# Retina-like Imaging and Its Applications: A Brief Review
## 2 Principle and Methods  
### Principle and Properties  
The principle of retina-like imaging includes two aspects, namely, space-variant sampling and log-polar transformation (LPT) [15], as shown in Figure 1. It is written as Equation (1) [16]:  
\[\left\{\begin{array}{l}\theta=\arctan\frac{y}{\lambda}\\ \xi=\log_{q}\sqrt{x^{2}+y^{2}}\end{array}\right., \tag{1}\]  
where (\(x\),\(y\)) are the Cartesian coordinates and (\(\xi\),\(\theta\)) are the log-polar coordinates.  
Some methods of optimizations are added on the basis of log-polar coordinates to ensure the coverage rate of the FOV. Circle spots are used to sample the image, and the sampling between rings is tangent. Accordingly, the space-variant retina sampling model is obtained. The mathematical model is written as Equation (2) [17]:  
\[\left\{\begin{array}{l}q=\frac{1+\sin(\pi/N)}{1-\sin(\pi/N)}\\ r_{1}=\frac{r_{0}}{1-\sin(\pi/N)}\\ R_{1}=\frac{r_{0}\sin(\pi/N)}{1-\sin(\pi/N)}\\ r_{i}=r_{1}\cdot q^{i-1}\\ R_{i}=q^{i-1}\cdot R_{1}\\ r_{\max}=r_{0}\cdot q^{M}\end{array}\right., \tag{2}\]  
where \(M\) and \(N\) are the number of rings and number of sectors per ring of the retina-like imaging, respectively. \(r_{0}\) is the radius of blind hole, \(q\) is the growth rate between rings,  
Figure 1: Mapping from Cartesian coordinates to log-polar coordinates.  
is the radius of the first ring, \(R_{1}\) is the radius of the spot in the first ring, and \(r_{max}\) is the maximum radius of the FOV, which is tangent with the spots in the last ring, as shown in Figure 2.  
Retina-like imaging has three properties. First, it is beneficial for balancing among high resolution, large FOV, and real-time performance. The two axes of log-polar coordinates represent the distance from a certain point to an original point \(\xi\) and an angle \(\theta\) from a certain point to the polar axis, respectively. When the ring number is low, the pixel is near the center of the FOV, and the resolution density is higher than the Cartesian coordinate that brings details to the target details. However, when the ring number is large, the resolution density becomes sparse, and it represents a lower resolution background area. Therefore, the retina-like structure allows for the efficient suppression of redundant data. However, this suppression image makes the pixels uncertain where a fuzzy edge detector would be necessary [18; 19]. Second, rotation and scaling invariance are considered. The rotation and scaling of the target in the Cartesian coordinate system only affects the axes \(\theta\) and \(\xi\), respectively, as represented by Equation (3) [20]:  
\[\left\{\begin{array}{l}\theta^{\prime}=\theta+\alpha\\ \xi^{\prime}=\log_{z}(\sqrt{\left(q\chi\right)^{2}+\left(qy\right)^{2}})=\log _{z}(qr)\\ =\log_{z}(q)+\log_{z}(r)=\xi+\log_{z}(r)\end{array}\right., \tag{3}\]  
where \(\alpha\) stands for the rotation and \(q\) stands for the scale change. \(\theta\) and \(\xi\) stand for the two axes of LPT, respectively. In this way, the relationship of rotation and scaling is rewritten by the shift in the \(\theta\) and \(\xi\) directions, and it is called the rotation and scale change invariance [20]. Third, the adjustable optical axis is given attention. On the basis of the space-variant resolution, we always pay attention to the most interesting region. Meanwhile, we can transfer the interesting region according to different environments. With the development of the theory on retina-like imaging, retina-like imaging models do not always strictly conform to LPT, but the space-variant resolution is the typical common feature of retina-like imaging. Therefore, space-variant resolution is the kernel character and is illustrated in the following methods (Section 2.2) and applications (Section 3).