A comparison between minimum variance control and other online compensation methods for specimen drift in transmission electron microscopy

Abstract

Transmission electron microscopes (TEMs) are the tools of choice in materials science, semiconductor, and biological research and it is expected that they will be increasingly used to autonomously perform high-volume, repetitive, nano-measurements in the near future. Thus, there is a clear need to develop automation strategies for these microscopes. In particular, an important feature in need of automation is specimen drift compensation, which is a common cause of image blurring in long-exposure TEM images, especially at high magnifications. In this paper, a systematic online approach to specimen drift compensation, called adaptive minimum variance control, is discussed in detail. The method makes use of an identified drift model, continuously updated from online drift measurements, to predict and ameliorate future drift values, significantly reducing their variance. The method’s performance, measured in terms of drift variance reduction, is illustrated using both experimental and simulated data, and it is then compared with the performance of two pragmatic model-free methods: last data point prediction and linear extrapolation prediction.

Appendix: Determination of stationarity properties of time series

Determining a time series’ stationarity properties is a well established subject of study, specially in econometrics. The best understood stationarity tests are those by Phillips and Perron (1988) (PP) and Kwiatkowski et al. (1992) (KPSS), which assume the time series to be generated by either model (2), (7), or (8) (with $d=1$) (for tests without this assumption see, e.g., Basu et al. 2009). Both tests use hypothesis testing and can discern a stationary time series with constant mean from one with a linear trend or one with a unit root. Moreover, these tests are complementary, since they have opposite null hypotheses (KPSS’s is that the time series is stationary, while PP’s is that the series has a unit root), and can be used to reinforce each other’s conclusions.

Both tests were applied to the experimental drift realizations described in Sect. 4 in three steps, making use of the fact that differencing a process with deterministic trend does not yield a stationary process. Thus, in the first step the test were applied, with a 99 % significance level, to the second difference of each experimental drift realization. In each of the 20 cases, the PP null hypothesis was rejected while the KPSS null hypothesis could not be rejected. This strongly suggested that the experimental drift realizations were generated by either an ARMA$(p,q)$, ARIMA$(p,1,q)$, or ARIMA$(p,2,q)$ process and not by a process with deterministic trend. To eliminate the ARIMA$(p,2,q)$ possibility, in the second step the tests were applied to the the first difference of each experimental drift realization. The tests results, shown in Table 5, strongly suggest that the drift is not generated by an ARIMA$(p,2,q)$ process (since most first-difference series seem to be stationary). The highlighted results (red solid boxes) in this table indicate instances where the PP and KPSS tests contradict each other, so no decision can be made. However, in each of these instances the companion pair of tests (i.e., those shown in the same row, adjacent column) applied to the same experimental drift realizations do allow for a decision. Note that in most cases, both the constant level and linear trend KPSS give positive results, which seems to indicate the absence of a linear trend in the first-difference data. To make the final determination, in the third step the tests were applied to the experimental drift realizations directly. The test results are shown in Table 6. It follows immediately from this table that realizations $RL_0^x,\, RL_0^y,\, RL_1^x,\, RL_2^x,\, RL_2^y,\, RL_3^x,\, RL_3^y,\, RL_4^x,\, RL_4^y,\, U\!D_0^x$, and $U\!D_3^x$ seem to come from an ARIMA$(p,1,q)$ process with constant mean. The results provide contradictory information for realizations $U\!D_0^y,\, U\!D_1^x,\, U\!D_2^y$ (green shaded boxes) that seems to suggest that these series may be barely stationary (i.e., they may have a root very close to the unitary circle). The results also suggest that realizations $U\!D_1^y$ and $U\!D_4^y$ may come from a stationary process and, finally, that no final decision can be reached for realizations $RL_1^y,\, U\!D_2^y,\, U\!D_3^y$, and $U\!D_4^x$ (black dashed boxes) based on the results of these tests (although Table 5 suggests that there is no loss in generality in assuming these realizations came from an ARIMA$(p,1,q)$ process). In conclusion, most realizations seem to come from an ARIMA$(p,1,q)$ process.

Table 5 Results of the PP and KPSS stationarity tests applied to the first difference of the experimental drift realizations (see text for details) (Color table online)

Table 6 Results of the PP and KPSS stationarity tests applied to the experimental drift realizations directly (see text for details) (Color table online)