## 3 Candidate Selection and Localization
### Robust Features  
As previously mentioned, the ring-projection transformation is used to describe the features of an image and to reduce computational complexity. However, the features should have unique and robust properties to correctly recognize the candidates. In despite of the ring-projection transformation having a rotation invariant property, it cannot describe the unique features of an image when the image contains a homogeneous region. For example, texture-less searching images containing different types of homogeneous regions are shown Figure 7a-d. The corresponding features that are described by ring-projection transformation are shown in Figure 7e. As can been seen in Figure 7e, the features of searching images 3 and 4 are easily distinguished; it also means the searching images 3 and 4 can be  
Figure 5: An image at two distinct angles of rotation. (**a**) The original image; (**b**) The rotated image (\(30^{\circ}\) counterclockwise, CCW); (**c**, **d**) Show the corresponding profiles of ring projection. Figure 6: Definition of the optimal step angle for image pyramid level \(l\). By contrast, the features of images 1 and 2 cannot be correctly recognized by evaluating the similarity of the features in the candidate selection and localization process. Consequently, the features must describe the homogeneous ring regions in a distinct rather than a uniform manner. To overcome this problem, the variance of the specific radius in the image is considered and the variance-projection transform is used [20]. The variance projection transformation in the template and searching images on image pyramid level \(l\) are defined as \(\sigma_{p,l}\) and \(\sigma_{s,l}\):  
\[\sigma_{p,l}(r)=\frac{1}{Q_{r}}\sum_{k}[P(x_{c}+r\text{cos}\theta_{k},y_{c}+r \text{sin}\theta_{k})-C_{P,l}(r)]^{2} \tag{5}\]  
\[\sigma_{S,l}(x,y,r)=\frac{1}{Q_{r}}\sum_{k}[P(x+r\text{cos}\theta_{k},y+r\text {sin}\theta_{k})-C_{S,l}(x,y,r)]^{2} \tag{6}\]  
In Figure 8, the features are described by the variance-projection transformation which actually reflects the variation for image 2 and other images (images 1, 3 and 4). However, this is not sufficient to describe the features of image 1, 3, and 4. Fortunately, the features of image 1, 3, and 4 can be distinguished by ring-projection transformation as shown in Figure 7e. For the sake of unique and robust features description, a hybrid-projection transformation that combines the ring- and variance-projection transformations is used in this paper. The hybrid-projection transformation in the template and searching images on image pyramid level \(l\) are defined as \(H_{p,l}\) and \(H_{s,l}\):  
\[H_{p,l}(r)=w_{m,l}(r)\times C_{P,l}(r)+w_{\sigma,l}(r)\times\sigma_{P,l}(r) \tag{7}\]  
\[H_{S,l}(x,y,r)=w_{m,l}(r)\times C_{S,l}(x,y,r)+w_{\sigma,l}(r)\times\sigma_{S, l}(x,y,r) \tag{8}\]  
Figure 7: (**a–d**) Textures-less searching images 1–4; (**e**) Corresponding ring projection features. where \(w_{mJ}(r)\) and \(w_{\sigma,l}(r)\) are weighting coefficients for the ring and variance-projection transformation, respectively. In Equations (7) and (8), the weighting coefficients at the specific radius on image pyramid level \(l\) are calculated by the concept of image entropy. The image entropy is a quantity which is used to describe the randomness of an image, the image entropy \(H\) with associated probabilities of occurrence \(pb(i)\) can be defined as:  
\[H=-1\times\sum_{i=0}^{g}pb(i)\text{log}(pb(i)) \tag{9}\]  
where \(pb(i)\) is the probability of gray level \(i\). In the case of an image, if all of the pixels have the same value, the image entropy of the image is zero. On the other hand, a high image entropy occurs when an image has chaotic gray levels. For the hybrid-projection transformation, the data along the specific radius having high variance should have high weights.