Real-time cross-correlation filtering of a one-dimensional grating position-encoded signal

Abstract

A cross-correlation technique is adopted and applied to the real-time signal filtering of a one-dimensional (1D) grating position-encoded signal scanned by a micro- or nano-probe. The position-encoded signal is used for both real-time displacement measurement and grating pitch measurement. The filtering technique applied is based on the 1D grating position-encoded signal which is passed over by a half period of sinusoidal waveform sequence as the template and is cross-correlated with it. Mathematical analysis and simulation experiment are performed not only to optimize the parameters of the template but also to prove the accuracy and credibility of real-time filtering when the template period is approximately equal to that of the 1D grating position-encoded signal. As an application, the motion displacements of a piezoelectric scanning stage are measured when it is controlled to move by a computer through its built-in capacitance sensor in open- and closed-loop modes respectively at different moving speeds.

Correlation and convolution filtering and image processing for pitch evaluation of 2D micro- and nano-scale gratings and lattices

“…Each group includes a constant item (I, J=0) and infinite numbers of 2D sinusoidal signals including a fundamental (I, J = 1) period Px and Py, and harmonic period Px/I and Py/J (I, J = 2, 3, 4,···). Since the amplitude AIJ decreases sharply with I and J increasing [25], the sinusoidal waveform in fundamental period (I,J=1) dominates equation (11). Concerning the square lattices with such 3D central symmetric features as shown in Fig.…”

“…Beside the center-of-gravity (CG) method [9,10] and Fouriertransform-based (FT) method [10], another pitch evaluation method of 1D gratings based on a 1D correlation filter has been previously published [11,12]. A half 1D sinusoidal waveform sequence with period PT is taken as a correlation filter.…”

“…From a half 1D sinusoidal waveform as 1D cross-correlation filter [11,12], when a half 2D sinusoidal waveform of Pq and Pr periods correlates or convolutes with the raster-scanned signal of a 2D sinusoidal grating of Px and Py periods with noise, we can deduce:…”

“…(a) and (b), equation(11) can be simplified as If a half 2D sinusoidal waveform with period Pq≈Px and Pr≈Py is used to filter a series of 2D sinusoidal signals of 2D lattice (with periods Px/I and Py/J expressed by equation(13)). The filtered 2D sinusoidal signal with fundamental periods (I, J=1) has the same period Px and Py.…”

A novel pitch evaluation of one-dimensional gratings based on a cross-correlation filter

“…In the previous investigation, real-time filtering of 1D sinusoidal grating position-encoded signal was performed [14]: a cross-correlation technique has been employed to filter a 1D grating position-encoded signal; a half sinusoidal waveform template was being proven to be very efficient and accurate to filter 1D sinusoidal grating-encoded signal by cross-correlating with it in real time. It is predicted that the cross-correlation filter, if applied for the pitch evaluation of the 1D arbitrarily structured grating, can achieve very similar accuracy to the FT method.…”