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Computational modeling of spectral data fitting with nonlinear distortions

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Abstract

Substances such as chemical compounds and biological agents are invisible to human eyes, they are usually captured by sensing equipments with their spectral fingerprints. Although spectra of pure substances can be identified by visual inspection, the spectra of their mixtures take a variety of complicated forms. Given the knowledge of spectral references of the constituent substances, the task of data fitting is to solve their weights, which usually can be solved by a least squares. Complications occur if the basis functions (reference spectra) may not be used directly to best fit the data. In fact, random spectral distortions such as shifting, compression, and expansion have been observed in some spectra. In this paper, we formulate mathematical model for such distortions and build them into data fitting algorithms. If minimal knowledge of the distortions is available, a deterministic approach termed augmented least squares is developed by fitting the spectral references together with their derivatives to the mixtures. If the distribution of the distortions is known a prior, we propose to solve the problem with maximum likelihood estimators which incorporate the distortions into the variance matrix. The proposed methods are substantiated with numerical examples including data from Raman spectroscopy, nuclear magnetic resonance, and differential optical absorption spectroscopy and show satisfactory results.

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References

  1. Zare, A., Ho, K.C.: Endmember variability in hyperspectral analysis: addressing spectral variability during spectral unmixing. IEEE Signal Process. Mag. 31(1), 95–104 (2014)

    Article  Google Scholar 

  2. Roberts, D.A., Gardner, M., Church, R., Ustin, S., Scheer, G., Green, R.O.: Mapping Chaparral in the Santa Monica Mountains using multiple endmember spectral mixture models. Remote Sens. Environ. 65, 267–279 (1998)

    Article  Google Scholar 

  3. Manolakis, D., Lockwood, R., Cooley, T., Jacobson, J.: Robust matched filters for target detection in hyperspectral imaging data. In: ICASSP 2007. IEEE International Conference, vol. 1, pp. I-529–I-532 (2007)

  4. Guo, Z., Osher, S.: Template matching via \(\ell _1\) minimization and its application to hyperspectral data. Inverse Probl. Imaging 5(1), 19–35 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Song, C.: Spectral mixture analysis for subpixel vegetation fractions in the urban environment: how to incorporate endmember variability? Remote Sens. Environ. 95(2), 248–263 (2005)

    Article  Google Scholar 

  6. Zhang, Y.J.: Adaptive region-based image fusion using energy evaluation model for fusion decision. SIViP 1(3), 215–223 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tonazzini, A., Savino, P., Salerno, E.: A non-stationary density model to separate overlapped texts in degraded documents. SIViP 9(Suppl. 1), 155–164 (2015)

    Article  Google Scholar 

  8. Thouvenin, P.-A., Dobigeon, N., Tourneret, J.-Y.: Hyper spectral unmixing with spectral variability using a perturbed linear mixing model. In: 23rd European Signal Processing Conference (EUSIPCO), Nice, France, Aug.31–Sept. 4, pp. 819–823 (2015)

  9. Stutz, J., Platt, U.: Numerical analysis and estimation of the statistical error of differential optical absorption spectroscopy measurements with least squares methods. Appl. Opt. 35, 6041–6053 (1996)

    Article  Google Scholar 

  10. Esser, E.: A convex model of image registration. UCLA CAM Report 10–04 (2010)

  11. http://grassmann.math.colostate.edu/ATD/home.html

Download references

Acknowledgements

The authors thank Professor Barbara Finlayson-Pitts and Dr. Lisa Wingen for the experimental DOAS data, and Naval Research Lab for the Raman Data. J. Xin acknowledges support of NSF Grant DMS-1222507. Y. Sun thanks support of Grant 800006981 from Simons Foundation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

    Yuanchang Sun & Wensong Wu

  2. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Jack Xin

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  1. Yuanchang Sun
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  2. Wensong Wu
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  3. Jack Xin
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Correspondence to Yuanchang Sun.

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Sun, Y., Wu, W. & Xin, J. Computational modeling of spectral data fitting with nonlinear distortions. SIViP 11, 651–658 (2017). https://doi.org/10.1007/s11760-016-1006-2

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  • DOI: https://doi.org/10.1007/s11760-016-1006-2

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