Enriching LiDAR data with partial derivatives and its uncertainty estimates

After some manipulations, raw LiDAR data can be filtered so as to describe the terrain surface or bare ground. The resulting set is composed of (x,y,z) triplets, with some uncertainty in all three dimensions. We want to describe a procedure able to enrich this 3D cloud adding also estimates of partial derivatives with its uncertainty, thus becoming a 7D point. Our theoretical framework assumes that the point cloud is itself error free, and that the uncertainty in the partial derivatives estimate will arise because of the finite spacing between points. Computing the enriched cloud will be a one-time operation. The resulting dataset could be used in many ways, including improving elevation estimates by using higher order Hermite interpolants intended to exploit the derivative values. The uncertainty information might be propagated to the interpolated elevation. The envisioned procedure has contacts with numerical methods for solving partial differential equations. Finite Difference methods rely on a regular grid like the one provided by standard DEMs to produce estimates of the partial derivatives which are functions of the grid size. Finite element methods use a non-regular grid, and its derivative estimates can also be related to some norm of the elements, defined through elaborated topological relationships. Meshless methods use cloud of points not linked in elements, an approach which offers enough flexibility to deal with selected problems in fluid mechanics, where moving boundaries and free surface precludes using rigid meshes. Tools already developed to estimate partial derivatives and its uncertainties can be conveniently reused with the LiDAR cloud dataset, even without a partial differential equation problem involved.