Digital Image Correlation with Self-Adaptive Gaussian Windows

Abstract

A novel subpixel registration algorithm with Gaussian windows is put forward for accurate deformation measurement in digital image correlation technique. Based on speckle image quality and potential deformation states, this algorithm can automatically minimize the influence of subset sizes by self-adaptively tuning the Gaussian window shapes with the aid of a so-called weighted sum-of-squared difference correlation criterion. Numerical results of synthetic speckle images undergoing in-plane sinusoidal displacement fields demonstrate that the proposed algorithm can significantly improve displacement and strain measurement accuracy especially in the case with relatively large deformation.

Appendices

Appendix A

For the Gaussian window function adopted in the manuscript, on the one hand, one may easily realize the following expressions

$$ \left\{ {\begin{array}{*{20}{c}} \hfill {\frac{{\partial {{W}_{G}}\left( {X,Y;{{D}_{0}}} \right)}}{{\partial {{p}_{7}}}} = \frac{{\partial {{W}_{G}}\left( {X,Y;{{D}_{0}}} \right)}}{{\partial {{D}_{0}}}} = {{W}_{G}}\left( {X,Y;{{D}_{0}}} \right) \times {{D}^{2}}\left( {X,Y} \right)/D_{0}^{3}} \\ \hfill {\frac{{\partial {{W}_{G}}\left( {x,y;{{D}_{0}}} \right)}}{{\partial {{p}_{7}}}} = \frac{{\partial {{W}_{G}}\left( {x,y;{{D}_{0}}} \right)}}{{\partial {{D}_{0}}}} = {{W}_{G}}\left( {x,y;{{D}_{0}}} \right) \times {{D}^{2}}\left( {x,y} \right)/D_{0}^{3}} \\ \end{array} } \right. $$

(A.1)

with $D\left( {X,Y} \right) = \sqrt {{{{{\left( {X - {{X}_0}} \right)}}^2} + {{{\left( {Y - {{Y}_0}} \right)}}^2}}}$, $D\left( {x,y} \right) = \sqrt {{{{{\left( {x - {{x}_0}} \right)}}^2} + {{{\left( {y - {{y}_0}} \right)}}^2}}}$, ${{x}_0} = {{X}_0} + u$ and ${{y}_0} = {{Y}_0} + v$. In combination with equation (2), on the other hand, one may find that

$$ \matrix{ {\frac{\partial }{{\partial {{p}_1}}}\left[ {{{W}_G}\left( {x,y;{{D}_0}} \right)} \right] = \frac{\partial }{{\partial u}}\left\{ {\exp \left[ {\frac{{ - {{D}^2}\left( {x,y} \right)}}{{2D_0^2}}} \right]} \right\}} \\ { = \frac{\partial }{{\partial u}}\left\{ {\exp \left[ { - \frac{{{{{\left( {x - {{x}_0}} \right)}}^2} + {{{\left( {y - {{y}_0}} \right)}}^2}}}{{2D_0^2}}} \right]} \right\}} \\ { = \frac{\partial }{{\partial u}}\left\{ {\exp \left[ { - \frac{{{{{\left[ {\left( {1 + \frac{{\partial u}}{{\partial X}}} \right)\left( {X - {{X}_0}} \right) + \frac{{\partial u}}{{\partial Y}}\left( {Y - {{Y}_0}} \right)} \right]}}^2} + {{{\left[ {\frac{{\partial v}}{{\partial X}}\left( {X - {{X}_0}} \right) + \left( {1 + \frac{{\partial v}}{{\partial Y}}} \right)\left( {Y - {{Y}_0}} \right)} \right]}}^2}}}{{2D_0^2}}} \right]} \right\}} \\ { = 0} \\ }<!end array> $$

(A.2)

$$ \matrix{ {\frac{\partial }{{\partial {{p}_2}}}\left[ {{{W}_G}\left( {x,y;{{D}_0}} \right)} \right] = \frac{\partial }{{\partial \left( {\frac{{\partial u}}{{\partial X}}} \right)}}\left\{ {\exp \left[ {\frac{{ - {{D}^2}\left( {x,y} \right)}}{{2D_0^2}}} \right]} \right\}} \\ { = \frac{{ - {{W}_G}\left( {x,y;{{D}_0}} \right)}}{{2D_0^2}}\frac{{\partial \left[ {{{D}^2}\left( {x,y} \right)} \right]}}{{\partial \left( {\frac{{\partial u}}{{\partial X}}} \right)}}} \\ { = \frac{{ - {{W}_G}\left( {x,y;{{D}_0}} \right)}}{{D_0^2}}\left[ {\left( {1 + \frac{{\partial u}}{{\partial X}}} \right)\left( {X - {{X}_0}} \right) + \frac{{\partial u}}{{\partial Y}}\left( {Y - {{Y}_0}} \right)} \right]\left( {X - {{X}_0}} \right)} \\ }<!end array> $$

(A.3)

$$ \matrix{ {\frac{\partial }{{\partial {{p}_3}}}\left[ {{{W}_G}\left( {x,y;{{D}_0}} \right)} \right] = \frac{\partial }{{\partial \left( {\frac{{\partial u}}{{\partial Y}}} \right)}}\left\{ {\exp \left[ {\frac{{ - {{D}^2}\left( {x,y} \right)}}{{2D_0^2}}} \right]} \right\}} \\ { = \frac{{ - {{W}_G}\left( {x,y;{{D}_0}} \right)}}{{2D_0^2}}\frac{{\partial \left[ {{{D}^2}\left( {x,y} \right)} \right]}}{{\partial \left( {\frac{{\partial u}}{{\partial Y}}} \right)}}} \\ { = \frac{{ - {{W}_G}\left( {x,y;{{D}_0}} \right)}}{{D_0^2}}\left[ {\left( {1 + \frac{{\partial u}}{{\partial X}}} \right)\left( {X - {{X}_0}} \right) + \frac{{\partial u}}{{\partial Y}}\left( {Y - {{Y}_0}} \right)} \right]\left( {Y - {{Y}_0}} \right)} \\ }<!end array> $$

(A.4)

In a similar way, one can further obtain that

$$ \frac{\partial }{{\partial {{p}_4}}}\left[ {{{W}_G}\left( {x,y;{{D}_0}} \right)} \right] = \frac{\partial }{{\partial v}}\left\{ {\exp \left[ {\frac{{ - {{D}^2}\left( {x,y} \right)}}{{2D_0^2}}} \right]} \right\} = 0 $$

(A.5)

$$ \matrix{ {\frac{\partial }{{\partial {{p}_5}}}\left[ {{{W}_G}\left( {x,y;{{D}_0}} \right)} \right] = \frac{\partial }{{\partial \left( {\frac{{\partial v}}{{\partial X}}} \right)}}\left\{ {\exp \left[ {\frac{{ - {{D}^2}\left( {x,y} \right)}}{{2D_0^2}}} \right]} \right\}} \\ { = \frac{{ - {{W}_G}\left( {x,y;{{D}_0}} \right)}}{{D_0^2}}\left[ {\frac{{\partial v}}{{\partial X}}\left( {X - {{X}_0}} \right) + \left( {1 + \frac{{\partial v}}{{\partial Y}}} \right)\left( {Y - {{Y}_0}} \right)} \right]\left( {X - {{X}_0}} \right)} \\ }<!end array> $$

(A.6)

$$ \matrix{ {\frac{\partial }{{\partial {{p}_6}}}\left[ {{{W}_G}\left( {x,y;{{D}_0}} \right)} \right] = \frac{\partial }{{\partial \left( {\frac{{\partial v}}{{\partial Y}}} \right)}}\left\{ {\exp \left[ {\frac{{ - {{D}^2}\left( {x,y} \right)}}{{2D_0^2}}} \right]} \right\}} \\ { = \frac{{ - {{W}_G}\left( {x,y;{{D}_0}} \right)}}{{D_0^2}}\left[ {\frac{{\partial v}}{{\partial X}}\left( {X - {{X}_0}} \right) + \left( {1 + \frac{{\partial v}}{{\partial Y}}} \right)\left( {Y - {{Y}_0}} \right)} \right]\left( {Y - {{Y}_0}} \right)} \\ }<!end array> $$

(A.7)

Appendix B

Around a specified point of interest (X _S0, Y _S0) located on the reference image as shown in Fig. 6, a square region with Q × Q pixels, usually called “strain window”, is selected, on which N × Ndiscrete displacement sampling points are beforehand chosen based on a regular sampling spacing L, while their displacement values are calculated via these above-mentioned iterative algorithms. For simplification, we employ a bilinear Lagrange polynomial [21] to mathematically fit the local displacement field involved in the specific strain window, which may be expressed as

$$ \left\{ {\begin{array}{*{20}{c}} {{{u}^{{fit}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right) = {{a}_{1}} + {{a}_{3}}{{X}_{{Sj}}} + {{a}_{5}}{{Y}_{{Sl}}} + {{a}_{7}}{{X}_{{Sj}}}{{Y}_{{Sl}}}} \hfill \\ {{{v}^{{fit}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right) = {{a}_{2}} + {{a}_{4}}{{X}_{{Sj}}} + {{a}_{6}}{{Y}_{{Sl}}} + {{a}_{8}}{{X}_{{Sj}}}{{Y}_{{Sl}}}} \hfill \\ \end{array} } \right. $$

(B.1)

where ${{\bf u}^{{fit}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right) = {{\left[ {{{u}^{{fit}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right), {{v}^{{fit}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right)} \right]}^T}$ denotes the fitted displacement vector on the discrete sampling position (X _Sj , Y _Sl ) with $- {{{\left( {N - 1} \right)}} \left/ {2} \right.} \leqslant j,l \leqslant {{{\left( {N - 1} \right)}} \left/ {2} \right.}$ (Notice that, for convenience, we here always set N to be an odd number greater than 1). In the sense of least squares, we can easily figure out the analytical expressions of the fitting coefficients, i.e., a ₁, a ₂, …, a ₈, with the help of the cost function [21]

$$ E\left( {{{a}_1},{{a}_2},...,{{a}_8}} \right) = \sum\limits_{{j,l = - \frac{{N - 1}}{2}}}^{{\frac{{N - 1}}{2}}} {{{{\left| {{{u}^{{fit}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right) - {{u}^{{dic}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right)} \right|}}^2}} $$

(B.2)

where ${{{{\bf u}}}^{{dic}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right) = {{\left[ {{{u}^{{dic}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right), {{v}^{{dic}}}\left( {{{X}_{{Sj}}},{{Y}_{{Sl}}}} \right)} \right]}^T}$ refers to the corresponding displacement vector numerically calculated by the conventional NR method or the proposed algorithm with self-adaptive Gaussian windows. In this way, the strain components on the specified point of interest (X _S0, Y _S0) can readily be derived from equation (B.1) as

$$ \left\{ {\begin{array}{*{20}{c}} {\left( {\frac{{\partial u}}{{\partial X}}} \right)_{{\left( {{{X}_{{Sj,}}}{{Y}_{{Sl}}}} \right)}}^{{fit}} = {{a}_{3}} + {{a}_{7}}{{Y}_{{Sl}}}} \\ {\left( {\frac{{\partial u}}{{\partial Y}}} \right)_{{\left( {{{X}_{{Sj,}}}{{Y}_{{Sl}}}} \right)}}^{{fit}} = {{a}_{5}} + {{a}_{7}}{{X}_{{Sj}}}} \\ {\left( {\frac{{\partial v}}{{\partial X}}} \right)_{{\left( {{{X}_{{Sj,}}}{{Y}_{{Sl}}}} \right)}}^{{fit}} = {{a}_{4}} + {{a}_{8}}{{Y}_{{Sl}}}} \\ {\left( {\frac{{\partial v}}{{\partial Y}}} \right)_{{\left( {{{X}_{{Sj,}}}{{Y}_{{Sl}}}} \right)}}^{{fit}} = {{a}_{6}} + {{a}_{8}}{{X}_{{Sj}}}} \\ \end{array} } \right. $$

(B.3)

By means of the strain-window method in which the parameters, L and N _, were specifically set as 2 pixels and 5, respectively, we recalculated the strain fields from the same synthetic speckle images as described in the preceding section, based upon the displacement data acquired by the traditional NR method and those obtained via the proposed algorithm with Gaussian windows. The corresponding results were shown in Fig. 7.