Regularized implicit surface reconstruction from points and normals

Topology-controllable Implicit Surface Reconstruction Based on Persistent Homology

Background: Implicit surface is a kind of surface modeling tool, which is widely used in point cloud reconstruction, deformation and fusion due to its advantages of good smoothness and Boolean operation. The most typical method is the surface reconstruction with Radial Basis Functions (RBF) under normal constraints. RBF has become one of the main methods of point cloud fitting because it has a strong mathematical foundation, an advantage of computation simplicity, and the ability of processing nonuniform points. Objective: Techniques and patents of implicit surface reconstruction interpolation with RBF are surveyed. Theory, algorithm, and application are discussed to provide a comprehensive summary for implicit surface reconstruction in RBF and Hermite Radial Basis Functions (HRBF) interpolation. Methods: RBF implicit surface reconstruction interpolation can be divided into RBF interpolation under the constraints of points and HRBF interpolation under the constraints of points and corresponding normals. Results: A total of 125 articles were reviewed, in which more than 30% were related to RBF in the last decade. The continuity properties and application fields of the popular global supported radial basis functions and compactly supported radial basis functions are analyzed. Different methods of RBF and HRBF implicit surface reconstruction are evaluated, and the challenges of these methods are discussed. Conclusion: In future work, implicit surface reconstruction via RBF and HRBF should be further studied in fitting accuracy, computation speed, and other fundamental problems. In addition, it is a more challenging but valuable research direction to construct a new RBF with both compact support and improved fitting accuracy.

Rapid advancement of 3D sensing techniques has led to dense and accurate point cloud of an object to be readily available. The growing use of such scanned point sets in product design, analysis, and manufacturing necessitates research on direct processing of point set surfaces. In this paper, we present an approach that enables the direct layered manufacturing of point set surfaces. This new approach is based on adaptive slicing of moving least squares (MLS) surfaces. Salient features of this new approach include the following: (1) It bypasses the laborious surface reconstruction and avoids model conversion induced accuracy loss. (2) The resulting layer thickness and layer contours are adaptive to local curvatures, and thus it leads to better surface quality and more efficient fabrication. (3) The curvatures are computed from a set of closed formula based on the MLS surface. The MLS surface naturally smoothes the point cloud and allows upsampling and downsampling, and thus it is robust even for noisy or sparse point sets. Experimental results on both synthetic and scanned point sets are presented.

Structuring finite sets of points is at the heart of computational geometry. Such point sets arise naturally in many applications. Examples in R3 are point sets sampled from the surface of a solid or the locations of atoms in a molecule. A first step in processing these point sets is to organize them in some data structure. Structuring a point set into a simplicial complex like the Delaunay triangulation has turned out to be appropriate for many modeling tasks. Here we introduce the flow complex which is another simplicial complex that can be computed efficiently from a finite set of points. The flow complex turned out to be well suited for surface reconstruction from a finite sample and for some tasks in structural biology. Here we study mathematical and algorithmic properties of the flow complex and show how to exploit it in applications.

In this paper, we address the problem of curve and surface reconstruction from sets of points. We introduce regular interpolants which are polygonal approximations of planar curves and surfaces verifying a local sampling criterion. Properties of regular interpolants lead to new polygonal reconstruction methods from sets of organized and unorganized points. These methods do not need any parameter of additional information apart from the original points and allow unorganized sets of points to be easily handled.

In this paper, we propose a new method for surface reconstruction from scattered point set based on least square radial basis function (LSRBF) and orthogonal least square forward selection procedure. Firstly, the traditional RBF formulation is rewritten into least square formula. A implicit surface can be represented with fewer centers. Then, the orthogonal least square procedure is utilized to select significant centers from original point data set. The RBF coefficients can be solved from the triangular matrix from OLS selection through backward substitution method. So, this scheme can offer a fast surface reconstruction tool and can overcome the numerical ill-conditioning of coefficient matrix and over-fitting problem. Some examples are presented to show the effectiveness of our algorithm in 2D and 3D cases.KeywordsRadial Basis FunctionSurface ReconstructionRadial Basis Function NetworkMove Less SquareImplicit SurfaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

We present a novel, physically-motivated method for surface reconstruction that can recover smooth surfaces from noisy and sparse data sets, without using orientation information. A new volumetric technique based on regularized-membrane potentials for aggregating the input sample points is introduced, which manages improved noise tolerability and outlier removal, without sacrificing much with respect to detail (feature) recovery. In this method, sample points are first aggregated on a volumetric grid. A labeling algorithm that relies on intrinsic properties of the smooth scalar field emerging after aggregation is used to classify grid points as exterior or interior to the surface. We also introduce a mesh-smoothing paradigm based on a mass-spring system, enhanced with a bending-energy minimizing term to ensure that the final triangulated surface is smoother than piecewise linear. The method compares favorably with respect to previous approaches in terms of speed and flexibility.

The data cube and iceberg cube computation problem has been studied by many researchers. There are three major approaches developed in this direction: (1) top-down computation, represented by MultiWay array aggregation (Zhao et. al., 1997) which utilizes shared computation and performs well on dense data sets; (2) bottom-up computation, represented by BUC (Beyer and Ramakrishnan, 1999), which takes advantage of Apriori Pruning and performs well on sparse data sets; and (3) integrated top-down and bottom-up computation, represented by Star-Cubing (Xin, et. al., 2003), which takes advantages of both and has high performance in most cases. However; the performance of Star-Cubing degrades in very sparse data sets due to the additional cost introduced by the tree structure. None of the three approaches achieves uniformly high performance on all kinds of data sets. In this paper; we present a new approach that compute Iceberg Cubes by factorizing the lattice space according to the frequency of values. This approach, different from all the previous dimension-based approaches where the importance of data distribution is not recognized, partitions the cube lattice into one dense subspace and several sparse subspaces. With this approach, a new method called MM-Cubing has been developed. MM-Cubing is highly adaptive to dense, sparse or skewed data sets. Our performance study shows that MM-Cubing is efficient and achieves high performance over all kinds of data distributions.

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalisation of the concept of the median of a (finite) set of points on the real line. In this paper, we present an algorithm for computing a centerpoint of a set of n points in the plane. The algorithm has complexity O(n) which significantly improves the O(n log3 n) complexity of the previously best known algorithm. We use suitable modifications of the ham-sandwich-cut algorithm and the prune-and-search technique to achieve this improvement.

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

Efficient algorithms exist to obtain a sparse 3D representation of the environment. Bundle adjustment (BA) and structure from motion (SFM) are techniques used to estimate both the camera poses and the set of sparse points in the environment. Many applications require such reconstruction to be performed online, while acquiring the data, and produce an updated result every step. Furthermore, using active feedback about the quality of the reconstruction can help selecting the best views to increase the accuracy as well as to maintain a reasonable size of the collected data. This paper provides novel and efficient solutions to solving the associated NLS incrementally, and to compute not only the optimal solution, but also the associated uncertainty. The proposed technique highly increases the efficiency of the incremental BA solver for long camera trajectory applications, and provides extremely fast covariance recovery.

Data fusion provides tools for solving problems which are characterized by distributed and diverse information sources. Many robotic applications need to retrieve particular properties from a scene; so it is necessary to use multiple knowledge sources since a single sensory modality cannot capture all of the physical causes of a given edge feature. In this paper we focus on the problem of extracting features such as image discontinuities from both synthetic and real images. Since edge detection and surface reconstruction are ill-posed problems according to Hadamard, Tikhonov's regularization paradigm is proposed as the basic tool for solving this inversion problem and restoring well-posedness. The proposed framework includes (1) a review of 2D regularization, (2) extension of the standard Tikhonov regularization method by allowing space-variant regularization parameters, and (3) further extension of the regularization paradigm by adding multiple data sources for different sensing modalities. The theoretical approach is complemented by developing a regularized hybrid fusion algorithm for solving the early vision problems of edge detection and surface reconstruction. An evaluation of these methods reveals that this new analytical data fusion technique reconstructs a smooth filtered surface in noisy regions while preserving edge characteristics needed for extracting object features. Results indicate the fusion technique is beneficial for combining edge features from different types of sensory data to locate and identify objects of interest.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

If $$\mathbb{X}$$ is a finite set of points in a multiprojective space $$\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } $$ withr ≥ 2, then $$\mathbb{X}$$ may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM

Computing the shape of a planar points set