Surface Reconstruction

A point cloud, however dense, is never quite the surface it describes. Between any two measured points the surface is unmeasured, and reconstruction makes a principled guess at what lies between. The guess matters: a volume computed on an open mesh can be off by tens of cubic metres, a 3D-printed replica of a heritage object fails to print where the algorithm left a hole, and a finite-element analysis on a non-watertight model returns garbage.

Reconstruction sits late in the pipeline and inherits everything done before it. It consumes the registered cloud of Chapter 8, cleaned of outliers by the filters of Chapter 6, and its most capable methods depend on the consistently oriented normals of Chapter 7, while the terrain meshes built here rest on the ground classification of Chapter 9. Its output feeds the parametric object models of Chapter 16 and the application workflows of Chapter 17, and the signed distance fields met near the end of the chapter return as the foundation of the neural surfaces of Chapter 19. The chapter works through the interpolating methods first (Delaunay triangulation, alpha shapes, ball pivoting), then the approximating Poisson family, then the Marching Cubes extraction and the signed distance fields that implicit methods rely on, and closes with a selection guide and the post-processing that every mesh needs.

The difficulty is that the guess must be made from poor materials. The input points are noisy, irregularly spaced, and may contain outliers, while the true surface may carry sharp edges, thin features, or topological complexity such as holes and tunnels. Every method in this chapter is a different answer to that mismatch, and the taxonomy we follow comes from the survey of Berger et al. (2017), extended with the screened Poisson formulation of Kazhdan and Hoppe (2013).

From points to surfaces: the reconstruction pipeline

The first fork in the road is the one stated in the definition above. Interpolating exactness reproduces the measurements faithfully, noise included. Approximating smoothness tolerates noise and closes gaps, at the cost of moving the measured points. The figure below maps the methods of this chapter onto the data conditions under which each is preferred: on the interpolating side, alpha shapes suit sparse data with simple topology, ball pivoting suits dense clean scans, and Delaunay-based or greedy projection methods serve when explicit control of the topology is needed. On the approximating side, screened Poisson is the default whenever oriented normals are available, Marching Cubes extracts a mesh from any signed distance field, and TSDF fusion with Marching Cubes covers the real-time GPU case.

Whichever branch you take, every method in this chapter shares one pipeline. The input must be cleaned and augmented with normals before reconstruction can succeed, and the output almost always requires post-processing before it is fit for downstream use. The figure below shows the common sequence.