On the benefits of correcting brightness and contrast in global digital image correlation: Monitoring cracks during curing and drying of a refractory castable

Abstract The gray level conservation is the underlying hypothesis of Digital Image Correlation (DIC). However, it may be challenging to enforce in some experimental configurations. Brightness and contrast corrections (BCCs) may be added to the registration procedure. Different types of BCCs were implemented for global DIC, and their benefits were analyzed for localized and diffuse sources of brightness changes. As a case study to apply BCCs, a refractory castable was placed inside a climatic chamber, and cracks were generated due to localized expansions during its curing and drying. To choose the best BCC for this case, two sets of images were considered. The first one allowed the noise floor levels to be evaluated. The second one dealt with the development of a crack network. The BCCs significantly reduced gray level residuals enabling cracks with small openings to be detected. The coarse discretization was effective in correcting lighting changes and avoided its coupling with the measured kinematic fields and other local phenomena.


On the benefits of correcting brightness and contrast in
fields on specimen surfaces [1,2] and in the bulk via digital volume correlation [3, 3 4]. The displacements may, for instance, be caused by mechanical loads [5] or 4 thermal histories [6]. The images can be 2D [7], generated not only by visible 5 light but also by infrared radiation [8,9] or electron beams as in Scanning 6 Electron Microscopy (SEM) [10,11]. They can also be 3D volumes reconstructed 7 from tomographic scans [12,13].
8 Two advantages of DIC over conventional extensometry are highlighted. It 9 provides full-field measurements (instead of point-data), and no physical con-10 tact is needed between the material and the probe [2,5]. The first advantage 11 represents an increase of the amount of gathered information. For example, dis-12 placement fields of entire surfaces are measured, and not only the crack mouth 13 opening displacement during fracture tests. The correlation between measured 14 fields and close-form solutions can be used to calibrate fracture mechanics pa-15 rameters [14,15]. The second advantage enables the technique to be applied to 16 soft materials [16,17]  Zero-Normalized Cross-Correlation (ZNCC) criterion is one of the most popular 31 corrections [29,21,2,22,23]. For instance, this type of correction was used to 32 investigate crack propagation in rocks [30]. Another known correction is a gray 33 level average to minimize high temperature effects [31]. 34 Gray level corrections were also introduced in global DIC via brightness and 35 contrast field changes [32]. Some specific cases used the gray level corrections, 36 for example, DIC analyses of infrared pictures [8,9], and distortion corrections 37 The hardware parameters of the optical setup are reported in Table 2. An 93 exposure time of 3.2 s was chosen to perform a physical average of the intensity 94 acquired by each pixel [49]. LED lights were put inside the climatic chamber 95 for lighting purposes.
where Ψ i (x) are, in the present case, finite element shape functions, and υ i 111 nodal displacements. The deformed image corrected by such displacement fields 112 where the column vector {υ} gathers all nodal displacements υ i . In global DIC, 114 the gray level differences are globally minimized over the ROI such that the sought displacement is the argument that minimizes T g ({υ}) where the additional assumption ∇g {υ} (x) ≈ ∇f (x) was used. With such lin-121 earization, the approximated least squares functional is quadratic in terms of 122 displacement corrections. Its minimization then leads to a linear system where [H] is the (symmetric semi-definite positive) Hessian, and {j} the righ-124 hand side term, whose components read contrast corrections (with no changes in displacements) where q(x,f) is the gray level correction field that depends on the pixel position 137 x, and the gray levels of f . In the present work, the first two terms are selected 138 Various spatial approximations were proposed [32,33,9,36]. Finite element 143 discretizations were selected for the sake of simplicity and adaptability. As for 144 the displacement field, the two correction fields are written as where θ k (x) are (scalar) finite element shape functions, b k nodal brightness and 146 c k nodal contrast corrections. The latter ones are obtained by minimizing the 147 L2-norm of ρ bcc with respect to the column vector {ς} gathering all corrections b k 148 8 and c k . In the present case, a standard least squares minimization is performed 149 (i.e., no iterations are needed). From these estimates, a corrected reference 150 imagef is computed and instead of using f in the first DIC step, the corrected reference imagef 152 is considered. Therefore, a staggered algorithm is used to minimize ρ bcc (x), 153 namely, first minimizing ρ {υ} (x) for given gray level corrections, and then 154 It is worth noting that the shape functions of BCCs (Equation (11)) do 162 not need to be identical to those used in the kinematic basis (Equation (1)). 163 Consequently, the scale at which the displacement field is discretized can be 164 different from that associated with brightness and contrast corrections. This 165 remark shows that the present framework is adaptable to various situations. 166 Two very different discretizations are studied hereafter.  BCCs were implemented (Table 3). Mesh regular (see Figure 1)

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Matching criterion see text

Interpolant cubic
Displacement noise-floor see Table 4 Strain calculation derivative of shape functions Strain noise-floor see Table 4 3. Results and compared to their counter-parts provided by DIC + BCC (i.e., ρ bcc ) using 199 Q8 (Figure 2(a)), or FM discretization (Figure 2(b)). If there is no effect using 200 the BCC procedure, then the data should lie on the 45°line. Further, the values 201 of ρ bcc resulting from the two discretizations used in the BCCs are compared in 202 Figure 2(c).

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The results for ρ bcc with the three corrections and the two discretizations 204 mostly lie below the 45°line, thereby proving that the BCCs reduced the overall 205 gray level residuals. This result shows that the BCC procedure is beneficial to 206 the reported DIC analyses. For the Q8 element, C and BC corrections achieved 207 lower levels in comparison to B corrections (Figure 2(a)). For the FM mesh, 208 the analyses using B or BC corrections led to lower residuals than that using C 209 correction only (Figure 2(b)). This difference in general trend is confirmed in 210 with a single Q8 element to be validated in addition to the FM discretization. 221 The same comparison procedure and permutation of images of set #1 was 222 used to study the standard displacement uncertainties, which are reported in 223 The case study discussed herein consists of monitoring and quantifying cracks 242 induced by curing and drying. Therefore, the maximum principal strain field 243 is one of the essential quantities to analyze since it can be related to the crack 244 opening displacement [45,48]. With the selected elements (i.e., T3), the strains 245 are uniform over each element. These values were considered with no filter-246 ing, and the in-plane principal strains were computed, of which the maximum 247 level was selected since cracks are detected with this quantity [13]. Figure 4 248 shows that the general trends are very close to those observed with displace-249 ment data (Figure 3) since strain uncertainties are proportional to displacement 250 uncertainties [57]. The previous results are summarized in Table 4. For both discretizations 252 studied herein, the BC correction leads to the lowest gray level residuals. For the 253 displacement and strain uncertainties, the coarse discretization provides slightly 254 lower levels. This observation means that, in the present case, the increase in 255 DOF (FM) is irrelevant for the gray level variations, which uniformly affect the 256 surface of interest, and the Q8 discretization is sufficient. The latter has a scale 257 separation with the kinematic basis that will avoid couplings between the two 258 steps of the registration procedure. The BCC procedure was applied to image set #2 during curing and drying 261 of the refractory cube. The discretizations and DIC parameters were the same 262 as those used in set #1 (Table 3 and Figure 1)  (b) Details for the sets of parameters with smaller RMS residuals. The BCC using coarse or fine discretizations reduced the residual levels when compared to DIC with no BCC In Figure 8(b), the three cases with higher RMS residuals (i.e., no BCC 322 and C corrections with both discretizations) are excluded to make easier the 323 comparison between B and BC results. The BC correction for Q8 and FM 324 reduced the residuals, and increased the difference between the results of the 325 two discretizations for B, which indicates that BC is more sensitive to the mesh 326 type than B. However, the mean difference of RMS residuals between B and BC 327 21 corrections remains small (i.e., ≈ 0.5% of the dynamic range).

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The maximum principal strain 1 fields for the last image of the experiment 329 (frame 200) were obtained using all BCC procedures (Figure 9). The levels of 1 are uniform, mainly for B and BC corrections and close to zero 331 in cluster regions, which is different for the grainy fields obtained using DIC 332 with no BCC (see Figure 5( easier between the fields of a pair of BCCs, their differences ∆ 1 are reported 352 in Figure 10.  The effect of the BCCs on the crack opening (or equivalently on the strain 367 fields) is analyzed in the sequel. In Figure 10 corrections. It is worth noting that finer meshes (e.g., of the characteristic size 433 that of pores) but not as fine as that used for capturing the complex kinematics 434 induced by crack networks could also be investigated in the future to confirm 435 this hypothesis.

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The above discussions showed that BCCs were desirable in the case studied 437 herein as they led to significant gains in terms of noise-floor levels and detection 438