Appendix A: Mathematical derivation of the expectation and variance of the displacement error for the IC-GN algorithm
On the noise contaminated speckle images, the incremental deformation vector ΔP
IC ‐ GN can be expressed as
$$ \Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla \tilde{f}\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l} f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\\ {}-\eta \left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\end{array}\right\} $$
(34)
The interpolation error of the intensity \( {\Delta}_p\left({x}_i^{\prime },{y}_j^{\prime}\right) \) at sub-pixel location \( \left({x}_i^{\prime },{y}_j^{\prime}\right) \) falls in the range of [−0.1, 0.1], as shown in Fig. 3. Its value is considerably smaller than the intensity \( \tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right) \) in the target subset (which varies from 0 to 255) if the image is not extremely dark and the high accuracy interpolation method is selected (e.g., cubic B-spline interpolation). The mean gray intensities of the reference subset and the target subset, i.e., \( {\overline{f}}_n \) and \( {\overline{g}}_n \), can be written as:
$$ \begin{array}{l}{\overline{f}}_n=\frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m\tilde{f}\left({x}_i,{y}_j\right)\\ {}\kern2em =\frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m\left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)\right]\\ {}\kern2em =\frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m f\left({x}_i,{y}_j\right)=\overline{f}\\ {}{\overline{g}}_n=\frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)\\ {}\kern2em =\frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m\left[\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_p\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]\\ {}\kern2em \approx \frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m\left[ G\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]\\ {}\kern2em =\frac{1}{ n m}\sum_{i=1}^n\sum_{j=1}^m G\left({x}_i^{\prime },{y}_j^{\prime}\right)={\overline{G}}_n\end{array} $$
(35)
where \( G\left({x}_i^{\prime },{y}_j^{\prime}\right) \) denotes the exact intensity without an interpolation error at the sub-pixel location \( \left({x}_i^{\prime },{y}_j^{\prime}\right) \) in the target subset. As location \( \left({x}_i^{\prime },{y}_j^{\prime}\right) \) in the target subset corresponds to (x
i
, y
j
) in the reference subset, \( {\overline{f}}_n={\overline{G}}_n={\overline{g}}_n \).
Assuming a constant level of the intensity interpolation error Δ
p
in the target subset, we have
$$ \begin{array}{l}\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left(\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right)}^2}\hfill \\ {}\approx \sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_p\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{G}}_n\right]}^2}\hfill \\ {}=\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{G}}_n\right]}^2+\sum_{i=1}^n\sum_{j=1}^m{\Delta}_p^2\left({x}_i^{\prime },{y}_j^{\prime}\right)+\sum_{i=1}^n\sum_{j=1}^m2{\Delta}_p\left({x}_i^{\prime },{y}_j^{\prime}\right)\left[\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{G}}_n\right]}\hfill \\ {}=\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{G}}_n\right]}^2+\sum_{i=1}^n\sum_{j=1}^m{\Delta}_p^2\left({x}_i^{\prime },{y}_j^{\prime}\right)}\hfill \end{array} $$
(36)
where \( {\Delta}_p^2\left({x}_i^{\prime },{y}_j^{\prime}\right) \) can be disregarded compared with \( {\left(\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{G}}_n\right)}^2 \) in most of real speckle images. Combining the statistical characteristics of Gaussian white noise, we have
$$ \begin{array}{l}\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]}^2}\approx \sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{G}}_n\right]}^2}\\ {}\kern17.5em =\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]}^2}\end{array} $$
(37)
Thus
$$ \eta =\frac{\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]}^2}}{\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]}^2}}\approx 1 $$
(38)
Then the expectation and variance of the incremental displacement vector ΔP can be obtained:
$$ \begin{array}{l} E\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)\\ {}= E\left\{-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\sum_{i=1}^n\sum_{j=1}^m\left\{{\left[\nabla \tilde{f}\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\right\}\right\}\right\}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1} E\left\{\sum_{i=1}^n\sum_{j=1}^m\left\{{\left[\nabla f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\right\}\right\}\right\}\\ {}\kern1.5em -{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1} E\left\{\sum_{i=1}^n\sum_{j=1}^m\left\{{\left[\nabla {\xi}_f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\right\}\right\}\right\}\end{array} $$
(39)
$$ Var\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)=\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\right) Var\left\{ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\right\} $$
(40)
where ∇ξ
f
(x
i
, y
j
) = [ξ
fx
, ξ
fy
]. According to equation (39), the expectation of ΔP
IC ‐ GN
can be divided into two parts, i.e.,
$$ E\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)={E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)+{E}_{\mathrm{noise}}\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right) $$
(41)
where
$$ \begin{array}{l}{E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m\left\{{\left[\nabla f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\right\}\right\}\right\}\\ {}{E}_{\mathrm{noise}}\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m\left\{{\left[\nabla {\xi}_f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\right\}\right\}\right\}\end{array} $$
(42)
E
int(ΔP
IC ‐ GN) is associated with the interpolation-induced system error and E
noise(ΔP
IC ‐ GN) is associated with the noise-induced system error. The two items can be further reduced:
$$ \begin{array}{l}{E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)\hfill \\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m\left\{{\left[\nabla f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l} f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-\overline{f}-{\overline{\xi}}_f\\ {}-\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-\overline{g}-{\overline{\xi}}_g\right]\end{array}\right\}\right\}\right\}\hfill \\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m\left\{\left[\nabla f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]\left[ f\left({x}_i,{y}_j\right)- g\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]\right\}\right\}\hfill \\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right)\right\}\hfill \end{array} $$
(43)
$$ \begin{array}{l}{E}_{\mathrm{noise}}\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)\hfill \\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n-\left(\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right)\right]\right\}\hfill \\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l} f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-\overline{f}-{\overline{\xi}}_f\\ {}-\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-\overline{g}-{\overline{\xi}}_g\right]\end{array}\right\}\right\}\hfill \\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\cdot {\xi}_f\left({x}_i,{y}_j\right)\right\}\hfill \end{array} $$
(44)
Figure 9 shows the distribution of the self-correlation function \( R(n)=\sum_{j=-\infty}^{\infty}\xi (j)\xi \left( j- n\right) \) of Gaussian white noise with a zero mean and different variances. This finding indicates a characteristic of R(n), i.e. R(n) = R(−n). The value of R(n) decreases sharply to a negligible level compared with R(0) when n ≥ 1. These characteristics eliminate E
noise(ΔP
IC ‐ GN) when ∇ξ
f
(x
i
, y
j
) is calculated using a central difference-based gradient operator. Therefore, equation (40) is simplified as follows:
$$ E\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)=-{\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla f\left({x}_i,{y}_j\right)\frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right)\right\} $$
(45)
According to the general properties of un-correlated signals (i.e.,Var(A ± B) = Var(A) + Var(B)), the variance of ΔP
IC ‐ GN can be reduced to
$$ \begin{array}{l} Var\left(\Delta {\mathbf{P}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}\right)=\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\right)\left[ Var\left( f- g\right)+ Var\left({\xi}_f-{\xi}_g\right)\right]\\ {}\kern9.5em =\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\right)\left[ Var\left( f- g\right)+ Var\left({\xi}_f\right)+ Var\left({\xi}_g\right)\right]\\ {}\kern9.5em =\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{IC}\hbox{-} \mathrm{GN}}^{-1}\right)\left({\sigma}_f^2+{\sigma}_g^2\right)\end{array} $$
(46)
Appendix B: Mathematical derivation of the expectation and variance of the displacement error for the FA-NR algorithm
On the noise contaminated speckle images, the ZNSSD correlation criterion for the FA-NR algorithm can be expressed as
$$ {C}_{\mathrm{ZNSSD}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)=\sum_{i=1}^n\sum_{j=1}^m\left\{\frac{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n}{\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]}^2}}-\frac{\tilde{g}\left({x}_i^{\prime }+\Delta {u}_i,{y}_j^{\prime }+\Delta {v}_j\right)-{\overline{g}}_n}{\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]}^2}}\right\} $$
(47)
Substituting the first-order Taylor expansion of \( \tilde{g}\left({x}_i^{\prime }+\Delta {u}_i,{y}_j^{\prime }+\Delta {v}_j\right) \) into equation (47) yields
$$ \begin{array}{l}{C}_{\mathrm{ZNSSD}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\hfill \\ {}=\sum_{i=1}^n\sum_{j=1}^m\left\{\frac{\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n}{\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]}^2}}-\frac{\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n+\Delta {u}_i{\tilde{g}}_x\left({x}_i^{\prime },{y}_j^{\prime}\right)+\Delta {v}_j{\tilde{g}}_y\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]}{\sqrt{\sum_{i=1}^n\sum_{j=1}^m{\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]}^2}}\right\}\hfill \end{array} $$
(48)
Minimizating C
ZNSSD(ΔP
FA ‐ NR) with respect to ΔP
FA ‐ NR, combined with equation (38), yields
$$ \begin{array}{l}\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla \tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial p}\right]}^T\left\{\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]-\eta \left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\right\}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla \tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial p}\right]}^T\left\{\begin{array}{l}\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\end{array} $$
(49)
In contrast to the IC-GN algorithm, in which the gradient at an integer pixel is directly calculated using a fourth-order central difference operator, the gradient \( \nabla g\left({x}_i^{\prime },{y}_j^{\prime}\right) \) in the FA-NR algorithm at sub-pixel location \( \left({x}_i^{\prime },{y}_j^{\prime}\right) \) needs to be estimated through interpolation. Considering the interpolation error, the gradients of the image and noise map at the sub-pixel location can be written as:
$$ \begin{array}{cc}\hfill {g}_x\left({x}_i^{\prime },{y}_j^{\prime}\right)={G}_x\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{px},\hfill & \hfill {g}_y\left({x}_i^{\prime },{y}_j^{\prime}\right)={G}_y\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{py}\hfill \\ {}\hfill {\xi}_{g x}\left({x}_i^{\prime },{y}_j^{\prime}\right)={\xi}_{G x}\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{nx},\hfill & \hfill {\xi}_{g y}\left({x}_i^{\prime },{y}_j^{\prime}\right)={\xi}_{G y}\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{ny}\hfill \end{array} $$
(50)
where G
x
, G
y
, ξ
Gx
and ξ
Gy
denote the exact gradients without the interpolation error at the sub-pixel location \( \left({x}_i^{\prime },{y}_j^{\prime}\right) \), and Δ
px
, Δ
py
, Δ
nx
and Δ
ny
are the corresponding interpolation errors.
The expectation and variance of ΔP
FA ‐ NR can be estimated through the procedure demonstrated in Appendix A:
$$ \begin{array}{l} E\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\\ {}= E\Big[-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla \tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l}\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1} E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla g\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l}\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\right\}\\ {}\kern1.5em -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1} E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l}\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\right\}\\ {}={E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)+{E}_{\mathrm{noise}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\end{array} $$
(51)
$$ \begin{array}{l} Var\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\\ {}= Var\left\{-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla \tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial p}\right]}^T\left\{\begin{array}{l}\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\right\}\\ {}= Var\left\{-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla \tilde{G}\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial p}\right]}^T\left\{\begin{array}{l}\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\right\}\\ {}\kern1.5em + Var\left\{-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\left[{\Delta}_{tx},{\Delta}_{ty}\right]\frac{\partial W}{\partial p}\right]}^T\left\{\begin{array}{l}\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\eta \left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\right\}\\ {}=\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\right) Var\left\{\begin{array}{l}\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\\ {}\kern1.5em +{E}^2\left({\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\left[{\Delta}_{tx},{\Delta}_{ty}\right]\frac{\partial W}{\partial p}\right]}^T\right)\cdot Var\left(\begin{array}{l}\left[ g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[ f\left({x}_i,{y}_j\right)+{\xi}_f\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right)\\ {}=\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\right)\left[ Var\left( g- f\right)+ Var\left({\xi}_f\right)+ Var\left({\xi}_g\right)\right]\\ {}\kern1.5em +{E}^2\left({\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\left[{\Delta}_{tx},{\Delta}_{ty}\right]\frac{\partial W}{\partial p}\right]}^T\right)\left[ Var\left( g- f\right)+ Var\left({\xi}_f\right)+ Var\left({\xi}_g\right)\right]\\ {}=\mathit{\operatorname{diag}}\left({\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\right)\left({\sigma}_f^2+{\sigma}_g^2\right)+{E}^2\left[{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\sum_{i=1}^n\sum_{j=1}^m{\left[\left[{\Delta}_{tx},{\Delta}_{ty}\right]\frac{\partial W}{\partial p}\right]}^T\right]\left({\sigma}_f^2+{\sigma}_g^2\right)\end{array} $$
(52)
E
int(ΔP
FA ‐ NR) can be reduced:
$$ \begin{array}{l}{E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla g\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\left\{\begin{array}{l}\left[\tilde{g}\left({x}_i^{\prime },{y}_j^{\prime}\right)-{\overline{g}}_n\right]\\ {}-\left[\tilde{f}\left({x}_i,{y}_j\right)-{\overline{f}}_n\right]\end{array}\right\}\right\}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\left[\begin{array}{l}{G}_x\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{px},\\ {}{G}_y\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{py}\end{array}\right]\frac{\partial W}{\partial P}\right]}^T\left[\begin{array}{l} g\left({x}_i^{\prime },{y}_j^{\prime}\right)- f\left({x}_i,{y}_j\right)\\ {}-\left({\overline{g}}_n-{\overline{f}}_n\right)\end{array}\right]\right\}\\ {}\approx -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right)\right\}\\ {}\kern1.5em -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\left[{\Delta}_{px},{\Delta}_{py}\right]\cdot \frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right)\right\}\end{array} $$
(53)
where \( {\left[\left[{\Delta}_{px},{\Delta}_{py}\right]\cdot \frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right) \) is a high minimum-order quantity compared with \( {\left[\nabla G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right) \) and can be neglected. Thus,
$$ {E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\approx -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right)\right\} $$
(54)
For E
noise(ΔP
FA ‐ NR), combining \( E\left({\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\right)=0 \) and \( \nabla {\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)=\left[{\xi}_{Gx}\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{nx},{\xi}_{Gx}\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_{ny}\right] \),
$$ \begin{array}{l}{E}_{\mathrm{noise}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\left[\begin{array}{l} g\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\xi}_g\left({x}_i^{\prime },{y}_j^{\prime}\right)- f\left({x}_i,{y}_j\right)\\ {}-{\xi}_f\left({x}_i,{y}_j\right)-\left({\overline{g}}_n-{\overline{f}}_n\right)\end{array}\right]\right\}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot \left[{\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_n\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]\right\}\\ {}\kern1.5em -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\left[{\Delta}_{n x},{\Delta}_{n y}\right]\frac{\partial W}{\partial P}\right]}^T\cdot \left[{\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)+{\Delta}_n\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]\right\}\\ {}=-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot {\Delta}_n\left({x}_i^{\prime },{y}_j^{\prime}\right)\right\}\\ {}\kern1.5em -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left(\left[{\Delta}_{n x},{\Delta}_{n y}\right]\frac{\partial W}{\partial P}\right)}^T\cdot \left[{\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\right]\right\}\\ {}\approx -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot {\Delta}_n\left({x}_i^{\prime },{y}_j^{\prime}\right)\right\}\end{array} $$
(55)
The expectation of ΔP
FA ‐ NR can be expressed as
$$ \begin{array}{l} E\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)={E}_{\mathrm{int}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)+{E}_{\mathrm{noise}}\left(\Delta {\mathbf{P}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}\right)\\ {}\kern8.5em =-{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot h\left({x}_i,{y}_j\right)\right\}\\ {}\kern10em -{\tilde{\mathbf{H}}}_{\mathrm{FA}\hbox{-} \mathrm{NR}}^{-1}\cdot E\left\{\sum_{i=1}^n\sum_{j=1}^m{\left[\nabla {\xi}_G\left({x}_i^{\prime },{y}_j^{\prime}\right)\frac{\partial W}{\partial P}\right]}^T\cdot {\Delta}_n\left({x}_i^{\prime },{y}_j^{\prime}\right)\right\}\end{array} $$
(56)
