http://library.utia.cas.cz/prace/20030125.pdf

Image registration methods: a survey

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.

Poisson surface reconstruction

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.

/pdf/deepsdf-learning-continuous-signed-distance-functions-for-43v0fu25wz.pdf

DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation

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?

/pdf/past-present-and-future-of-simultaneous-localization-and-27i1pn9khc.pdf

Past, Present, and Future of Simultaneous Localization and Mapping: Toward the Robust-Perception Age

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.

/pdf/reconstruction-and-representation-of-3d-objects-with-radial-1col9c83sl.pdf

Reconstruction and representation of 3D objects with radial basis functions

Solving large radial basis function (RBF) interpolation problems with non‐customised methods is computationally expensive and the matrices that occur are typically badly conditioned. For example, using the usual direct methods to fit an RBF with N centres requires O(N 2) storage and O(N 3) flops. Thus such an approach is not viable for large problems with N ≥10,000.

Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration

A generalized multiquadric radial basis function is a function of the form $ s(x) = \sum_{i=1}^N d_i \phi(|x -t_i|), $ where $ \phi(r)= ( r^2+\tau^2 )^{k/2}$, $x \in \mathbb{R}^n$, and $k\in\mathbb{Z}$ is odd. The direct evaluation of an N center generalized multiquadric radial basis function at m points requires $\mathcal{O}(m N)$ flops, which is prohibitive when m and N are large. Similar considerations apparently rule out fitting an interpolating N center generalized multiquadric to N data points by either direct or iterative solution of the associated system of linear equations in realistic problems.
In this paper we will develop far field expansions, recurrence relations for efficient formation of the expansions, error estimates, and translation formulas for generalized multiquadric radial basis functions in n-variables. These pieces are combined in a hierarchical fast evaluator requiring only $\mathcal{O}((m+N)\log N|\log\epsilon|^{n+1})$ flops for evaluation of an N center generalized multiquadric at m points. This flop count is significantly less than that of the direct method. Moreover, used to compute matrix-vector products, the fast evaluator provides a basis for fast iterative fitting strategies.

Fast Evaluation of Radial Basis Functions: Methods for Generalized Multiquadrics in $\RR^\protectn$

As is now well known for some basic functions $\phi$, hierarchical and fast multipole-like methods can greatly reduce the storage and operation counts for fitting and evaluating radial basis functions. In particular, for spline functions of the form \begin{equation*} s(x) = p(x) + \sum_{k=1}^{N} d_k \phi(|x - x_k|), \end{equation*} where p is a low degree polynomial and with certain choices of $\phi$, the cost of a single extra evaluation can be reduced from \Order{N} to \Order{\log N}, or even \Order{1}, operations and the cost of a matrix-vector product (i.e., evaluation at all centers) can be decreased from \Order{N^2} to \Order{N\log N}, or even \Order{N}, operations. This paper develops the mathematics required by methods of these types for polyharmonic splines in $\mathbb{R}^4$. That is, for splines s built from a basic function from the list $\phi(r) = r^{-2}$ or $\phi(r)= r^{2n}\ln(r)$, $n = 0,1,\ldots$. We give appropriate far and near field expansions, together with corresponding error estimates...

/pdf/fast-evaluation-of-radial-basis-functions-methods-for-four-3fl1xecezf.pdf

Fast Evaluation of Radial Basis Functions: Methods for Four-Dimensional Polyharmonic Splines

Model-based calibration tools can be used to prototype a process for an advanced engine type, reducing the burden of developing optimal calibrations for complex modern engines. This paper is a practical example of using simulation to prototype a calibration process before the engine hardware is available. We produce optimized tables for spark and cam timings for a 2.2 L naturally aspirated 4-valve overhead-cam spark ignition engine with twin-independent variable valve timing. We use design of experiments techniques to minimize testing, and a novel boundary modeling technique to describe the surface of the engine operating envelope and to constrain optimizations to realistic bounds. Testing simulated prototypes rather than the real hardware can save time and money by catching process errors and clearly defining measurement data requirements early in the design cycle. Offline optimization can be carried out independently of the test bench, limiting expenditure. This can help obtain consistent results faster, strengthen the predictive capability of the design process, and increase test bed productivity.

J. B. Cherrie

Papers

Reconstruction and representation of 3D objects with radial basis functions

Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration

Fast Evaluation of Radial Basis Functions: Methods for Generalized Multiquadrics in $\RR^\protectn$

Fast Evaluation of Radial Basis Functions: Methods for Four-Dimensional Polyharmonic Splines

Defining a model-based calibration process for a twin-independent valve timing engine