Editing Poisson's equation (section)

=== Surface reconstruction ===
Surface reconstruction is an [[inverse problem]]. The goal is to digitally reconstruct a smooth surface based on a large number of points ''p<sub>i</sub>'' (a [[point cloud]]) where each point also carries an estimate of the local [[surface normal]] '''n'''<sub>''i''</sub>.<ref>{{cite journal |first1=Fatih |last1=Calakli |first2=Gabriel |last2=Taubin |title=Smooth Signed Distance Surface Reconstruction |journal=Pacific Graphics |year=2011 |volume=30 |number=7 |url=http://mesh.brown.edu/ssd/pdf/Calakli-pg2011.pdf }}</ref> Poisson's equation can be utilized to solve this problem with a technique called Poisson surface reconstruction.<ref name="Kazhdan06">{{cite book |first1=Michael |last1=Kazhdan |first2=Matthew |last2=Bolitho |first3=Hugues |last3=Hoppe |year=2006 |chapter=Poisson surface reconstruction |title=Proceedings of the fourth Eurographics symposium on Geometry processing (SGP '06) |publisher=Eurographics Association, Aire-la-Ville, Switzerland |pages=61–70 |isbn=3-905673-36-3 |chapter-url=https://dl.acm.org/doi/abs/10.5555/1281957.1281965 }}</ref>

The goal of this technique is to reconstruct an [[implicit function]] ''f'' whose value is zero at the points ''p<sub>i</sub>'' and whose gradient at the points ''p<sub>i</sub>'' equals the normal vectors '''n'''<sub>''i''</sub>. The set of (''p<sub>i</sub>'', '''n'''<sub>''i''</sub>) is thus modeled as a continuous [[Euclidean vector|vector]] field '''V'''.  The implicit function ''f'' is found by [[Integral|integrating]] the vector field '''V'''. Since not every vector field is the [[gradient]] of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field '''V''' to be the gradient of a function ''f'' is that the [[Curl (mathematics)|curl]] of '''V''' must be identically zero. In case this condition is difficult to impose, it is still possible to perform a [[least-squares]] fit to minimize the difference between '''V''' and the gradient of ''f''.

In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field '''V'''. The basic approach is to bound the data with a [[finite-difference]] grid. For a function valued at the nodes of such a grid, its gradient can be represented as valued on staggered grids, i.e. on grids whose nodes lie in between the nodes of the original grid. It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data. On each staggered grid we perform [[trilinear interpolation]] on the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of ''n<sub>i</sub>'' onto the nodes of the particular staggered grid cell containing ''p<sub>i</sub>''. Kazhdan and coauthors give a more accurate method of discretization using an adaptive finite-difference grid, i.e. the cells of the grid are smaller (the grid is more finely divided) where there are more data points.<ref name="Kazhdan06"/> They suggest implementing this technique with an adaptive [[octree]].