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

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 finite-element 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, higher-quality surface reconstructions.

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Screened poisson surface reconstruction

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.This idea leads to a simple algorithm with theoretical guarantees comparable to those of other surface reconstruction and medial axis approximation algorithms. It also comes with a guarantee that does not depend in any way on the quality of the input point sample. Any input gives an output surface which is the `watertight' boundary of a three-dimensional polyhedral solid: the solid described by the approximate MAT. This unconditional guarantee makes the algorithm quite robust and eliminates the polygonalization, hole-filling or manifold extraction post-processing steps required in previous surface reconstruction algorithms.In this paper, we use the theory to develop a power crust implementation which is indeed robust for realistic and even difficult samples. We describe the careful design of a key subroutine which labels parts of the MAT as inside or outside of the object, easy in theory but non-trivial in practice. We find that we can handle areas in which the input sampling is scanty or noisy by simply discarding the unreliable parts of the MAT approximation. We demonstrate good empirical results on inputs including models with sharp corners, sparse and unevenly distributed point samples, holes, and noise, both natural and synthetic.We also demonstrate some simple extensions: intentionally leaving holes where there is no data, producing approximate offset surfaces, and simplifying the approximate MAT in a principled way to preserve stable features.

The power crust

Automatic reconstruction of as-built building information models from laser-scanned point clouds: A review of related techniques

Building information models (BIMs) are maturing as a new paradigm for storing and exchanging knowledge about a facility. BIMs constructed from a CAD model do not generally capture details of a facility as it was actually built. Laser scanners can be used to capture dense 3D measurements of a facility's as-built condition and the resulting point cloud can be manually processed to create an as-built BIM — a time-consuming, subjective, and error-prone process that could benefit significantly from automation. This article surveys techniques developed in civil engineering and computer science that can be utilized to automate the process of creating as-built BIMs. We sub-divide the overall process into three core operations: geometric modeling, object recognition, and object relationship modeling. We survey the state-of-the-art methods for each operation and discuss their potential application to automated as-built BIM creation. We also outline the main methods used by these algorithms for representing knowledge about shape, identity, and relationships. In addition, we formalize the possible variations of the overall as-built BIM creation problem and outline performance evaluation measures for comparing as-built BIM creation algorithms and tracking progress of the field. Finally, we identify and discuss technology gaps that need to be addressed in future research.

/pdf/automatic-reconstruction-of-as-built-building-information-58eqr9ulay.pdf

Automatic Reconstruction of As-Built Building Information Models from Laser-Scanned Point Clouds: A Review of Related Techniques | NIST

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 sufficiently 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 polyhedral solid, so we avoid the polygonization, 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, converges to the medial axis.

/pdf/the-power-crust-unions-of-balls-and-the-medial-axis-mhmmyz0jdt.pdf

The power crust, unions of balls, and the medial axis transform

We present an algorithm for computing the exact interior medial axis of a union of balls in R d . Our algorithm combines the simple characterization of this medial axis given by Attali and Montanvert with the combinatorial information provided by Edelsbrunner's α-shape. This leads to a simple algorithm, which we have implemented for d=3.

The Medial Axis of a Union of Balls.

We present an algorithm for computing the exact interior medial axis of a union of balls in R d  Our algorithm combines the simple characterization of this medial axis given by Attali and Montanvert with the combinatorial information provided by Edelsbrunner's α-shape This leads to a simple algorithm, which we have implemented for d=3

/pdf/the-medial-axis-of-a-union-of-balls-3e8hea4y8g.pdf

The Medial axis of a union of balls

Given a sample of points from the boundary of an object in IR 3, we construct a representation of the object as a union of balls. We use many fewer balls than previous constructions, but our shape representation is better. We bound the distance from the surface of the union to the original object surface, and show that when the sampling is sufficiently dense the two are homeomorphic. This implies a topological relationship between the true medial axis of the object and both the medial axis, and the a-shape, of the union of balls. We show that the set of ball centers in our construction converges to the true medial axis as the sampling density increases.

/pdf/accurate-and-efficient-unions-of-balls-1znb6l3rtg.pdf

Ravi Krishna Kolluri

Papers

The power crust

The power crust, unions of balls, and the medial axis transform

The Medial Axis of a Union of Balls.

The Medial axis of a union of balls

Accurate and efficient unions of balls