Fine-Tuning a Deconvolution Algorithm to Restore Displacement and Strain Maps Obtained with LSA

Abstract

Background

Reliably measuring sharp details in displacement and strain maps returned by full-field measurement techniques remains an open question in the photomechanics community.

Objective

The primary objective of this study is to improve and fine-tune a deconvolution algorithm in order to limit the blur that obscures the details in displacement and strain maps.

Methods

Checkerboard patterns are used and processed with a spectral method, namely the Localized Spectrum Analysis (LSA), and the raw maps returned by this technique are deconvolved. The influence of various settings on the quality of the results is studied by using synthetic images deformed through a well-vetted reference displacement field.

Results

It is shown that linking the size of the analysis window used in LSA on the one hand, and the size of the second derivative kernel employed in the deconvolution algorithm on the other hand, ensures the convergence of the deconvolution algorithm in all cases. This was not the case with the initial version. The ratio between these sizes, which optimizes the metrological performance of LSA followed by deconvolution, is identified. The influence of the sampling density of the checkerboard pattern in the images is also examined. The efficiency of the deconvolution algorithm employed with optimized settings is illustrated with strain maps obtained on two specimens, one in shape memory alloy, and the other in wood.

Conclusions

It is shown in this study that deconvolution with optimized settings is an effective tool to enhance small and sharp details in strain maps obtained with LSA.

Appendix: Metrological Parameters

The metrological performance of the full-field measurement techniques discussed in this paper are estimated with four parameters, namely the measurement resolution $\sigma _u$, the matching bias, the spatial resolution and the metrological efficiency indicator. These quantities are defined as follows:

• The displacement resolution $\sigma _u$ quantifies the random noise level which affects these maps. It is mainly caused by sensor noise propagation. According to [47], $\sigma _u$ is equal to the standard deviation of the noise in the displacement maps.

• The systematic error (or bias) reflects that the amplitude of small details is generally lower than the real one. This change in amplitude is expressed as a percentage $\lambda$ of the amplitude of a sine wave characterized by a given period $p_{wave}$. Thus, considering the reference displacement field of amplitude 0.5 [px] defined in 'Synthetic Displacement Field', and denoting by “$\text {ampl}$" the amplitude of the sine wave returned by the measuring system for a certain period $p_{wave}$, we have:

  $$\begin{aligned} \lambda = \dfrac{\text {ampl}-0.5}{0.5}\times 100~[\%{]} \end{aligned}$$

  (23)

  It is worth noting that contrary to the maps returned by DIC or by LSA for which $\lambda$ is negative since we have a loss of amplitude, deconvolution may return an amplitude of the signal greater than the reference one, $\lambda$ becoming positive in this case.

• The spatial resolution reflects the ability of the technique to distinguish close features in a displacement/strain map. According to [48], it is defined as the period of the sine wave for which the value of $\lambda$ equal to a threshold value arbitrarily fixed to -10 %, as in previous studies dealing with the metrological performance of full-field measurement techniques [4, 5, 19, 20]. The inverse of this period is referred to as the cutoff frequency. It is denoted by $f_{c(\lambda )}$. Changing the threshold value $\lambda$ also changes the value of this frequency.

• These three quantities are not independent but linked. They depend in particular on some settings made arbitrarily by the user, the main one being the size over which calculations are performed around a given pixel to obtain a measurement at this pixel. With the present approach, this is the size of the analysis window used in LSA. With a Gaussian analysis window, this size is governed by the standard deviation $\ell _{\textsf{LSA}}$. A quantity independent of this size has been introduced in [4] to overcome this shortcoming, namely the Metrological Efficiency Indicator denoted by MEI. It is merely defined as the product between the spatial resolution and the displacement resolution. The advantage of using the MEI is that it is independent of $\ell _{\textsf{LSA}}$ before deconvolution, as discussed in [4]. The MEI is, therefore, a parameter that is intrinsic to the technique and thus enables a fair comparison between different techniques [4, 5], or between different versions of the same techniques like DIC [19, 20]. An issue addressed in this study is to know if MEI is still independent of $\ell _{\textsf{LSA}}$ after deconvolution, and to what extent the sampling density $\rho$ influences this quantity. These questions are addressed in 'Seeking the Optimal Ratio' and 'Influence of the Sampling Density', respectively.

Note that, in this paper, $f_{c(\lambda )}$ and MEI are normalized with respect to the sampling density of the periodic pattern denoted by $\rho$, see 'Influencing Parameters'. The corresponding normalized quantities are denoted by $\widetilde{f}_{c(\lambda )} = f_{c(\lambda )} \times \rho$ and $\widetilde{\text {MEI}} = \frac{\text {MEI}}{\rho }$, respectively.