Differential Geometry in Neural Implicits
- URL: http://arxiv.org/abs/2201.09263v2
- Date: Wed, 26 Jan 2022 23:31:57 GMT
- Title: Differential Geometry in Neural Implicits
- Authors: Tiago Novello, Vinicius da Silva, Helio Lopes, Guilherme Schardong,
Luiz Schirmer, Luiz Velho
- Abstract summary: We introduce a neural implicit framework that bridges discrete differential geometry of triangle meshes and continuous differential geometry of neural implicit surfaces.
It exploits the differentiable properties of neural networks and the discrete geometry of triangle meshes to approximate them as the zero-level sets of neural implicit functions.
- Score: 0.6198237241838558
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We introduce a neural implicit framework that bridges discrete differential
geometry of triangle meshes and continuous differential geometry of neural
implicit surfaces. It exploits the differentiable properties of neural networks
and the discrete geometry of triangle meshes to approximate them as the
zero-level sets of neural implicit functions.
To train a neural implicit function, we propose a loss function that allows
terms with high-order derivatives, such as the alignment between the principal
directions, to learn more geometric details. During training, we consider a
non-uniform sampling strategy based on the discrete curvatures of the triangle
mesh to access points with more geometric details. This sampling implies faster
learning while preserving geometric accuracy.
We present the analytical differential geometry formulas for neural surfaces,
such as normal vectors and curvatures. We use them to render the surfaces using
sphere tracing. Additionally, we propose a network optimization based on
singular value decomposition to reduce the number of parameters.
Related papers
- Neural varifolds: an aggregate representation for quantifying the geometry of point clouds [2.2474167740753557]
We propose a new surface geometry characterisation, namely a neural varifold representation of point clouds.
The varifold representation quantifies the surface geometry of point clouds through the manifold-based discrimination.
The proposed neural varifold is evaluated on three different sought-after tasks -- shape matching, few-shot shape classification and shape reconstruction.
arXiv Detail & Related papers (2024-07-05T20:08:16Z) - ParaPoint: Learning Global Free-Boundary Surface Parameterization of 3D Point Clouds [52.03819676074455]
ParaPoint is an unsupervised neural learning pipeline for achieving global free-boundary surface parameterization.
This work makes the first attempt to investigate neural point cloud parameterization that pursues both global mappings and free boundaries.
arXiv Detail & Related papers (2024-03-15T14:35:05Z) - Parameterization-driven Neural Surface Reconstruction for Object-oriented Editing in Neural Rendering [35.69582529609475]
This paper introduces a novel neural algorithm for parameterizing neural implicit surfaces to simple parametric domains like spheres and polycubes.
It computes bi-directional deformation between the object and the domain using a forward mapping from the object's zero level set and an inverse deformation for backward mapping.
We demonstrate the method's effectiveness on images of human heads and man-made objects.
arXiv Detail & Related papers (2023-10-09T08:42:40Z) - From Complexity to Clarity: Analytical Expressions of Deep Neural Network Weights via Clifford's Geometric Algebra and Convexity [54.01594785269913]
We show that optimal weights of deep ReLU neural networks are given by the wedge product of training samples when trained with standard regularized loss.
The training problem reduces to convex optimization over wedge product features, which encode the geometric structure of the training dataset.
arXiv Detail & Related papers (2023-09-28T15:19:30Z) - NeuralMeshing: Differentiable Meshing of Implicit Neural Representations [63.18340058854517]
We propose a novel differentiable meshing algorithm for extracting surface meshes from neural implicit representations.
Our method produces meshes with regular tessellation patterns and fewer triangle faces compared to existing methods.
arXiv Detail & Related papers (2022-10-05T16:52:25Z) - Minimal Neural Atlas: Parameterizing Complex Surfaces with Minimal
Charts and Distortion [71.52576837870166]
We present Minimal Neural Atlas, a novel atlas-based explicit neural surface representation.
At its core is a fully learnable parametric domain, given by an implicit probabilistic occupancy field defined on an open square of the parametric space.
Our reconstructions are more accurate in terms of the overall geometry, due to the separation of concerns on topology and geometry.
arXiv Detail & Related papers (2022-07-29T16:55:06Z) - Primal-Dual Mesh Convolutional Neural Networks [62.165239866312334]
We propose a primal-dual framework drawn from the graph-neural-network literature to triangle meshes.
Our method takes features for both edges and faces of a 3D mesh as input and dynamically aggregates them.
We provide theoretical insights of our approach using tools from the mesh-simplification literature.
arXiv Detail & Related papers (2020-10-23T14:49:02Z) - Geometric Attention for Prediction of Differential Properties in 3D
Point Clouds [32.68259334785767]
In this study, we present a geometric attention mechanism that can provide such properties in a learnable fashion.
We establish the usefulness of the proposed technique with several experiments on the prediction of normal vectors and the extraction of feature lines.
arXiv Detail & Related papers (2020-07-06T07:40:26Z) - CNNs on Surfaces using Rotation-Equivariant Features [10.259432250871997]
Transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface.
We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features.
We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.
arXiv Detail & Related papers (2020-06-02T12:46:00Z) - PUGeo-Net: A Geometry-centric Network for 3D Point Cloud Upsampling [103.09504572409449]
We propose a novel deep neural network based method, called PUGeo-Net, to generate uniform dense point clouds.
Thanks to its geometry-centric nature, PUGeo-Net works well for both CAD models with sharp features and scanned models with rich geometric details.
arXiv Detail & Related papers (2020-02-24T14:13:29Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.