On the Propagation of Camera Sensor Noise to Displacement Maps Obtained by DIC - an Experimental Study

Abstract

This paper focuses on one of the metrological properties of DIC, namely displacement resolution. More specifically, the study aims to validate, in the environment of an experimental mechanics laboratory, a recent generalized theoretical prediction of displacement resolution. Indeed, usual predictive formulas available in the literature neither take into account sub-pixel displacement, nor have been validated in an experimental mechanics laboratory environment, nor are applicable to all types of DIC (Global as well as Local). Here, the formula used to account for sub-pixel displacements is first recalled, and an accurate model of the sensor noise is introduced. The hypotheses required for the elaboration of this prediction are clearly stated. The formula is then validated using experimental data. Since rigid body motion between the specimen and the camera impairs the experimental data, and since sensor noise is signal-dependent, particular tools need to be introduced in order to ensure the consistency between the observed image noise and the model on which prediction hypotheses are based. Pre-processing tools introduced for another full-field measurement approach, namely the Grid Method, are employed to address these issues.

Appendices

Appendix A: DIC Minimization Scheme

This appendix is a brief digest of the Gauss-Newton scheme in the context of DIC optimization.

A.1 Notations

We use the same notations as in Section “Digital Image Correlation and Displacement Resolution Prediction”; that is, (f) and (g) are the two images, Ω is the region of interest (composed of N _p pixels $x_{1},\dots , x_{N_{\mathsf {p}}}$), $\underline {u}(\underline {x},{\boldsymbol {\lambda }})$ is the displacement field at pixel $\underline {x}$, with the N _DoF DoF λ = (λ ₁,…, λ _N ) such that:

$$ \underline{u}(\underline{x},{\boldsymbol{\lambda}}) = \sum\limits_{i=1}^{N_{\text{DoF}}} \lambda_{i} \underline{\varphi}_{i}(\underline{x}) $$

(38)

Note that $\underline {\varphi }_{i}(\underline {x})$ is a first-rank tensor, like $\underline {u}(\underline {x},{\boldsymbol {\lambda }})$.

The aim of DIC-based algorithms is to find λ minimizing

$$ \sum\limits_{{\underline{x}_{\mathsf{p}}} \in {\Omega}} \left( f({\underline{x}_{\mathsf{p}}}) - g({\underline{x}_{\mathsf{p}}}+\underline{u}({\underline{x}_{\mathsf{p}}},{\boldsymbol{\lambda}}))\right)^{2} $$

(39)

Let $r_{p}({\boldsymbol {\lambda }}) = f({\underline {x}_{\mathsf {p}}}) - g({\underline {x}_{\mathsf {p}}}+\underline {u}({\underline {x}_{\mathsf {p}}},{\boldsymbol {\lambda }}))$ for any ${\underline {x}_{\mathsf {p}}} \in {\Omega }$, and $r({\boldsymbol {\lambda }}) = (r_{1}({\boldsymbol {\lambda }}),\dots ,r_{N_{\mathsf {p}}}({\boldsymbol {\lambda }}))$. Minimizing equation (39) is thus equivalent to minimizing the squared Euclidean norm of r(λ), i.e., ∥r(λ)∥².

A.2 A Modified Gauss-Newton Method

Let J _r (λ) be the N _p ×N _DoF Jacobian matrix of r at λ. Its coefficient at row i and column j is:

$$\begin{array}{@{}rcl@{}} [{\boldsymbol{J_{r}}}({\boldsymbol{\lambda}})]_{ij} &=& \frac{\partial r_{i}}{\partial \lambda_{j}}({\boldsymbol{\lambda}}) \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} &=& \underline{\nabla} g({\underline{x}_{\mathsf{i}}} +\underline{u}({\underline{x}_{\mathsf{i}}},{\boldsymbol{\lambda}})) \cdot \frac{\partial \underline{u}}{\partial \lambda_{j}}({\underline{x}_{\mathsf{i}}},{\boldsymbol{\lambda}}) \end{array} $$

(41)

$$\begin{array}{@{}rcl@{}} &=& \underline{\nabla} g^{T}({\underline{x}_{\mathsf{i}}} + \underline{u}({\underline{x}_{\mathsf{i}}},{\boldsymbol{\lambda}})) \cdot \underline{\varphi_{j}({\underline{x}_{\mathsf{i}}})} \end{array} $$

(42)

equation (42) holds from equation (38). Here $\underline {\nabla } g^{T}$ is the transpose of the gradient of g.

Assuming that Δλ is small enough so that r(λ + Δλ) can be approximated by its first-order Taylor expansion:

$$ {\boldsymbol{r}}({\boldsymbol{\lambda}} + {\boldsymbol{\Delta \lambda}}) = {\boldsymbol{r}}({\boldsymbol{\lambda}}) + {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}}) {\boldsymbol{\Delta \lambda}} $$

(43)

The Gauss-Newton algorithm consists of iterating λ ^it+1 = λ ^it + δλ ^it from an initial guess λ ⁰, with δλ ^it minimizing ∥r(λ ^it + δλ ^it)∥² under the approximation given by equation (43).

Minimizing ∥r(λ) + J _r (λ)Δλ∥² is an ordinary least squares problem, which gives Δλ as a solution to the normal equation:

$$ {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}})^{T} {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}}) {\boldsymbol{\Delta\lambda}} = {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}})^{T} {\boldsymbol{ \lambda}} $$

(44)

This can be solved in:

$$ {\boldsymbol{\Delta\lambda}} = ({\boldsymbol{J_{r}}}({\boldsymbol{\lambda}})^{T} {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}}))^{-1} {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}})^{T} {\boldsymbol{r}}({\boldsymbol{\lambda}}) $$

(45)

Consequently,

$$ {\boldsymbol{\Delta\lambda}} = \widehat{{\boldsymbol{M}}}^{-1} \hat{{\boldsymbol{b}}} $$

(46)

with

$$\begin{array}{@{}rcl@{}} \widehat{[{\boldsymbol{M}}]}_{ij} &=& \sum\limits_{k} \left[ {\boldsymbol{J_{r}}} ({\boldsymbol{\lambda}} ) \right]_{ki} \left[ {\boldsymbol{J_{r}}}({\boldsymbol{\lambda}}) \right]_{kj} \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} &=& \sum\limits_{{\underline{x}_{\mathsf{p}}} \in {\Omega}} \left( \underline{\nabla} g^{T}({\underline{x}_{\mathsf{p}}}+\underline{u}({\underline{x}_{\mathsf{p}}},{\boldsymbol{\lambda}})) \cdot \underline{\varphi}_{i}({\underline{x}_{\mathsf{p}}}) \right)\\ &&\times\left( \underline{\nabla} g^{T}({\underline{x}_{\mathsf{p}}} + \underline{u}({\underline{x}_{\mathsf{p}}},{\boldsymbol{\lambda}})) \cdot \underline{\varphi}_{j}({\underline{x}_{\mathsf{p}}}) \right) \end{array} $$

(48)

and

$$\begin{array}{@{}rcl@{}} [\hat{{\boldsymbol{b}}}]_{i} & =& \sum\limits_{k} [{\boldsymbol{J_{r}}}({\boldsymbol{\lambda}})]_{ki} r_{k} ({\boldsymbol{\lambda}}) \end{array} $$

(49)

$$\begin{array}{@{}rcl@{}} &=& \sum\limits_{{\underline{x}_{\mathsf{p}}} \in {\Omega}} \left( \underline{\nabla} g^{T}({\underline{x}_{\mathsf{p}}}+\underline{u}({\underline{x}_{\mathsf{p}}},{\boldsymbol{\lambda}})) \cdot \underline{\varphi}_{i}({\underline{x}_{\mathsf{p}}}) \right) \\ &&\times\left( f({\underline{x}_{\mathsf{p}}}) - g({\underline{x}_{\mathsf{p}}}+\underline{u}({\underline{x}_{\mathsf{p}}},\underline{\lambda})) \right) \end{array} $$

(50)

However, the term $\underline {\nabla } g({\underline {x}_{\mathsf {p}}}+\underline {u}({\underline {x}_{\mathsf {p}}},\underline {\lambda }))$ necessitates a numerical interpolation scheme, since g is known only on the pixel grid. Instead of repeating this procedure for each iteration, it is more convenient to identify $\underline {\nabla } g({\underline {x}_{\mathsf {p}}}+\underline {u}({\underline {x}_{\mathsf {p}}},\underline {\lambda }))$ with $\underline {\nabla } f({\underline {x}_{\mathsf {p}}})$, which is a licit approximation when λ is close to minimizing equation (39) if a small displacement hypothesis is assumed (cf [25]).

With this modified Gauss-Newton scheme, the following formula holds:

$$ {\boldsymbol{\Delta\lambda}} = {\boldsymbol{M}}^{-1} {\boldsymbol{b}} $$

(51)

with

$$ [{\boldsymbol{M}}]_{ij} = \sum\limits_{{\underline{x}_{\mathsf{p}}} \in {\Omega}} \left( \underline{\nabla} f^{T}({\underline{x}_{\mathsf{p}}}) \cdot \underline{\varphi}_{i}({\underline{x}_{\mathsf{p}}} )\right) \left( \underline{\nabla} f^{T}(\underline{x}_{\mathsf{p}}) \cdot \varphi_{j}({\underline{x}_{\mathsf{p}}}) \right) $$

(52)

and

$$ [{\boldsymbol{b}}]_{i} = \sum\limits_{{\underline{x}_{\mathsf{p}}} \in {\Omega}} \left( \underline{\nabla} f^{T}({\underline{x}_{\mathsf{p}}}) \cdot \underline{\varphi}_{i}({\underline{x}_{\mathsf{p}}} )\right) \left( f({\underline{x}_{\mathsf{p}}}) - g({\underline{x}_{\mathsf{p}}}+\underline{u}({\underline{x}_{\mathsf{p}}},{\boldsymbol{\lambda}})) \right) $$

(53)

Note that M is a symmetrical matrix.

Appendix B: DIC on a Grid

A grid was deposited using the procedure described in [34] on a similar specimen as the one in Section “Experimental Validation: Vertical Translation”, as illustrated in Fig. 21(a). The same experimental set-up was used and the same experimental methodology was applied. In order to allow statistical analysis, N _img = 100 pictures were also acquired at each step of the tests, i.e., $((\mathrm {f}_{t})_{1\leq t \leq N_{\text {img}}})$ describing the reference state and $((\mathrm {g}_{t})_{1\leq t \leq N_{\text {img}}})$ the current one. Figure 21(a) illustrates the grid (f), which was used throughout this paper. Figure 21(b) presents a close-up of this grid; the lines are clearly visible, and the pitch of 5 pixels can be distinguished. Because the grid used in this study is composed of black lines over a white background, it presents darker intensity levels than for a usual speckle, as used in the study above. It is clearly visible in Fig. 21(a), which can be compared with the speckle, cf Fig. 1(b).

Fig. 21

Illustration of the imaged grid. (a) Image of the grid; the red box specifies the location of the close-up. (b) Close-up of one spot of the RoI and the mesh plotted on top of it, illustrating the regular definition of the grid pattern

The main difficulty in applying DIC to a grid is the initial guess that should be made to initiate the method. This difficulty can easily be overcome using a pyramidal DIC approach, as introduced in [10]. Here, the few defects impairing the grid drive the DIC solution at the coarsest scales towards the right solution and thus feed the final fine DIC solving correctly. This initialization functions work correctly in both examples illustrating this article, i.e., the vertical translation and the open-hole problem. The finest DIC mesh is plotted over the grid in Fig. 21(b), highlighting the arbitrary choice for the element size of 13×13 pixels. Such a choice thus ensures a mismatch between the support of each nodal function, and by consequence between each local residual that the DIC method minimizes (cf 1).

Figure 22 summarizes the main statistical characteristics associated with the grid images that were acquired. The maps of the average pixel value 〈f〉 (cf Fig. 22(a), close-up Fig. 22(c)) and of the standard deviation σ _f (cf Fig. 22(b), close-up Fig. 22(d)) are shown. The heteroscedatic nature of this noise is clearly visible in Fig. 22(b & d), in which the blue lines (from Fig. 22(d)), which correspond to low standard deviation) correspond to the black lines of the grid (which are of low light intensity, cf Fig. 22(b)).

Fig. 22

Main characteristics of the statistical analysis applied to grid images, namely the pixel mean value 〈f〉 (a), the pixel standard deviation σ _{f, p} (b), and two close-ups of 〈f〉 (c) and σ _{f, p} (d)

A DIC calculation is carried out using the same displacement description as in Section “Experimental Set-Up and Methods”. On the one hand, the formula in equation (13) can of course be applied here, and we obtain the predicted standard deviation at each node. This is depicted in Fig. 23(a). The ratio between predicted measurements of the displacement standard deviation when DIC is applied to a grid or a random speckle is also presented in Fig. 23(b). This ratio lying between 0 and 1, it a priori highlights interesting properties for applying DIC to grids. This is due to the higher values of the image gradient of a grid than of usual speckles like that used in this study. Since the square of the image gradient is involved in the correlation matrix M, and since predicted formulas are based on M ⁻¹, higher gradients lead to a lower displacement resolution standard deviation.

Fig. 23

Predicted standard deviation of the vertical displacement $\left (\sigma ^{p}_{\mathrm {u}_{2}}\right )$ and its ratio to that obtained when DIC is applied to the speckle

On the other hand, Fig. 24 illustrates the empirical displacement standard deviation. Indeed, since N _img images were acquired at both time steps, the reference image and the current one, N _img DIC calculations can be performed, and, as in Section “Experimental Validation: Vertical Translation”, a statistical analysis can also be performed on the DIC outputs. Figure 24(a) thus maps the observed displacement standard deviation $\sigma ^{p}_{\mathrm {u}_{2}}$ (only the vertical component is mapped) whereas Fig. 24(b) illustrates the ratio between observed resolutions when DIC is applied to a grid or to a speckle. The ratio here is higher than with the predictive formulas, but is still slightly smaller than 1.

Fig. 24

Empirical standard deviation of the vertical displacement $\left (\sigma ^{e}_{\mathrm {u}_{2}}\right )$ and its ratio to that obtained when DIC is applied to the speckle

As a conclusion to this pre-study, we can say that applying DIC to a grid image does not preset any drawbacks, provided that the DIC algorithm has been adequately initialized. Moreover, the particularly high gradients that are present in the grid images can offer interesting properties in terms of mastering displacement resolution.

Finally, Fig. 25 presents the ratio between the empirical and predicted displacement resolution. As with the speckle-based pattern, a heterogeneous spatial distribution of the ratio is visible in Fig. 25(a), and its mean value is greater than 1. There is an underestimation of the actual measurement resolution, which is due to both the RBM that occurs between the camera and the specimen, and to the assumed homoscedasticity hypothesis for the image noise.

Fig. 25

Ratio between observed and predicted standard deviation of the vertical displacement, $\rho _{\mathrm {u}} = \sigma _{\mathrm {u}_{2}}^{e} / \sigma _{\mathrm {u}_{2}}^{p}$