu et al., ; Shi et al., ).
Our method leverages the Normal-Depth diffusion model used in (Qiu et al., ) for text-to-3D shape generation, which is trained on the large-scale LAION dataset, but replaces the geometry representation with our TetSphere.

## Peer Reviews

confidence: 5
contribution: 2
initial_rating: 3
presentation: 3
questions: - can you explain why did you choose to not compare to [Nicolet et al. 2021] and methods that adopted it later? 
- you do not cite "Continuous remeshing for inverse rendering" - how do you think your method would fair in respect to it?
- are there other advantages to using a tetrahedral mesh as opposed to triangle, except the two regularization losses?
soundness: 3
strengths: The paper revisits the idea of using meshes for 3D reconstruction, and reaffirms that an explicit representation is extremely efficient in producing high quality 3D geometry from images. The paper is written in a clear way and is easy to understand.
summary: This paper proposes a method for reconstructing 3D geometry from multiview images. The geometry is represented by Tetrahedral meshes of sphere topology that are optimized in standard fashion w.r.t a rendering loss, along with two geometric regularizes - the biharmonic energy to ensure a smooth deformation, and a barrier term that prevents tetrahedra from inverting.
weaknesses: 1. My main concern with the paper is its novelty and contribution: the paper essentially proposes to deform a mesh w.r.t a visual loss which is standard (e.g., [Nicloet et al. 2021]), while using two standard regularizes to guarantee a "good" deformation (smooth and without inversions). This is a very standard approach. The only part that seems less explored is the use of tetrahedra instead of triangles, however this of course has been explored extensively outside of the context of differentiable rendering, hence the only part I can deem truly novel is "using tetrahedra instead of triangles in tandem with multiview reconstruction". This feels like a marginal contribution. It could be argued to be a very practical approach which may convince me to champion it, however that leads me to my second concern:
2. Evaluation seems somewhat selective, and the methods compared to do not strike me as the immediate alternatives. Namely, I find [Nicolet et al. 2021] as a main contender, as well as "Continuous remeshing for inverse rendering" which is uncited. Many other techniques that cite [Nicolet et al. 2021] can be considered. 
3. The argumentation for the use of a volumetric mesh instead of a triangle mesh needs to be justified further through experiments. It seems the only argument for a full discretization of the *volume* (as opposed to the surface via a triangle mesh), is to regularize the volumetric deformation with the two losses. This seems like a significant overkill, as simpler regularizers could be employed on the triangle mesh (again, e.g., Nicolet et al. 2021) to ensure it behaves well.

---

confidence: 2
contribution: 3
initial_rating: 8
presentation: 4
questions: When does the method break? At which point does the proposed machinery fail to fit a reasonable geometry to data, and how do the problems look like / introduce themselves? Which conceptual or technical challenges would be needed to overcome to fix this?

Does the method have fundamental advantages over previous representations? If so, can you make a formal or experimental argument to convince the reader? If the differences are more nuanced, what are the key trade-offs one has to make here?
soundness: 3
strengths: I think that the main strength of the paper is the idea itself: Regularization is important to avoid bad mesh structure, and a volumetric representation intuitively gives much more leeway to regularization over a pure thin-shell / surface model (the inductive bias of this being a solid is stronger than regularizing details on thin surface). The proposed setup with inversion constraints and smoothness looks plausible and seems to work very well in practice. I should add that I am not actively working in this area, so I might miss some related work; but if this kind of approach has not yet been tried, the approach itself seems worth publishing.

The results are also convincing in terms of quality and versatility. The paper is very well written, nicely illustrated and enjoyable to read. Comparisons against some related recent method also underline the quality of the results quantitatively.
summary: The paper proposes a shape reconstruction method that takes a set of images as input and fits a 3D model with reflectance properties to them. In line with a large body of recent work, it combines a rendering loss with shape regularization to directly optimize a 3D scene representation by "simple" numerical descent. The problem is particularly challenging as the paper uses a "Lagrangian" representation, specifically, a tetrahedral mesh that co-moves with the surface points obtained.

As recent work, such as [Nicolet 2021] has pointed out, the success of such an approach critically hinges upon suitable regularization, as the rendering loss is ambiguous and it is easy to get stuck in a bad local minimum with nonsensical mesh structures (which has been an issue for a long time in attempting such direct shape optimization against relatively weak data constraints).

The key contribution of this paper is to use a volumetric representation, a tet-mesh, with a penalty against inversion of elements along with an incrementally applied smoothness regularizer (a bi-harmonic energy applied to deformation gradients of the current mesh) that promotes changes in low-frequency shape first.

The paper provides very convincing results and shows a number of interesting applications, some of which only made it into the appendix.
weaknesses: In my opinion, the main downside is that the paper is rather light on analysis: Why does it use the bi-harmonic energy on deformation gradients? What does this actually mean (application to derivatives should create higher-order smoothness, or in terms of a Fourier-perspective, add an additional high-pass filter in front of the regularizer)? What happens, if other regularizers are used? How important is the non-inversion term, and how brittle are parameter choices? It would also be interesting to discuss the relation to other attempts at stricter regularization, such as Nilcolet et al.'s work. I would guess that the volumetric approach is strictly superior, if fully exploited; showing something like this experimentally would make the paper much stronger.

Some of the limitations could also been discussed more clearly. For example, the paper only produces piecewise meshes, which are overlayed; it does not solve the problem of globally fitting a single tet-mesh. Comparisons in mesh quality should take this into account. Further, this also shows implicitly that the method is still to brittle to handle strong deformations during optimization; otherwise, one could just fit a single sphere globally. In this context, as well as in general, it would be instructive to show how to break the method by slowly stepping outside its "convergence radius" where good results can be obtained into cases where results are unsatisfactory. My criticism here is not that there are limitations but that the paper could be strengthened by studying them more in depth.

Finally, the paper seems like a better fit to a conference on computer vision or graphics in terms of methods and problems, but I would consider the topic close enough not to exclude it.

---

confidence: 2
contribution: 3
initial_rating: 8
presentation: 3
questions: -	When supervised purely based on depth/mask, the interior TetSphere not visible to any images will not receive gradients, except for regularization. Is this the main reason manifoldness is maintained?

-	Mosaic SDF [a] is one missing work that also follows the Lagrangian framework, where a set of volumes are moved in space. 

[a] Yariv, L., Puny, O., Gafni, O., & Lipman, Y. (2024). Mosaic-sdf for 3d generative models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 4630-4639).

-	Clarification needed, below variables are not properly defined:
 - L265: what is N?
 - L266: what is T?
 - L309: what is n?
soundness: 3
strengths: -	The problem is of interest to the research community.
-	The formulation of TetSphere Splatting is novel, and an interesting theoretical perspective from prior work.
-	The experiments setup is diverse, including many applications.
summary: The manuscript introduces a framework named TetSphere Splatting, which can be used to reconstruct 3D meshes from multi-view images. TetSphere uses tetrahedron spheres to represent an object. During optimization, deformation field is predicted to relocate vertex points to minimize cost. Experiments in multi-view reconstruction, image/text to 3D shows potential applications of TetSphere Splatting.
weaknesses: Overall, the paper is quite interesting. However, my concern about the paper is its claim, where experimental results cannot back up.

-	TetSphere is claimed to have superior surface quality due to the fact Marching Cubes is not needed. However, all qualitative results seem to suggest that the TetSphere needs to be sufficiently dense to produce detailed structures. This leads to unnecessarily dense triangles, which can also be achieved by very dense marching cubes resolution. What is the advantage in this case?

-	Moreover, while the manifoldness of TetSphere is guaranteed, the baselines compared are all Eulerian representations without sufficient regularization. Approaches such as NeuS and VolSDF use volume rendering to achieve manifold meshes. While I understand the faster optimization process aspect, can authors comment on TetSphere’s advantage over NeuS/VolSDF from the quality perspective?

-	Lastly, the claimed superior surface quality is not backed up by smaller Chamfer Distance, such as in Table 2. Can authors provide more explanation?

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: The proposed optimisation does not appear to prevent non-uniform deformation of TetSpheres. However, the resulting reconstruction Aspect Loss Ratio (ALR) is higher than in other methods, suggesting that the tetrahedra within each TetSphere largely preserve their original volume. Is there any factor specifically contributing to the high ALR?


What are the values of N (number of tetrahedra) and M (number of TetSpheres) used in the experiments? How does the trade-off between these values affect reconstruction quality and computational complexity?

More details on the image-to-3D and text-to-3D generation pipelines are needed to ensure the experiments can be reproduced.
Figures 8 to 10 would benefit from captions with additional information about the experimental setup.
soundness: 4
strengths: The paper is generally well-written, conveying complex concepts and methods in a concise style. 

The state-of-the-art is clearly explained and appears to be up-to-date.

The proposed primitive is original and effectively addresses common issues in other Lagrangian geometry representations, such as irregular triangles, non-manifoldness, and floating artifacts.

The energy optimization process for splatting is well-designed and effective, with a convincing initialization algorithm.

The accuracy in monocular and multiview reconstruction applications is competitive with current methods while achieving higher surface quality. This approach is also lighter than other Lagrangian representations in terms of memory and computational complexity.
summary: This paper presents a Lagrangian geometry representation based on TetSpheres, volumetric tetrahedral spheres that deform to fit the desired geometry. Key applications of TetSphere splatting are demonstrated in monocular and multiview reconstruction. The deformation of TetSpheres is formulated as an energy optimization problem with geometric constraints that prevent the generation of irregular surfaces. Experiments are proposed to compare TetSpheres with state-of-the-art methods based on both Eulerian and Lagrangian geometry representations.
weaknesses: The paper proposes a new Lagrangian primitive that is notably more complex than the primitives used in previous methods, such as 3D Gaussians or triangles. Each TetSphere consists of a collection of N tetrahedra, with an apparent trade-off between N and the total number of TetSpheres needed to represent the surface. This trade-off seems to be overlooked in the experimental section, as specific values for the number of TetSpheres and tetrahedra in each experiment are not provided.

The surface is represented as a union of TetSpheres. Due to the proposed splatting optimization, TetSpheres may intersect, leading to artifacts at the intersection points. However, the paper does not specify how intersections between tetrahedra are managed.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: My main doubts lie in the topological and manifold guarantees claimed by the authors, and I hope to see more detailed analyses and experiments to prove their method. I am also curious about the mesh extraction method and the number and quality of triangles. And if the code is attached, it will also increase the credibility of this paper.

Overall, I like this idea, but it seems to have a lot of minor issues. If the authors can address these issues during rebuttal, I will increase my rating.
soundness: 3
strengths: I think it’s a good idea to use tetrahedrons to represent shapes, which is very common in classical geometry processing. I’m excited to see that the authors were able to combine computer vision and geometry processing and make good progress on multiple tasks.

This new method surpasses or catches up with Sota methods on multiple data sets, and because it is based on the tetrahedron geometric primitive, it seems to be able to achieve better topology preservation, manifold preservation and better triangulation quality.
summary: This paper uses tetrahedral spheres to represent three-dimensional shapes and constructs its corresponding differentiable inverse rendering process, which improves tasks such as single-view, multi-view reconstruction, and text to 3d shapes.
weaknesses: The implementation details of this paper are not enough for me, and many questions come to my mind when I read this paper.

1. The author does mention the problem of tetrahedron inversion, and cites theories from other papers to show that it is useful. I think it would be easier to understand if there is a simple ablation experiment.
2. The paper does not seem to mention how to extract the surface mesh from the tetrahedral mesh, which I think is also an important part of the whole pipeline. At the same time, how to ensure that there is no non-manifold when extracting the surface is also a problem. 
3. How to ensure that the topology is correct? If I understand correctly, this seems to be strongly related to "silhouette coverage", which must be consistent with the topology of GT. I would like to see an analysis of the Euler characteristic (Genus) in the tables.
4. This paper introduces many metrics to evaluate the quality of the mesh, but no corresponding references or mathematical definitions are given, like Manifoldness Rate and Connected Component Discrepancy. I hope to see their mathematical definitions in the appendix, which will make it easier for readers to understand. 
5. Regarding mesh quality, first of all, do all compared methods have similar resolution/triangle numbers? If mesh quality is to be compared, giving the triangle numbers is a must. It would be great if the authors could provide a metric for triangle quality, such as the Edge Chamfer Distance in NMC [Chen and Zhang. 2021] and TriangleQ in CWF [Xu et al. 2024].
6. In addition, since your initialization is multiple tetrahedralized spheres, what is the connection between the spheres? Are they multiple independent connected branches? If so, how to deal with the intersection between them, and how to ensure that the final mesh is an independent and connected one?
7. Citation problem, I found multiple citations of same paper in several places, such as One-2-3-45 line 648-653, Syncdreamer line 657-662 etc.

Meta-review: The submission received positive reviews from all the reviewers. The reviewers generally appreciate the clarity, recognize the novelty of the method, and are convinced by the positive experimental results. After reading the paper, the reviewers' comments and the authors' rebuttal, the AC agrees with the decision by the reviewers and recommends acceptance.

Decision: Accept (Oral)