In some measurement techniques the profile, <i>f</i>(<i>x</i>), of a function should be obtained from the data on measured slope <i>f</i><sup>′</sup>(<i>x</i>) 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 <i>et al., Numerical Recipes: The Art of Scientific Computing</i> (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.
© 2002 Optical Society of America
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure
(200.3050) Optics in computing : Information processing
J. Campos, L. P. Yaroslavsky, A. Moreno, and M. J. Yzuel, "Integration in the Fourier domain for restoration of a function from its slope: comparison of four methods," Opt. Lett. 27, 1986-1988 (2002)