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Statistical estimation of the structural similarity index for image quality assessment

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Abstract

The structural similarity (SSIM) index has been studied from different perspectives in the last decade. Most of the developments consider its parameters fixed. Because each of these parameters corresponds to the weight of a factor in the final SSIM coefficient, the usual assumption that all parameters are equal to one is questionable. In this article, a new estimation method is proposed from a statistical perspective. The approach we develop is a model-based estimation method so that the usual assumption that all parameters are equal to one can be handled via approximate hypothesis-testing techniques that are properly developed in the context of regression. The method considers nonlinear models with multiplicative noise to explain the root mean square error as a function of the SSIM index. A numerical experiment based on a Monte Carlo simulation is carried out to test whether the parameters are all equal to one and to gain more insight into the performance of the estimates in practice. Our analysis showed that the assumption that the parameters are equal to one is not supported by the data and may lead to a misconception of the closeness between two images.

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Notes

  1. https://www.mathworks.com/matlabcentral/answers/9217-need-ssim-m-code.

  2. https://gist.github.com/Bibimaw/8873663.

  3. URL: https://github.com/faosorios/SSIM.

  4. URL: https://www.iceye.com/downloads/datasets.

  5. URL:  http://sipi.usc.edu/database.

References

  1. Mittal, A., Moorthy, A.K., Bovik, A.C.: No-reference image quality assessment in the spatial domain. IEEE Trans. Image Process. 21, 4695–4708 (2012)

    Article  MathSciNet  Google Scholar 

  2. Wang, K., Yong, B., Gu, X., Xiao, P., Zhang, X.: Spectral similarity measure using frequency spectrum for hyperspectral image classification. IEEE Geosci. Remote Sens. 12, 130–134 (2015)

    Article  Google Scholar 

  3. Wang, Z., Bovik, A.C.: A universal image quality index. IEEE Signal Proc. Lett. 9, 81–84 (2002)

    Article  Google Scholar 

  4. Zhang, L., Zhang, L., Mou, X., ZHang, D.: FSIM: A feature similarity index for image quality assessment. IEEE Trans. Image Process. 20, 2378–2386 (2011)

  5. Brunet, D., Vrscay, E.R., Wang, Z.: On the mathematical properties of the structural similarity index. IEEE Trans. Image Process. 21, 1488–1499 (2012)

    Article  MathSciNet  Google Scholar 

  6. Vallejos, R., Mancilla, D., Acosta, J.: Image similarity assessment based on coefficients of spatial association. J. Math. Imaging Vis. 56, 77–98 (2016)

    Article  MathSciNet  Google Scholar 

  7. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)

    Article  Google Scholar 

  8. Sheikh, H.R., Bovik, C.: Image information and visual quality. IEEE Trans. Image Process. 15, 430–444 (2006)

    Article  Google Scholar 

  9. Chandler, D.M., Hemami, S.S.: VSNR: A wavelet-based visual signal-to-noise-ratio for natural images. IEEE T. Image Process. 16, 2284–2298 (2007)

    Article  MathSciNet  Google Scholar 

  10. Ma, K., Zeng, K., Wang, Z.: Perceptual quality assessment for multi-exposure image fusion. IEEE Trans. Image Process. 24, 3345–3356 (2015)

    Article  MathSciNet  Google Scholar 

  11. Wang, Y.K., Li, L., Zhou, X.Y., Cui, T.J.: Supervised automatic detection of UWB ground-penetrating radar targets using the regression SSIM measure. IEEE Geosci. Remote Sens. 13, 621–625 (2016)

    Article  Google Scholar 

  12. Ojeda, S., Vallejos, R., Lamberti, P.: Measure of similarity between images based on the codispersion coefficient. J. Electron. Imaging 21, 023019 (2012)

  13. Wang, H., Maldonado, D., Silwal, S.: A nonparametric-test-based structural similarity measure for digital images. Comput. Stat. Data Anal. 55, 2925–2936 (2011)

    Article  MathSciNet  Google Scholar 

  14. Rehuman, A., Wang, Z.: Reduced-reference SSIM estimation. In: Proceedings of 2010 IEEE 17th International Conference on Image Processing. Hong Kong, pp. 26–29 (2010)

  15. Wang, Z., Li, L., Wu, S., Xia, Y., Wan, Z., Cai, C.: A new image quality assessment algorithm based on SSIM and multiple regressions. Int. J. Signal Process. 8, 221–230 (2015)

    Google Scholar 

  16. Dosselmann, R., Yang, X.D.: A comprehensive assessment of the structural similarity index. SIViP 5, 81–91 (2011)

    Article  Google Scholar 

  17. Yeo, C., Tan, H.L., Tan, Y.H.: On rate distortion optimization using SSIM. IEEE Trans. Circuits Syst. Video 23, 1170–1181 (2013)

    Article  Google Scholar 

  18. Kim, S., Pak, D., Lee, S.: SSIM-based distortion metric for film grain noise in HEVC. SIViP 12, 489–496 (2018)

    Article  Google Scholar 

  19. Davidian, M., Carroll, R.J.: Variance function estimation. J. Am. Stat. Assoc. 82, 1079–1091 (1987)

    Article  MathSciNet  Google Scholar 

  20. Frery, A.C., Müller, H.J., Yanasse, C.C.F., Sant’Anna, S.J.S.: A model for extremely heterogeneous clutter. IEEE Trans. Geosci. Remote 35, 648–659 (1997)

    Article  Google Scholar 

  21. Goldfeld, S.M., Quandt, R.E.: Nonlinear Methods in Econometrics. North-Holland, Amsterdam (1972)

    MATH  Google Scholar 

  22. Seber, G.A.F., Wild, C.J.: Nonlinear Regression. Wiley, New York (1988)

    MATH  Google Scholar 

  23. Cribari-Neto, F., Frery, A.C., Silva, M.F.: Improved estimation of clutter properties in speckled imagery. Comput. Stat. Data Anal. 40, 801–824 (2002)

    Article  MathSciNet  Google Scholar 

  24. Carroll, R.J., Ruppert, D.: Transformation and Weighting in Regression. Chapman and Hall, New York (1988)

    Book  Google Scholar 

  25. Brent, R.P.: Algorithms for Minimization without Derivatives. Dover, New York (1973)

    MATH  Google Scholar 

  26. Nash, J.C.: Compact Numerical Methods for Computers. Linear Algebra and Function Minimization. Adam Hilger, Bristol (1990)

    MATH  Google Scholar 

  27. Gorieroux, C., Monfort, A.: Statistics and Econometrics Models: General Concepts, Estimation, Prediction, and Algortihms. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  28. Terrell, G.R.: The gradient statistic. Comput. Sci. Stat. 34, 206–215 (2002)

    Google Scholar 

  29. Osorio, F., Vallejos, R.: SpatialPack: Tools for assessment the association between two spatial processes. R package version 0.3-8196. CRAN.R-project.org/package=SpatialPack (2020)

  30. Vallejos, R., Osorio, F., Bevilacqua, M.: Spatial Relationships Between Two Georeferenced Variables: With Applications in R. Springer, Cham (2020)

    Book  Google Scholar 

  31. Loader, C., Pilla, R.S.: Iteratively reweighted generalized least squares for estimation and testing with correlated data: an inference function framework. J. Computat. Graph. Stat. 16, 925–945 (2007)

    Article  MathSciNet  Google Scholar 

  32. Buonaccorsi, J.P.: Measurement error in the response in the general linear model. J. Am. Stat. Assoc. 91, 633–642 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the suggestions and comments from the Associate Editor and anonymous referees that led to a significant improvement of the manuscript. Felipe Osorio and Ronny Vallejos acknowledge financial support from CONICYT through the MATH-AMSUD program, Grant 20-MATH-03, from UTFSM, Grants PI_LI_19_11 and P_LIR_2020_20. Ronny Vallejos was also partially funded by the Advanced Center for Electrical and Electronic Engineering (AC3E), Grant FB-0008. Silvia M. Ojeda and Marcos Landi were supported by Secretaría de Ciencia y Tecnología (SeCyT) from Universidad Nacional de Córdoba. Proyecto Consolidar 2018-2121 (P.I.D. No. 33620180100055CB) y CIEM (Córdoba), CONICET.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile

    Felipe Osorio & Ronny Vallejos

  2. U-Planner, Avenida Tupungato 3850, Curauma, Valparaíso, Chile

    Wilson Barraza

  3. Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria (X5000HUA), Córdoba, Argentina

    Silvia María Ojeda

  4. Instituto de Diversidad y Ecología Animal, CONICET y Facultad de Ciencias Exactas Físicas y Naturales, Universidad Nacional de Córdoba, 5016, Córdoba, Argentina

    Marcos Alejandro Landi

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  1. Felipe Osorio
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  2. Ronny Vallejos
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  3. Wilson Barraza
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  4. Silvia María Ojeda
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  5. Marcos Alejandro Landi
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Corresponding author

Correspondence to Felipe Osorio.

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Osorio, F., Vallejos, R., Barraza, W. et al. Statistical estimation of the structural similarity index for image quality assessment. SIViP 16, 1035–1042 (2022). https://doi.org/10.1007/s11760-021-02051-9

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  • DOI: https://doi.org/10.1007/s11760-021-02051-9

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