Surface smoothing and partial spatial derivatives computation using a regressive discrete Fourier series

Abstract

The smoothing of surfaces which can be described as scalar functions of a two-dimensional domain is classical in image processing. Smoothing techniques can be used in experimental modal analysis applications to smooth spatially dense measured operating shapes. When transformation filtering techniques are used, a mathematical model of the smoothed surface is obtained. The mathematical model of the out-of-plane displacement operating shape surface can be used to calculate in-plane angular displacements, thus increasing by threefold the number of degrees-of-freedom obtained from the experimental measurements. In this paper, a surface smoothing method consisting of estimating by least-squares the coefficients of a two-dimensional discrete Fourier series with arbitrary period and arbitrary frequency resolution is presented. It is shown that the proposed method called Regressive Discrete Fourier Series (RDFS) minimises the leakage problem and can be used with non-equally-spaced and non-rectangular-domain data. The RDFS method is compared to a more classical approach consisting of using the two-dimensional discrete Fourier transform (DFT). With the Fourier series model it is straightforward to calculate partial derivatives. Numerical results are shown to illustrate the proposed method and to compare it to the two-dimensional DFT method, which was enhanced with spline padding to reduce leakage.