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70
Voronoibased Variational Reconstruction of Unoriented Point Sets
, 2007
"... We introduce an algorithm for reconstructing watertight surfaces from unoriented point sets. Using the Voronoi diagram of the input point set, we deduce a tensor field whose principal axes and eccentricities locally represent respectively the most likely direction of the normal to the surface, and t ..."
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Cited by 62 (10 self)
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We introduce an algorithm for reconstructing watertight surfaces from unoriented point sets. Using the Voronoi diagram of the input point set, we deduce a tensor field whose principal axes and eccentricities locally represent respectively the most likely direction of the normal to the surface, and the confidence in this direction estimation. An implicit function is then computed by solving a generalized eigenvalue problem such that its gradient is most aligned with the principal axes of the tensor field, providing a bestfitting isosurface reconstruction. Our approach possesses a number of distinguishing features. In particular, the implicit function optimization provides resilience to noise, adjustable fitting to the data, and controllable smoothness of the reconstructed surface. Finally, the use of simplicial meshes (possibly restricted to a thin crust around the input data) and (an)isotropic Laplace operators renders the numerical treatment simple and robust.
Consolidation of Unorganized Point Clouds for Surface Reconstruction
"... We consolidate an unorganized point cloud with noise, outliers, nonuniformities, and in particular interference between closeby surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optim ..."
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Cited by 47 (10 self)
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We consolidate an unorganized point cloud with noise, outliers, nonuniformities, and in particular interference between closeby surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optimal projection operator produces a set of denoised, outlierfree and evenly distributed particles over the original dense point cloud, so as to improve the reliability of local PCA for initial estimate of normals. Next, an iterative framework for robust normal estimation is introduced, where a prioritydriven normal propagation scheme based on a new priority measure and an orientationaware PCA work complementarily and iteratively to consolidate particle normals. The priority setting is reinforced with front stopping at thin surface features and normal flipping to enable robust handling of the closeby surface sheet problem. We demonstrate how a point cloud that is wellconsolidated by our method steers conventional surface generation schemes towards a proper interpretation of the input data. 1
Manifold reconstruction in arbitrary dimensions using witness complexes
 In Proc. 23rd ACM Sympos. on Comput. Geom
, 2007
"... It is a wellestablished fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, und ..."
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Cited by 39 (11 self)
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It is a wellestablished fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling assumptions. Unfortunately, these results do not extend to higherdimensional manifolds, even under stronger sampling conditions. In this paper, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between both complexes still hold on higherdimensional manifolds. We also use our structural results to devise an algorithm that reconstructs manifolds of any arbitrary dimension or codimension at different scales. The algorithm combines a farthestpoint refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample W to be sparse. Its time complexity is bounded by c(d)W  2, where c(d) is a constant depending solely on the dimension d of the ambient space. 1
Screened Poisson Surface Reconstruction
"... 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 Poi ..."
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Cited by 36 (1 self)
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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 finiteelement 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, higherquality surface reconstructions.
Triangulating Point Set Surfaces with Bounded Error
, 2005
"... We introduce an algorithm for constructing a highquality triangulation directly from Point Set Surfaces. Our algorithm requires no intermediate representation and no postprocessing of the output, and naturally handles noisy input data, typically in the form of a set of registered range scans. It c ..."
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Cited by 29 (6 self)
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We introduce an algorithm for constructing a highquality triangulation directly from Point Set Surfaces. Our algorithm requires no intermediate representation and no postprocessing of the output, and naturally handles noisy input data, typically in the form of a set of registered range scans. It creates a triangulation where triangle size respects the geometry of the surface rather than the sampling density of the range scans. Our technique does not require normal information, but still produces a consistent orientation of the triangles, assuming the sampled surface is an orientable twomanifold. Our work is based on using Moving LeastSquares (MLS) surfaces as the underlying representation. Our technique is a novel advancing front algorithm, that bounds the Hausdorff distance to within a userspecified limit. Specifically, we introduce a way of augmenting advancing front algorithms with global information, so that triangle size adapts gracefully even when there are large changes in surface curvature. Our results show that our technique generates highquality triangulations where other techniques fail to reconstruct the correct surface due to irregular sampling on the point cloud, noise, registration artifacts, and underlying geometric features, such as regions with high curvature gradients.
Reconstruction using witness complexes
 In Proceedings 18th ACMSIAM Symposium: Discrete Algorithms
, 2007
"... We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a oneparameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable differen ..."
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Cited by 20 (9 self)
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We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a oneparameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions. 1
Cone Carving for Surface Reconstruction
"... We present cone carving, a novel space carving technique supporting topologically correct surface reconstruction from an incomplete scanned point cloud. The technique utilizes the point samples not only for local surface position estimation but also to obtain global visibility information under th ..."
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Cited by 14 (2 self)
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We present cone carving, a novel space carving technique supporting topologically correct surface reconstruction from an incomplete scanned point cloud. The technique utilizes the point samples not only for local surface position estimation but also to obtain global visibility information under the assumption that each acquired point is visible from a point lying outside the shape. This enables associating each point with a generalized cone, called the visibility cone, that carves a portion of the outside ambient space of the shape from the inside out. These cones collectively provide a means to better approximate the signed distances to the shape specifically near regions containing large holes in the scan, allowing one to infer the correct surface topology. Combining the new distance measure with conventional RBF, we define an implicit function whose zero level set defines the surface of the shape. We demonstrate the utility of cone carving in coping with significant missing data and raw scans from a commercial 3D scanner as well as synthetic input.
A Streaming Algorithm for Surface Reconstruction
, 2007
"... We present a streaming algorithm for reconstructing closed surfaces from large nonuniform point sets based on a geometric convection technique. Assuming that the sample points are organized into slices stacked along one coordinate axis, a triangle mesh can be efficiently reconstructed in a streamab ..."
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Cited by 9 (0 self)
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We present a streaming algorithm for reconstructing closed surfaces from large nonuniform point sets based on a geometric convection technique. Assuming that the sample points are organized into slices stacked along one coordinate axis, a triangle mesh can be efficiently reconstructed in a streamable layout with a controlled memory footprint. Our algorithm associates a streaming 3D Delaunay triangulation datastructure with a multilayer version of the geometric convection algorithm. Our method can process millions of sample points at the rate of 50k points per minute with 350 MB of main memory.