# Fractal Dimension and Retinal Pathology: A Meta-Analysis
## 1 Introduction
### Common Methods for Measuring Retinal Fractal Dimension  
After image acquisition, retinal images need to be processed first to extract the features of the vascular tree, a process known as _vessel segmentation_, before image binarization and then fractal analysis. Several algorithms exist for this purpose and the reader is referred to the literature [9; 10; 11]. We will now briefly discuss commonly used methods for calculating retinal fractal dimension.  
Box-counting (capacity) dimension: The simplest and most common method used in the literature is the box-counting method [7] for fractal dimension. Given a binarized image of the retinal vascular tree, we overlay the image with a grid of boxes of side-length \(\varepsilon\) and count how many boxes contain a part of the tree. By decreasing \(\varepsilon\), we capture more and more fine details of the tree from the covering. Taking \(N(\varepsilon)\) to be the box-count as a function of \(\varepsilon\), the box-counting (capacity) dimension [12; 13] is defined as  
\[D_{box}=\lim_{\varepsilon\to 0}\frac{\log N(\varepsilon)}{\log 1/\varepsilon} \tag{1}\]  
It should be noted that the base of the logarithm does not affect the calculated value.  
Information dimension: Similar to the method for determining the box-counting dimension, we overlay the retinal image with a grid of boxes of side-length \(\varepsilon\). Instead of a box count however, we assign to each box a weight based on its contribution to the tree's information entropy and sum up the weights for each box, defining the information dimension [13; 14] as  
\[D_{info}=\lim_{\varepsilon\to 0}\frac{\sum_{i=1}^{N(\varepsilon)}p_{i}\log p_{i}}{ \log 1/\varepsilon} \tag{2}\]  
where \(N(\varepsilon)\) is the total number of boxes that contain a part of the tree and \(p_{i}=m_{i}/M\) is the proportion of retinal tree contained in the \(i\)-th box; \(m_{i}\) is the number of pixels in the i-th box and \(M\) is the total number of pixels in the tree.  
Correlation dimension: We overlay the retinal image with a grid of side-length \(\varepsilon\) boxes and define the correlation integral \(C(\varepsilon)\)[13] as  
\[C(\varepsilon)=\frac{1}{N(\varepsilon)^{2}}\sum_{i=1,j=1,i\
eq j}^{N( \varepsilon)}H(\varepsilon-||\mathbf{x_{i}}-\mathbf{x_{j}}||)\approx\sum_{i= 1}^{N(\varepsilon)}p_{i}^{2}, \tag{3}\]  
where \(H\) is the Heaviside step function and counts the number of tree pixel pairs such that the distance between them is less than \(\varepsilon\). The correlation dimension is then defined as  
\[D_{corr}=\lim_{\varepsilon\to 0}\frac{\log C(\varepsilon)}{\log(\varepsilon)} \tag{4}\]Generalized dimensions: When discussing multifractal structures such as the retinal vasculature, they are more accurately described by an infinite hierarchy of fractal dimensions [15]. For any real \(q\), the generalized fractal dimension is defined as  
\[D_{q}=\frac{1}{1-q}\lim_{\varepsilon\to 0}\frac{\log I(q,\varepsilon)}{\log 1/\varepsilon} \tag{5}\]  
where  
\[I(q,\ \varepsilon)=\sum_{i=1}^{N(\varepsilon)}p_{i}^{q} \tag{6}\]  
It can be seen and verified mathematically that \(D_{0},\ D_{1},\ D_{2}\) correspond to the capacity, information, and correlation dimensions respectively as previously discussed. If \(q_{1}>q_{2}\) then \(D_{q_{1}}\leq D_{q_{2}}\)[16], so we have that \(D_{0}\geq D_{1}\geq D_{2}\), which is a useful check when computing fractal dimensions in practice.