## 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.

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### Corresponding author

## Additional information

Michel GREDIAC is a Fellow of the Society for Experimental Mechanics.

## Appendices

### Appendix 1: Localized Spectrum Analysis Applied to Checkerboard Images

This section is a brief reminder on the Localized Spectrum Analysis applied to checkerboard images in order to retrieve displacement fields. Full detail can be found in [18]. The first step of LSA consists in calculating the Windowed Fourier Transform (WFT) of the image. This calculation is performed in a particular case since the frequency is set to the value of the nominal frequency of the quasi-periodic marking. This leads to the following expression for the WFT, which is defined, for any (*x*_{1},*x*_{2}) ∈ *ℝ*^{2} and *𝜃* ∈ [0, 2*π*], by:

where w is a 2D window function. We use here a Gaussian window characterized by its standard deviation *σ*:

It has been shown in [19] that *σ* should be greater or equal to the pitch *p* of the quasi-periodic pattern to process correctly the images. In this study, we consider patterns which are optimal in terms of sensor propagation, namely checkerboards [10, 45]. A typical checkerboard is shown in Fig. 19. For checkerboards, it is shown in [18] that Ψ shall be calculated along the diagonals \(x^{\prime \prime }_{1}, x^{\prime \prime }_{2}\) of the natural symmetry axes \(x^{\prime }_{1}, x^{\prime }_{2}\) of the checkerboard (see the axes shown in Fig. 19), thus for \(\theta =\alpha +\frac {\pi }{4}\) and \(\theta =\alpha +\frac {3\pi }{4}\) in equation (28). *x*_{1},*x*_{2} correspond to the natural coordinate system of the camera sensor. It is different from \(x^{\prime }_{1}, x^{\prime }_{2}\) because images of regular patterns may be prone to aliasing problems if these two coordinate systems ((*x*_{1},*x*_{2}) and (\(x^{\prime }_{1},x^{\prime }_{2}\))) are aligned [32]. The WFT being applied twice: once along direction \(x^{\prime \prime }_{1}\) and once along direction \(x^{\prime \prime }_{2}\), two complex numbers are available at each pixel of coordinates (*x*_{1},*x*_{2}): \({\Psi }(x_{1},x_{2},\alpha + \frac {\pi }{4})\) and \({\Psi }(x_{1},x_{2},\alpha + \frac {3\pi }{4})\).

The second step of the method consists in extracting and unwrapping the two phases of both the reference and the deformed images along the \(x_{1}^{\prime \prime }\)- and \(x_{2}^{\prime \prime }\)-directions. These quantities are generally considered to be equal to the arguments of the WFT (here \({\Psi }(x_{1},x_{2},\alpha + \frac {\pi }{4})\) and \({\Psi }(x_{1},x_{2},\alpha + \frac {3\pi }{4})\)), [15, 46, 47]. They are then expressed in the *x*_{1},*x*_{2} coordinate system by a change of basis, and the sought displacements along the natural symmetry axes of the camera sensor *x*_{1},*x*_{2} are given by

\(\underline {u}\) is the displacement at any point of coordinates \(\underline {x}\). \(\underline {\Phi }_{g}\) and \(\underline {\Phi }_{f}\) are the phases of the periodic pattern of the current (or deformed) and reference images, respectively. The unknown displacement is involved in both parts of equation (30), so it can be found by using a fixed-point algorithm, which generally converges after one iteration only [15]. It has been recently demonstrated in [4] that this result is only an approximation. Indeed, regardless of noise, the arguments of the WFTs discussed above are equal to the sought phases *convolved* by the window *w* used in the WFT.

### Appendix 2: Definition of the Metrological Parameters used in this Study

Three metrological parameters are first discussed in this paper, namely the *measurement resolution*, the *bias* and the *spatial resolution*. A fourth one named *metrological efficiency indicator* is also introduced. It is defined by the product of the first and last quantities. All these parameters are throughly defined in [5]. Their definitions are recalled below:

Measurement resolution: in Ref. [36], the measurement resolution is defined by the *smallest change in a quantity being measured that causes a perceptible change in the corresponding indication*. More precisely, it is proposed in [37] to define it as the *change in quantity being measured that causes a change in the corresponding indication greater than one standard deviation of the measurement noise*, which enables us to quantify the measurement resolution. This definition is quite arbitrary, any other (reasonable) multiple of the standard deviation being also potentially acceptable, but the idea is that the resolution quantifies the smallest change not likely to be caused by measurement noise [37].

Bias: there are several causes for the systematic error observed with full-field measurement techniques. We consider here the so-called matching bias, which concerns both DIC and LSA. A classic way to assess it is to consider a synthetic reference sine function with a given amplitude, and to consider that the relative loss of amplitude quantifies this bias, as in Ref. [38,39,40] for DIC or in [22, 41] for LSA. The bias is denoted by *λ*. The systematic error due to the interpolation function used to have both the reference and the deformed images in the same coordinate system [42,43,44] is not considered here because it concerns only DIC and not LSA [15].

Spatial resolution: the spatial resolution denoted *ℓ* is defined here by *the lowest period of a sinusoidal deformation that the technique is able to reproduce before losing a certain percentage of amplitude*, in other words before the bias reaches a certain value, this quantity being chosen *a priori* [39]. The advantage of this definition is that it is not based on an arbitrary value for the subset size in DIC or for the window used while processing a periodic pattern with LSA. This makes it possible to compare the spatial resolution between these two techniques, whose principle is totally different. This definition of the spatial resolution holds here for the phase, and consequently for the displacement. It also holds for the phase derivatives and the strain components if no smoothing is performed before differentiating the phases and the displacements. Otherwise the spatial resolution is all the more impaired as the width of the filter increases.

Metrological efficiency indicator : For LSA, it has been proven that if the noise impairing the images is homoscedastic, the product between the displacement resolution and the spatial resolution is constant whatever the value of the size of the Gaussian window used to find the displacement [5, 15]. This quantity is defined for a value of the bias *λ* and denoted by *α*_{λ}. It is observed that considering a more representative heteroscedastic noise makes *α*_{λ} nearly constant. Simulations also show that *α*_{λ} as defined here is nearly constant for DIC whatever the choice of the subset size. In conclusion, *α*_{λ} represents an indicator of the metrological performance of the measurement system, which is independent of the choice of the size of the window with LSA and of the subset with DIC.

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Grédiac, M., Blaysat, B. & Sur, F. A Robust-to-Noise Deconvolution Algorithm to Enhance Displacement and Strain Maps Obtained with Local DIC and LSA.
*Exp Mech * **59, **219–243 (2019). https://doi.org/10.1007/s11340-018-00461-4

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### Keywords

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