Topology-Aware Latent Diffusion for 3D Shape Generation

TL;DR

This work tackles the lack of topology control in 3D shape generation by integrating persistent homology with latent diffusion. It encodes implicit 3D shapes into a latent space via a transformer-based AutoEncoder, extracts topological features as Betti numbers and persistence diagrams, and conditions the diffusion process on these features to produce topology-aware shapes. The method enables direct control of topology (e.g., loop count via $β_1$) and supports varied inputs such as sparse point clouds and sketches, with topology editable through the persistence diagrams. Evaluations on ShapeNet and ABC datasets show competitive quality with improved diversity when topology is incorporated, highlighting the practical potential for topology-guided 3D synthesis in design and visualization tasks.

Abstract

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

Figures (9)

• Figure 1: Our framework primarily comprises three steps: (a). The process starts with a persistent homology (PH) analysis applied to the signed distance field (SDF) of the input mesh. This analysis yields topological features, which are represented by persistence diagrams (PDs). (b). Subsequently, dense point clouds sampled on the mesh surface are input to train a shape AutoEncoder. This training phase is supervised by the occupancy field, resulting in a set of latent vectors that represent the 3D shapes. (c). The final step involves training the diffusion model within the previously learned latent space. The diffusion process is made aware of the topology by conditioning each denoising block with features derived from the PDs.
• Figure 2: Taking the implicit representations of 3D shapes (SDFs) as filtration values, we utilize the cubical complex to perform the persistent homology analysis for the shapes. The multi-scale topological features can be depicted by the persistence diagrams, where each point indicates the birth value and death value of the corresponding topological feature, respectively.
• Figure 3: The architecture of Shape AutoEncoder (a), Topology Encoder (b), and Denoising Block in Diffusion process (c). We mainly utilize cross-attention (Cross-Attn) and self-attention (Self-Attn) mechanisms for the extraction and transformation of features.
• Figure 4: Distinct from the vectorization approaches of persistence images and persistence landscapes, our method interprets the birth-death pairs from persistence homology (PH) analysis as discrete points within a 2D space. This representation serves as the underpinning for our topological feature characterization.
• Figure 5: We can generate 3D shapes with different topologies by specifying the 1-dimension Betti number $\beta_1$.