## 2 Materials and Methods
### Mathematics for wED  
As wED is a new approach to the ED reconstruction process, the OCTA scanning protocol for wED is the same as the aED [11; 12]. The B scans were applied at the same locations a number of times. Therefore, each A-scan signal, X, would contain both the depth and temporal information, which can be presented by:  
\[\mathrm{X}=\left((\mathrm{X}_{\mathrm{m},\mathrm{n}})_{\mathrm{m}\in[1, \mathrm{L}]}\right)_{\mathrm{n}\in[1,\mathrm{N}]}, \tag{1}\]  
where L is the total depth of each A-scan, N is the number of acquisitions at the same scanning location, and m and n are both integers. After applying a window, the windowed signal W can be represented by:  
\[\mathrm{W}_{\mathrm{k}}=\left((\mathrm{X}_{\mathrm{m},\mathrm{n}})_{\mathrm{m }\in[\mathrm{k},\mathrm{k}+\mathrm{s})}\right)_{\mathrm{n}\in[1,\mathrm{N}]}; \mathrm{k}=1,2,3,\ldots,\ \mathrm{L}-\mathrm{s}, \tag{2}\]where k is the window's position, and s is the window size. The applied window would move from the first pixel towards the \((\mathrm{L-s})\)th pixel. Since the temporal information is contained in W, the windowed signal can also be presented as:  
\[\mathrm{W_{k}}=\big{[}\mathrm{W_{kstr}},\ \mathrm{W_{k_{flow}}},\ \mathrm{W_{k_{noise}}} \big{]}, \tag{3}\]  
where \(\mathrm{W_{kstr}}\) is the structure component, \(\mathrm{W_{k_{flow}}}\) is the flow component, and \(\mathrm{W_{k_{noise}}}\) is the noise component, e.g., white noise. The aim of ED analysis is to extract \(\mathrm{W_{k_{flow}}}\). When \(\mathrm{W_{k}}\) is assumed to be Gaussian, the raw signals can be completely characterized by its correlation matrix, \(\mathrm{R_{k}}\), given by:  
\[\mathrm{R_{k}}=\mathrm{R_{str}}+\mathrm{R_{flow}}+\sigma_{\mathrm{noise}}^{2} \mathrm{I}, \tag{4}\]  
where \(\mathrm{R_{str}}\) is the structure correlation matrix, \(\mathrm{R_{flow}}\) is the flow correlation matrix, \(\sigma_{\mathrm{noise}}^{2}\) is the noise variance, and \(\mathrm{I}\) is the identity matrix [11; 36; 41]. The correlation matrix of the raw data can be estimated by:  
\[\widehat{\mathrm{R_{k}}}=\frac{1}{\mathrm{N}}\mathrm{W_{k}}\mathrm{W_{k}^{H}}, \tag{5}\]  
where H is the Hermitian transpose operation [11; 12]. In this study, the structure component \(\mathrm{W_{kstr}}\) was assumed to be the dominant component within the input data \(\mathrm{W_{k}}\). The dominant component can be separated by ED with the estimated raw data correlation matrix, which can be estimated by:  
\[\widehat{\mathrm{R_{k}}}=\mathrm{E}\Lambda\mathrm{E^{H}}=\sum_{i=1}^{\mathrm{ N}}\lambda(\mathrm{i})\mathrm{e}(\mathrm{i})\mathrm{e}(\mathrm{i})^{H}\, \tag{6}\]  
where \(\mathrm{E}=[\mathrm{e}(1),\ \mathrm{e}(2),\ldots,\ \mathrm{e}(\mathrm{N})]\) is the unitary matrix of N eigenvectors, and \(\Lambda=[\lambda(1),\ \lambda(2),\ldots,\ \lambda(\mathrm{N})]\) is the diagonal matrix of N eigenvalues. The eigenvalues were sorted in descending order as the output of eigen-decomposition, pairing with the eigenvectors. More dominant components contribute to larger eigenvalues. Hence, the structure component was corresponding to the first Jth eigenvectors. Then, the flow component was extracted by:  
\[\mathrm{W_{kflow}}=\Big{[}\mathrm{I-\sum_{i=1}^{J}\mathrm{e}(\mathrm{i}) \mathrm{e}(\mathrm{i})^{H}}\Big{]}\mathrm{W_{k}}, \tag{7}\]  
where J is an integer between 1 and N [11; 12]. The value of J can be manually set to a fixed value or automatically set based on the eigenvalues. In this study, the total number of acquisitions at the same location, N, was 4, while J was set to be a fixed value, 2. Lastly, the windowed flow signals were combined into one A-scan flow signal, which was calculated by:  
\[\mathrm{X_{flow}}=\frac{\sum_{1}^{L-s}\mathrm{W_{k_{flow}}}}{\mathrm{L-s}}. \tag{8}\]  
By averaging the results from all windows, the final flow component was the wED OCTA result for each A-scan. To summarise, wED applies a moving window to the OCTA data then the ED analysis to extract blood flow signals from static tissue signals.