Abstract
Imprecise measurements present universally due to variability in the measurement error. We devised a very simple membership function to evaluate fuzzily the quality of optical sensing with a small dataset, where a normal distribution cannot be assumed. The proposed membership function was further used as a weighting function for non-linear curve fitting under expected mathematical model constraints, namely the membership function-weighted Levenberg–Marquardt (MFW-LM) algorithm. The robustness and effectiveness of the MFW-LM algorithm were demonstrated by an optical-sensing simulation and two practical applications. (1) In laser-absorption spectroscopy, molecular spectral line modeling was greatly improved by the method. The measurement uncertainty of temperature and pressure were reduced dramatically, by 53.3% and 43.5%, respectively, compared with the original method. (2) In imaging, a laser beam-profile reconstruction from heavy distorted observations was improved by the method. As the dynamic range of the infrared camera increased from 256 to 415, the detailed resolution of the laser-beam profiles increased by an amazing 360%, achieving high dynamic-range imaging to capture optical signal details. Therefore, the MFW-LM algorithm provides a robust and effective tool for fitting a proper physical model and precision parameters from low-quality data.
Highlights
Curve fitting is one of the most powerful and most widely used analysis tools to pre- and post-process data [1,2], to remove outliers [3,4], to compare candidate models [5], and to examine the relationship between one or more predictors [6,7,8]
The membership function-weighted Levenberg–Marquardt (MFW-LM) algorithm is extremely robust and effective in minimizing the contribution of outliers and distorted signals and accurately reconstructs the laser-beam profile when dealing with optical sensing with small datasets; that is, the proposed MFW-LM algorithm is suitable for curve fitting that requires high-precision measurement
To improve the accuracy of optical absorption spectroscopy analysis, which is often frustrated by imperfect response, interference, and electronic noise [24,25,36,37,38] in signal acquisition and processing, we applied the MFW-LM algorithm to molecular absorption line modeling of H2 O and compared the measurement uncertainty of the temperature and pressure calculated by the fitting results of the MFW-LM and the LM algorithms
Summary
Curve fitting is one of the most powerful and most widely used analysis tools to pre- and post-process data [1,2], to remove outliers [3,4], to compare candidate models [5], and to examine the relationship between one or more predictors [6,7,8]. Outliers present universally due to variability in measurement error, including a transient malfunction of apparatuses, errors in data transmission or transcription, changes in system behavior, operation error, instrument error or through natural deviations, and a flaw in the assumed theory These outliers always lower the goodness of fit and the precision of fitted parameters. Optical sensing is often performed in situ or in online applications, where small datasets are common for the variability of working conditions or fast response time These results in outliers buried in observations being more difficult to distinguish from low-quality data distorted by non-linear transmission, interference, electrical noise, dark current fluctuation (DCF), and dark-current non-uniformity and photoresponse non-uniformity between pixels for optical image sensors [21,22,23]. (1) laser-absorption-spectroscopy analysis and (2) laser-beam-profile measurement
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