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# 3DGeoMeshNet: A multi-scale graph auto-encoder for 3D mesh reconstruction and completion

Saqib Nazir, Olivier Lézoray, Sébastien Bougleux

► To cite this version:

Saqib Nazir, Olivier Lézoray, Sébastien Bougleux. 3DGeoMeshNet: A multi-scale graph auto-encoder for 3D mesh reconstruction and completion. Neurocomputing, 2026, 671, pp.132652. 10.1016/j.neucom.2026.132652. hal-05453145

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# 3DGeoMeshNet: A multi-scale graph auto-encoder for 3D mesh reconstruction and completion

Saqib Nazir*, Olivier Lézoray, Sébastien Bougleux

Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ, GREYC UMR 6072, Caen, F-14000, Normandie, France

## ARTICLE INFO

Communicated by Y. Long

**Keywords:**

3D mesh reconstruction

Completion

Deep learning

Graph neural networks

## ABSTRACT

We propose 3D Geometric Mesh Network (3DGeoMeshNet), a method for 3D mesh reconstruction and completion. 3DGeoMeshNet is a novel Graph Convolutional Network (GCN)-based framework that leverages anisotropic convolution layers to effectively learn multi-scale global and local features directly in the spatial domain. Unlike traditional approaches that convert meshes into voxel grids or point clouds, our method operates directly on the polygonal mesh structure, preserving geometric fidelity throughout the learning and reconstruction process. In addition to 3D mesh reconstruction, we extend our framework to tackle the 3D mesh completion task, where missing or incomplete regions of the mesh are accurately recovered. The completion results are further refined through a set of pre- and post-processing steps. We extensively evaluate our approach on two benchmark datasets. COMA and DFAUST, and achieve SOTA results for 3D mesh reconstruction on both datasets. Additionally, our mesh completion experiments on the COMA dataset demonstrate the promising capability of 3DGeoMeshNet in recovering incomplete geometries. We further showcase the versatility of our method through additional applications, including mesh denoising, interpolation, and extrapolation, highlighting the robustness and generalization ability of our framework across various 3D mesh processing tasks.

## 1. Introduction

3D meshes have become a crucial representation for capturing the geometric structure of complex objects, particularly in fields like computer vision, graphics, and animation. These meshes, composed of vertices, edges, and faces, provide an efficient and scalable way to represent surfaces, making them highly suitable for applications such as 3D shape reconstruction, shape modeling, object recognition, and shape segmentation [1]. However, processing 3D meshes presents unique challenges due to their non-Euclidean nature, making the direct application of traditional neural network architectures, such as Convolutional Neural Networks (CNNs), difficult. In Euclidean domains like images, CNNs have proven to be highly effective for tasks like image classification and segmentation due to the structured nature of the data. However, 3D meshes are typically represented as graphs, where the number and orientation of neighbors for each vertex vary. This irregularity, combined with the non-Euclidean domain, complicates the use of standard convolutional architectures, which rely on consistent grid-like data structures. Early approaches, such as voxel-based representations and point clouds, were proposed to circumvent these challenges [2,3]. However, they struggle to capture the fine geometric details of the meshes due to their

high memory and computational costs. Graph Convolutional Networks (GCNs) [4] have emerged as a promising solution to this problem, as they can operate directly on graph-structured data, allowing the learning of meaningful geometric features from 3D meshes. State-of-the-art (SOTA) GCN methods are typically classified into spectral and spatial approaches [5]. Spectral methods, such as ChebNet [4], apply convolutions in the frequency domain but rely on isotropic filters that are limited in their ability to capture both local and global features of complex shapes. Spatial methods, such as SpiralNet [6], attempt to address this by defining a specific order of neighbors for each vertex, but these methods struggle with inconsistencies across different mesh topologies.

One of the earliest and most notable GCN-based approaches for 3D mesh reconstruction is the Convolutional Mesh Auto-Encoder (COMA) by Ranjan et al. [5], which uses spectral convolution and mesh sampling to encode and decode 3D meshes. COMA utilizes spectral convolution layers and a quadric-based mesh up-and-down sampling [7] approach to encode and decode 3D meshes. The spectral approach transforms the mesh into the frequency domain using the Laplacian operator, which allows for localized and efficient representations, but it is topology dependent. While effective for aligned facial meshes, COMA struggles

<footnote>* Corresponding author.
Email addresses: saqib.nazir@edgehill.ac.uk (S. Nazir), olivier.lezoray@unicaen.fr (O. Lézoray), sebastien.bougleux@unicaen.fr (S. Bougleux).</footnote>
with larger-scale deformations and global variations, which are crucial for more complex human body reconstructions. To address this limitation, Bouritsas et al. [6] introduced the Neural 3D Morphable Model (Neural3DMM), which replaced spectral convolution with a novel spiral convolution operator. In order to introduce anisotropic filters in graph convolutions, Bouritsas et al. [6] formulated a Spiral Convolution operator (SpiralNet) that defines an explicit order of the neighbors via a spiral scan for each vertex on 3D meshes with fixed topology. However, serializing the local neighbors of vertices by following a spiral cannot resolve the inconsistency between different nodes. Furthermore, this method requires manually assigning a starting point to determine the order of neighbors, which makes it difficult to maintain a consistent local coordinate system across meshes. The selection of the starting point may affect the performance of the spiral convolution operator. Gong et al. proposed SpiralNet++ [8] to cope with these issues. Recent efforts, such as Local Structure-Aware Anisotropic Convolution (LSA-Conv), have sought to address these limitations by dynamically adjusting filter weights based on local geometry [9]. However, these methods often focus too heavily on local features, lacking critical global topology, and require significant computational resources, especially for dense meshes. Other convolutional operators like FeaStConv [10] have achieved promising results by learning to dynamically aggregate neighboring features, but still face challenges in striking a balance between preserving local details and maintaining overall structural consistency.

While 3D mesh reconstruction deals with generating complete meshes from various inputs, 3D mesh completion addresses a related but distinct problem of recovering the complete structure of partial or damaged meshes. This task is particularly relevant in scenarios where data acquisition is imperfect due to occlusions, sensor limitations, or physical constraints [11]. The applications of mesh completion span numerous domains, including archaeological artifact restoration, medical imaging for prosthetic design, industrial quality control, and augmented reality [12]. Traditional approaches to mesh completion have relied on geometric priors and template fitting [13], but these methods often struggle with complex topologies and require extensive domain expertise. More recently, learning-based approaches have gained attention, with methods like PoinTr [14] and ShapeInversion [15] demonstrating impressive results for point cloud completion. However, translating these successes into mesh completion introduces additional challenges due to the need to maintain mesh connectivity and preserve fine structural details. The fundamental challenge in mesh completion lies in understanding both the local geometric properties and the global structural context of the object, which is essential for generating plausible completions that maintain geometric consistency with the partial input. This dual requirement of local-global understanding makes mesh completion an ideal application for our approach.

In this paper, we present 3DGeoMeshNet,<fnref n="1" /> a novel GCN-based framework designed to overcome the limitations of both spectral and spatial approaches for 3D mesh reconstruction and completion. Our network uses anisotropic convolution layers that operate directly in the spatial domain, effectively capturing both global and local features without the need to convert the data into intermediate formats like voxel grids or point clouds. Our multi-scale encoder–decoder architecture consists of two distinct pathways, local and global, enabling efficient feature extraction and high-fidelity mesh reconstruction and completion at different scales. To thoroughly evaluate the effectiveness of different convolutional operators in our framework, we conducted extensive experiments comparing several SOTA graph convolution layers, including FeaStConv [10], GC [16], and GAT [17]. This comparative analysis provides valuable insights into the strengths and limitations of each operator in the context of mesh reconstruction and completion tasks. This paper presents an extension of our initial model [18] to multiple levels. First, the training loss has been modified: this reduces the mean reconstruction

error from 0.171 to 0.125, demonstrating the effectiveness of the proposed improvements and resulting in superior performance compared to the original model. Second, we provide a more in-depth analysis of the proposed architecture by testing it with several datasets, performing rigorous ablation studies that demonstrate the benefits of each of the components used. Third, in addition to reconstruction, we also show how the proposed auto-encoder architecture can be used for mesh completion, which has received very little attention in the literature.

We validated our approach on two datasets that span different scales and complexities: the COMA dataset [5], which contains human facial expressions, and the DFAUST dataset [19], which includes full human body shapes with various poses and movements. This evaluation demonstrates the versatility and robustness of our approach across different anatomical structures and scales. The primary contributions of this work are:

* We propose a unified framework for both 3D mesh reconstruction and completion that leverages a weighted fusion mechanism within a multi-scale auto-encoder architecture, effectively capturing both local and global features.

* Our method operates entirely in the spatial domain, avoiding the complexity and limitations of spectral-domain methods, and directly processes 3D meshes without requiring conversion to intermediate representations.

* We conduct a comprehensive comparative analysis of different graph convolutional operators within our framework, providing insights into their relative strengths and limitations for mesh processing tasks.

* Our network achieves higher reconstruction and completion accuracy compared to isotropic filters or spectral decomposition-based approaches, offering a more efficient and scalable solution that generalizes well across different anatomical structures.

* We demonstrate the effectiveness of our approach through extensive experiments on both the COMA [5] and the DFAUST [19] datasets, showcasing its applicability across varying scales and complexities.

* In addition to the reconstruction tasks previously explored (mesh denoising, interpolation, and extrapolation), we extend our evaluation to include mesh completion tasks, further demonstrating the versatility of our approach.

## 2. Related work

Our approach is situated at the intersection of two major research areas in computer graphics. The first area focuses on mesh reconstruction. The second area concerns the use of inpainting techniques to fill in large missing parts of meshes. Both of these research areas have received considerable attention in recent years, and as such, we provide a thorough literature review of mesh reconstruction and completion in the following subsections.

### 2.1. Mesh reconstruction

In this section, we review previous research efforts on 3D mesh reconstruction, focusing on both linear and nonlinear methods. 3D representations of human bodies and faces are particularly challenging due to non-linear movements and varying expressions. Existing works are broadly categorized into linear statistical models and non-linear deep learning-based approaches.

#### 2.1.1. Linear approaches

The pioneering approach to 3D representation was introduced through 3D Morphable Models (3DMMs), which use statistical priors and dimensionality reduction techniques to model 3D faces, bodies, and hands. Blanz et al. [20] introduced the first linear parametric 3DMM based on Principal Component Analysis (PCA), which became the foundation for subsequent models. The widely adopted Basel Face Model (BFM) [21] expanded on this work by incorporating a larger

<footnote n="1">¹ Code is available at https://github.com/saqibnaziir/3DGeoMeshNet.</footnote>
dataset of 200 subjects, but it was still limited to neutral facial expressions. Li et al. [22] proposed FLAME to learn a unified model of facial shape and expression by analyzing 4D facial scan data, enabling the reconstruction and synthesis of dynamic facial expressions with high precision.

Later, Zhu et al. [23] merged BFM with the FaceWarehouse dataset [24], adding dynamic facial expressions and creating a more comprehensive model. For full-body reconstruction, the Skinned Multi-Person Linear Model (SMPL) [25] became a prominent solution, using PCA to represent various body shapes in natural human poses. The MANO model [26] is based on similar principles for modeling hands, using a dataset of more than 1,000 high-resolution hand scans.

Recent advances in linear methods, such as the Large-Scale Facial Model (LSFM) introduced by Booth et al. [27], and Learning-by-Synthesis (LBS) [22], demonstrate the enduring relevance of these approaches. Despite the rise of non-linear methods, linear models are still viable due to their simplicity, interpretability, and ability to efficiently capture overall variations in shape.

### 2.1.2. Non-linear approaches

The rapid growth of deep learning applications on graph-structured data has led to significant advancements in non-linear methods for 3D mesh reconstruction. These approaches are particularly effective in capturing the complex and non-linear variations present in 3D meshes, such as human facial expressions and body deformations. The two dominant categories of Graph Neural Networks (GNNs) used for mesh reconstruction are spectral-based and spatial-based approaches.

Spectral-based methods define convolutional operations based on graph signal processing. One of the earliest models in this category is Spectral CNN [28], which generalizes the concept of convolution to graphs using Laplacian eigenvectors. Building on this, ChebNet [4] and GCN [17] significantly reduce the computational cost by using fast localized filters based on Chebyshev polynomials. Yuan et al. [29] also used spectral graph convolutions to form a complete neural network architecture and introduced a new mesh pooling and unpooling method by repeated edge contraction. With the help of new pooling operations and convolutional kernels, they reported far fewer parameters than those of the original mesh Variational Auto-encoder (VAE) and thus can handle denser models. Jiang et al. [30] proposed an attribute decomposition framework for 3D face meshes that separates identity and expression components in a non-linear manner using a vertex deformation-based representation. Similarly, Zhang et al. [31] introduced a variational auto-encoder with a GCN, called Mesh-Encoder, to disentangle identity and expression components. These methods allow convolutions to be performed in the spectral domain, providing localized representations of a 3D mesh. However, a major limitation of spectral-based approaches is the need for eigen-decomposition, which is computationally expensive and difficult to adapt to large meshes.

Spatial-based methods define graph convolutions based on the spatial relationships between vertices, making them more intuitive for 3D mesh reconstruction. One notable method is GraphSage [32], which samples a fixed number of neighbors for each vertices and aggregates neighboring features to compute new vertex representations. This method excels in capturing local geometry but may struggle with global feature extraction on large and complex meshes. Graph Attention Networks (GAT) [17] introduced attention mechanisms that learn relative weights between connected vertices, improving the ability to capture varying levels of importance between neighbors. This approach allows for dynamic weight adjustment, making it effective in handling non-uniform mesh topologies. MoNet [33] introduced pseudo-coordinates to define the relative spatial position between a vertex and its neighbors, allowing the network to assign different weights to neighboring vertices based on their positions. This flexibility makes MoNet particularly effective for 3D meshes with irregularly distributed geometric features. A key contribution in the spatial-based category is FeaStNet [10], which learns a dynamic weighting matrix for each convolutional

operation based on vertex features. This allows FeaStNet to better handle irregular mesh structures by dynamically adjusting the influence of each neighboring vertex during the convolution process.

Several mesh-based auto-encoders have been developed to specifically address the reconstruction of 3D shapes. COMA [5], one of the first auto-encoder-based approaches for 3D meshes, used spectral convolutions for encoding and decoding facial meshes. While effective for small-scale facial reconstructions, COMA is sensitive to significant distortions and overall variations in shape due to its use of spectral methods. SpiralNet [6] introduced a spiral convolution operator to capture local features by scanning neighboring vertices in a spiral order. Although SpiralNet improved upon COMA's local feature extraction, it faced challenges with mesh consistency and required manually defining the starting point for each spiral, which impacted the generalizability of the approach. SpiralNet++ [8] addressed this problem by introducing a simple operator that captures local geometric structure from serializing the local neighborhood of vertices. Instead of randomly generating sequences per epoch [6], SpiralNet++ generates spiral sequences only once in order to employ the prior knowledge of fixed meshes, which improves robustness. Olivier et al. [34] presented FaceTuneGAN, a neural architecture adapting image-to-image translation networks for 3D face geometry. By leveraging SpiralNet++ [8] and large-scale face scan datasets, it achieves superior identity decomposition, facial expression transfer, and face neutralization. Sun et al. [35] proposed a VAE-based framework with dual decoders for disentangling identity and expression in 3D facial models. A mutual information regularizer enhances the balance between expressive and neutral face representations. Local Structure-Aware Anisotropic Convolution (LSA-Conv) [9] built upon these approaches by dynamically learning anisotropic filters based on local geometry, similar to attention mechanisms. This allows for better handling of irregular mesh data by adapting filter weights to capture local directional dependencies. However, LSA-Conv's focus on local geometry often misses global topology, which is critical for reconstructing large-scale 3D meshes. Zhang et al. [36] proposed a continuous mesh representation, called Spherical Implicit Surface (SIS), which models 3D shapes as implicit functions of spherical coordinates. By leveraging spherical conformal parameterization for closed manifold meshes, SIS enables mesh reconstruction at arbitrary resolutions using either global or local features. Their mesh encoder is built with the convolutional operation of LSA-Conv [9]. In parallel with anisotropic and graph-attention methods, recent works have explored lightweight attention and knowledge distillation strategies for efficient 3D representation learning. For instance, Miao et al. [37] introduced LASER, which employs adapter-based attention and CLIP-guided distillation to transfer semantic priors from vision language models into compact 3D neural fields. Such approaches demonstrate how distilled attention modules can achieve high expressiveness with minimal additional parameters. This idea is conceptually related to our vertex-level weighted fusion, which aims for efficiency and adaptivity within mesh-based architectures.

Some methods have also been proposed that are not mesh-specific. Gu et al. [38] used a PointNet-based VAE and a discriminator for reconstruction, disentangling 3D face identities and expressions. Chandran et al. [39] proposed Shape Transformers, a new non-linear parametric 3D shape model based on a transformer architecture.

### 2.2. Mesh completion

3D mesh completion, which addresses the challenge of filling holes or removing artifacts in both simple and complex objects, has been extensively studied. Classical approaches broadly fall into two categories. The first focuses on volumetric representations, where data is stored as grids and often requires full re-meshing after the processing. These methods are particularly effective for filling complex holes. For example, [13] represented surfaces using a grid of Signed Distance Function (SDF) values and propagated them via a diffusion process involving alternating
<figure><img src="image_4.png" /><figcaption>Fig. 1. The architecture of 3DGeoMeshNet used for training.</figcaption></figure>

steps of blurring and compositing. Similarly, [40] employed diffusion to propagate a voxel grid of quadruplets encoding distance, direction, weight, and changeability.

The second category leverages surface-based repair, utilizing geometric information near holes to reconstruct missing regions without altering the rest of the mesh. While effective for small holes, these methods struggle with larger or irregular gaps. For instance, [41] introduced a hole-filling technique that triangulates missing regions and minimizes the resulting surface area using a weighted function, followed by refinement to align triangle density with surrounding areas. Context-aware methods extend this idea by incorporating patterns from the existing mesh. For example, [42] prioritized coherence between the completed region and the original mesh’s local neighborhoods, ensuring geometric consistency. A similar approach was proposed by Sharf et al. in [43].

With advancements in Deep Neural Networks (DNNs), data-driven methods have emerged for 3D mesh completion. These approaches learn to infer missing geometry by extracting features from partial inputs. Many employ encoder-decoder architectures to map partial shapes to latent representations of complete shapes [44]. SDFs are widely used to represent noisy inputs and recover missing regions [45,46]. Methods like those of [46,47] utilize VAEs to achieve accurate completions on datasets such as KITTI and ShapeNet. Coarse-to-fine strategies are common, as seen in [44,48], where initial rough predictions are iteratively refined. For instance, [44] augmented training with a 3D shape classification network, though this limits applicability to domains with labeled data.

Generative Adversarial Networks (GANs) have also shown promise. In [49], two GANs share a generator to repair holes: one focuses on local hole generation, while the other ensures global consistency when merging the repaired region with the original mesh. Graph-based methods further enhance geometric reasoning. For example, Kim et al. [50] introduced Anisotropic SpiralNet, which uses anisotropic spiral convolutions to handle complex features, unifying completion and denoising tasks to improve robustness against noise and incompleteness.

Recent work emphasizes domain-specific geometric priors for high-fidelity results. Hernández-Bautista and Melero [51] proposed a curvature-driven framework where multi-scale curvature descriptors (e.g., mean curvature) guide the network to preserve ridges, valleys, and other fine details. By explicitly integrating curvature consistency into training, this approach complements purely data-driven methods. Similarly, Li et al. [48] trained an encoder-decoder to jointly model global facial topology (e.g., symmetry) and local details (e.g., eyes, nose). Weak supervision from a pre-trained 3D inpainting model guides the decoder, while optional test-time optimization (e.g., non-rigid ICP) enhances adaptability to unseen inputs.

### 3. Proposed method

We propose a framework called 3DGeoMeshNet, a GCN designed for 3D mesh reconstruction. The architecture of 3DGeoMeshNet is

illustrated in Fig. 1. The architecture is based on an auto-encoder structure that focuses on capturing local and global geometric features of the mesh. This is achieved through a combination of spatial Graph Convolutions and hierarchical encoder-decoder networks.

**Mesh representation.** The input of 3DGeoMeshNet is a 3D mesh ($V_{in}$, $E_{in}$, $F_{in}$, $X_{in}$), where $V$ is a set of $N$ vertices, $E$ is the set of edges, $F$ is the set of faces, and $X_{in} \in \mathbb{R}^{N \times f}$ is a matrix of vertex features. Classically $f=3$ and $X_{in}$ denotes the vertex positions. As for most other mesh auto-encoders, 3DGeoMeshNet is trained to reconstruct meshes representing the same object type (e.g., scanned human heads), with the same number of vertices but with possibly different edges and faces. Moreover, the vertex set of each mesh is aligned with the vertex set of a template mesh $M_T$, so that the input vertex positions are normalized as $\hat{X}_{in} = (X_{in} - X_T)/\Sigma$, with $X_T$ and $\Sigma$ being the mean and variance of the position of each vertex over the training set (both $X_T$ and $\Sigma$ are in $\mathbb{R}^{N \times f}$). As in [51], to enhance the ability of the network to capture detailed geometric features, we add to $\hat{X}_{in}$ the mean principal curvatures for each vertex (so $f=4$). It is defined as:

$$H(v) = \frac{1}{2}(k_1(v) + k_2(v)) \qquad (1)$$

with $v \in V_{in}$, and $k_1$ and $k_2$ being the estimated maximum and minimum principal curvatures of the surface at a given vertex $v$ [52], respectively. As the mean principal curvature is invariant to rotation, it helps to understand the overall bending behavior of the surface at each vertex, as shown in [53]. As illustrated in Figs. 1 and 3, the auto-encoder reconstructs vertex features from $\hat{X}_i$ as $\hat{X}_{out} = D(E(\hat{X}_{in}))$, with $E: \mathbb{R}^{N \times f} \to \mathbb{R}^Z$ being the encoder and $D: \mathbb{R}^Z \to \mathbb{R}^{N \times f}$ being the decoder. The final mesh positions are $X_{out} = \Sigma \odot \hat{X}_{out} + X_T$. As in FaceCom [48], the graph auto-encoder merges a global encoder-decoder and a local encoder-decoder (Fig. 4) to capture large-scale features while maintaining fine-grained details. The encoders and decoders are GCNs based on Feature-Steered Graph Convolution (FeaStConv) [10], which demonstrated excellent feature extraction capabilities compared to the other GC models [24,54]. FeaStConv aggregates features using the graph structure of the mesh equipped with learned dynamic weights on edges [10]. In the following, a GC layer is defined as FeaStConv followed by LeakyReLU activation. The encoder $E$ and decoder $D$ are detailed below; the differences with FaceCom are discussed.

**Encoder.** The encoder $E$ maps the input vertex features to a latent vector $z = E(\hat{X}_{in}) \in \mathbb{R}^Z$ by concatenating global and local features extracted by a global encoder $E_G$ and a local encoder $E_L$, respectively, before applying a linear transformation with a Fully Connected (FC) layer:

$$E(\hat{X}_{in}) = FC(\text{concat}(E_G(\hat{X}_{in}), E_L(\hat{X}_{in}))) \qquad (2)$$

Both encoders output a feature vector of size $Z/2$ before concatenation. The global encoder $D_G$ extracts features with a sequence of GC layers interleaved with down-sampling layers. Finally, the features are averaged
across the vertices and rectified with LeakyReLU to produce a global feature vector. In contrast, the local encoder $D_L$ extracts features with a sequence of GC layers, with, contrary to [48], residual connections. Each residual connection skips a GC layer by linking its input directly to its output. This improves feature learning and mesh reconstruction, as shown in our ablation studies. The resulting vertex features are flattened and non-linearly transformed (Linear with LeakyReLU activation) into a local feature vector before concatenation with the global feature vector.

**Decoder.** It generates vertex features $\hat{\mathbf{x}}_{\text{out}} = D(\mathbf{z})$ from a latent code $\mathbf{z}$ by combining the features provided by the global decoder $D_G$ and the local decoder $D_L$:

$$D(\mathbf{z}) = \operatorname{diag}(\mathbf{w}_G) D_G(\mathbf{z}_G) + \operatorname{diag}(\mathbf{w}_L) D_L(\mathbf{z}_L) \quad (3)$$

where $[\mathbf{z}_L] = \text{FC}(\mathbf{z})$ is the low-level feature vector decoded by an FC layer, split into two equal-length vectors ($\mathbf{z}_G$ and $\mathbf{z}_L$). The weight vectors $\mathbf{w}_G, \mathbf{w}_L \in \mathbb{R}_{>0}^N$ control the contribution of each decoder at each vertex. Whereas these weights are empirically fixed scalars in FaceCom [48], we propose adapting them to each vertex using a weighted fusion mechanism. Symmetrically to the global encoder, the global decoder $D_G$ generates vertex features $\mathbf{X}_G = D_G(\mathbf{z}_G)$ with a sequence of Up-Sampling and GC layers. Similarly, symmetrically to the local encoder, the local decoder $D_L$ generates vertex features $\mathbf{X}_L = D(\mathbf{z}_L)$ by a Linear layer followed by three GC layers with residual connections.

**Weighted fusion block.** As illustrated in Fig. 2, the weights $\mathbf{w}_G$ and $\mathbf{w}_L$ are adapted to the decoded vertex features ($\mathbf{X}_G, \mathbf{X}_L$) using a small neural network:

$$[\mathbf{w}_G, \mathbf{w}_L] = \text{FusionNet}(\mathbf{X}_G, \mathbf{X}_L) \quad (4)$$

FusionNet concatenates features for each vertex and defines attention weights through two linear layers with ReLU activation for the first layer and Softmax for the second. Combined with the decoder $D$, the resulting adaptive weighted fusion allows the model to learn the optimal balance from the training set, rather than relying on a pre-fixed ratio as in FaceCom [48]. The key difference is that FaceCom [48] used fixed scalar weights, whereas our block learns them dynamically. Our ablation studies show that the attention network improves model performance.

**Down- and up-sampling.** We considered the strategies presented in [48], improved from [5] based on Quadratic-Edge Collapse for mesh down-sampling. Each down-sampling operation reduces the number of vertices by a factor of 2, and conversely, each up-sampling operation doubles the number of vertices. This pooling operation is mesh-specific, and it ensures that the pooled mesh remains a triangular mesh.

## 4. Loss function

Our network is trained using a combination of three losses to ensure accurate mesh reconstruction while maintaining smoothness and structural coherence. During the training, the reconstruction loss is defined

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Weighted FusionNet block for concatenation of global and local decoded features.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Global Encoder-Decoder. The down/up arrows respectively indicate down-sampling and up-sampling layers, always by a factor of 2.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Local Encoder-Decoder does not include down/up sampling, but each layer is followed by a residual connection where the input features are either directly added (when dimensions match) or linearly projected (when dimensions differ) before being aggregated with the transformed features through element-wise addition.</figcaption></figure>

as a combination of three losses. The Mean Squared Error loss (MSE) enforces closeness to the input mesh:

$$L_{\text{MSE}} = \frac{1}{N} \| \mathbf{X}_{\text{out}} - \mathbf{X}_{\text{in}} \|_{F}^{2} \quad (5)$$

where $\|\mathbf{X}\|_F = \sqrt{\sum_i \sum_j |X_{ij}|^2}$ is the Frobenius Norm, $\mathbf{X}_{\text{out}}$ are the predicted features and $\mathbf{X}_{\text{in}}$ the ground truth ones.

To complement the MSE loss, we also incorporate the $L_1$ loss (in contrast to our previous work [18]), which encourages sparsity and robustness to outliers:

$$L_1 = \frac{1}{N} \| \mathbf{X}_{\text{out}} - \mathbf{X}_{\text{in}} \|_1 \quad (6)$$

where $\|\mathbf{X}\|_1 = \sum_{ij} |X_{ij}|$ denotes the element-wise $L_1$ norm.

To enforce a smooth latent space and maintain a spherical distribution for the latent representations, we use a custom spherical regularization loss [48].

$$L_{\text{Reg}} = (\|\mathbf{z}\|_2 - 1)^2 \quad (7)$$

This ensures that the latent features $\mathbf{z}$ lie close to a spherical manifold, promoting better generalization. This loss is particularly effective in preventing vanishing KL-divergence and non-decreasing $L_{\text{MSE}}$ and maintaining structural integrity in the reconstructed meshes. Our experimental results further support the notion that this approach facilitates optimization-based inference procedures. A previous study [48,55] has illustrated that this approach can significantly enhance auto-encoder generative capacity and overall stability.

The overall reconstruction loss is defined as:

$$\text{Total Loss} = L_{\text{MSE}} + L_1 + \lambda_{\text{reg}} \cdot L_{\text{Reg}} \quad (8)$$

where $\lambda_{\text{reg}} = 0.0001$ is a weighting factor for the spherical regularization loss. This combination ensures a balance between accurate reconstruction, robustness to outliers, and smooth latent representations.
# 5. Experiments

In this section, we evaluate the proposed model on the COMA and DFAUST 3D shape datasets and compare it to SOTA approaches for auto-encoder-based reconstruction. In addition, ablation tests are conducted to demonstrate the effectiveness of the different components of our model. We also propose an approach for mesh completion that relies on the proposed 3D Mesh auto-encoder. Finally, we show the results of two applications, 3D interpolation/extrapolation and mesh denoising, to demonstrate the effectiveness of our method for real-world problems.

## 5.1. Datasets and training details

We evaluate our model on two datasets: COMA [5] and DFAUST [19]. For the meshes of each dataset, the mesh topology is fixed (always the same number of vertices and edges). COMA is a human facial dataset that consists of 12 classes of extreme expressions from 12 different subjects. The dataset contains 20466 3D meshes that were registered to a common reference template with $N = 5023$ vertices. We split the COMA dataset into training and test sets with a ratio of 9:1 and randomly selected 100 samples from the training set for validation, as in [5]. DFAUST contains a set of more than 40000 full human body meshes from 10 different subjects, for both male and female. A mesh registration method using both 3D geometry and texture information enabled the registration of all scans in a sequence to a common reference topology with $N = 6890$ vertices. Similar to COMA, we split the training and test samples with a ratio of 9:1 and randomly selected 100 samples from the training set for validation.

We initialized the learning rate at 0.0005 and halved it every 50 epochs. The training process utilized the Adam optimizer within the PyTorch framework, and ran for approximately 24 hours on an NVIDIA GeForce RTX 4090 GPU, with a batch size of 32 and 300 epochs. The local convolutional layers were configured with feature sizes of 8, 16, and 32, while the global convolutional layers were designed with feature sizes of 32, 64, and 128. For the quantitative results, we computed the mean, median, and $L_2$ errors in millimeters (mm), as in previous works.

## 5.2. Mesh reconstruction results

For the evaluation of 3D mesh reconstruction, we compared the performance of our method, 3DGeoMeshNet, against several SOTA approaches, including PCA [20], FLAME [22], Jiang et al. [30], Zhang et al. [31], FaceTuneGAN [34], Sun et al. [35], COMA [5], SpiralNet++ [8], Yuan et al. [29], Gu et al. [38], FaceCom [48], and LSA-Conv [9]. These methods vary significantly in their underlying architectures: PCA represents a linear model, while all other methods employ DNNs with non-linear mechanisms. Moreover, Gu et al. [38] utilize PointNet, whereas the remaining methods adopt GCNs. FaceCom [48] is designed for 3D in-painting, but we modified it so that it also reconstructs 3D meshes.

Table 1 presents the quantitative comparison of reconstruction errors for our method and related approaches on the COMA dataset. Mean and Median reconstruction errors are provided to compare the methods. The Median error is small in cases where most of the vertices are well reconstructed. The Mean error captures more significant errors. The results were taken from the associated papers. With a latent dimension of $z = 256$, 3DGeoMeshNet achieves the lowest reconstruction error, outperforming all the competing methods by a significant margin. Only LGP-Conv [56] and LSA-Conv [9] have the advantage of a smaller latent space size. Notably, all DNN-based methods consistently outperform PCA. Indeed, linear models like PCA are inherently limited to capturing global features, while DNN-based non-linear models leverage convolutional operations to capture both global and local features. These results demonstrate the efficacy of incorporating global and local encoder-decoder networks, which enhance the expressive power of 3D shape representation compared to ChebNet-based COMA and SpiralNet.

<table><caption>**Table 1**
Reconstruction Errors on COMA dataset.</caption><thead><tr><th>Method</th><th>Latent z size</th><th>Mean</th><th>Median</th></tr></thead><tbody><tr><td>PCA [20]</td><td>-</td><td>1.639 ± 1.638</td><td>1.101</td></tr><tr><td>FLAME [22]</td><td>-</td><td>1.451 ± 1.64</td><td>0.87</td></tr><tr><td>Jiang et al. [30]</td><td>-</td><td>1.413 ± 1.639</td><td>1.017</td></tr><tr><td>COMA [5]</td><td>8</td><td>0.845 ± 0.994</td><td>0.496</td></tr><tr><td>FaceTuneGAN [34]</td><td>-</td><td>0.83 ± 0.21</td><td>0.77</td></tr><tr><td>Zhang et al. [31]</td><td>-</td><td>0.665 ± 0.748</td><td>0.434</td></tr><tr><td>Sun et al. [35]</td><td>-</td><td>0.663 ± 0.215</td><td>0.643</td></tr><tr><td>Gu et al. [38]</td><td>256</td><td>0.651 ± 0.208</td><td>0.625</td></tr><tr><td>Yuan et al. [29]</td><td>128</td><td>0.583 ± 0.436</td><td>-</td></tr><tr><td>FaceCom [48]</td><td>256</td><td>0.516 ± 0.708</td><td>0.255</td></tr><tr><td>SpiralNet++ [8]</td><td>-</td><td>0.426 ± 0.538</td><td>0.238</td></tr><tr><td>ShapeTransformers [39]</td><td>-</td><td>0.413 ± 0.313</td><td>0.323</td></tr><tr><td>LGP-Conv [56]</td><td>64</td><td>0.177</td><td>-</td></tr><tr><td>LSA-Conv [9]</td><td>32</td><td>0.153 ± 0.217</td><td>0.077</td></tr><tr><td><strong>3DGeoMeshNet</strong></td><td>256</td><td><strong>0.125 ± 0.149</strong></td><td><strong>0.066</strong></td></tr></tbody></table>

<table><caption>**Table 2**
Reconstruction errors on the DFAUST dataset.</caption><thead><tr><th>Method</th><th>Latent z size</th><th>Mean</th><th>Median</th></tr></thead><tbody><tr><td>SpiralNet++ [8]</td><td>32</td><td>4.667</td><td>0.227</td></tr><tr><td>LSA-Conv [9]</td><td>16</td><td>3.492</td><td><strong>0.117</strong></td></tr><tr><td>FaceCom [48]</td><td>256</td><td>2.920</td><td>0.292</td></tr><tr><td>LGP-Conv [56]</td><td>64</td><td><strong>2.170</strong></td><td>-</td></tr><tr><td><strong>3DGeoMeshNet</strong></td><td>256</td><td>2.370</td><td>0.197</td></tr></tbody></table>

When comparing 3DGeoMeshNet to Yuan et al. [29] and Gu et al. [38], with latent dimensions of $z = 128$ and $z = 256$, respectively, our method achieves superior reconstruction accuracy, particularly when using the same latent dimension as Gu et al. In our experiments, FaceCom [48] can be considered as the baseline model, as our approach is an extension of their architecture. The quantitative comparison with FaceCom demonstrates that our modified network significantly outperformed this baseline. Specifically, the modified FaceCom achieved a mean error of 0.516, whereas our approach achieved a substantially lower mean error of 0.171. These results highlight the effectiveness of incorporating the weighted fusion module, residual connections, and principal curvature into the network design.

Fig. 6 illustrates qualitative results of the reconstruction errors for selected examples from the COMA test set with varying latent space size. Using a latent size of $z = 256$, our method consistently produces smaller reconstruction errors across various test cases, underscoring its robustness and accuracy.

We also used the full human body dataset, DFAUST [19], for mesh reconstruction. Table 2 and Fig. 5 show the results for the full human body using 3DGeoMeshNet. In contrast with the COMA, the DFAUST contains the full human body and a higher number of vertices of 6890. Also, since it contains full-body complex meshes, the training takes longer time. In comparison with FaceCom [48], 3DGeoMeshNet achieves competitive performance on the DFAUST dataset, achieving a much lower mean error of 2.370 in comparison with other approaches. As for the COMA dataset, our approach outperforms all the competing methods by a significant margin, except for LGP-Conv [56] that provides slightly better results. An improvement of our approach could consist in enhancing the convolution operator in the same way as LGP-Conv [56] at the price of using larger convolution hops. Indeed, when they use 1-hop for convolution as in our approach, their reconstruction error increases to a mean error of 2.66, much higher than our results. Their results become better than ours when using 3-hop convolutions with a mean error of 2.24. The qualitative results are shown in Fig. 5. Our method was able to reconstruct most of the parts correctly, except for a few very complex parts, such as the areas near the armpits in the first example. Fig. 7 shows the $L_1$ testing errors on the COMA and DFAUST datasets.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Reconstruction results on different body poses from the DFAUST dataset.</figcaption></figure>

<table><caption>**Table 3**

Ablation studies on the COMA dataset, we increased the latent size from 16 to 256.</caption><thead><tr><td>Latent z size</td><td>Mean Error</td><td>Median Error</td><td>L<sub>2</sub></td><td>Parameters</td></tr></thead><tbody><tr><td>16</td><td>0.258 ± 0.238</td><td>0.173</td><td>0.202</td><td>5,333,885</td></tr><tr><td>32</td><td>0.198 ± 0.203</td><td>0.121</td><td>0.164</td><td>10,483,117</td></tr><tr><td>64</td><td>0.145 ± 0.167</td><td>0.075</td><td>0.127</td><td>20,784,653</td></tr><tr><td>128</td><td>0.130 ± 0.154</td><td><b>0.064</b></td><td>0.116</td><td>41,400,013</td></tr><tr><td><b>256</b></td><td><b>0.125 ± 0.149</b></td><td>0.066</td><td><b>0.113</b></td><td>82,679,885</td></tr></tbody></table>

### 5.3. Ablation study on mesh reconstruction

**Latent space size:** Table 3 provides an ablation analysis of 3DGeoMeshNet under different configurations. Initially, we evaluated the model using a latent size of $z = 16$ and progressively increased it to $z = 256$. As the latent dimension increased, the number of model parameters grew accordingly, leading to a reduction in reconstruction error. The parameters increased from 5333885 at $z = 16$ to 82679885 at $z = 256$. This trend illustrates the trade-off between model complexity and performance. Specifically, the mean reconstruction error has been halved when the latent size increased from $z = 16$ to $z = 256$. Fig. 6 shows the qualitative results of our ablation studies on different latent sizes. It can be seen that as the latent size increases, the error map decreases. This is especially evident in areas with strong variations, such as the mouth and the chin. For the rest of our experiments, we have kept a latent size $z = 256$.

**Removing the regularization loss:** To assess the influence of the spherical regularization loss $L_{Reg}$, we conducted an ablation study by removing it and setting $\lambda_{reg} = 0$. The removal of this term resulted in a

degradation of the reconstruction accuracy, with the mean error increasing from 0.125±0.149 mm to 0.387±0.554 mm, and the median error rising from 0.066 mm to 0.155 mm. These results confirm that the spherical regularization loss plays a critical role in maintaining geometric consistency and reducing reconstruction variance across samples.

**Training with one path:** To evaluate the significance of our dual-path architecture, we conducted an ablation study by removing either the global or local path and, consequently, the weighted fusion module from our model. As shown in Table 4, utilizing only the local path resulted in a substantial performance drop, with the mean error increasing from 0.125 to 0.688. Notably, when employing only the global path, performance deteriorated even further, with a mean error of 1.520. It is worth mentioning that while employing only the local path, the results are better than when employing the global path only. The reason behind this is that the local path includes the residual connections, which let the information from the earlier layers to pass to the later layers, while the global path does not include the residual connections. These results demonstrate the importance of integrating both global and local paths with a weighted fusion mechanism. The global path appears to capture high-level structural information that complements the local geometric features processed by the local path, with the weighted fusion module effectively combining these complementary representations to achieve optimal reconstruction quality. As most SOTA auto-encoders use a structure similar to our global path (as it was initially proposed in [5]), this highlights the interest of our proposal that combines feature extraction with (global path) and without (local path) mesh pooling.

**3D Convolution Layers:** To assess the impact of the convolutional operator on our 3D mesh reconstruction performance, we conducted ablation studies by replacing the FeaStConv layers in 3DGeoMeshNet with alternative graph convolution layers, namely Graph Attention Networks
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Ablation results with different latent size on the COMA dataset. We increased the latent size from 16 to 256.</figcaption></figure>

(GAT [17] and Graph Convolution (GC) [16]. Table 5 summarizes the quantitative results obtained on the COMA dataset.

Using FeaStConv consistently achieved the best performance, with a mean error of 0.125. In comparison, the version employing GAT layers resulted in slightly degraded performance (mean error: 0.129), indicating a reduced capability in capturing the finer geometric details. The network with GC layers demonstrated the highest errors (mean error: 0.232), suggesting that this conventional graph convolution approach is less effective for the complex geometric modeling required by the reconstruction task.

These results show that the adaptive nature of FeaStConv in learning spatial vertex relationships is crucial for high-fidelity 3D mesh reconstruction. In contrast, while GAT provides a moderate improvement over traditional GC layers by leveraging attention mechanisms, it still falls short of the performance achieved by FeaStConv. As the latter has been specifically designed to process meshes, it appears natural that it obtains better results than generic convolution operators.

**The 3DGeoMeshNet components:** Finally, we investigate the importance of the components of the proposed 3DGeoMeshNet model. To do so, we study the impact of suppressing the curvature $κ$ at the input and the residual connections in the local path. Table 6 summarizes these results on the COMA and DFAUST datasets. It can be seen that the

residual connections have a strong influence on the reconstruction error, and removing them results in a significant drop in reconstruction performance. As stacking several graph convolution layers is prone to the over-smoothing effect, adding these residual connections between layers mitigates this effect, facilitates gradient flow, and maintains stable training. Removing the curvature at the input to the network results in some loss of performance. This confirms the importance of curvature for learning an accurate 3D mesh representation, as the curvature provides a local measure of surface bending.

### 5.4. Mesh completion

To the best of our knowledge, there have been few works [48,50,57] specifically addressing mesh completion with mesh auto-encoders. In addition, the experiments of point cloud completion [58] methods for facial datasets have yielded unsatisfactory results. In this section, we present how to use our auto-encoder 3DGeoMeshNet to perform mesh completion that effectively addresses the challenges of completing partial facial meshes.

As in [50,57], our mesh completion algorithm follows a template-based approach where a complete template mesh guides the reconstruction of missing regions in the partial input mesh. The whole process is
<figure><img src="image_10.png" /><figcaption>Fig. 7. Ablation results on the COMA and DFAUST datasets. (a) L1 errors on the datasets. (b) We increased the latent size from 16 to 256 on COMA. (c) Training network with one head only on COMA. (d) results with different convolutional layers on COMA.</figcaption></figure>

<table><caption>**Table 4**

Ablation studies of 3DGeoMeshNet on the COMA dataset while using only one head.</caption><thead><tr><th>Head</th><th>Mean Error</th><th>Median Error</th><th>L<sub>2</sub></th></tr></thead><tbody><tr><td>Local only</td><td>0.688 ± 0.527</td><td>0.563</td><td>0.797</td></tr><tr><td>Global only</td><td>1.520 ± 0.705</td><td>1.425</td><td>0.973</td></tr><tr><td>Both</td><td><strong>0.125 ± 0.149</strong></td><td><strong>0.066</strong></td><td><strong>0.113</strong></td></tr></tbody></table>

<table><caption>**Table 5**

Ablation studies of 3DGeoMeshNet on the COMA dataset with different convolution layers.</caption><thead><tr><th>Conv. Layer</th><th>Mean Error</th><th>Median Error</th><th>L<sub>2</sub></th></tr></thead><tbody><tr><td>GC</td><td>0.232 ± 0.395</td><td>0.188</td><td>0.125</td></tr><tr><td>GAT</td><td>0.162 ± 0.138</td><td>0.112</td><td>0.123</td></tr><tr><td>FeaStConv</td><td><strong>0.125 ± 0.149</strong></td><td><strong>0.066</strong></td><td><strong>0.113</strong></td></tr></tbody></table>

illustrated in Fig. 8 and Algorithm 1. A template mesh $M_{template}$ and a partial face mesh $M_{partial}$ are provided. First, following [48,50,57], we compute a rigid alignment through Iterative Closest Point (ICP) registration [59] to align the template with the partial mesh. Then, both meshes are decomposed into connected components using topological analysis.

<figure><img src="image_11.png" /><figcaption>**Table 6**

Ablation study of components of 3DGeoMeshNet on the COMA and DFAUST datasets.</figcaption></figure>

<table><thead><tr><th rowspan="2">Components</th><th colspan="2">COMA</th><th colspan="2">DFAUST</th></tr><tr><th>Mean Error</th><th>Median Error</th><th>Mean Error</th><th>Median Error</th></tr></thead><tbody><tr><td>FusionNet</td><td>0.177 ± 0.217</td><td>0.114</td><td>2.664 ± 1.325</td><td>0.220</td></tr><tr><td>FusionNet + Res</td><td>0.131 ± 0.191</td><td>0.071</td><td>2.484 ± 0.803</td><td>0.212</td></tr><tr><td>FusionNet + Res + κ</td><td><strong>0.125 ± 0.149</strong></td><td><strong>0.066</strong></td><td><strong>2.370 ± 0.766</strong></td><td><strong>0.197</strong></td></tr></tbody></table>

This decomposition facilitates localized processing of mesh structures, allowing for more precise handling of distinct anatomical regions. In particular, in the COMA dataset, the original faces are composed of three distinct components: the face and two eyes. The proposed method effectively handles such complex cases where multiple components may be missing or partially present. A distance-based assignment is performed to match components of the partial mesh to their corresponding components in the template. For each template component $M_{template}^j$, we identify the closest partial component $M_{partial}^j$ based on their centroid distances:

$$d_{ij} = \| c_{template}^j - c_{partial}^j \|_2 \qquad (9)$$
<figure><img src="image_12.png" /><figcaption>Fig. 8. The pipeline of our proposed mesh completion algorithm. See text for details.</figcaption></figure>

**Algorithm 1 Template-based mesh completion.**

Inputs: Template mesh $M_{template}$, Partial mesh $M_{partial}$

Align $M_{template}$ to $M_{partial}$ using ICP

Decompose both meshes into connected components: $\{M_{template}^i\}$ and $\{M_{partial}^j\}$

Construct $M_{completed}$ from the matching between the connected components $\{M_{template}^i\}$ and $\{M_{partial}^j\}$

Apply inverse ICP to return to original coordinates

Run 3DGeoMeshNet on $M_{completed}$ to get $M_{reconstructed}$

Match $M_{reconstructed}$ with $M_{partial}$ and restore the original vertex positions into $M_{prepro}$

Smooth $M_{prepro}$ with Taubin filtering to obtain the final completion $M_{final}$

where $c_i^l$ is the center point of the i-th connected component of the mesh. Components are matched when their distance falls below a threshold $\tau$. This distance threshold $\tau$ controls the sensitivity of component matching, with a default value of 0.05 in normalized coordinate space. If no matching component exists for a template component (indicating a missing region in the input mesh), the template component is used directly to fill the gap in the partial mesh. For instance, as shown in Fig. 8, an eye does not appear in the partial mesh $M_{partial}$ (as half of the face is missing in the input mesh), the connected component representing the eye from $M_{template}$ has no corresponding component in $M_{partial}$ and is copied directly into $M_{partial}$.

For the other cases (a matching component exists for a template component), for each pair of matched components ($M_{template}^i, M_{partial}^j$), we establish a vertex-level correspondence using bipartite matching through the Hungarian algorithm [60]. Given the two point sets $V_{template}^i$ and $V_{input}^j$ of the matched components, we compute a cost matrix $C$ where:

$$C_{kl} = \|v_{partial}^{i,k} - v_{template}^{j,l}\|_2 \qquad (10)$$

with $v_{input}^{i,k}$ the k-th vertex of the i-th connected component of the mesh. The Hungarian algorithm solves the assignment problem:

$$\min_{\pi} \sum_{k=1}^{|V_{template}^i|} C_{k,\pi(k)} \qquad (11)$$

where $\pi$ represents the one-to-one assignment function that maps vertices from $M_{partial}^i$ to the template vertices from $M_{template}^j$.

This vertex-level correspondence between the partial mesh and the template directly enables identifying the unmatched vertices in the template. The latter correspond to missing regions in the partial mesh with respect to the template. To perform the completion, we simply copy the unmatched vertices of the template into the partial mesh, and obtain a filled mesh $M_{completed}$. After this step, the partial mesh has exactly the same number of vertices as the template, but the faces can be different. To ensure that the topology of the completed mesh is the same, we perform a vertex-to-vertex matching again between $M_{completed}$ and $M_{template}^j$, and map the template's faces to the completed mesh. The completed mesh $M_{filled}$ is then translated back to its original coordinates by applying the inverse of the initial ICP transformation. The completed mesh is provided as input to our proposed 3DGeoMeshNet auto-encoder, and its output provides a reconstructed mesh $M_{reconstructed}$. As can be seen in Fig. 8, the reconstructed mesh has gained in coherence (see at the mesh nose and mouth). This reconstructed mesh is of good quality, but it has a drawback: the original vertices from the partial mesh $M_{partial}$ do not have exactly the same coordinates. However, a completion algorithm should only complete the missing parts and not alter the existing parts of $M_{partial}$. To cope with this problem, we again apply a Hungarian matching between $M_{partial}$ and $M_{reconstructed}$, and restore the original vertices' coordinates, giving a new mesh $M_{prepro}$. As this can introduce consistency issues around the boundary of the completed part, Taubin filtering is applied [61] to reduce these artifacts and obtain the final result $M_{final}$. As shown in the following experiments, these final steps result in a more coherent final reconstructed mesh.

### 5.4.1. Mesh completion results

In this section, we present mesh completion results on the COMA dataset using 3DGeoMeshNet. We show the results on different facial expressions with missing parts ranging from small to large. Fig. 9 shows completion results for a natural facial expression. For better evaluation, we removed different parts from the facial mesh and tried to complete the missing regions using the proposed model, such as plain, without eyes, without mouth, without ear, and half meshes. The first row of Fig. 9 shows the partial meshes $M_{partial}$, and the second row represents the completed meshes $M_{completed}$ using our mesh completion process. After simple mesh completion, these meshes are fed to
<figure><img src="image_13.png" /><figcaption>Fig. 9. Mesh Completion results on natural expression (the rows show $M_{partial}^j$, $M_{completed}$, and $M_{reconstructed}$).</figcaption></figure>

the trained auto-encoder model for reconstruction. The third row of Fig. 9 shows the reconstruction results $M_{reconstructed}$. Our model trained on normal meshes was able to generate promising results on completed meshes. In most cases, e.g., plain, no eyes, no mouth, and no ear, the model was able to produce accurate results, but when half of the mesh is missing, the results are less accurate, especially at the completion boundary. To correct these artifacts, we further performed two processing steps on the reconstructed meshes. As explained in Section 5.4, the pre-processing step restores the coordinates of $M_{partial}$ in $M_{reconstructed}$, and the post-processing step applies Taubin filtering. Fig. 10 shows the results of both these steps along with the error maps. In [48], an alternative Laplacian smoothing was used to filter the final result, but this tends to cause volume shrinkage, and we prefer Taubin smoothing as it effectively removes high-frequency noise while preserving medium-frequency features essential for facial expressions.

Rows 3 and 5 of Fig. 10 show the effects of post-processing, confirming the improvements made over pre-processing results. Note that the pre-processing and post-processing steps primarily serve as an engineering approach to enhance the conformity between input and completed meshes.

While our method achieves accurate and visually coherent completions for most facial configurations, certain challenging cases still produce boundary artifacts. These boundary completion defects, highlighted in Fig. 10 (bottom rows, left-most), typically arise when the missing region exceeds 40% of the facial surface or when crucial geometric cues (such as the eye or mouth contours) are absent from the partial mesh. In such cases, the network struggles to infer plausible curvature continuity across large missing spans, leading to mild discontinuities or over-smoothed transitions. To mitigate these issues, we plan to explore topology-aware feature propagation and adaptive local patch refinement strategies that explicitly model boundary context and neighborhood consistency. Despite these limitations, the proposed fusion-based reconstruction pipeline remains stable and produces coherent results in most practical scenarios.

Table 7 summarizes the quantitative evaluation of the mesh completion results across the different removed facial configurations. The Chamfer Distance (CD) values indicate that the proposed model achieves

consistent reconstruction quality across all partial conditions, with notably lower errors for smaller missing regions (plain, no eyes, no mouth, no ear). In contrast, large missing areas such as half-face removals yield higher CD values due to the increased geometric ambiguity.

Table 8 presents a quantitative comparison of mesh completion performance with several SOTA network architectures (COMA [5], SpiralNet++ [8], and Anisotropic SpiralNet [50]). We followed the test protocol proposed in [50] and removed the half-left of the mesh vertices for the 12 distinct actors from the COMA dataset (as in the last removal case of Fig. 9). The reported metrics represent per-vertex reconstruction errors averaged across all vertices for each actor. Among the baseline methods, CoMA [5] exhibits the highest mean error of 1.071 mm, while SpiralNet++ [8] achieves substantial improvements with a mean error of 0.603 mm. The SOTA Anisotropic SpiralNet method [50] demonstrates superior performance across nearly all actors, obtaining the lowest mean error of 0.541 mm and achieving the best results on 10 out of 12 actors. The proposed 3DGeoMeshNet yields a mean error of 0.975 mm, which is notably higher than the competing methods. This performance gap is expected, as our network architecture is primarily designed and optimized for mesh reconstruction rather than mesh completion. Consequently, our model was not specifically trained for the completion task, which requires different geometric priors and feature representations to effectively predict missing regions. In particular, both SpiralNet++ and Anisotropic SpiralNet have considered an overfitting process to finetune the decoder for each mesh to be completed. The results nonetheless demonstrate that our geometric processing framework can be adapted to completion scenarios, though similar per-mesh overfitting training would be necessary to achieve competitive performance with specialized completion methods.

We do not provide completion results on the DFAUST dataset. Indeed, the vertex-vertex matching used in our completion approach cannot be directly used on the full 3D mesh bodies of this dataset. The Hungarian matching that we have considered will not be sufficiently efficient for the body deformations encountered in this dataset. However, in case a more robust matching is considered, such as MoNet [10], our completion strategy could be used on the DFAUST dataset to achieve similar results to those in [57].
<figure><img src="image_14.png" /><figcaption>Fig. 10. Mesh Completion results using pre-processing and post-processing (the first three rows show $M_{reconstructed}$, $M_{prepro}$, and $M_{final}$).</figcaption></figure>

<table><caption>**Table 7**
Chamfer Distance (CD) in (mm) between the ground truth meshes and the completion result across different removed facial regions.</caption><thead><tr><th>Variant</th><th>Plain</th><th>No Eyes</th><th>No Mouth</th><th>No Ear</th><th>Half Right</th><th>Half Left</th></tr></thead><tbody><tr><td>Partial</td><td>2.168</td><td>2.936</td><td>2.097</td><td>2.391</td><td>28.909</td><td>28.251</td></tr><tr><td>Completed</td><td>2.147</td><td>2.576</td><td>2.194</td><td>2.214</td><td>4.446</td><td>3.270</td></tr><tr><td>Reconstruction</td><td>2.058</td><td>2.077</td><td>2.048</td><td>2.060</td><td>3.066</td><td>2.591</td></tr><tr><td>Preprocess</td><td>2.003</td><td>2.012</td><td>1.993</td><td>1.989</td><td>3.414</td><td>3.427</td></tr><tr><td>Postprocess</td><td>1.971</td><td>1.999</td><td>1.915</td><td>1.901</td><td>3.053</td><td>2.548</td></tr></tbody></table>

### 5.5. Other applications

**Interpolation & Extrapolation:** To evaluate the interpolation capabilities of our model, we selected two significantly different samples, $x_1$ and $x_2$, from the test set. These samples are encoded into their respective latent representations, $z_1$ and $z_2$, similarly to [6]. Intermediate latent encodings are then generated by interpolating along the line segment

connecting $z_1$ and $z_2$, defined as:

$$z = a \cdot z_1 + (1-a) \cdot z_2 \quad (12)$$

where $a \in [0, 1]$ with a variation of 0.2. Fig. 11(a) shows the interpolation results.

Similarly to interpolation, we evaluated the extrapolation capabilities of our model. We selected two significantly different samples and encoded them into their respective latent representations, $z_1$ and $z_2$. Extrapolated latent encodings are then generated by extending beyond the line segment connecting $z_1$ and $z_2$, defined as:

$$z = z_1 + a \cdot (z_2 - z_1) \quad (13)$$

where $a$ can be any real number, including values outside the interval $[0, 1]$. This allows for the exploration of the latent space beyond the original samples. Fig. 11(b) shows the extrapolation results.

<table><caption>**Table 8**
Per-actor reconstruction error comparison of different networks on the COMA mesh completion task. The best results are highlighted in bold.</caption><thead><tr><th>Network</th><th>0137</th><th>3272</th><th>0024</th><th>0138</th><th>3274</th><th>3275</th><th>0128</th><th>3276</th><th>3277</th><th>3278</th><th>3279</th><th>0223</th></tr></thead><tbody><tr><td>CoMA [5]</td><td>1.073</td><td>1.179</td><td>1.057</td><td>1.107</td><td>1.161</td><td>0.890</td><td>1.306</td><td>1.059</td><td>0.999</td><td>0.926</td><td>1.040</td><td>1.071</td></tr><tr><td>SpiralNet++ [8]</td><td>0.489</td><td><strong>0.568</strong></td><td>0.592</td><td>0.675</td><td>0.699</td><td>0.460</td><td>0.760</td><td>0.594</td><td>0.558</td><td>0.516</td><td>0.611</td><td>0.603</td></tr><tr><td>Anisotropic [50]</td><td><strong>0.447</strong></td><td>0.574</td><td><strong>0.495</strong></td><td><strong>0.593</strong></td><td><strong>0.625</strong></td><td><strong>0.389</strong></td><td><strong>0.759</strong></td><td><strong>0.533</strong></td><td><strong>0.474</strong></td><td><strong>0.437</strong></td><td><strong>0.530</strong></td><td><strong>0.541</strong></td></tr><tr><td>3DGeoMeshNet</td><td>0.528</td><td>0.732</td><td>0.911</td><td>0.818</td><td>0.805</td><td>1.061</td><td>0.905</td><td>1.555</td><td>1.048</td><td>1.148</td><td>0.908</td><td>0.975</td></tr></tbody></table>
<figure><img src="image_15.png" /><figcaption>**Fig. 11.** A. Interpolation results between expressions on the COMA dataset. B. Extrapolation results on the COMA dataset (Left: neutral expression).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Mesh denoising results for various expressions from the COMA dataset. The PSNR values between the ground truth, noisy, and denoised meshes are displayed below each corresponding sample. The last row illustrates the error maps comparing the ground truth and denoised meshes.</figcaption></figure>

**Mesh Denoising:** For the last application of our model, we introduced synthetic Gaussian noise to the input mesh data by sampling from a Gaussian distribution with a mean of 0.0 and a standard deviation of 0.001 to assess its ability to remove noise. The denoising results, illustrated in Fig. 12, shows that our method successfully removed Gaussian noise while preserving the original geometric details. Notably, the model was trained exclusively on noise-free meshes, yet it exhibited robust

denoising capabilities across all tested samples. Obviously, better results could be obtained for this specific denoising task by fine-tuning the auto-encoder on a dataset of noisy meshes.

### 5.6. Limitations and future work

Although 3DGeoMeshNet can achieve good performance in mesh reconstruction for both facial and full human body meshes, as well as facial mesh completion, there are some limitations. Firstly, the proposed method needs a uniform sampling standard across the 3D meshes (with the same topology). This is similar to image-based CNNs; the computations of convolution and pooling need to be performed consistently on the 3D shapes under a uniformly defined structure. In future works, we plan to explore the generalization of our framework to meshes with variable vertex counts and dynamic topologies, broadening the applicability of our method to more diverse and complex real-world scenarios. Secondly, the current work is limited to non-textured, monochromatic meshes. Extending the framework to handle textured 3D meshes represents an important future direction, which would enable more comprehensive and visually realistic reconstructions. Additionally, while our mesh completion approach shows promising results on facial meshes, its performance on full human body meshes remains a challenge. Finally, building upon the promising results in interpolation and extrapolation applications, future research will explore advanced applications, including identity interpolation for biometric systems, facial animation, facial morphing, aging simulation, and emotion modeling.

## 6. Conclusion

In this study, we presented 3DGeoMeshNet, a multi-scale graph auto-encoder framework for 3D mesh representation, reconstruction, and completion. Our architecture combines complementary local and global pathways to effectively capture both fine-grained details and large-scale geometric structures. The integration of anisotropic graph convolutions and weighted fusion mechanisms enables our model to process complex 3D geometries directly on the polygonal mesh structure, without relying on intermediate representations such as voxel grids or point clouds. Through extensive experiments, we achieved SOTA results for 3D mesh reconstruction on both the COMA (human facial) and DFAUST (full human body) datasets, significantly improving reconstruction accuracy over existing methods. In addition to mesh reconstruction, our framework successfully addresses 3D mesh completion, recovering missing regions through a dedicated optimization process. The mesh completion results are further refined using a carefully designed sequence of pre- and post-processing operations, enhancing the quality of the completed meshes. Moreover, we showed the generalization capabilities of 3DGeoMeshNet through additional tasks, including mesh denoising, interpolation, and extrapolation.

### CRediT authorship contribution statement

**Saqib Nazir:** Writing – review & editing, Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Conceptualization. **Olivier Lézoray:** Supervision. **Sébastien Bougleux:** Supervision.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This project has received funding from the Normandy region under the COSURIA project. Part of this work was performed using the computing resources of CRIANN (Normandy, France).
# Data availability

Data will be made available on request.

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## Author biography

<figure><img src="image_17.png" /></figure>

Saqib Nazir received the B.Sc. and M.Sc. degrees in Computer Science from COMSATS University Islamabad, Pakistan, and the Ph.D. degree from the University Politehnica of Bucharest, Romania, where he was a member of the CEOSpaceTech Laboratory. During P.h.D. (2021-2024), his role as an Early Stage Researcher with the MENELOAS-NT Project was to acquire depth from focus images using state-of-the-art deep learning methods. He is currently a Postdoctoral Research Associate with the IMAGE Group, GREYC Laboratory, CNRS, University of Caen Normandy, France. His research focuses on developing generative models for geometric and colour completion of partial 3D colour meshes of human scans, with applications in AR/VR/XR and 3D avatar editing. His broader research interests include 3D computer vision, generative modeling, and deep learning.

<figure><img src="image_18.png" /></figure>

Olivier Lézoray (Senior Member, IEEE) received the M.Sc., Ph.D., and Habilitation degrees in computer science from the University of Caen Normandy, France, in 1996, 2000, and 2007, respectively. From 1999 to 2000, he was an Assistant Professor with the Computer Science Department, University of Caen. In 2000, he joined the Communication Networks and Services Department, West Normandy Institute of Technology, as an Associate Professor of computer science, and was the Director of Graduate Studies (2000-2009). In 2010, he became a Full Professor at the Multimedia and Internet (MMI) Department, West Normandy Institute of Technology. He was the Head of the MMI Department, from 2015 to 2021. Since 2022, he has been the Deputy Director of the GREYC CNRS Research Laboratory. Applications of his works include computational photography, computer-aided diagnosis, and computer vision. He is the author of two books, 11 book chapters, 50 articles in international journals, and more than 120 in international conferences. His research interests include graph-based signal processing, adaptive and multidimensional mathematical morphology, and machine learning. He is a member of SPIE, IAPR, and EURASIP. He was an Associate Editor of the IEEE Transactions on Signal Processing and the IET Image Processing. He guest-edited for the following journals: Signal Processing: Image Communication, Journal of Electronic Imaging, Signal, Image and Video Processing, Computerized Medical Imaging and Graphics, Signal Processing, and the EURASIP Journal on Advances in Signal Processing.

<figure><img src="image_19.png" /></figure>

Sébastien Bougleux received the M.Sc. and Ph.D. degrees in computer science from the University of Caen Normandie, France, in 2002 and 2007, respectively. From 2005 to 2007, he was an Assistant Professor with the University of Caen Normandie, ENSICAEN, and ENSTC Caen, and a Postdoctoral Researcher with CEREMADE, University of Paris Dauphine, France, in 2008. In 2008, he became an Associate Professor of computer science at the University of Caen Normandie and a member of the GREYC CNRS Laboratory. He was the Director of Graduate Studies (2009-2013 and 2021-2025) and the Training Director of the D2PC bachelor's degree (2013-2017) at the Multimedia and Internet Department, West Normandy Institute of Technology. His research interests include statistical and structural pattern recognition, computer vision, and image processing, with a focus on graph-based models.