Abstract

In this paper, a pretreatment method for a matrix in convex optimization is proposed to optimize the spectral reconstruction process of a disordered dispersion spectrometer. Unlike the reconstruction process of traditional spectrometers using Fourier transforms, the reconstruction process of disordered dispersion spectrometers involves solving a large-scale matrix equation. However, since the matrices in the matrix equation are obtained through measurement, they contain uncertainties due to out of band signals, background noise, rounding errors, temperature variations and so on. It is difficult to solve such a matrix equation by using ordinary nonstationary iterative methods, owing to instability problems. Although the smoothing Tikhonov regularization approach has the ability to approximatively solve the matrix equation and reconstruct most simple spectral shapes, it still suffers the limitations of reconstructing complex and irregular spectral shapes that are commonly used to distinguish different elements of detected targets with mixed substances by characteristic spectral peaks. Therefore, we propose a special pretreatment method for a matrix in convex optimization, which has been proved to be useful for reducing the condition number of matrices in the equation. In comparison with the reconstructed spectra gotten by the previous ordinary iterative method, the spectra obtained by the pretreatment method show obvious accuracy.

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Other (22)

J. M. Pearce and R. A. Komoroski2015Analysis of phospholipid molecular species in brain by 31P NMR spectroscopyMag. Resonan. Med.44215223

Y. Zhou, C. H. Liu, Y. Pu, B. Wu, T. A. Nguyen, G. Cheng, L. Zhou, K. Zhu, J. Chen, Q. Li, and R. R. Alfano2019Combined spatial frequency spectroscopy analysis with visible resonance Raman for optical biopsy of human brain metastases of lung cancersJ. Innovative Opt. Health Sci.121950010

A. Kuze, H. Suto, M. Nakajima, and T. Hamazaki2009Thermal and near infrared sensor for carbon observation Fourier-transform spectrometer on the greenhouse gases observing satellite for greenhouse gases monitoringAppl. Opt.4867166733

C. Hu, L. Feng, Z. Lee, C. O. Davis, A. Mannino, C. R. McClain, and B. A. Franz2012Dynamic range and sensitivity requirements of satellite ocean color sensors: learning from the pastAppl. Opt.5160456062

P. Mouroulis, B. V. Gorp, R. O. Green, H. Dierssen, D. W. Wilson, M. Eastwood, J. Boardman, B.-C. Gao, D. Cohen, B. Franklin, F. Loya, S. Lundeen, A. Mazer, I. McCubbin, D. Randall, B. Richardson, J. I. Rodriguez, C. Sarture, E. Urquiza, R. Vargas, V. White, and K. Yee2014Portable remote imaging spectrometer coastal ocean sensor: design, characteristics, and first flight resultsAppl. Opt.5313631380

Y. Zhou, Q. W. Zhang, X. J. Luo, P. F. Li, H. Song, and B. L. Zhang2014Identification of some piper crude drugs based on Fourier transform infrared spectrometrySpectrosc. Spectral Anal.3424192423

M. Zhang, J. Liang, Z. Liang, J. Lv, Y. Qin, and W. Wang2018Fabrication and flatness error analysis of a low-stepped mirror in a static Fourier transform infrared spectrometerJ. Opt. Technol.85582589

R. A. Soref, F. D. Leonardis, V. M. N. Passaro, and Y. Fainman2018On-chip digital Fourier-transform spectrometer using a thermo-optical Michelson grating interferometerJ. Lightwave Technol.3651605167

T. Yang, J.-X. Peng, X.-A. Li, X. Shen, X.-H. Zhou, X.-L. Huang, W. Huang, and H.-P. Ho2017Compact broadband spectrometer based on upconversion and downconversion luminescenceOpt. Lett.4243754378

T. Yang, C. Xu, H.-P. Ho, Y.-Y. Zhu, X.-H. Hong, Q.-J. Wang, Y.-C. Chen, X.-A. Li, X.-H. Zhou, M.-D. Yi, and W. Huang2015Miniature spectrometer based on diffraction in a dispersive hole arrayOpt. Lett.4032173220

M. Tostado-Véliz, S. Kamel, and F. Jurado2019Development of different load flow methods for solving large-scale ill-conditioned systemsInt. Trans. Electr. Energ. Syst.29e2784

K. Ito, B. Jin, and T. Takeuchi2014Multi-parameter Tikhonov regularization - an augmented approachChin. Annal. Math.35B383398

J. Cheng and B. HofmannO. Scherzer2015Regularization Methods for Ill-Posed ProblemsHandbook of Mathematical Methods in Imaging2nd edSpringerNY, USA87109

V. P. Zhuravlev2016Ill-posed problems in mechanicsMech. Solids51538541

X. T. Xiong, X. Y. Fan, and M. Li2015Spectral method for ill-posed problems based on the balancing principleInverse Probl. Sci. Eng.23292306

J. J. Liu, G. Q. He, and C. G. Kang2009Nonlinear implicit iterative method for solving nonlinear ill-posed problemsAppl. Math. Mech.3011831192

M. Tanaka and K. Nakata2014Positive definite matrix approximation with condition number constraintOptim. Lett.8939947

X. Wu2007Error analysis for the predictor-corrector process relating to ill-conditioned linear system of equationsAppl. Math. Comput.186530534

H. Li and S. Wang2018On the partial condition numbers for the indefinite least squares problemAppl. Numer. Math.123200220

N. H. Godoge and E. Gionfriddo2019A critical outlook on recent developments and applications of matrix compatible coatings for solid phase microextractionTrends Anal. Chem.111220228

D. A. Guimaraes, G. H. F. Floriano, and L. S. Chaves2015A tutorial on the CVX system for modeling and solving convex optimization problemsIEEE Lat. Am.Trans.1312281257

A. B. Malinowska2006Nonessential objective functions in linear multi-objective optimization problemsControl Cybern.35873880

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