Abstract
Interpolation is a big topic in its own right. In this chapter, we discuss how interpolation can be applied to surface reconstruction. The simplest case is when we have a set of points in 2D with associated height values. Assuming these points are scattered (i.e. not aligned with a grid) a good way of interpolating the height values is using radial basis functions (RBF). Using the RBF method, we can obtain a smooth surface from points plus height. The RBF method has the attractive property that it is simple to implement, since it reduces to solving a single (albeit dense) linear system.
Reconstructing 3D surfaces is more challenging, but again the RBF method can be applied if we have points which are known to be both inside and outside as well as on the surface. The result is an implicit representation of the reconstructed object where the object is the set of points that have value zero with respect to the function produced by the RBF method.
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Notes
- 1.
Since w is also a function of point distance, one might say that the functions used in Shepard’s method are also “radial”. Sometimes nomenclature can be a bit misleading.
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Bærentzen, J.A., Gravesen, J., Anton, F., Aanæs, H. (2012). Surface Reconstruction using Radial Basis Functions. In: Guide to Computational Geometry Processing. Springer, London. https://doi.org/10.1007/978-1-4471-4075-7_16
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DOI: https://doi.org/10.1007/978-1-4471-4075-7_16
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