Reconstruction and representation of 3D objects with radial basis functions
01 Aug 2001-Vol. 67, pp 67-76
TL;DR: It is shown that the RBF representation has advantages for mesh simplification and remeshing applications, and a greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages.
Abstract: We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data.
Citations
More filters
26 Jun 2006
TL;DR: A spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model, and which reduces to a well conditioned sparse linear system.
Abstract: We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.
2,712 citations
15 Jun 2019
TL;DR: DeepSDF as mentioned in this paper represents a shape's surface by a continuous volumetric field: the magnitude of a point in the field represents the distance to the surface boundary and the sign indicates whether the region is inside (-) or outside (+) of the shape.
Abstract: Computer graphics, 3D computer vision and robotics communities have produced multiple approaches to representing 3D geometry for rendering and reconstruction. These provide trade-offs across fidelity, efficiency and compression capabilities. In this work, we introduce DeepSDF, a learned continuous Signed Distance Function (SDF) representation of a class of shapes that enables high quality shape representation, interpolation and completion from partial and noisy 3D input data. DeepSDF, like its classical counterpart, represents a shape's surface by a continuous volumetric field: the magnitude of a point in the field represents the distance to the surface boundary and the sign indicates whether the region is inside (-) or outside (+) of the shape, hence our representation implicitly encodes a shape's boundary as the zero-level-set of the learned function while explicitly representing the classification of space as being part of the shapes interior or not. While classical SDF's both in analytical or discretized voxel form typically represent the surface of a single shape, DeepSDF can represent an entire class of shapes. Furthermore, we show state-of-the-art performance for learned 3D shape representation and completion while reducing the model size by an order of magnitude compared with previous work.
2,247 citations
TL;DR: Simultaneous localization and mapping (SLAM) as mentioned in this paper consists in the concurrent construction of a model of the environment (the map), and the estimation of the state of the robot moving within it.
Abstract: Simultaneous localization and mapping (SLAM) consists in the concurrent construction of a model of the environment (the map ), and the estimation of the state of the robot moving within it. The SLAM community has made astonishing progress over the last 30 years, enabling large-scale real-world applications and witnessing a steady transition of this technology to industry. We survey the current state of SLAM and consider future directions. We start by presenting what is now the de-facto standard formulation for SLAM. We then review related work, covering a broad set of topics including robustness and scalability in long-term mapping, metric and semantic representations for mapping, theoretical performance guarantees, active SLAM and exploration, and other new frontiers. This paper simultaneously serves as a position paper and tutorial to those who are users of SLAM. By looking at the published research with a critical eye, we delineate open challenges and new research issues, that still deserve careful scientific investigation. The paper also contains the authors’ take on two questions that often animate discussions during robotics conferences: Do robots need SLAM? and Is SLAM solved?
2,039 citations
TL;DR: What is now the de-facto standard formulation for SLAM is presented, covering a broad set of topics including robustness and scalability in long-term mapping, metric and semantic representations for mapping, theoretical performance guarantees, active SLAM and exploration, and other new frontiers.
Abstract: Simultaneous Localization and Mapping (SLAM)consists in the concurrent construction of a model of the environment (the map), and the estimation of the state of the robot moving within it. The SLAM community has made astonishing progress over the last 30 years, enabling large-scale real-world applications, and witnessing a steady transition of this technology to industry. We survey the current state of SLAM. We start by presenting what is now the de-facto standard formulation for SLAM. We then review related work, covering a broad set of topics including robustness and scalability in long-term mapping, metric and semantic representations for mapping, theoretical performance guarantees, active SLAM and exploration, and other new frontiers. This paper simultaneously serves as a position paper and tutorial to those who are users of SLAM. By looking at the published research with a critical eye, we delineate open challenges and new research issues, that still deserve careful scientific investigation. The paper also contains the authors' take on two questions that often animate discussions during robotics conferences: Do robots need SLAM? and Is SLAM solved?
1,828 citations
Additional excerpts
...functions [39], signed-distance function [56], and truncated...
[...]
TL;DR: This work extends Poisson surface reconstruction to explicitly incorporate the points as interpolation constraints and presents several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster, higher-quality surface reconstructions.
Abstract: Poisson surface reconstruction creates watertight surfaces from oriented point sets. In this work we extend the technique to explicitly incorporate the points as interpolation constraints. The extension can be interpreted as a generalization of the underlying mathematical framework to a screened Poisson equation. In contrast to other image and geometry processing techniques, the screening term is defined over a sparse set of points rather than over the full domain. We show that these sparse constraints can nonetheless be integrated efficiently. Because the modified linear system retains the same finite-element discretization, the sparsity structure is unchanged, and the system can still be solved using a multigrid approach. Moreover we present several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster, higher-quality surface reconstructions.
1,712 citations
References
More filters
01 Aug 1987
TL;DR: In this paper, a divide-and-conquer approach is used to generate inter-slice connectivity, and then a case table is created to define triangle topology using linear interpolation.
Abstract: We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.
13,231 citations
Book•
01 Mar 1990
TL;DR: In this paper, a theory and practice for the estimation of functions from noisy data on functionals is developed, where convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework.
Abstract: This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework. Methods for including side conditions and other prior information in solving ill posed inverse problems are provided. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals.
6,120 citations
TL;DR: An algorithm is presented for the rapid evaluation of the potential and force fields in systems involving large numbers of particles whose interactions are Coulombic or gravitational in nature, making it considerably more practical for large-scale problems encountered in plasma physics, fluid dynamics, molecular dynamics, and celestial mechanics.
5,003 citations
"Reconstruction and representation o..." refers methods in this paper
...Fast evaluation of RBF’s is performed via the Fast Multipole Method (FMM) of Greengard & Rokhlin [15]....
[...]
01 Jul 1992
TL;DR: A general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points that is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.
Abstract: This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects.
Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.
The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data.
Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, including meshes, NURBS surfaces, CSG models, and implicit surfaces.
3,119 citations
"Reconstruction and representation o..." refers methods in this paper
...However, unlike other methods [ 16 ] which also rely on forming a signed-distance function, it is not critical to estimate normals everywhere....
[...]
01 Jan 1977
TL;DR: A family of semi-norms is defined, subject to some interpolating conditions, providing interpolation methods that preserve polynomials of degree≤m−1 and converge in Sobolev spaces Hm+s(Ω).
Abstract: We define a family of semi-norms ‖μ‖m,s=(∫ℝ n∣τ∣2s∣ℱ Dmu(τ)∣2 dτ)1/2 Minimizing such semi-norms, subject to some interpolating conditions, leads to functions of very simple forms, providing interpolation methods that: 1°) preserve polynomials of degree≤m−1; 2°) commute with similarities as well as translations and rotations of ℝn; and 3°) converge in Sobolev spaces Hm+s(Ω).
1,494 citations
Additional excerpts
...Duchon [12] showed that the smoothest interpolant,i....
[...]