On the Optimal Pattern for Displacement Field Measurement: Random Speckle and DIC, or Checkerboard and LSA?

Abstract

This paper deals with the optimal pattern that can be used to retrieve displacement fields by minimizing the optical residual calculated over small regions of contrasted images. This minimization is generally performed in the spatial domain by processing speckle patterns with DIC. Another option is also considered here. It consists in switching this minimization to the Fourier domain. The benefit is that periodic patterns can be processed, which is generally not possible with DIC. It turns out that the optimal pattern in terms of sensor noise propagation is theoretically the checkerboard if it is correctly sampled, and this pattern is periodic. The reason why checkerboard is optimal is that the image gradient is maximum in this case. In addition, the minimization of the image residual in this case has a quasi-direct solution, which considerably speeds up the calculations. We first recall the basics of the different techniques used in the paper, namely classic subset-based DIC, and a spectral method called Localized Spectrum Analysis (LSA). A recent deconvolution procedure introduced to enhance

Comparing several spectral methods used to extract displacement fields from checkerboard images

“…No aliasing was observed in the present case probably because we considered synthetic checkerboard images obtained with pure sine functions in the examples, so this issue is not addressed here. The interested reader can find in [6,11,24] typical real and synthetic checkerboard images as well as close-up views to more easily visualize such patterns. As explained above, the bias as defined here depends on the frequency of the signal.…”

“…Second, the argument of the WFT directly provides the phase, so no inverse Fourier transform shall be applied before extracting the phase, which is not the case with WGPA, see Appendix 1. Note finally that the computation times observed for LSA and Digital Image Correlation are compared in [6].…”

“…In [3], it is recalled that checkerboards are theoretically optimal patterns for DIC in terms of sensor noise propagation because their images maximize the sum of square of subset intensity gradient (SSSIG), and noise in final displacement and strain maps is inversely proportional to this quantity [4,5]. This conclusion is drawn if the nearest neighbor forward difference scheme (and not the central difference scheme) is used to numerically estimate the gradient [3,6]. In [7], it is also shown that checkerboards, as other periodic patterns [8], induce lower pattern-induced bias than random speckles usually employed with DIC.…”

“…In [7], it is also shown that checkerboards, as other periodic patterns [8], induce lower pattern-induced bias than random speckles usually employed with DIC. Checkerboards are however quasi-periodic patterns and as such, they cannot be processed by DIC, except if the iterative minimization starts close to the solution to avoid convergence toward a local minimum [6], or if displacement continuity is enforced from a given point where the displacement can be considered as reliable. Consequently, checkerboards are optimal but not easily manageable (if not unmanageable) patterns for DIC in practice.…”

“…The phases along these diagonals are extracted and the displacement along the x and y-directions are then deduced by applying successively Equation 1 and a change of basis. In [11,6], it is also experimentally verified that displacement and strain fields extracted from checkerboard images are less noisy than their counterparts obtained at iso-bias and iso-spatial resolution, either from 2D grid images processed with the same spectral method, [11], or from random speckle images processed by classic subset-based DIC, [6]. In this context, it is of prime importance to check the efficiency of various spectral methods developed hitherto to process 2D grids when they are employed to process the optimal pattern for optical residual minimization, namely the checkerboard.…”