Neural Shortest Path for Surface Reconstruction from Point Clouds

In this paper, we propose the neural shortest path (NSP), a vector-valued implicit neural representation (INR) that approximates a distance function and its gradient. The key feature of NSP is to learn the exact shortest path (ESP), which directs an arbitrary point to its nearest point on the target surface. The NSP is decomposed into its magnitude and direction, and a variable splitting method is used that each decomposed component approximates a distance function and its gradient, respectively. Unlike to existing methods of learning the distance function itself, the NSP ensures the simultaneous recovery of the distance function and its gradient. We mathematically prove that the decomposed representation of NSP guarantees the convergence of the magnitude of NSP in the $H^1$ norm. Furthermore, we devise a novel loss function that enforces the property of ESP, demonstrating that its global minimum is the ESP. We evaluate the performance of the NSP through comprehensive experiments on diverse datasets, validating its capacity to reconstruct high-quality surfaces with the robustness to noise and data sparsity. The numerical results show substantial improvements over state-of-the-art methods, highlighting the importance of learning the ESP, the product of distance function and its gradient, for representing a wide variety of complex surfaces.

$p$-Poisson surface reconstruction in curl-free flow from point clouds

The aim of this paper is the reconstruction of a smooth surface from an unorganized point cloud sampled by a closed surface, with the preservation of geometric shapes, without any further information other than the point cloud. Implicit neural representations (INRs) have recently emerged as a promising approach to surface reconstruction. However, the reconstruction quality of existing methods relies on ground truth implicit function values or surface normal vectors. In this paper, we show that proper supervision of partial differential equations and fundamental properties of differential vector fields are sufficient to robustly reconstruct high-quality surfaces. We cast the $p$-Poisson equation to learn a signed distance function (SDF) and the reconstructed surface is implicitly represented by the zero-level set of the SDF. For efficient training, we develop a variable splitting structure by introducing a gradient of the SDF as an auxiliary variable and impose the $p$-Poisson equation directly on the auxiliary variable as a hard constraint. Based on the curl-free property of the gradient field, we impose a curl-free constraint on the auxiliary variable, which leads to a more faithful reconstruction. Experiments on standard benchmark datasets show that the proposed INR provides a superior and robust reconstruction. The code is available at \url{https://github.com/Yebbi/PINC}.