Reconstruction of the surface f(x, y) when only partial derivative/partial derivative xf(x, y) and partial derivative/partial derivative yf(x, y) are known

摘要

This is an investigation into the reconstruction of three-dimensional surfaces f(x, y) when only the first derivatives, or slopes, in the x- and y-directions, respectively partial derivative/partial derivative xf(x, y) and partial derivative/partial derivative yf(x, y), are known at sampling points (x, y). The algorithms are based upon Taylor's series expansion of a function, and a simple phase corrected low-pass filter obtained by the Fourier coefficient filter design and an appropriate window. These algorithms produce essentially surface profiles. Using only Taylor's series expansion results in a phase-corrected algorithm that can reconstruct surface profiles successfully as long as the minimum surface wavelength is longer than 10 sampling intervals. When shorter surface wavelengths occur a low-pass filter has to be introduced. The influence of noise is also investigated. It is shown that by applying the surface profile reconstruction algorithm in the forward and reverse directions, one can obtain an estimate of the overall influence of noise. This enables the user to accept or reject a result. [References: 2]