Abstract A number of techniques have been developed for reconstructing surfaces by integrating groups of aligned range images. A desirable set of properties for such algorithms includes: incremental updating, representation of directional uncertainty, the ability to fill gaps in the reconstruction, and robustness in the presence of outliers. Prior algorithms possess subsets of these properties. In this paper, we present a volumetric method for integrating range images that possesses all of these properties. Our volumetric representation consists of a cumulative weighted signed distance function. Working with one range image at a time, we first scan-convert it to a distance function, then combine this with the data already acquired using a simple additive scheme. To achieve space efficiency, we employ a run-length encoding of the volume. To achieve time efficiency, we resample the range image to align with the voxel grid and traverse the range and voxel scanlines synchronously. We generate the final manifold by extracting an isosurface from the volumetric grid. We show that under certain assumptions, this isosurface is optimal in the least squares sense. To fill gaps in the model, we tessellate over the boundaries between regions seen to be empty and regions never observed. Using this method, we are able to integrate a large number of range images (as many as 70) yielding seamless, high-detail models of up to 2.6 million triangles.

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We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in R³. The algorithm is the first for this problem with provable guarantees. Given a “good sample” from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling density increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points. Our algorithm is based on the three-dimensional Voronoi diagram. Given a good program for this fundamental subroutine, the algorithm is quite easy to implement.

We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on "local feature size", the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We describe an implementation of the algorithm and show example outputs. 1 Introduction The problem of reconstructing a surface from scattered sample points arises in many applications such as computer graphics, medical imaging, and cartography. In this paper we consider the specific reconstruction problem in which the input is a set of sample points S drawn from a smooth two-dimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...

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.

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The power crust is a construction which takes a sample of points from the surface of a three-dimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT.

The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a three-dimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewise-linear approximation to the object surface, which we call the power crust. We assume that we are given as input a suficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls representation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a solid, so we avoid the hole-filling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewise-linear dual representations, which in some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes to infinity, the set of poles, the centers of the polar balls, converge to the medial axis.

In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, and/or curves, and/or surface patches. We do not assume any knowledge of the ordering, connectivity or topology of the data sets or of the true shape. Only the location of each point or general Hausdorff distance to the data set is known. The key idea in our approach is to find an implicit nonparametric representation of the curve or surface on a fixed rectangular grid. With this representation of surfaces we can easily (a) find the closest point and distance from any point to the surface (useful in illumination and many other applications), (b) find the intersection curve of two surfaces which is guaranteed to lie on both surfaces in our representation, and (c) perform any Boolean operatio...

In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surface-like model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly non-uniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.