Surface reconstruction with data-driven exemplar priors

In this paper, we propose a framework to reconstruct 3D models from raw scanned points by learning the prior knowledge of a specific class of objects. Unlike previous work that heuristically specifies particular regularities and defines parametric models, our shape priors are learned directly from existing 3D models under a framework based on affinity propagation. Given a database of 3D models within the same class of objects, we build a comprehensive library of 3D local shape priors. We then formulate the problem to select as-few-as-possible priors from the library, referred to as exemplar priors. These priors are sufficient to represent the 3D shapes of the whole class of objects from where they are generated. By manipulating these priors, we are able to reconstruct geometrically faithful models with the same class of objects from raw point clouds. Our framework can be easily generalized to reconstruct various categories of 3D objects that have more geometrically or topologically complex structures. Comprehensive experiments exhibit the power of our exemplar priors for gracefully solving several problems in 3D shape reconstruction such as preserving sharp features, recovering fine details and so on.     

Implicit meshes for surface reconstruction

Deformable 3D models can be represented either as traditional explicit surfaces, such as triangulated meshes, or as implicit surfaces. Explicit surfaces are widely accepted because they are simple to deform and render, but fitting them involves minimizing a nondifferentiable distance function. By contrast, implicit surfaces allow fitting by minimizing a differentiate algebraic distance, but are harder to meaningfully deform and render. Here, we propose a method that combines the strength of both approaches. It relies on a technique that can turn a completely arbitrary triangulated mesh, such as one taken from the Web, into an implicit surface that closely approximates it and can deform in tandem with it. This allows both automated algorithms to take advantage of the attractive properties of implicit surfaces for fitting purposes and people to use standard deformation tools they feel comfortable for interaction and animation purposes. We demonstrate the applicability of our technique to modeling the human upper-body, including face, neck, shoulders, and ears, from noisy stereo and silhouette data.     

Computational topology for isotopic surface reconstruction

New computational topology techniques are presented for surface reconstruction of 2-manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope   (as defined herein) of the original surface. For any compact C²$C^2$-manifold M$M$ embedded in R³${\mathbf{R}}^3$, it is shown that its envelope is C^1,1$C^{1,1}$. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M$M$, whenever M$M$ is orientable. The emphasis of this paper is upon the formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to non-orientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.     

The ball-pivoting algorithm for surface reconstruction

The Ball-Pivoting Algorithm (BPA) computes a triangle mesh interpolating a given point cloud. Typically, the points are surface samples acquired with multiple range scans of an object. The principle of the BPA is very simple: Three points form a triangle if a ball of a user-specified radius p touches them without containing any other point. Starting with a seed triangle, the ball pivots around an edge (i.e., it revolves around the edge while keeping in contact with the edge's endpoints) until it touches another point, forming another triangle. The process continues until all reachable edges have been tried, and then starts from another seed triangle, until all points have been considered. The process can then be repeated with a ball of larger radius to handle uneven sampling densities. We applied the BPA to datasets of millions of points representing actual scans of complex 3D objects. The relatively small amount of memory required by the BPA, its time efficiency, and the quality of the results obtained compare favorably with existing techniques     

Functional networks for B-spline surface reconstruction

Recently, a new extension of the standard neural networks, the so-called functional networks, has been described [E. Castillo, Functional networks, Neural Process. Lett. 7 (1998) 151–159]. This approach has been successfully applied to the reconstruction of a surface from a given set of 3D data points assumed to lie on unknown Bézier [A. Iglesias, A. Gálvez, Applying functional networks to CAGD: the tensor-product surface problem, in: D. Plemenos (Ed.), Proceedings of the International Conference on Computer Graphics and Artificial Intelligence, 3IA’2000, 2000, pp. 105–115; A. Iglesias, A. Gálvez, A new artificial intelligence paradigm for computer-aided geometric design, in: Artificial Intelligence and Symbolic Computation, J.A. Campbell, E. Roanes-Lozano (Eds.), Lectures Notes in Artificial Intelligence, Berlin, Heidelberg, Springer-Verlag, vol. 1930, 2001, pp. 200–213] and B-spline tensor-product surfaces [A. Iglesias, A. Gálvez, Applying functional networks to fit data points from B-spline surfaces, in: H.H.S. Ip, N. Magnenat-Thalmann, R.W.H. Lau, T.S. Chua (Eds.), Proceedings of the Computer Graphics International, CGI’2001, IEEE Computer Society Press, Los Alamitos, CA, 2001, pp. 329–332]. In both cases the sets of data were fitted using Bézier surfaces. However, in general, the Bézier scheme is no longer used for practical applications. In this paper, the use of B-spline surfaces (by far the most common family of surfaces in surface modeling and industry) for the surface reconstruction problem is proposed instead. The performance of this method is discussed by means of several illustrative examples. A careful analysis of the errors makes it possible to determine the number of B-spline surface fitting control points that best fit the data points. This analysis also includes the use of two sets of data (the training and the testing data) to check for overfitting, which does not occur here.     

Ordered subsets Bayesian tomographic reconstruction using 2-D smoothing splines as priors

The ordered subsets expectation maximization (OS-EM) algorithm has enjoyed considerable interest for accelerating the well-known EM algorithm for emission tomography. The OS principle has also been applied to several regularized EM algorithms, such as nonquadratic convex minimization-based maximum a posteriori (MAP) algorithms. However, most of these methods have not been as practical as OS-EM due to their complex optimization methods and difficulties in hyperparameter estimation. We note here that, by relaxing the requirement of imposing sharp edges and using instead useful quadratic spline priors, solutions are much easier to compute, and hyperparameter calculation becomes less of a problem. In this work, we use two-dimensional smoothing splines as priors and apply a method of iterated conditional modes for the optimization. In this case, step sizes or line-search algorithms necessary for gradient-based descent methods are avoided. We also accelerate the resulting algorithm using the OS approach and propose a principled way of scaling smoothing parameters to retain the strength of smoothing for different subset numbers. Our experimental results show that the OS approach applied to our quadratic MAP algorithms provides a considerable acceleration while retaining the advantages of quadratic spline priors.     

Virtual reconstruction of archeological vessels using expert priors and intrinsic differential geometry information

This paper presents a method to assist in the tedious task of reconstructing ceramic vessels from unearthed archeological shards (fragments) using 3D computer vision-enabling technologies. The method exploits the shards surface intrinsic differential geometry information (one of many possible tools) coupled with a series of generic models to produce a virtual reconstruction and rendition of what the original vessel may have looked like. Generic models are constructed based on a host of factors including expert historical knowledge of the period, provenance of the artifact, and site location. The generic models need not to be identical to the original vessel, but they must be within a geometric transformation of it in most of its parts. The method is suited for ceramic vessels with some relief (i.e., surface with molding, carving, or stamping), as this method exploits surface intrinsic features for alignment. The alignment of the shards against the generic model uses a novel set of 3D weighted curve moments. The transformation is computed from corresponding parabolic contours on the shard and the generic model. A distance error metric is used to access the accuracy of alignment of a fragment to a given generic vessel model. The method is also extendable to surface markings. For a vessel that has no relief, color information or surface breaks can be used for the alignment. The method is tested on a subset of 3D scanned Independence National Historical Park (INDE) ceramic artifacts and the generic models created by experts. This work is part of an ongoing research activity in computational archeology that exploits many different tools and features to help in the mending process. Only the use of surface differential geometry information is reported here. This aspect is complementary to various other tools reported elsewhere by us and others, such as surface breaks, texture, color, etc.     

Progressive surface reconstruction for heart mapping procedure

The composite imaging of measured cardiac properties like electrical activation and contractile properties on a reconstructed endocardial surface allows for the diagnosis of cardiac arrhythmia and myocardial infarct. However, it is difficult for an interventionalist to acquire heart surface contacts by navigating a catheter to the desired region of interest under minimal visual aid. This paper discusses a new method for the progressive reconstruction of an endocardial surface during a heart mapping procedure. A generic mesh is first aligned with a set of anchor points to obtain a first approximation of the surface. Subsequent deformations are constrained by the preservation of local surface characteristics and the fidelity of new contact points. The mesh is refined by local subdivision and its geometrical shape is further improved by edge swapping. Compared to prior art, the new method can reconstruct a realistic surface from a set of sparse and random data. It can advantageously provide a smooth reconstruction at initial acquisition and ensure a geometrical consistency between consecutive reconstructions. The accurate reconstruction of a heart chamber provides important visual cues for an interventionalist to decide on the next mapping site, thus constructively influencing the final diagnosis.     

A surface reconstruction algorithm for topology optimization

For mechanical structural design, topology optimization is often utilized. During this process, a topologically optimized model must be converted into a parametric CAD solid model. The key point of conversion is that a discretized shape of a topologically optimized model must be smoothed, but features such as creases and corners must be retained. Thus, a surface reconstruction algorithm to produce the parametric CAD solid model from a topologically optimized model is proposed in this paper. Our presented algorithm consists of three parts: (1) an enclosed isosurface geometry from which the topologically optimized model is generated, (2) features detected and (3) the parametric CAD solid model reconstructed as biquartic surface splines. In order to generate an enclosed isosurface model effectively, we propose an algorithm based upon the marching cubes method to detect elements intersected by an isosurface. After generating an enclosed isosurface model, we produce biquartic surface splines. By applying our algorithm to an enclosed isosurface model, it is possible to produce smoothed biquartic surface splines with features retained. Some examples are shown and the effectiveness of our algorithm is discussed in this paper.     

Surface and normal ensembles for surface reconstruction

The majority of the existing techniques for surface reconstruction and the closely related problem of normal reconstruction are deterministic. Their main advantages are the speed and, given a reasonably good initial input, the high quality of the reconstructed surfaces. Nevertheless, their deterministic nature may hinder them from effectively handling incomplete data with noise and outliers. An ensemble is a statistical technique which can improve the performance of deterministic algorithms by putting them into a statistics based probabilistic setting. In this paper, we study the suitability of ensembles in normal and surface reconstruction. We experimented with a widely used normal reconstruction technique [Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W. Surface reconstruction from unorganized points. Computer Graphics 1992;71-8] and Multi-level Partitions of Unity implicits for surface reconstruction [Ohtake Y, Belyaev A, Alexa M, Turk G, Seidel H-P. Multi-level partition of unity implicits. ACM Transactions on Graphics 2003;22(3):463-70], showing that normal and surface ensembles can successfully be combined to handle noisy point sets.     

Efficient surface reconstruction method for distributed CAD

This paper describes a new fast Reverse Engineering (RE) method for creating a 3D computerized model from an unorganized cloud of points. The proposed method is derived directly from the problems and difficulties currently associated with remote design over the Internet, such as accuracy, transmission time and representation at different levels of abstraction. With the proposed method, 3D models suitable for distributed design systems can be reconstructed in real time. The mesh reconstruction approach is based on aggregating very large scale 3D scanned data into a Hierarchical Space Decomposition Model (HSDM), realized by the Octree data structure. Then, a Connectivity Graph (CG) is extracted and filled with facets. The HSDM can represent both the boundary surface and the interior volume of an object. Based on the proposed volumetric model, the surface reconstruction process becomes robust and stable with respect to sampling noise. Moreover, the data received from different surface/volume sampling devices can be handled naturally. The hierarchical structure of the proposed volumetric model enables data reduction, while preserving significant geometrical features and object topology. As a result, reconstruction and transmission over the network are efficient. Furthermore, the hierarchical representation provides a capability for extracting models at desired levels of detail, thus enabling designers to collaborate at any product development stage: draft or detailed design.     

Optimization-based approach for curve and surface reconstruction

This paper introduces an optimization-based approach for the curve reconstruction problem, where piecewise linear approximations are computed from sets of points sampled from target curves. In this approach, the problem is formulated as an optimization problem. To be more concrete, at first the Delaunay triangulation for the sample points is computed, and a weight is assigned with each Delaunay edge. Then the problem becomes minimization or maximization of the total weights of the edges that constitute the reconstruction. This paper proposes one exact method and two approximate methods, and shows that the obtained results are improved both theoretically and empirically. In addition, the optimization-based approach is further extended to three dimensions, where surfaces are to be reconstructed, and the quality of the reconstructions is examined.     

On compatible priors for Bayesian networks

Given a Bayesian network of discrete random variables with a hyper-Dirichlet prior, a method is proposed for assigning Dirichlet priors to the conditional probabilities of structurally different networks. It defines a distance measure between priors which is to be minimized for the assignment process. Intuitively one would expect that if two models priors are to qualify as being `close' in some sense, then their posteriors should also be nearby after an observation. However one does not know in advance what will be observed next. Thus we are led to propose an expectation of Kullback-Leibler distances over all possible next observations to define a measure of distance between priors. In conjunction with the additional assumptions of global and local independence of the parameters, a number of theorems emerge which are usually taken as reasonable assumptions in the Bayesian network literature. A simple example is given to illustrate the technique     

Localized Cocone surface reconstruction

We introduce a surface reconstruction algorithm suitable for large point sets. The algorithm is an octree-based version of the Cocone reconstruction algorithm [4], allowing independent processing of small subsets of the total input point set. When the points are sufficiently sampled from a smooth surface, the global guarantee of topological correctness of the original algorithm is preserved, together with the geometric accuracy guarantees.     

On the information-based complexity of optimal reconstruction for nonlinear systems

The information-based complexity of optimal reconstruction problems for general nonlinear systems is studied. Both lower and upper bounds for the worst-case reconstruction-error functional are given. The existence of an optimal reconstruction algorithm which achieves the infimum of the reconstruction-error is established.     

Higher-order communications for concurrent programming

In Pettorossi and Skowron (1983) a recursive-equations language is introduced. Its operational semantics is specified by means of computing agents which communicate and exchange messages. Those communications are, so to speak, zero-order, in the sense that the exchanged messages are values of a data structure, possibly defined by the programmer.

In this paper we extend that approach and we consider also ‘higher-order’ communications by allowing the exchange of agents behaviours, i.e. sets of computations, among computing agents. This extension leads to a new programming methodology which makes use of proofs of computing agents behaviours and their related strategies.     

Minimal area for surface reconstruction from cross sections

Surface reconstruction from cross-sectional data is important in a variety of applications. It is usually possible to generate a surface in many ways, but only reasonable ones are acceptable. A surface of minimal area has been considered as one of the most natural optimal criteria for the original tiling method of surface reconstruction from cross sections. In the paper, we consider minimal surfaces for continuous generalization of the tiling approach and in the general situation of reconstruction from cross sections. We show that in these cases the minimal area criterion leads to defective surfaces and is thus unacceptable. Published online: 23 July 2002 Correspondence to: D. Berzin     

A new mesh-growing algorithm for fast surface reconstruction

This paper presents a new high-performance method for triangular mesh generation based on a mesh-growing approach. Starting from a seed triangle, the algorithm grows the triangular mesh by selecting a new point based on the Gabriel 2—Simplex criterion. This criterion can be considered to be a good approximation of the 2D Delaunay if the point cloud is well-sampled and not too rough. The performance of the proposed method is compared with that of the Cocone family and that of Ball Pivoting as regards the tessellation rate and the quality of the surface being generated from some benchmark point clouds and artificially noised test cases. The results are analysed and critically discussed.