Digital Image Correlation with Enhanced Accuracy and Efficiency: A Comparison of Two Subpixel Registration Algorithms

Abstract

The two major subpixel registration algorithms, currently being used in subset-based digital image correlation, are the classic Newton-Raphson (FA-NR) algorithm with forward additive mapping strategy and the recently introduced inverse compositional Gauss-Newton (IC-GN) algorithm. Although the equivalence of these two algorithms has been proved in existing studies, practical implementations of the two subpixel registration algorithms do involve differences, and therefore lead to different performance. In the present work, detailed theoretical error analyses of the two algorithms are performed. Based on the simple sum of squared difference criterion and the practical first-order shape function, analytic formulae that can quantify both the bias error (systematic error) and the variability (random error) in the displacements measured by IC-GN and FA-NR algorithms with various interpolation methods (i.e., cubic convolution interpolation, cubic polynomial interpolation, cubic B-spline interpolation and quintic B-spline interpolation) are derived. It is shown that, compared with FA-NR algorithm, IC-GN algorithm leads to reduced bias error in displacement estimation by eliminating noise-induced bias error, and gives rise on the average to smaller random errors in displacement estimation in the cases of high noise levels or using small subsets. Numerical tests with precisely controlled subpixel displacements confirm the correctness of the theoretical derivations. The results reveal that IC-GN algorithm outperforms the classic FA-NR algorithm not only in terms of computational efficiency, but also in respect of subpixel registration accuracy and noise-proof performance, and is strongly recommended as a standard subpixel registration algorithm for practical DIC applications instead of FA-NR algorithm.

Appendices

Appendix A: Rigorous Mathematical Derivation of the Variance of Displacement Error for IC-GN Algorithm

By taking imperfect intensity interpolation into consideration, more general formula regarding the variance of deformation error vector can be deduced for IC-GN algorithm as follows

$$ \begin{array}{l}\operatorname{Var}\left({\mathbf{p}}_{\mathbf{e}}\right)=\operatorname{Var}\left[{\left({\mathbf{H}}^{\hbox{-} 1}\right)}_{6\times 6}\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla {f}^{\prime}\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot \left({f}^{\prime }-{g}^{\prime}\right)}\right]\\ {}\kern3.7em =\kern0.7em {\operatorname{E}}^2\left[{\left({\mathbf{H}}^{\hbox{-} 1}\right)}_{6\times 6}\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla {f}^{\prime}\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}}\right]\cdot \operatorname{Var}\left[{\displaystyle \sum_{i,j\in \varOmega}\left({f}^{\prime }-{g}^{\prime}\right)}\right]\\ {}\kern3.7em +\operatorname{Var}\left[{\left({\mathbf{H}}^{\hbox{-} 1}\right)}_{6\times 6}\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla {f}^{\prime}\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}}\right]\cdot \left\{{\operatorname{E}}^2\left[{\displaystyle \sum_{i,j\in \varOmega}\left({f}^{\prime }-{g}^{\prime}\right)}\right]+\operatorname{Var}\left[{\displaystyle \sum_{i,j\in \varOmega}\left({f}^{\prime }-{g}^{\prime}\right)}\right]\right\}\end{array} $$

(A1)

where the general rules of variance calculation, i.e., Var(XY) = [E(X)]²Var(Y) + [E(Y)]²Var(X) + Var(X)Var(Y), are utilized.

According to equation (B3) in Ref. [32], the variance of deformation error vector can be simplified as

$$ \begin{array}{l}\operatorname{Var}\left({\mathbf{p}}_{\mathbf{e}}\right)\cong \mathrm{diag}{\left[{\left({\mathbf{J}}_f\right)}^T\left({\mathbf{J}}_f\right)\right]}^{-1}\cdot {\sigma}^2\left[1+\frac{1}{N^2}{\displaystyle \sum_{i,j\in \varOmega }K\left(\varDelta {x}_i,\varDelta {y}_j\right)}\right]\\ {}\kern4em +\mathrm{diag}{\left[{\left({\mathbf{J}}_f\right)}^T\left({\mathbf{J}}_f\right)\right]}^{-2}\cdot {\sigma}^2\cdot \frac{130}{144}\cdot \left\{{\left({f}^{\prime }-{g}^{\prime}\right)}^2-{\sigma}^2\left[{N}^2+{\displaystyle \sum_{i,j\in \varOmega }K\left(\varDelta {x}_i,\varDelta {y}_j\right)}\right]\right\}\end{array} $$

(A2)

It can be clearly seen that equation (A2) can be simplified to equation (13) by omitting the second term and removing interpolation error. Due to the relatively low SD of image noise, the following approximation can be obtained based on the use of the Barron operator

$$ \mathrm{diag}{\left[{\left({\mathbf{J}}_f\right)}^T\left({\mathbf{J}}_f\right)\right]}^{-1}\cong \mathrm{diag}{\left[{\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla f\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}^{\mathrm{T}}\cdot \left(\nabla f\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)\cdot \frac{130}{144}}\right]}^{-1} $$

(A3)

where (Δx _i , Δy _j ) is the fractional part of interpolation position in deformed subset; N is the subset size used in DIC. For the four aforementioned interpolation methods in 1D case, K(Δx _i , Δy _j ) is the sum-of-squares of the coefficients for the intensity, which can be expressed as

$$ \left\{\begin{array}{l}{C}_4:\kern0.6em K\left(\varDelta {x}_i\right)=\frac{1}{2}\left(10\varDelta {x}_i^6-30\varDelta {x}_i^5+21\varDelta {x}_i^4+8\varDelta {x}_i^3-9\varDelta {x}_i^2+2\right)\hfill \\ {}{L}_4:\kern0.6em K\left(\varDelta {x}_i\right)=\frac{1}{18}\left(10\varDelta {x}_i^6-30\varDelta {x}_i^5+\varDelta {x}_i^4+48\varDelta {x}_i^3-11\varDelta {x}_i^2-18\varDelta {x}_i+18\right)\hfill \\ {}{B}_4:\kern0.6em K\left(\varDelta {x}_i\right)=\frac{1}{18}\left(10\varDelta {x}_i^6-30\varDelta {x}_i^5+21\varDelta {x}_i^4+8\varDelta {x}_i^3-9\varDelta {x}_i^2+9\right)\hfill \\ {}{B}_6:\kern0.6em K\left(\varDelta {x}_i\right)=\frac{1}{7200}\left(126\varDelta {x}_i^{10}-630\varDelta {x}_i^9+735\varDelta {x}_i^8+840\varDelta {x}_i^7-1770\varDelta {x}_i^6-276\varDelta {x}_i^5+1350\varDelta {x}_i^4-375\varDelta {x}_i^2+2855\right)\hfill \end{array}\right. $$

(A4)

For clarity, function K(Δx _i ) is plotted as a function of subpixel positions for the four different interpolation methods in Fig. 9(a).

Fig. 9

(a) K(△x _i ), (b) D(△x _i ) and (c) W(△x _i ) as a function of subpixel position in 1D case for various interpolation methods

Appendix B: Theoretical Error Assessment of DIC with FA-NR Algorithm

Similar to equation (6), the SSD criterion for DIC using FA-GN algorithm can be written as

$$ {C}_{\mathrm{SSD}}\left({\mathbf{p}}_{\mathbf{e}}\right)={\displaystyle \sum_{i,j\in \varOmega }{\left[{f}^{\prime}\left({x}_i,{y}_j\right)-{g}^{\prime}\left({x}_i+{u}_i+{u}_{ie},{y}_j+{v}_j+{v}_{je}\right)\right]}^2} $$

(B1)

Substituting the first-order Taylor expansion of g′(x ^′ _i , y ^′ _j ) into SSD function, one can obtain

$$ \begin{array}{l}{C}_{\mathrm{SSD}}\left({\mathbf{p}}_{\mathbf{e}}\right)={\displaystyle \sum_{i,j\in \varOmega }{\left[{f}^{\prime}\left({x}_i,{y}_j\right)-{g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{u}_{ie}\cdot {g}_x^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{v}_{je}\cdot {g}_y^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}^2}\\ {}\kern4.8em \cong {\displaystyle \sum_{i,j\in \varOmega}\left[f\left({x}_i,{y}_j\right)+{n}_f\left({x}_i,{y}_j\right)-g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{n}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{u}_{ie}\cdot {g}_x^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{v}_{je}\cdot {g}_y^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}\end{array} $$

(B2)

Minimization of the C _SSD function with respect to p _e gives

$$ \begin{array}{l}{\left({\mathbf{p}}_{\mathbf{e}}\right)}_{6\times 1}=-{\left({\mathbf{H}}^{\hbox{-} 1}\right)}_{6\times 6}\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left[\nabla {g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right]}_{6\times 1}^{\mathrm{T}}\cdot \left[{f}^{\prime}\left({x}_i,{y}_j\right)-{g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}\\ {}\kern3.1em =-{\left({\mathbf{H}}^{\hbox{-} 1}\right)}_{6\times 6}\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left[\nabla {g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right]}_{6\times 1}^{\mathrm{T}}\cdot \left[f\left({x}_i,{y}_j\right)+{n}_f\left({x}_i,{y}_j\right)-g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{n}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}\end{array} $$

(B3)

where H can be written as

$$ {\mathbf{H}}_{6\times 6}={\displaystyle \sum_{i,j\in \varOmega }{\left[\nabla {g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right]}^{\mathrm{T}}\cdot \left[\nabla {g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right]} $$

(B4)

with $\nabla {g}^{\prime}\left({x}_i^{\prime },{y}_j^{\prime}\right)\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}=\left({g}_x^{\prime },{g}_x^{\prime}\varDelta {x}_i,{g}_x^{\prime}\varDelta {y}_j,{g}_y^{\prime },{g}_y^{\prime}\varDelta {x}_i,{g}_y^{\prime}\varDelta {y}_j\right)$.

Analogously, the variance matrix of deformation vector errors p _e can be simplified as

$$ \operatorname{Var}\left({\mathbf{p}}_{\mathbf{e}}\right)\cong 2{\sigma}^2\cdot \mathrm{diag}\left({\mathbf{H}}_{6\times 6}^{-1}\right) $$

(B5)

Based on the above equation, the variance of u _e can be extracted as

$$ \operatorname{Var}\left({u}_e\right)\cong 2{\sigma}^2\cdot {\left({\mathbf{H}}^{-1}\right)}_{11}\approx \frac{2{\sigma}^2}{{\displaystyle \sum {\displaystyle \sum {\left({g}_x^{\prime}\right)}^2}}} $$

(B6)

To better approximate practical experiment, intensity interpolation should be considered with respect to equation (B6). According to Ref [32], more general governing formula can be expressed as

$$ \begin{array}{l}\operatorname{Var}\left({\mathbf{p}}_{\mathbf{e}}\right)\cong \operatorname{Var}\left[{\left({\mathbf{H}}^{\hbox{-} 1}\right)}_{6\times 6}\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla {g}^{\prime}\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot \left({f}^{\prime }-{g}^{\prime}\right)}\right]\\ {}\kern4em \cong \mathrm{diag}{\left[{\left({\mathbf{J}}_g\right)}^T\left({\mathbf{J}}_g\right)\right]}^{-1}\cdot {\sigma}^2\left[1+\frac{1}{N^2}{\displaystyle \sum_{i,j\in \varOmega }K\left(\varDelta {x}_i,\varDelta {y}_j\right)}\right]\\ {}\kern4em +\mathrm{diag}{\left[{\left({\mathbf{J}}_g\right)}^T\left({\mathbf{J}}_g\right)\right]}^{-2}\cdot \frac{\sigma^2}{N^2}{\displaystyle \sum_{i,j\in \varOmega }D\left(\varDelta {x}_i,\varDelta {y}_j\right)}\cdot \left\{{\left({f}^{\prime }-{g}^{\prime}\right)}^2-{\sigma}^2\left[{N}^2+{\displaystyle \sum_{i,j\in \varOmega }K\left(\varDelta {x}_i,\varDelta {y}_j\right)}\right]\right\}\end{array} $$

(B7)

Due to the relatively low SD of image noise, the following approximation can be explained

$$ \mathrm{diag}{\left[{\left({\mathbf{J}}_g\right)}^T\left({\mathbf{J}}_g\right)\right]}^{-1}\cong \mathrm{diag}{\left[{\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla {g}^{\prime}\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}^{\mathrm{T}}\cdot \left(\nabla {g}^{\prime}\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)\cdot D\left(\varDelta {x}_i,\varDelta {y}_j\right)}\right]}^{-1} $$

(B8)

where (Δx _i , Δy _j ) is the fractional part of interpolation position in deformed subset. For the four aforementioned interpolation methods in 1D case, D(Δx _i , Δy _j ) is the sum-of-squares of the coefficients for the intensity gradient. Figure 9(b) plots function D(Δx _i ) against the fractional part of interpolation position for the four different interpolation methods. Due to the concave quadratic distribution of D ^‐ 1(Δx _i , Δy _j ) and K(Δx _i , Δy _j ), the quadratic shape of SD error in FA-NR algorithm is theoretically predictable.

Also, the mathematical expectation of deformation vector errors p _e can be described as

$$ \begin{array}{l}\operatorname{E}\left({\mathbf{p}}_{\mathbf{e}}\right)\cong -\left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot \operatorname{E}\left[{\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla g\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot \left({f}^{\prime }-{g}^{\prime}\right)}\right]-\left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot \operatorname{E}\left[{\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla {n}_g\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot \left({f}^{\prime }-{g}^{\prime}\right)}\right]\\ {}\kern3.2em ={\mathrm{E}}_{\mathrm{int}}\left({\mathbf{p}}_{\mathbf{e}}\right)+{\mathrm{E}}_{\mathrm{noise}}\left({\mathbf{p}}_{\mathbf{e}}\right)\end{array} $$

(B9)

The interpolation-induced bias error E_int(p _e ) can be represented as

$$ \begin{array}{l}{\operatorname{E}}_{\mathrm{int}}\left({\mathbf{p}}_{\mathbf{e}}\right)=-\left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla g\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot \operatorname{E}\left[f\left({x}_i,{y}_j\right)+{n}_f\left({x}_i,{y}_j\right)-g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{n}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}\\ {}\kern3.5em =-\left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla g\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot \mathrm{E}\left[f\left({x}_i,{y}_j\right)-g\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}\\ {}\kern3.4em =-\left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\nabla g\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot h\left({x}_i,{y}_j\right)}\end{array} $$

(B10)

Similarly, the interpolation-induced bias error of displacement component u _e can be extracted as

$$ {\mathrm{E}}_{\mathrm{int}}\left({u}_e\right)\cong \frac{{\displaystyle \sum {\displaystyle \sum \nabla g\cdot h}}}{{\displaystyle \sum {\displaystyle \sum {\left({g}_x^{\prime}\right)}^2}}} $$

(B11)

And the noise-induced bias error E_noise(p _e ) can be further written as

$$ \begin{array}{l}{\operatorname{E}}_{\mathrm{noise}}\left({\mathbf{p}}_{\mathbf{e}}\right)\cong \left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot \operatorname{E}\left\{{\displaystyle \sum_{i,j\in \varOmega }{\left[\nabla {n}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)\cdot \frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right]}_{6\times 1}^{\mathrm{T}}\cdot {n}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)}\right\}\\ {}\kern4.5em \cong \left({\mathbf{H}}_{6\times 6}^{-1}\right)\cdot {\sigma}^2\cdot {\displaystyle \sum_{i,j\in \varOmega }{\left(\frac{\partial \mathbf{W}}{\partial \mathbf{p}}\right)}_{6\times 1}^{\mathrm{T}}\cdot W\left(\varDelta {x}_i,\varDelta {y}_j\right)}\end{array} $$

(B12)

where (Δx _i , Δy _j ) is the fractional part of interpolation position in deformed subset; W(Δx _i , Δy _j ) denotes weight function at certain interpolation position. For example, in the case of one-dimension, weight functions of four aforementioned interpolation strategies have the following forms

$$ \left\{\begin{array}{l}{C}_4:\kern0.6em W\left(\varDelta {x}_i\right)=\frac{1}{2}\left(-30\varDelta {x}_i^5+75\varDelta {x}_i^4-42\varDelta {x}_i^3-12\varDelta {x}_i^2+9\varDelta {x}_i\right)\hfill \\ {}{L}_4:\kern0.6em W\left(\varDelta {x}_i\right)=\frac{1}{18}\left(-30\varDelta {x}_i^5+75\varDelta {x}_i^4-2\varDelta {x}_i^3-72\varDelta {x}_i^2+11\varDelta {x}_i+9\right)\hfill \\ {}{B}_4:\kern0.5em W\left(\varDelta {x}_i\right)=\frac{1}{18}\left(-30\varDelta {x}_i^5+75\varDelta {x}_i^4-42\varDelta {x}_i^3-12\varDelta {x}_i^2+9\varDelta {x}_i\right)\hfill \\ {}{B}_6:\kern0.6em W\left(\varDelta {x}_i\right)=\frac{1}{480}\left(-42\varDelta {x}_i^9+189\varDelta {x}_i^8-196\varDelta {x}_i^7-196\varDelta {x}_i^6+354\varDelta {x}_i^5+46\varDelta {x}_i^4-180\varDelta {x}_i^3+25\varDelta {x}_i\right)\hfill \end{array}\right. $$

(B13)

Specifically, the noise-induced bias error of displacement component u _e can be extracted as

$$ {\operatorname{E}}_{\mathrm{noise}}\left({u}_e\right)\cong \frac{\sigma^2\cdot {\displaystyle \sum {\displaystyle \sum W}}}{{\displaystyle \sum {\displaystyle \sum {\left({g}_x^{\prime}\right)}^2}}} $$

(B14)

It can be intuitively seen that mean bias errors due to FA-GN algorithm have direct relationship with image noise level, thus confirming its noise susceptibility. Figure 9(c) plots the weight functions as a function of the fractional part of interpolation position for various interpolation strategies.