Neural Implicit Surfaces in Higher Dimension (original) (raw)
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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 opt...
Exploring differential geometry in neural implicits
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We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions. To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches. We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface.
1999
We introduce a new method of creating smooth implicit surfaces of arbitrary manifold topology. These surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is created from these constraints using a variational scattered data interpolation approach. We call the iso-surface of this function a variational implicit surface. Like other implicit surface descriptions, these surfaces can be used for CSG and interference detection, may be interactively manipulated, are readily approximated by polygonal tilings, and are easy to ray trace. A key strength is that variational implicit surfaces allow the direct specification of both the location of points on the surface and surface normals. These are two important manipulation techniques that are difficult to achieve using other implicit surface representations such as sums of spherical or ellipsoidal Gaussian functions ("blobbies"). We show that these properties make variational implicit surfaces particularly attractive for interactive sculpting using the particle sampling technique introduced by Witkin and Heckbert in [30]. Our formulation also yields a simple method for converting a polygonal model to a smooth implicit model. 2 Background and Related Work Variational implicit surfaces draw upon two areas of modeling: implicit surfaces and thin-plate interpolation. In this section we briefly review work in these two sub-areas. 2.1 Implicit Surfaces An implicit surface is defined by an implicit function, a continuous scalar-valued function over the domain R 3. The implicit surface of
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The generation of triangle meshes from point clouds, i.e. meshing, is a core task in computer graphics and computer vision. Traditional techniques directly construct a surface mesh using local decision heuristics, while some recent methods based on neural implicit representations try to leverage data-driven approaches for this meshing process. However, it is challenging to define a learnable representation for triangle meshes of unknown topology and size and for this reason, neural implicit representations rely on non-differentiable post-processing in order to extract the final triangle mesh. In this work, we propose a novel differentiable meshing algorithm for extracting surface meshes from neural implicit representations. Our method produces the mesh in an iterative fashion, which makes it applicable to shapes of various scales and adaptive to the local curvature of the shape. Furthermore, our method produces meshes with regular tessellation patterns and fewer triangle faces compared to existing methods. Experiments demonstrate the comparable reconstruction performance and favorable mesh properties over baselines.
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