Integration in the Fourier domain for restoration of a function from its slope: comparison of four methods

Abstract

In some measurement techniques the profile, f ͑x͒, of a function should be obtained from the data on measured slope f 0 ͑x͒ by integration. The slope is measured in a given set of points, and from these data we should obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods [Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987)] that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function in the Fourier domain, by which the most-accurate interpolation is automatically carried out. Analysis of the integration methods in the Fourier domain permits us to easily study and compare the methods' behavior.

Improving wavefront reconstruction accuracy by using integration equations with higher-order truncation errors in the Southwell geometry

“…For example, using the method, we can deduce the first three Newton-Cotes quadrature rules, which are the trapezoidal, the Simpson, and the 3/8 Simpson rules [11,12]. The trapezoidal rule is given by Eq.…”

“…The third IE (IE3) contains four adjacent function values and four adjacent slopes that are symmetrical at the same intermediate point (2.5), and given by 11 27…”

“…Much literature has been published in the past concerning these two factors. The discrete Fourier transform (DFT)-based integration algorithms compute the wavefront values in the frequency domain [8][9][10][11][12][13][14]. Fast speed can be achieved for DFT-based algorithms using fast Fourier transforms (FFTs), since the FFTs need only On log 2 n operations [15], where n is the number of wavefront values to be calculated.…”

“…The IE used in Southwell's LS-based integration algorithm is the first Newton-Cotes quadrature rule (the trapezoidal rule) [11,12], which indicates the finite difference relationship between the function values and the slopes at two adjacent points in one dimension. As will be shown in Section 2.A, this IE has the truncation error (TE) of Oh 3 (where h is the interval of the two adjacent points).…”

Frequency responses and resolving power of numerical integration of sampled data

“…This deviation contains the slope data information of the profile of the test surface. For this purpose, we begin investigating a one-dimensional integration operation [9,10].…”