{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:06:04Z","timestamp":1760241964368,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,3]],"date-time":"2018-11-03T00:00:00Z","timestamp":1541203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Scientific Instruments and Equipment Development of China","award":["2012YQ06016501"],"award-info":[{"award-number":["2012YQ06016501"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>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\u2013Marquardt (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.<\/jats:p>","DOI":"10.3390\/s18113762","type":"journal-article","created":{"date-parts":[[2018,11,5]],"date-time":"2018-11-05T10:43:45Z","timestamp":1541414625000},"page":"3762","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Membership Function-Weighted Non-Linear Fitting Method for Optical-Sensing Modeling and Reconstruction"],"prefix":"10.3390","volume":"18","author":[{"given":"Shuo","family":"Meng","sequence":"first","affiliation":[{"name":"State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3102-8773","authenticated-orcid":false,"given":"Zhenhui","family":"Du","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China"}]},{"given":"Liming","family":"Yuan","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China"}]},{"given":"Shuanke","family":"Wang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China"}]},{"given":"Ruiyan","family":"Han","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China"}]},{"given":"Xiaoyu","family":"Wang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,3]]},"reference":[{"key":"ref_1","unstructured":"Barajas, L.G., and Xiao, G. (2008). Curve Fitting for Signal Estimation, Prediction, and Parametrization. (7,324,924), U.S. Patent."},{"key":"ref_2","first-page":"159","article-title":"Application of piece-wise curve fitting to post-processing of synchronous distances","volume":"3","author":"Ma","year":"2004","journal-title":"Tech. Acoust."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1016\/S0169-7161(05)80059-8","article-title":"24 Identifying Outliers and influential Observations in Econometric Models","volume":"Volume 11","author":"Donald","year":"1993","journal-title":"Handbook of Statistics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/0169-2070(95)00625-7","article-title":"Outliers in Statistical Data","volume":"12","author":"Ord","year":"1996","journal-title":"Int. J. Forecast."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"S177","DOI":"10.1365\/s10337-003-0182-8","article-title":"Comparison of Curve Fitting Models for Ligand Binding Assays","volume":"59","author":"Little","year":"2004","journal-title":"Chromatographia"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Motulsky, H., and Christopoulos, A. (2004). Fitting Models to biological Data Using Linear and Nonlinear Regression: A Practical Guide to Curve Fitting, Oxford University Press.","DOI":"10.1093\/oso\/9780195171792.001.0001"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Salkind, N.J. (2010). Encyclopedia of Research Design, Sage Publications Ltd.","DOI":"10.4135\/9781412961288"},{"key":"ref_8","unstructured":"Arlinghaus, S. (1994). Practical Handbook of Curve Fitting, CRC Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.matcom.2017.12.007","article-title":"Analytical investigations for the design of fast approximation methods for fitting curves and surfaces to scattered data","volume":"147","author":"Brakhage","year":"2018","journal-title":"Math. Comput. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.bspc.2016.06.009","article-title":"A modified approach to objective surface generation within the Gauss-Newton parameter identification to ignore outlier data points","volume":"30","author":"Gray","year":"2016","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Glynn, J.E. (1989). Numerical Recipes: The Art of Scientific Computing, Cambridge University Press.","DOI":"10.1016\/0098-3004(89)90134-9"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1090\/qam\/10666","article-title":"A Method for the Solution of Certain Problems in Least Squares","volume":"2","author":"Levenberg","year":"1944","journal-title":"Q. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/0111030","article-title":"An algorithm for least-squares estimation of nonlinear parameters","volume":"11","author":"Marquardt","year":"1963","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.knosys.2012.05.003","article-title":"Hybridization of evolutionary Levenberg\u2013Marquardt neural networks and data pre-processing for stock market prediction","volume":"35","author":"Asadi","year":"2012","journal-title":"Knowl.-Based Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.protcy.2013.12.157","article-title":"A New Levenberg Marquardt based Back Propagation Algorithm Trained with Cuckoo Search","volume":"11","author":"Nawi","year":"2013","journal-title":"Procedia Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"015204","DOI":"10.1088\/0957-0233\/24\/1\/015204","article-title":"An improved Levenberg\u2013Marquardt algorithm for extracting the features of Brillouin scattering spectrum","volume":"24","author":"Zhang","year":"2013","journal-title":"Meas. Sci. Technol."},{"key":"ref_17","first-page":"573","article-title":"A new statistic for detecting outliers in exponential case","volume":"32","author":"Shadrokh","year":"2010","journal-title":"Aust. J. Basic Appl. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1080\/00401706.1972.10488930","article-title":"Rejecting Outliers in Factorial Designs","volume":"14","author":"Stefansky","year":"1972","journal-title":"Technometrics"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"459","DOI":"10.18637\/jss.v016.i03","article-title":"Programs to Compute Distribution Functions and Critical Values for Extreme Value Ratios for Outlier Detection","volume":"16","author":"Mcbane","year":"2006","journal-title":"J. Stat. Softw."},{"key":"ref_20","first-page":"38","article-title":"Peirce\u2019s criterion for the elimination of suspect experimental data","volume":"20","author":"Ross","year":"2003","journal-title":"J. Eng. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1109\/TIM.2002.807803","article-title":"Modeling, calibration, and correction of nonlinear illumination-dependent fixed pattern noise in logarithmic CMOS image sensors","volume":"51","author":"Joseph","year":"2002","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9452","DOI":"10.3390\/s91209452","article-title":"Non-Linearity in Wide Dynamic Range CMOS Image Sensors Utilizing a Partial Charge Transfer Technique","volume":"9","author":"Shafie","year":"2009","journal-title":"Sensors"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"C02012","DOI":"10.1088\/1748-0221\/10\/02\/C02012","article-title":"Non-linear responsivity characterisation of a CMOS Active Pixel Sensor for high resolution imaging of the Jovian system","volume":"10","author":"Soman","year":"2015","journal-title":"J. Instrum."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1063\/1.4935920","article-title":"Modulation index optimization for optical fringe suppression in wavelength modulation spectroscopy","volume":"86","author":"Xiong","year":"2015","journal-title":"Rev. Sci. Instrum."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Li, J., Luo, G., Du, Z., and Ma, Y. (2017). Hollow Waveguide Enhanced Dimethyl Sulfide Sensor Based on a 3.3 \u03bcm Interband Cascade Laser. Sens. Actuators B Chem., 3550\u20133557.","DOI":"10.1016\/j.snb.2017.09.190"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1186\/1475-925X-4-50","article-title":"Removal of power-line interference from the ECG: A review of the subtraction procedure","volume":"4","author":"Levkov","year":"2005","journal-title":"Biomed. Eng. Online"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_28","unstructured":"Ren, H.J. (2009). About the Accuracy of the Electric Instrument Reading. J. Mianyang Normal Univ., 39\u201342."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1017\/S0373463313000131","article-title":"Image Processing Algorithms For Deep-Space Autonomous Optical Navigation","volume":"66","author":"Li","year":"2013","journal-title":"J. Navig."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1088\/0266-5611\/23\/3\/002","article-title":"Levenberg\u2013Marquardt methods for parameter estimation problems in the radiative transfer equation","volume":"23","author":"Feng","year":"2007","journal-title":"Inverse Probl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s00607-009-0055-6","article-title":"Distance regression by Gauss\u2013Newton-type methods and iteratively re-weighted least-squares","volume":"86","author":"Aigner","year":"2009","journal-title":"Computing"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00340-012-4891-2","article-title":"Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic\/dielectric waveguides","volume":"108","author":"Patimisco","year":"2012","journal-title":"Appl. Phys. B Lasers Opt."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1329","DOI":"10.3390\/s130101329","article-title":"Low-Loss Hollow Waveguide Fibers for Mid-Infrared Quantum Cascade Laser Sensing Applications","volume":"13","author":"Patimisco","year":"2013","journal-title":"Sensors"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3114","DOI":"10.1364\/AO.51.003114","article-title":"Loss and modal properties of Ag\/AgI hollow glass waveguides","volume":"51","author":"Bledt","year":"2012","journal-title":"Appl. Opt."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Patimisco, P., Sampaolo, A., Mihai, L., Giglio, M., Kriesel, J., Sporea, D., Scamarcio, G., Tittel, F.K., and Spagnolo, V. (2016). Low-Loss Coupling of Quantum Cascade Lasers into Hollow-Core Waveguides with Single-Mode Output in the 3.7\u20137.6 m Spectral Range. Sensors, 16.","DOI":"10.3390\/s16040533"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"19140","DOI":"10.1063\/1.4955411","article-title":"Dynamic spectral characteristics measurement of DFB interband cascade laser under injection current tuning","volume":"109","author":"Du","year":"2016","journal-title":"Appl. Phys. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.jqsrt.2017.03.027","article-title":"Ultrahigh-resolution spectroscopy for methyl mercaptan at the \u03bd 2-band by a distributed feedback interband cascade laser","volume":"196","author":"Du","year":"2017","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_38","unstructured":"Li, J., Du, Z., Ma, Y., and Liu, J. (July, January 28). Indoor carbon dioxide monitoring with diode laser absorption at 2 \u03bcm. Proceedings of the International Society for Optics and Photonics, Shanghai, China."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/j.snb.2018.04.035","article-title":"In situ, multiparameter optical sensor for monitoring the selective catalytic reduction process of diesel engines","volume":"267","author":"Du","year":"2018","journal-title":"Sens. Actuators B Chem."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/S0022-4073(02)00377-1","article-title":"Wavelength modulation diode laser absorption signals from Doppler broadened absorption profiles","volume":"83","author":"Kluczynski","year":"2004","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.infrared.2018.04.014","article-title":"Resolving multiple combustion parameters from a single transition using degraded deconvolution of Voigt lineshape","volume":"92","author":"Li","year":"2018","journal-title":"Infrared Phys. Technol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4058","DOI":"10.1364\/AO.26.004058","article-title":"The HITRAN database: 1986 edition","volume":"26","author":"Rothman","year":"1987","journal-title":"Appl. Opt."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/11\/3762\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:27:50Z","timestamp":1760196470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/11\/3762"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,3]]},"references-count":42,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["s18113762"],"URL":"https:\/\/doi.org\/10.3390\/s18113762","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2018,11,3]]}}}