Einblick - Computer Graphics - # Mesh Generation

Generating Manifold Surface Meshes with Continuous Connectivity Representations

Q: How could this continuous mesh representation be extended to handle open surfaces or non-manifold connectivity?

To extend the continuous mesh representation to handle open surfaces or non-manifold connectivity, several modifications could be implemented. First, the representation could incorporate a mechanism to explicitly identify boundary edges. This could be achieved by introducing a binary flag for each edge, indicating whether it is a boundary edge or part of a closed surface. This flag would allow the model to differentiate between edges that contribute to manifold connectivity and those that define the limits of an open surface. Additionally, the representation could be adapted to allow for non-manifold structures by relaxing the constraints on edge-manifoldness. This could involve redefining the twin and next relationships to accommodate edges that may connect more than two faces or vertices. By allowing for more complex relationships among halfedges, the representation could effectively model non-manifold geometries, such as those found in CAD models or complex organic shapes. Furthermore, the training process could be adjusted to include examples of open and non-manifold meshes, enabling the model to learn the unique characteristics of these structures. This would involve augmenting the training dataset with a diverse range of mesh types, ensuring that the model can generalize to various topological configurations.

Q: What are the theoretical limits on the expressiveness of this representation, and how does the dimensionality of the vertex embeddings affect its ability to represent diverse mesh structures?

The theoretical limits on the expressiveness of the continuous mesh representation are closely tied to the dimensionality of the vertex embeddings. In general, a higher-dimensional embedding space allows for a more nuanced representation of complex relationships among vertices, edges, and faces. However, there is a trade-off between dimensionality and computational efficiency. As the dimensionality increases, the complexity of the optimization process also rises, potentially leading to slower convergence and increased risk of overfitting. The expressiveness of the representation is also influenced by the capacity of the embedding to capture the manifold structure of the mesh. If the dimensionality is too low, the embedding may not be able to encode all necessary topological features, resulting in a loss of fidelity in the generated mesh. Conversely, if the dimensionality is sufficiently high, the representation can effectively capture a wide variety of mesh structures, including those with intricate connectivity and diverse polygonal configurations. In practice, the authors found that low-dimensional embeddings (e.g., (k < 10)) were sufficient to represent the diverse mesh structures encountered in their experiments. This suggests that while there are theoretical limits to expressiveness based on dimensionality, practical applications can often achieve satisfactory results with relatively low-dimensional embeddings, provided that the training data is rich and varied.

Q: Could this continuous mesh representation be combined with unsupervised learning techniques to fit mesh distributions without relying on large labeled datasets?

Yes, the continuous mesh representation could be effectively combined with unsupervised learning techniques to fit mesh distributions without the need for large labeled datasets. One potential approach is to utilize generative models, such as Variational Autoencoders (VAEs) or Generative Adversarial Networks (GANs), which can learn to capture the underlying distribution of mesh structures from unlabeled data. By training a generative model on a diverse collection of meshes, the continuous representation can learn to produce new meshes that adhere to the learned distribution. The model could leverage the continuous embeddings to represent the latent space of mesh connectivity and geometry, allowing for the generation of novel meshes that maintain manifold properties. Additionally, techniques such as clustering or dimensionality reduction could be employed to identify patterns and structures within the mesh dataset, facilitating the learning process. For instance, clustering algorithms could group similar mesh structures, enabling the model to learn representative embeddings for each cluster without explicit labels. Moreover, self-supervised learning strategies could be implemented, where the model generates meshes based on partial inputs or reconstructs missing parts of a mesh. This would allow the model to learn meaningful representations of mesh structures in an unsupervised manner, further enhancing its ability to generate high-quality meshes from diverse input conditions. In summary, the continuous mesh representation holds significant potential for integration with unsupervised learning techniques, enabling the generation of diverse mesh structures without the reliance on extensive labeled datasets.

Kernkonzepte

A continuous representation for manifold polygonal meshes that can be optimized and learned to generate diverse mesh outputs.

Zusammenfassung

The paper presents SpaceMesh, a continuous representation for manifold polygonal meshes that can be used for learning-based mesh generation. The key innovation is a parameterization of mesh connectivity using continuous vertex embeddings, which guarantees the output meshes will be manifold by construction.

The representation consists of two main components:

Adjacency embeddings: Each vertex is associated with a continuous embedding that defines its adjacency to other vertices. A spacetime distance metric is used to define edges between sufficiently close vertices.
Permutation embeddings: Each vertex also has a set of continuous embeddings that define the cyclic ordering of its incident edges, allowing the representation of general polygonal faces.

The authors demonstrate that this continuous representation can be effectively optimized to fit individual meshes, as well as learned to generate diverse mesh outputs conditioned on input geometry. Compared to alternatives, the SpaceMesh representation shows faster convergence during optimization and the ability to generate high-quality meshes with complex connectivity.

The authors further showcase applications of the learned mesh generation model, including conditional mesh repair, where the model can be used to regenerate problematic regions of an input mesh while preserving the overall geometry.

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arxiv.org

Statistiken

"Meshes are ubiquitous in visual computing and simulation, yet most existing machine learning techniques represent meshes only indirectly, e.g. as the level set of a scalar field or deformation of a template, or as a disordered triangle soup lacking local structure."
"Our key innovation is to define a continuous latent connectivity space at each mesh vertex, which implies the discrete mesh."
"We first explore the basic properties of this representation, then use it to fit distributions of meshes from large datasets. The resulting models generate diverse meshes with tessellation structure learned from the dataset population, with concise details and high-quality mesh elements."

Zitate

"Meshes are ubiquitous in visual computing and simulation, yet most existing machine learning techniques represent meshes only indirectly, e.g. as the level set of a scalar field or deformation of a template, or as a disordered triangle soup lacking local structure."
"Our key innovation is to define a continuous latent connectivity space at each mesh vertex, which implies the discrete mesh."

Wichtige Erkenntnisse aus

SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes

by Tianchang Sh... um arxiv.org 10-01-2024

https://arxiv.org/pdf/2409.20562.pdf

SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes

Tiefere Fragen

How could this continuous mesh representation be extended to handle open surfaces or non-manifold connectivity?

To extend the continuous mesh representation to handle open surfaces or non-manifold connectivity, several modifications could be implemented. First, the representation could incorporate a mechanism to explicitly identify boundary edges. This could be achieved by introducing a binary flag for each edge, indicating whether it is a boundary edge or part of a closed surface. This flag would allow the model to differentiate between edges that contribute to manifold connectivity and those that define the limits of an open surface.
Additionally, the representation could be adapted to allow for non-manifold structures by relaxing the constraints on edge-manifoldness. This could involve redefining the twin and next relationships to accommodate edges that may connect more than two faces or vertices. By allowing for more complex relationships among halfedges, the representation could effectively model non-manifold geometries, such as those found in CAD models or complex organic shapes.
Furthermore, the training process could be adjusted to include examples of open and non-manifold meshes, enabling the model to learn the unique characteristics of these structures. This would involve augmenting the training dataset with a diverse range of mesh types, ensuring that the model can generalize to various topological configurations.

What are the theoretical limits on the expressiveness of this representation, and how does the dimensionality of the vertex embeddings affect its ability to represent diverse mesh structures?

The theoretical limits on the expressiveness of the continuous mesh representation are closely tied to the dimensionality of the vertex embeddings. In general, a higher-dimensional embedding space allows for a more nuanced representation of complex relationships among vertices, edges, and faces. However, there is a trade-off between dimensionality and computational efficiency. As the dimensionality increases, the complexity of the optimization process also rises, potentially leading to slower convergence and increased risk of overfitting.
The expressiveness of the representation is also influenced by the capacity of the embedding to capture the manifold structure of the mesh. If the dimensionality is too low, the embedding may not be able to encode all necessary topological features, resulting in a loss of fidelity in the generated mesh. Conversely, if the dimensionality is sufficiently high, the representation can effectively capture a wide variety of mesh structures, including those with intricate connectivity and diverse polygonal configurations.
In practice, the authors found that low-dimensional embeddings (e.g., (k < 10)) were sufficient to represent the diverse mesh structures encountered in their experiments. This suggests that while there are theoretical limits to expressiveness based on dimensionality, practical applications can often achieve satisfactory results with relatively low-dimensional embeddings, provided that the training data is rich and varied.

Could this continuous mesh representation be combined with unsupervised learning techniques to fit mesh distributions without relying on large labeled datasets?

Yes, the continuous mesh representation could be effectively combined with unsupervised learning techniques to fit mesh distributions without the need for large labeled datasets. One potential approach is to utilize generative models, such as Variational Autoencoders (VAEs) or Generative Adversarial Networks (GANs), which can learn to capture the underlying distribution of mesh structures from unlabeled data.
By training a generative model on a diverse collection of meshes, the continuous representation can learn to produce new meshes that adhere to the learned distribution. The model could leverage the continuous embeddings to represent the latent space of mesh connectivity and geometry, allowing for the generation of novel meshes that maintain manifold properties.
Additionally, techniques such as clustering or dimensionality reduction could be employed to identify patterns and structures within the mesh dataset, facilitating the learning process. For instance, clustering algorithms could group similar mesh structures, enabling the model to learn representative embeddings for each cluster without explicit labels.
Moreover, self-supervised learning strategies could be implemented, where the model generates meshes based on partial inputs or reconstructs missing parts of a mesh. This would allow the model to learn meaningful representations of mesh structures in an unsupervised manner, further enhancing its ability to generate high-quality meshes from diverse input conditions.
In summary, the continuous mesh representation holds significant potential for integration with unsupervised learning techniques, enabling the generation of diverse mesh structures without the reliance on extensive labeled datasets.