## 2 Materials and Methods
### Fovea Localization  
The main retinal vessels distribute as a parabolic curve, and the symmetry axis of retina vasculature passes through the VO. The fovea lies on the symmetry axis in a distance of 2.5 OD diameters away from the VO [6,21]. Here, a Hough transform is employed to search for the fittest parabola and symmetry axis of the retinal vessel. In order to reduce the dimension of parameter space of Hough transform, the origin of the coordinate system is shifted to the VO, and then the coordinate axes are rotated about the VO. Therefore, an SR-Hough transform is proposed, which combines the Hough transform with coordinate transformation. In the meantime, to speed up the process of the Hough transform, the small vessels of vasculature on the binary vessel image \(I_{o}\) are pruned by the morphological opening processing before applying the SR-Hough transform. A coordinate transformation integrating the shift and rotation of coordinates is performed. As shown in Figure 5, a Cartesian coordinate system, denoted by _x-y_ and originating at the upper-leftcorner in the image, is given. A second Cartesian coordinate system, denoted by \(x^{\prime}\)-\(y^{\prime}\), is obtained by shifting the origin of the \(x\)-\(y\) plane to the VO location (_i.e._, taking it as the origin of the \(x^{\prime}\)-\(y^{\prime}\) coordinate system) and rotating it in a counter-clockwise direction through an angle \(\theta\) around the new origin. The coordinate transform is defined as:  
\[ \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}\begin{bmatrix}\cos\alpha&\sin\alpha\\ -\sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}x-h\\ y-k\end{bmatrix}  \tag{10}\]  
Where (\(h\),\(k\)) denotes the origin of the \(x^{\prime}\)-\(y^{\prime}\) plane. The coordinate (\(x\),\(y\)) is an arbitrary point relative to the \(x\)-\(y\) coordinate system, while the coordinates (\(x^{\prime}\)-\(y^{\prime}\)) are relative to the \(x^{\prime}\)-\(y^{\prime}\) coordinate system. After rotating through an angle \(\alpha\), the symmetry axis of the fittest parabola is exactly closed with the \(x^{\prime}\) axis of the \(x^{\prime}\)-\(y^{\prime}\) coordinate system. Therefore, the fittest parabola can be expressed as the following,  
\[y^{\prime 2}=4 cx^{\prime} \tag{11}\]  
Where \(c\) is the focal length of the parabola. Substituting \(x^{\prime}\) and \(y^{\prime}\) in Equation (10) into Equation (11) leads to the Equation (12),  
\[[-(x-h)\sin\alpha+(y-k)\cos\alpha]^{2}=4c[(x-h)\cos\alpha+(y-k)\sin\alpha] \tag{12}\]  
This parabola is utilized to estimate the parameters (\(\alpha\),\(c\)) with the Hough transform. In principle, the Hough transform [19] maps the points in the \(x\)-\(y\) plane to the \(\alpha\)-\(c\) plane. All the points on this parabola have the same values of \(\alpha\) and \(c\) in the \(\alpha\)-\(c\) parameter space. For every vessel point in the \(x\)-\(y\) plane, let \(\alpha\) equal to each of the subdivision value on the \(\alpha\)-axis (here, \(0\)\(\leq\)\(a\)\(<\)\(359^{\circ}\) for \(1^{\circ}\) step) and solve for the  
Figure 5: **(a)** The proposed parabolic model; **(b)** the fittest parabola superimposing on the retinal vessel structure. corresponding \(c\) that satisfies Equation (12). Then, the \(c\) value is rounded off to the nearest integer. The cell at coordinates \((i\!j)\), with accumulator value \(A(i\!j)\), corresponds to the square associated with parameter space coordinates \((a_{i}\!,\!c_{j})\). While a choice of \(a_{l}\) yields solution \(c_{m}\), then let \(A(l,m)=A(l,m)+1\). A value of \(n\) in \(A(i,j)\) means that \(n\) points in the \(x\)-\(y\) plane lie on a parabola, of which \(a=a_{i}\), and \(c=c_{j}\). Once all the vessel points have been processed, the fittest parabola can be determined by finding the \((a,c)\) fittest of the parabola according to the following equations.