Zonal reconstruction of surfaces from gradient fields

Abstract

A method is given to reconstruct a surface from measured surface gradient values located on a two-dimensional, square grid using a zonal method. Reconstruction is done square by square, using a surface elevation value calculated when processing the previous neighboring square as a reference elevation value and then calculating the difference from that value for the other three locations of the square. The curved surface area element defined by a grid square is, in general, a tilted, toric surface that is decomposed into four component surfaces; a tilted plane, a surface of revolution, and two cross-cylinder surfaces. Only the tilted plane and the oblique cross-cylinder surface contribute to elevation differences at the square corners so only they need to be included in the calculations. This unique use of available information allows the surface to be reconstructed without the iterative steps that are generally needed in zonal reconstruction methods. Methods to deal with boundary conditions and with measurement noise are also included in the implementation of the method.