STITCH: Surface reconstrucTion using Implicit neural representations with Topology Constraints and persistent Homology

TL;DR

STITCH addresses topology-preserving surface reconstruction from sparse point clouds by integrating a differentiable topology loss based on persistent homology with an implicit neural representation of the surface. The method optimizes a unified loss $\\mathcal{L}(\\theta;\\mathbf{P})=\\mathcal{L}_{g}(\\theta;\\mathbf{P})+\\mathcal{L}_{c}(\\theta;\\mathbf{P})$, where $\\mathcal{L}_{g}$ promotes closeness of pulled query points to the surface and $\\mathcal{L}_{c}=\\lambda_1\\mathcal{L}_{\\mathcal{S}}+\\lambda_2\\mathcal{L}_{\\mathcal{N}}$ enforces topology via the persistence diagram. A differentiable pathway built on cubical complexes and cofaces enables end-to-end gradient flow and SGD convergence to a solution with a single connected component, with theoretical guarantees and a connectivity theorem based on alpha-beta connectivity. Empirically, STITCH achieves competitive geometric accuracy while substantially reducing significant-feature loss across diverse datasets, demonstrating robust topology preservation for thin and sparse geometries and enabling watertight, simulation-ready meshes.

Abstract

We present STITCH, a novel approach for neural implicit surface reconstruction of a sparse and irregularly spaced point cloud while enforcing topological constraints (such as having a single connected component). We develop a new differentiable framework based on persistent homology to formulate topological loss terms that enforce the prior of a single 2-manifold object. Our method demonstrates excellent performance in preserving the topology of complex 3D geometries, evident through both visual and empirical comparisons. We supplement this with a theoretical analysis, and provably show that optimizing the loss with stochastic (sub)gradient descent leads to convergence and enables reconstructing shapes with a single connected component. Our approach showcases the integration of differentiable topological data analysis tools for implicit surface reconstruction.

Figures (12)

• Figure 1: Current reconstruction approaches prioritize accuracy of the reconstruction over the topology of the reconstructed object. This leads to cases where the method has to be tuned for the point cloud spacing, otherwise leading to gaps or isolated islands in the reconstruction. In our approach, we apply topological constraints to obtain a single connected 2-manifold surface.
• Figure 2: We propose a neural implicit surface reconstruction approach, STITCH, that leverages persistent homology and builds upon Neural-Pull ma2020neural. We begin with a point cloud, predict its signed distance field through an implicit neural representation, and compute a cubical complex. Through the filtration process, we generate various topological features, which are penalized using our differentiable topological loss terms. The topological losses encourage significant features to persist and simultaneously diminish the persistence of noisy features, leading to a single connected component in the reconstruction.
• Figure 3: A visual representation of topological features represented by the persistence diagram. The arrows on the persistence diagram showcase the direction of movement for each set of features. Features with green arrows are considered significant ($\mathcal{S}$), and we want to preserve them, while features with a light blue arrow are considered as noise ($\mathcal{N}$), and we minimize them.
• Figure 4: Comparison of the reconstructions of the SRB dataset by our proposed method with Neural Pull.
• Figure 5: Comparison of different methods for the reconstruction of four models from the DFAUST dataset.

Theorems & Definitions (18)

• Definition 1
• Theorem 1
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• Definition 5
• Lemma 1
• proof
• proof