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 the metrological performance of DIC and LSA is also briefly recalled and used in this study. Synthetic images are considered to assess in different cases the displacement resolution, as well as other sources of spurious spatial fluctuations observed in the displacement fields such as the pattern-induced bias with DIC. The main conclusion is that using checkerboards instead of random speckles leads to measurements featuring a better compromise between spatial resolution and measurement resolution.

Appendix

The aim of this section is to show how to determine the value of ℓ used in LSA, which is equivalent to the value of 2M + 1 used in DIC. “Equivalent” means here that DIC used with a subset size equal to 2M + 1 features the same bias and spatial resolution as LSA used with a standard deviation equal to ℓ. As mentioned earlier, it does not mean that the outputs of DIC and LSA with these equivalent parameters give the same estimates of a real displacement field featuring strain gradients.

Predicting the bias for given values of 2M + 1 and ℓ can be made as follows. Assuming the real (and sought) displacement is a sine function of amplitude A, the bias as defined in “Definitions and Techniques Used in the Present Study” can be directly estimated with the transfer function of the filter characterizing the measuring technique. This filter is the Savitzky-Golay filter for DIC [1] and a Gaussian filter for LSA [18]. By definition, the transfer function of the filter is equal to its Fourier transform. Indeed, in linear translation-invariant signal processing, the transfer function of any linear filter is defined as the Fourier transform of its impulse response.

Case of DIC

With DIC, we deal by definition with the discrete form of the transfer function, which can be written as follows as a function of the frequency f of the prescribed displacement

$$ \widehat{h}^{DIC}(f)=h(0)+2\sum\limits_{i=1}^{M} h(i)\cos \left( 2i \pi f\right) $$

(13)

Since we deal with images, we have sampled signals. Hence f lies in this equation between 0 and the Nyquist frequency, equal here to 0.5 pixel^− 1. The h(i) coefficients, i ∈ [0 M], are the Savitzky-Golay coefficients defined in [43] for DIC, as explained in [1] and demonstrated in [47]. These coefficients depend on M and on the degree of the matching function used to express the displacement field within the subset. In the common case, for which the matching function is linear, the Savitzky-Golay coefficients are all equal to $\frac {1}{2M+1}$ [43].

Case of LSA

With LSA, the same discrete form could be chosen for the Fourier transform, but a continuous form of the Fourier transform is also available, which is not the case for DIC. This continuous form can therefore be used for the sake of convenience. Since a Gaussian window of standard deviation ℓ is used here in the WFT, the Fourier transform of the filter used with LSA is also a Gaussian function. Indeed, the Fourier transform of a Gaussian function is also a Gaussian function, which is a classic result in signal processing. This transfer function can be written as follows [31]:

$$ \widehat{h}^{LSA}(f)=e^{-2\pi^{2}\ell^{2}f^{2}} $$

(14)

Equivalence Between 2M + 1 and ℓ for a Given Value of λ

As a consequence of the two preceding equations, finding the standard deviation of a Gaussian window used in LSA, which is equivalent, in terms of bias and spatial resolution, to a subset size 2M + 1 used in DIC, consists of the following steps:

• 1. choosing the same value for the bias λ for both DIC and LSA, which gives:

  $$ \widehat{h}^{DIC}(f)=\widehat{h}^{LSA}(f)=1-\lambda $$

  (15)

• 2. for a given subset size 2M + 1, finding numerically (no analytical solution is available) the highest frequency f, which is the solution of Eq. 13. Thus

  $$ h(0)+2\sum\limits_{i=1}^{M} h(i)\cos \left( 2i \pi f\right)=1-\lambda $$

  (16)

• 3. inverting Eq. 14 in order to deduce the corresponding standard deviation ℓ for the Gaussian window used in the WFT. Since $d=\frac {1}{f}$, we have

  $$ \ell=\frac{d}{\pi}\sqrt{\frac{-\log\left( 1-\lambda\right)}{2}}, \quad \text{with}~0<\lambda<1 $$

  (17)

Influence of λ

Choosing a bias λ = 10% for both techniques is somewhat arbitrary. Changing the value of the bias, for instance by considering λ = 5% and λ = 20% instead of λ = 10%, has a great impact on the spatial resolution d but does not really influence the standard deviation ℓ of the Gaussian window used in the WFT, as illustrated in the examples shown in Table 3 in a particular (but representative) case, namely 2M + 1 = 21 pixels. d and ℓ are calculated for various values of λ fixed a priori. It can be seen that the value of ℓ equivalent to 2M + 1 is nearly unaffected when the bias λ changes. This makes the results nearly independent of this arbitrary choice.

Particular Case of Small Values of λ and DIC Used with Linear Matching Functions

We demonstrate here that if linear matching functions are considered in DIC, a closed-form expression giving ℓ as a function of M is available. In addition, ℓ does not depend, at first approximation, on the bias λ when λ is small. We consider first again Eq. 16. Assuming that 2Mπf is small enough, the cosines involved in this equation can be replaced by their third-order approximation. Thus Eq. 16 becomes

$$ h(0)+2\sum\limits_{i=1}^{M} h(i)\left( 1-\frac{(2i \pi f)^{2}}{2}\right)=1-\lambda $$

(18)

Since $h(0)+2\sum \limits _{i=1}^{M} h(i) =1$ [43], we have

$$ 4\pi^{2} f^{2}\sum\limits_{i=1}^{M} i^{2}h(i)=\lambda $$

(19)

Let us now focus on linear matching functions. In this case the Savitzky-Golay coefficients are all equal to $h(i)=\frac {1}{2M+1},~\forall i \in \{1{\dots } M\}$ [43]. Hence

$$ \sum\limits_{i=1}^{M} i^{2}h(i)=\frac{1}{2M+1}\sum\limits_{i=1}^{M} i^{2} $$

(20)

According to the Faulhaber’s formula, we have $\sum \limits _{i=1}^{M} i^{2}=\frac {2M^{3}+3M^{2}+M}{6}$, so after simplification, we can deduce from Eq. 19

$$ f=\frac{1}{2\pi }\sqrt{\frac{6\lambda }{M(M+1)}} $$

(21)

It can be checked that the third-order approximation of the cosines made in Eq. 18 still remains acceptable for significant values of M. For instance, with M = 10, which is the case considered in the examples discussed in this paper, the error made by substituting the quantity corresponding to the greatest value of the index i of the sum is equal to 11%. The error made on the other terms of this approximated sum is automatically lower. In addition, the weight of these terms in the sum is higher than the weight of the last term, the cosine being a decreasing function just after 0.

Plugging Eq. 21 into Eq. 17, where $f=\frac {1}{d}$, leads to

$$ \ell=\sqrt{\frac{M(M+1)}{3}}\times \sqrt{\frac{-log\left( 1-\lambda\right)}{\lambda}}, \quad \text{with}~0<\lambda<1 $$

(22)

If λ is small, then $\frac {-log\left (1-\lambda \right )}{\lambda }\simeq 1$ and Eq. 22 reduces to

$$ \ell=\sqrt{\frac{M(M+1)}{3}} $$

(23)

For instance, in the case 2M + 1 considered in Table 3 above, Eq. 23 gives ℓ = 6.05 pixels, which is very close to the values of ℓ given in this table for various values of λ. This result shows that in the particular case of DIC used with linear matching functions, ℓ does not really depend on λ when λ is small. ℓ and thus the width of the Gaussian window used in LSA only depend on the size of the subset 2M + 1 used in DIC.