# A Robust-to-Noise Deconvolution Algorithm to Enhance Displacement and Strain Maps Obtained with Local DIC and LSA

## Abstract

Digital Image Correlation (DIC) and Localized Spectrum Analysis (LSA) are two techniques available to extract displacement fields from images of deformed surfaces marked with contrasted patterns. Both techniques consist in minimizing the optical residual. DIC performs this minimization iteratively in the real domain on random patterns such as speckles. LSA performs this minimization nearly straightforwardly in the Fourier domain on periodic patterns such as grids or checkerboards. The particular case of local DIC performed pixelwise is considered here. In this case and regardless of noise, local DIC and LSA both provide displacement fields equal to the actual one convolved by a kernel known a priori. The kernel corresponds indeed to the Savitzky-Golay filter in local DIC, and to the analysis window of the windowed Fourier transform used in LSA. Convolution reduces the noise level, but it also causes actual details in displacement and strain maps to be returned with a damped amplitude, thus with a systematic error. In this paper, a deconvolution method is proposed to retrieve the actual displacement and strain fields from their counterparts given by local DIC or LSA. The proposed algorithm can be considered as an extension of Van Cittert deconvolution, based on the small strain assumption. It is demonstrated that it allows enhancing fine details in displacement and strain maps, while improving the spatial resolution. Even though noise is amplified after deconvolution, the present procedure can be considered as robust to noise, in the sense that off-the-shelf deconvolution algorithms do not converge in the presence of classic levels of noise observed in strain maps. The sum of the random and systematic errors is also lower after deconvolution, which means that the proposed procedure improves the compromise between spatial resolution and measurement resolution. Numerical and real examples considering deformed speckle images (for DIC) and checkerboard images (for LSA) illustrate the efficiency of the proposed approach.

## Keywords

Checkerboard Digital image correlation Displacement Deconvolution Full-field measurement Grid method Localized spectrum analysis Metrology Periodic pattern Speckle Strain## References

- 1.Réthoré J (2010) A fully integrated noise robust strategy for the identification of constitutive laws from digital images. Int J Numer Methods Eng 84(6):631–660zbMATHGoogle Scholar
- 2.Schreier HW, Sutton M (2002) Systematic errors in digital image correlation due to undermatched subset shape functions. Exp Mech 42(3):303–310Google Scholar
- 3.Savitzky A, Golay MJE (1964) Smoothing and differentiation of data by simplified least-squares procedures. Anal Chem 36(3):1627–1639Google Scholar
- 4.Sur F, Grédiac M (2014) Towards deconvolution to enhance the grid method for in-plane strain measurement. Inverse Probl Imaging 8(1):259–291. American Institute of Mathematical SciencesMathSciNetzbMATHGoogle Scholar
- 5.Grédiac M, Blaysat B, Sur F (2017) A critical comparison of some metrological parameters characterizing local digital image correlation and grid method. Exp Mech 57(3):871–903Google Scholar
- 6.Sutton M, Orteu JJ, Schreier H (2009) Image Correlation for Shape, Motion and Deformation Measurements. Basic Concepts, Theory and Applications. Springer, BerlinGoogle Scholar
- 7.Pan B, Lu Z, Xie H (2010) Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation. Opt Lasers Eng 48(4):469–77Google Scholar
- 8.Neggers J, Blaysat B, Hoefnagels JPM, Geers MGD (2016) On image gradients in digital image correlation. Int J Numer Methods Eng 105(4):243–260MathSciNetGoogle Scholar
- 9.Surrel Y, Zhao B (1994) Simultaneous u-v displacement field measurement with a phase-shifting grid method. In: Stupnicki J, Pryputniewicz RJ (eds) Proceedings of the SPIE, the International Society for Optical Engineering, vol 2342. SPIEGoogle Scholar
- 10.Bomarito GF, Hochhalter JD, Ruggles TJ, Cannon AH (2017) Increasing accuracy and precision of digital image correlation through pattern optimization. Opt Lasers Eng 91:73–85Google Scholar
- 11.Hytch MJ, Snoeck E, Kilaas R (1998) Quantitative measurement of displacement and strain fields from HREM micrographs. Ultramicroscopy 74:131–146Google Scholar
- 12.Zhu RH, Xie H, Dai X, Zhu J, Jin A (2014) Residual stress measurement in thin films using a slitting method with geometric phase analysis under a dual beam (fib/sem) system. Measur Sci Technol 25(9):095003Google Scholar
- 13.Dai X, Xie H, Wang H, Li C, Liu Z, Wu L (2014) The geometric phase analysis method based on the local high resolution discrete Fourier transform for deformation measurement. Measur Sci Technol 25(2):025402Google Scholar
- 14.Dai X, Xie H, Wang H (2014) Geometric phase analysis based on the windowed Fourier transform for the deformation field measurement. Opt Laser Technol 58(6):119–127Google Scholar
- 15.Grédiac M, Sur F, Blaysat B (2016) The grid method for in-plane displacement and strain measurement: a review and analysis. Strain 52(3):205–243Google Scholar
- 16.Kemao Q (2004) Windowed Fourier transform for fringe pattern analysis. Appl Opt 43(13):2695–2702Google Scholar
- 17.Kemao Q, Wang H, Gao W (2010) Windowed fourier transform for fringe pattern analysis: theoretical analyses. Appl Opt 47(29):5408–5419Google Scholar
- 18.Grédiac M, Blaysat B, Sur F (2018) Extracting displacement and strain fields from checkerboard images with the localized spectrum analysis. Exp Mech. AcceptedGoogle Scholar
- 19.Sur F, Grédiac M (2016) Influence of the analysis window on the metrological performance of the grid method. J Math Imaging Vis 56(3):472–498MathSciNetzbMATHGoogle Scholar
- 20.Starck JL, Pantin E, Murtagh F (2002) Deconvolution in astronomy: a review. Publ Astron Soc Pac 114(800):1051–1069Google Scholar
- 21.Gonzalez RC, Woods RE (2006) Digital Image Processing, 3rd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
- 22.Grédiac M, Sur F, Badulescu C, Mathias J-D (2013) Using deconvolution to improve the metrological performance of the grid method. Opt Lasers Eng 51(6):716–734Google Scholar
- 23.Murtagh F, Pantin E, Starck J-L (2007) Deconvolution and blind deconvolution in astronomy. In: Campisi P, Egiazarian K (eds) Blind image deconvolution: theory and applications. Taylor and Francis, pp 277–316Google Scholar
- 24.Sagaut P (2002) Structural modeling. In: Large Eddy simulation for incompressible flows: an introduction. Springer, Berlin, pp 183–240Google Scholar
- 25.Sur F, Blaysat B, Grédiac M (2018) Rendering deformed speckle images with a Boolean model. J Math Imaging Vis 60(5):634–650MathSciNetzbMATHGoogle Scholar
- 26.Reu P (2014) All about speckles: Aliasing. Exp Tech 38(5):1–3Google Scholar
- 27.Grédiac M, Sur F (2014) Effect of sensor noise on the resolution and spatial resolution of the displacement and strain maps obtained with the grid method. Strain 50(1):1–27. Paper invited for the 50th anniversary of the journalGoogle Scholar
- 28.Lehoucq RB, Reu PL, Turner DZ (2017) The effect of the ill-posed problem on quantitative error assessment in digital image correlation. Experimental Mechanics, Accepted, onlineGoogle Scholar
- 29.Schafer RW (2011) What is a Savitzky-Golay filter? (lecture notes). IEEE Signal Proc Mag 28(4):111–117Google Scholar
- 30.Grafarend EW (2006) Linear and nonlinear models: Fixed Effects, Random Effects, and Mixed Models. Walter de Gruyter, RoslynGoogle Scholar
- 31.Piro JL, Grédiac M (2004) Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods. Exp Tech 28(4):23–26Google Scholar
- 32.Sur F, Blaysat B, Grédiac M (2016) Determining displacement and strain maps immune from aliasing effect with the grid method. Opt Lasers Eng 86:317–328Google Scholar
- 33.Pierron F, Grédiac M (2012) The virtual fields method. Springer, Berlin. 517 pages, ISBN 978-1-4614-1823-8zbMATHGoogle Scholar
- 34.Richardson WH (1972) Bayesian-based iterative method of image restoration. J Opt Soc Am 62(1):55–59MathSciNetGoogle Scholar
- 35.Lucy LB (1974) An iterative technique for the rectification of observed distributions. Astron J 79(6):745–754Google Scholar
- 36.International vocabulary of metrology. Basic and general concepts and associated terms (2008) Third editionGoogle Scholar
- 37.Chrysochoos A, Surrel Y (2012) Chapter 1. Basics of metrology and introduction to techniques Grédiac M, Hild F (eds), WileyGoogle Scholar
- 38.Bornert M, Brémand F, Doumalin P, Dupré J-c, Fazzini M, Grédiac M, Hild F, Mistou S, Molimard J, Orteu J-J, Robert L, Surrel Y, Vacher P, Wattrisse B (2009) Assessment of digital image correlation measurement errors: methodology and results. Exper Mech 49(3):353–370Google Scholar
- 39.Wittevrongel L, Lava P, Lomov SV, Debruyne D (2015) A self adaptive global digital image correlation algorithm. Exp Mech 55(2):361–378Google Scholar
- 40.Blaber J, Adair B, Antoniou A (2015) Ncorr: open-source 2d digital image correlation matlab software. Experimental Mechanics. https://doi.org/10.1007/s11340-015-0009-1
- 41.Badulescu C, Bornert M, Dupré J-c, Equis S, Grédiac M, Molimard J, Picart P, Rotinat R, Valle V (2013) Demodulation of spatial carrier images: Performance analysis of several algorithms. Exper Mech 53(8):1357–1370Google Scholar
- 42.Wang YQ, Sutton M, Bruck H, Schreier HW (2009) Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements. Strain 45(2):160–178Google Scholar
- 43.Su Y, Zhang Q, Gao Z, Xu X, Wu X (2015) Fourier-based interpolation bias prediction in digital image correlation. Opt Express 23(15):19242–19260Google Scholar
- 44.Su Y, Zhang Q, Xu X, Gao Z (2016) Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors. Opt Lasers Eng 86:132–142Google Scholar
- 45.Bomarito GF, Hochhalter JD, Ruggles TJ (2017) Development of optimal multiscale patterns for digital image correlation via local grayscale variation. Experimental Mechanics, accepted, onlineGoogle Scholar
- 46.Surrel Y (1994) Moiré and grid methods: a signal-processing approach. In: Stupnicki J Pryputniewicz RJ, editor, Interferometry’94: photomechanics. SPIE, vol 2342Google Scholar
- 47.Surrel Y (2000) Photomechanics, Topics in Applied Physic, volume 77, chapter Fringe Analysis. Springer, pp 55–102Google Scholar