# American Institute of Mathematical Sciences

February  2014, 8(1): 259-291. doi: 10.3934/ipi.2014.8.259

## Towards deconvolution to enhance the grid method for in-plane strain measurement

 1 LORIA - projet Magrit, Université de Lorraine, Cnrs, Inria, Umr 7503, Campus Scientifique BP 239, 54506 Vanduvre-lès-Nancy cedex, France 2 Institut Pascal, Clermont Université, Cnrs Umr 6602, Université Blaise Pascal BP 10448, 63000 Clermont-Ferrand, France

Received  November 2012 Revised  June 2013 Published  March 2014

The grid method is one of the techniques available to measure in-plane displacement and strain components on a deformed material. A periodic grid is first transferred on the specimen surface, and images of the grid are compared before and after deformation. Windowed Fourier analysis-based techniques permit to estimate the in-plane displacement and strain maps. The aim of this article is to give a precise analysis of this estimation process. It is shown that the retrieved displacement and strain maps are actually a tight approximation of the convolution of the actual displacements and strains with the analysis window. The effect of digital image noise on the retrieved quantities is also characterized and it is proved that the resulting noise can be approximated by a stationary spatially correlated noise. These results are of utmost importance to enhance the metrological performance of the grid method, as shown in a separate article.
Citation: Frédéric Sur, Michel Grédiac. Towards deconvolution to enhance the grid method for in-plane strain measurement. Inverse Problems and Imaging, 2014, 8 (1) : 259-291. doi: 10.3934/ipi.2014.8.259
##### References:
 [1] P. Abrahamsen, A Review of Gaussian Random Fields and Correlation Functions, Technical report, Norwegian Computing Center, Oslo, 1997. [2] T. M. Atanackovic and A. Guran, Theory of Elasticity for Scientists and Engineers, Springer, 2000. doi: 10.1007/978-1-4612-1330-7. [3] F. Auger, E. Chassande-Mottin and P. Flandrin, On phase-magnitude relationships in the short-time Fourier transform, IEEE Signal Processing Letters, 19 (2012), 267-270. doi: 10.1109/LSP.2012.2190279. [4] C. Badulescu, M. Bornert, J.-C. Dupré, S. Equis, M. Grédiac, J. Molimard, P. Picart, R. Rotinat and V. Valle, Demodulation of spatial carrier images: Performance analysis of several algorithms using a single image, Experimental Mechanics, 53 (2013), 1357-1370. doi: 10.1007/s11340-013-9741-6. [5] C. Badulescu, M. Grédiac and J.-D. Mathias, Investigation of the grid method for accurate in-plane strain measurement, Measurement Science and Technology, 20 (2009), 095102. doi: 10.1088/0957-0233/20/9/095102. [6] C. Badulescu, M. Grédiac, J.-D. Mathias and D. Roux, A procedure for accurate one-dimensional strain measurement using the grid method, Experimental Mechanics, 49 (2009), 841-854. doi: 10.1007/s11340-008-9203-8. [7] P. Balazs, D. Bayer, F. Jaillet and P. Søndergaard, The phase derivative around zeros of the short-time Fourier transform, e-prints, arXiv:1103.0409, March 2011. [8] J. Boulanger, C. Kervrann, P. Bouthemy, P. Elbau, J.-B. Sibarita and J. Salamero, Patch-based nonlocal functional for denoising fluorescence microscopy image sequences, IEEE Transaction on Medical Imaging, 29 (2010), 442-454. doi: 10.1109/TMI.2009.2033991. [9] L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995. [10] N. Delprat, B. Escudié, Ph. Guillemain, R. Kronland-Martinet, P. Tchamitchian and B. Torrésani, Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies, IEEE Transactions on Information Theory, 38 (1992), 644-664. doi: 10.1109/18.119728. [11] J.-C. Dupré, F. Brémand and A. Lagarde, Numerical spectral analysis of a grid: Application to strain measurements, Optics and Lasers in Engineering, 18 (1993), 159-172. doi: 10.1016/0143-8166(93)90025-G. [12] H. Faraji and W. J. MacLean, CCD noise removal in digital images, IEEE Transactions on Image Processing, 15 (2006), 2676-2685. doi: 10.1109/TIP.2006.877363. [13] M. Grédiac and F. Sur, Effect of sensor noise on the resolution and spatial resolution of displacement and strain maps estimated with the grid method, Strain, 50 (2014), 1-27. doi: 10.1111/str.12070. [14] M. Grédiac, F. Sur, C. Badulescu and J.-D. Mathias, Using deconvolution to improve the metrological performance of the grid method, Optics and Lasers in Engineering, 51 (2013), 716-734. doi: 10.1016/j.optlaseng.2013.01.009. [15] E. Héripré, M. Dexet, J. Crépin, L. Gélébart, A. Roos, M. Bornert and D. Caldemaison, Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials, International Journal of Plasticity, 23 (2007), 1512-1539. doi: 10.1016/j.ijplas.2007.01.009. [16] F. Jaillet, P. Balazs, M. Dörfler and N. Engelputzeder, On the structure of the phase around the zeros of the short-time Fourier transform, in Proceedings of NAG/DAGA International Conference on Acoustics, Rotterdam, Netherlands, (2009), 1584-1587. [17] Q. Kemao, Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations, Optics and Lasers in Engineering, 45 (2007), 304-317. doi: 10.1016/j.optlaseng.2005.10.012. [18] S. Mallat, A Wavelet Tour of Signal Processing, (2nd edition) Academic Press, 1998. [19] F. Murthagh, J. L. Starck and A. Bijaoui, Image restoration with noise suppression using a multiresolution support, Astronomy and Astrophysics, 112 (1995), 179-189. [20] R. D. Rajaona and P. Sulmont, A method of spectral analysis applied to periodic and pseudoperiodic signals, Journal of Computational Physics, 61 (1985), 186-193. doi: 10.1016/0021-9991(85)90067-1. [21] F. Sur and M. Grédiac, Enhancing with deconvolution the metrological performance of the grid method for in-plane strain measurement, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, British Columbia, Canada, (2013). doi: 10.1109/ICASSP.2013.6637914. [22] Y. Surrel, Photomechanics, Vol. 77 of Topics in Applied Physics, chapter Fringe Analysis, Springer, 2000, 55-102. doi: 10.1007/3-540-48800-6_3. [23] M. Sutton, J.-J. Orteu and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements, Springer, 2009. doi: 10.1007/978-0-387-78747-3. [24] M. Takeda, H. Ina and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, Journal of the Optical Society of America, 72 (1982), 156-160. doi: 10.1364/JOSA.72.000156.

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##### References:
 [1] P. Abrahamsen, A Review of Gaussian Random Fields and Correlation Functions, Technical report, Norwegian Computing Center, Oslo, 1997. [2] T. M. Atanackovic and A. Guran, Theory of Elasticity for Scientists and Engineers, Springer, 2000. doi: 10.1007/978-1-4612-1330-7. [3] F. Auger, E. Chassande-Mottin and P. Flandrin, On phase-magnitude relationships in the short-time Fourier transform, IEEE Signal Processing Letters, 19 (2012), 267-270. doi: 10.1109/LSP.2012.2190279. [4] C. Badulescu, M. Bornert, J.-C. Dupré, S. Equis, M. Grédiac, J. Molimard, P. Picart, R. Rotinat and V. Valle, Demodulation of spatial carrier images: Performance analysis of several algorithms using a single image, Experimental Mechanics, 53 (2013), 1357-1370. doi: 10.1007/s11340-013-9741-6. [5] C. Badulescu, M. Grédiac and J.-D. Mathias, Investigation of the grid method for accurate in-plane strain measurement, Measurement Science and Technology, 20 (2009), 095102. doi: 10.1088/0957-0233/20/9/095102. [6] C. Badulescu, M. Grédiac, J.-D. Mathias and D. Roux, A procedure for accurate one-dimensional strain measurement using the grid method, Experimental Mechanics, 49 (2009), 841-854. doi: 10.1007/s11340-008-9203-8. [7] P. Balazs, D. Bayer, F. Jaillet and P. Søndergaard, The phase derivative around zeros of the short-time Fourier transform, e-prints, arXiv:1103.0409, March 2011. [8] J. Boulanger, C. Kervrann, P. Bouthemy, P. Elbau, J.-B. Sibarita and J. Salamero, Patch-based nonlocal functional for denoising fluorescence microscopy image sequences, IEEE Transaction on Medical Imaging, 29 (2010), 442-454. doi: 10.1109/TMI.2009.2033991. [9] L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995. [10] N. Delprat, B. Escudié, Ph. Guillemain, R. Kronland-Martinet, P. Tchamitchian and B. Torrésani, Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies, IEEE Transactions on Information Theory, 38 (1992), 644-664. doi: 10.1109/18.119728. [11] J.-C. Dupré, F. Brémand and A. Lagarde, Numerical spectral analysis of a grid: Application to strain measurements, Optics and Lasers in Engineering, 18 (1993), 159-172. doi: 10.1016/0143-8166(93)90025-G. [12] H. Faraji and W. J. MacLean, CCD noise removal in digital images, IEEE Transactions on Image Processing, 15 (2006), 2676-2685. doi: 10.1109/TIP.2006.877363. [13] M. Grédiac and F. Sur, Effect of sensor noise on the resolution and spatial resolution of displacement and strain maps estimated with the grid method, Strain, 50 (2014), 1-27. doi: 10.1111/str.12070. [14] M. Grédiac, F. Sur, C. Badulescu and J.-D. Mathias, Using deconvolution to improve the metrological performance of the grid method, Optics and Lasers in Engineering, 51 (2013), 716-734. doi: 10.1016/j.optlaseng.2013.01.009. [15] E. Héripré, M. Dexet, J. Crépin, L. Gélébart, A. Roos, M. Bornert and D. Caldemaison, Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials, International Journal of Plasticity, 23 (2007), 1512-1539. doi: 10.1016/j.ijplas.2007.01.009. [16] F. Jaillet, P. Balazs, M. Dörfler and N. Engelputzeder, On the structure of the phase around the zeros of the short-time Fourier transform, in Proceedings of NAG/DAGA International Conference on Acoustics, Rotterdam, Netherlands, (2009), 1584-1587. [17] Q. Kemao, Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations, Optics and Lasers in Engineering, 45 (2007), 304-317. doi: 10.1016/j.optlaseng.2005.10.012. [18] S. Mallat, A Wavelet Tour of Signal Processing, (2nd edition) Academic Press, 1998. [19] F. Murthagh, J. L. Starck and A. Bijaoui, Image restoration with noise suppression using a multiresolution support, Astronomy and Astrophysics, 112 (1995), 179-189. [20] R. D. Rajaona and P. Sulmont, A method of spectral analysis applied to periodic and pseudoperiodic signals, Journal of Computational Physics, 61 (1985), 186-193. doi: 10.1016/0021-9991(85)90067-1. [21] F. Sur and M. Grédiac, Enhancing with deconvolution the metrological performance of the grid method for in-plane strain measurement, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, British Columbia, Canada, (2013). doi: 10.1109/ICASSP.2013.6637914. [22] Y. Surrel, Photomechanics, Vol. 77 of Topics in Applied Physics, chapter Fringe Analysis, Springer, 2000, 55-102. doi: 10.1007/3-540-48800-6_3. [23] M. Sutton, J.-J. Orteu and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements, Springer, 2009. doi: 10.1007/978-0-387-78747-3. [24] M. Takeda, H. Ina and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, Journal of the Optical Society of America, 72 (1982), 156-160. doi: 10.1364/JOSA.72.000156.
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