## 2 Methodology
### Weighting Functions
\begin{array}{ll}\left\langle[\alpha_{1}(\theta_{2})-\alpha_{2}(\theta_{1}) ]^{2}\right\rangle&=\left(\frac{32\lambda}{\pi^{2}D^{4}}\right)^{2}\left[ \int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}(\mathbf{r}-\mathbf{r}_{1}) \cdot\left(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{1}\right)W(\mathbf{r}- \mathbf{r}_{1})W(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{1})\left\langle\phi( \mathbf{r},\theta_{2})\phi(\mathbf{r}^{{}^{\prime}},\theta_{2})\right\rangle \right.\\ &-\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}(\mathbf{r}-\mathbf{r}_{2}) \cdot\left(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{1}\right)W(\mathbf{r}- \mathbf{r}_{2})W(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{1})\left\langle\phi( \mathbf{r},\theta_{1})\phi(\mathbf{r}^{{}^{\prime}},\theta_{2})\right\rangle \right.\\ &-\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}(\mathbf{r}-\mathbf{r}_{1}) \cdot\left(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{2}\right)W(\mathbf{r}- \mathbf{r}_{1})W(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{2})\left\langle\phi( \mathbf{r},\theta_{2})\phi(\mathbf{r}^{{}^{\prime}},\theta_{1})\right\rangle \right.\\ &+\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}(\mathbf{r}-\mathbf{r}_{2}) \cdot\left(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{2}\right)W(\mathbf{r}- \mathbf{r}_{2})W(\mathbf{r}^{{}^{\prime}}-\mathbf{r}_{2})\left\langle\phi( \mathbf{r},\theta_{1})\phi(\mathbf{r}^{{}^{\prime}},\theta_{1})\right\rangle \right].\end{array}  \tag{5}\]  
Equation (5) can be equivalently expressed as,  
\[ \begin{array}{ll}\left\langle[\alpha_{1}(\theta_{2})-\alpha_{2}(\theta_{1}) ]^{2}\right\rangle&=\left(\frac{32\lambda}{\pi^{2}D^{4}}\right)^{2}\left[ \int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}\left(\mathbf{r}\cdot\mathbf{r} ^{{}^{\prime}}\right)W(\mathbf{r})W(\mathbf{r}^{{}^{\prime}})\left\langle\phi( \mathbf{r}+\mathbf{r}_{1},\theta_{2})\phi(\mathbf{r}^{{}^{\prime}}+\mathbf{r} _{1},\theta_{2})\right\rangle\right.\\ &-\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}\left(\mathbf{r}\cdot\mathbf{r} ^{{}^{\prime}}\right)W(\mathbf{r})W(\mathbf{r}^{{}^{\prime}})\left\langle\phi( \mathbf{r}+\mathbf{r}_{2},\theta_{1})\phi(\mathbf{r}^{{}^{\prime}}+\mathbf{r} _{1},\theta_{2})\right\rangle\right.\\ &-\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}\left(\mathbf{r}\cdot\mathbf{r} ^{{}^{\prime}}\right)W(\mathbf{r})W(\mathbf{r}^{{}^{\prime}})\left\langle\phi( \mathbf{r}+\mathbf{r}_{1},\theta_{2})\phi(\mathbf{r}^{{}^{\prime}}+\mathbf{r} _{2},\theta_{1})\right\rangle\right.\\ &+\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}\left(\mathbf{r}\cdot\mathbf{r} ^{{}^{\prime}}\right)W(\mathbf{r})W(\mathbf{r}^{{}^{\prime}})\left\langle\phi( \mathbf{r}+\mathbf{r}_{2},\theta_{1})\phi(\mathbf{r}^{{}^{\prime}}+\mathbf{r} _{2},\theta_{1})\right\rangle\right].\\ \end{array}  \tag{6}\]  
Since \(\int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}\left(\mathbf{r}\cdot\mathbf{r} ^{{}^{\prime}}\right)W(\mathbf{r})W(\mathbf{r}^{{}^{\prime}})=0\), terms which are functions of either \(\mathbf{r}\) or \(\mathbf{r}^{{}^{\prime}}\) and not both, can be added without changing the result of the integration. Figure 1: Imaging path geometries. (**a**) Sensing paths cross; (**b**) Sensing paths do not cross; (**c**) Sensing at a single aperture. Hence, Equation (6) becomes,  
\[ \begin{array}{ll}\left\langle\left[\alpha_{1}(\theta_{2})-\alpha_{2}(\theta_{1}) \right]^{2}\right\rangle&=-\left(\frac{32\lambda}{\pi^{2}D^{4}}\right)^{2}\! \int\int d\mathbf{r}d\mathbf{r}^{{}^{\prime}}\left(\mathbf{r}\cdot\mathbf{r}^{{} ^{\prime}}\right)W(\mathbf{r})W(\mathbf{r}^{{}^{\prime}})\left[\mathrm{D}_{ \phi}\left(\mathbf{r}-\mathbf{r}^{{}^{\prime}}\right)\right.\\ &\left.-0.5\left\{\mathrm{D}_{\phi}\left(\mathbf{r}-\mathbf{r}^{{}^{\prime}}, \theta_{1}-\theta_{2},\mathbf{s}\right)+\mathrm{D}_{\phi}\left(\mathbf{r}- \mathbf{r}^{{}^{\prime}},\theta_{2}-\theta_{1},-\mathbf{s}\right)\right\} \right],\end{array}  \tag{7}\]  
where  
\[