MIP-plicits: Level of Detail Factorization of Neural Implicits Sphere Tracing (original) (raw)

Differential Geometry in Neural Implicits

2022

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...

Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces

Sphere tracing is a new technique for rendering implicit surfaces that uses geometric distance. Sphere tracing marches along the ray toward its first intersection in steps guaranteed not to penetrate the implicit surface. It is particularly adept at rendering pathological surfaces. Creased and rough implicit surfaces are defined by functions with discontinuous or undefined derivatives. Sphere tracing requires only a bound on the magnitude of the derivative, robustly avoiding problems where the derivative jumps or vanishes. It is an efficient direct visualization system for the design and investigation of new implicit models. Sphere tracing efficiently approximates cone tracing, supporting symbolicprefiltered antialiasing. Signed distance functions for a variety of primitives and operations are derived.

Neural Implicit Surfaces in Higher Dimension

2022

This work investigates the use of neural networks admitting high-order derivatives for modeling dynamic variations of smooth implicit surfaces. For this purpose, it extends the representation of differentiable neural implicit surfaces to higher dimensions, which opens up mechanisms that allow to exploit geometric transformations in many settings, from animation and surface evolution to shape morphing and design galleries. The problem is modeled by a kkk-parameter family of surfaces ScS_cSc, specified as a neural network function f:mathbbR3timesmathbbRkrightarrowmathbbRf : \mathbb{R}^3 \times \mathbb{R}^k \rightarrow \mathbb{R}f:mathbbR3timesmathbbRkrightarrowmathbbR, where ScS_cSc is the zero-level set of the implicit function f(cdot,c):mathbbR3rightarrowmathbbRf(\cdot, c) : \mathbb{R}^3 \rightarrow \mathbb{R} f(cdot,c):mathbbR3rightarrowmathbbR, with cinmathbbRkc \in \mathbb{R}^kcinmathbbRk, with variations induced by the control variable ccc. In that context, restricted to each coordinate of mathbbRk\mathbb{R}^kmathbbRk, the underlying representation is a neural homotopy which is the solution of a general partial differential equation.

OctField: Hierarchical Implicit Functions for 3D Modeling

arXiv (Cornell University), 2021

Recent advances in localized implicit functions have enabled neural implicit representation to be scalable to large scenes. However, the regular subdivision of 3D space employed by these approaches fails to take into account the sparsity of the surface occupancy and the varying granularities of geometric details. As a result, its memory footprint grows cubically with the input volume, leading to a prohibitive computational cost even at a moderately dense decomposition. In this work, we present a learnable hierarchical implicit representation for 3D surfaces, coded OctField, that allows high-precision encoding of intricate surfaces with low memory and computational budget. The key to our approach is an adaptive decomposition of 3D scenes that only distributes local implicit functions around the surface of interest. We achieve this goal by introducing a hierarchical octree structure to adaptively subdivide the 3D space according to the surface occupancy and the richness of part geometry. As octree is discrete and non-differentiable, we further propose a novel hierarchical network that models the subdivision of octree cells as a probabilistic process and recursively encodes and decodes both octree structure and surface geometry in a differentiable manner. We demonstrate the value of OctField for a range of shape modeling and reconstruction tasks, showing superiority over alternative approaches.

NeuralMeshing: Differentiable Meshing of Implicit Neural Representations

Lecture Notes in Computer Science, 2022

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.

Exploring differential geometry in neural implicits

Computers & Graphics

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.

HyperCube: Implicit Field Representations of Voxelized 3D Models

2021

Recently introduced implicit field representations offer an effective way of generating 3D object shapes. They leverage implicit decoder trained to take a 3D point coordinate concatenated with a shape encoding and to output a value which indicates whether the point is outside the shape or not. Although this approach enables efficient rendering of visually plausible objects, it has two significant limitations. First, it is based on a single neural network dedicated for all objects from a training set which results in a cumbersome training procedure and its application in real life. More importantly, the implicit decoder takes only points sampled within voxels (and not the entire voxels) which yields problems at the classification boundaries and results in empty spaces within the rendered mesh. To solve the above limitations, we introduce a new HyperCube architecture based on interval arithmetic network, that enables direct processing of 3D voxels, trained using a hypernetwork paradig...

Iterative methods for visualization of implicit surfaces on gpu

2007

The ray-casting of implicit surfaces on GPU has been explored in the last few years. However, until recently, they were restricted to second degree (quadrics). We present an iterative solution to ray cast cubics and quartics on GPU. Our solution targets efficient implementation, obtaining interactive rendering for thousands of surfaces per frame. We have given special attention to torus rendering since it is a useful shape for multiple CAD models. We have tested four different iterative methods, including a novel one, comparing them with classical tessellation solution. Fig. 1. The faces of two bounding boxes are used to trigger the fragment shader responsible for rendering the tori.

A neural network scheme for transparent surface modelling

Conference: Proceedings of the 3rd International Conference on Computer Graphics and Interactive Techniques in Australasia and Southeast Asia 2005, Dunedin, New Zealand, November 29 - December 2, 2005, 2005

This paper presents a new neural network (NN) scheme for recovering three dimensional (3D) transparent surface. We view the transparent surface modeling, not as a separate problem, but as an extension of opaque surface modeling. The main insight of this work is we simulate transparency not only for generating visually realistic images, but for recovering the object shape. We construct a formulation of transparent surface modeling using ray tracing framework into our NN. We compared this ray tracing method, with a texture mapping method that simultaneously map the silhouette images and smooth shaded images (obtained form our NN), and textured images (obtained from the teacher image) to an initial 3D model. By minimizing the images error between the output images of our NN and the teacher images, observed in multiple views, we refine vertices position of the initial 3D model. We show that our NN can refine the initial D model obtained by polarization images and converge into more accurate surface.