Error caused by sampling in Fourier transform profilometry

Abstract

The phase error caused by sampling the fringe patterns in practical Fourier transform profilometry is discussed. In the FTP method, a grating pattern is projected onto an object surface and the deformed fringe is Fourier-transformed and processed in its spatial frequency domain as well as in its space-signal domain. It has been stated that the fundamental spectrum must separate from zero and higher order spectra, because only the fundamental component is needed for phase retrieval. Here we discuss another kind of spectrum overlapping that is caused by the discrete Fourier transform (DFT). The DFT of a digitized fringe pattern results in periodically equi-spaced frequency islands in the spectrum domain. The higher order spectra from adjacent islands may overlap the fundamental spectrum that is filtered out for phase reconstruction. This kind of overlapping also introduces noise into the reconstructed phase. The conclusion is that to obtain a correct reconstruction of the measured object, the fundamental spectrum does not alias with zero order or higher order spectra in the same frequency island nor with the higher order spectra from adjacent frequency islands. Theoretical analysis is given and the criteria of selecting sampling frequency are discussed. Computer simulations and experiments verified our theories.

Error analysis about CCD sampling in Fourier transform profilometry

“…In the first step, the actual sensor structure (the size of light-sensitive areas), which is ignored generally, has to be considered. In this paper, we focus on discussing the influence of CCD sampling in FTP, especially the influence from the lightsensitive area size variation, and deduce the mathematical description of spectra considering the size of sensor-sensitive area, which is closer to practical case than the previous sampling analysis [4]. Some experimental demonstrations verify our analysis.…”

“…Fourier transform profilometry (FTP) is one of the popular 3-D sensing methods, in which, a grating pattern is projected onto an object and the deformed fringe captured by CCD camera is Fourier transformed and processed in its frequency domain as well as in its space domain. After Takeda et al [1], the FTP method has been extensively studied [2][3][4][5] focusing on how to improve the measurement range and accuracy of the method. Due to the discrete characteristic of DFT algorithm, the Fourier spectrum of the discrete fringe is made of the infinitely replicated spectra of the original continuous fringe.…”

Nanocomposites obtained by embedding of conjugated polymers in porous silicon and silica

“…Conjugated polymers are well known for their large third-order optical nonlinearity [1,2], which make s them excellent candidates for optical modulators, couplers, as well as for all optical switches [3]. Nevertheless, such components have to satisfy high standards concerning for instance, mechanical rigidity and molecule orientation that those organic materials generally do not have intrinsically.…”

Measurement of the third order nonlinear properties of conjugated polymers embedded in porous silicon and silica

“…In order to enhance nonlinear properties even more, pores can be filled up with other more nonlinear materials. Conjugated polymers as poly(diacetylene-bis-toluene-sulfonide) (PTS), poly(3-dodecyl thiophene) (PT12), and poly(p phenylene vinylene) (PPV) present large third-order optical nonlinearity [3,4]. However, potential use of such polymers in signal processing, due to their very promising nonlinear optical properties, is conditioned by high standards concerning optical quality, homogeneity and mechanical rigidity which are missing intrinsically [5].…”