## 4 Estimation of the Rotation  
The candidate-selection process, as described in the previous section, selects the most likely candidates. A new rotation-estimation technique known as the ring-shifted technique is proposed to detect the rotation of only the best of the output of the candidates. Although, the original ring-projection \(C_{P,l}(r)\) shown in Equation (2) is rotation invariant, the ring-shifted technique will only inherit the ring intensity profile \(P(x_{c}+r\text{cos}\theta_{k},y_{c}+r\text{sin}\theta_{k})\) for each specific radius \(r\). So, it does not need any operation for preparing the ring intensity profile in the rotation estimation process, which considerably reduces the computation time. For the rotation invariance, the hybrid-projection value is constructed along circular rings with successively increasing radii. In the ring-shifted technique, the ring intensity profile at each radius is used to estimate the rotation. To illustrate the ring-shifted technique, Figure 11 shows images with different rotation angles and the corresponding ring intensity profiles \(p_{r}\) and \(p_{r}^{\prime}\) at the specific radius of \(r=17\). It can be seen that the intensity of point \(p_{17}(n^{*})\) is equal to the point \(p_{17}^{\prime}(n^{*}+k\left(17\right))\) when the consequent point shifts by \(k\left(17\right)\), as shown in Figure 11a. Actually, the shift value \(k\left(17\right)\) represents the ring that is rotated by \(\phi_{17}\) at a radius of 17 in the spatial domain. Furthermore, we can define the relationship between the rotation angle \(\phi_{r}\) and the shift value \(k_{opt}\left(r\right)\) at the specific radius, which is defined as follows:  
\[\phi_{r}=k_{opt}\left(r\right)\times\phi_{opt,0}(r) \tag{13}\]  
where \(k_{opt}\left(r\right)\) is the shift value at radius \(r\) and \(\phi_{opt,0}(r)\) is the optimal step angle described in Equation (4). Figure 10: The image pyramid search technique. Before the rotation estimation, the shift value \(k_{opt}\left(r\right)\) at radius \(r\) should be calculated. The cross correlation measurement is employed to find the maximum correlation of the two ring intensity profiles by a shift value, the so-called ring-shifted correlation. Therefore, the shift value can be derived; with the measurement with maximum correlation, the shift value can be defined as:  
\[k_{opt}\left(r\right)=\underset{k\in\left[0,N_{r}\right]}{\operatorname{argmax}} \left(\delta_{r}(k)\right) \tag{14}\]  
where \(N_{r}\) represents the number of pixels at radius \(r\). \[\delta_{r}(k)=\frac{\sum\limits_{n=0}^{N_{r}}\left(p_{r}(n)-\overline{p}_{r}( n)\right)\cdot\left(p_{r}^{\prime}(n+k)-\overline{p}_{r}^{\prime}(n+k)\right)}{ \sqrt{\sum\limits_{n=0}^{N_{r}}\left(p_{r}(n)-\overline{p}_{r}(n)\right)^{2} \cdot\sum\limits_{n=0}^{N_{r}}\left(p_{r}^{\prime}(n+k)-\overline{p}_{r}^{ \prime}(n)\right)^{2}}} \tag{15}\]  
Based on the definitions given from Equations (13)-(15), the rotation angle of the candidate can be estimated at only a single radius. According to Equation (4), the accuracy of the rotation angle depends on the optimal step angle and the radius. Obviously, the optimal step angle is inversely proportional to the radius. Therefore, the rotation angle can be accurately estimated by integrating all ring weighting coefficients \(w_{r}\) in the range between \(R_{\text{min}}\) and \(R_{\text{max}}\). Hence, the rotation angle of the most likely candidate is defined as follows:  
\[\theta_{obj}=\sum\limits_{r=R_{\text{min}}}^{R_{\text{max}}}w_{r}\cdot k_{opt }\left(r\right)\cdot\Phi_{opt,0}(r) \tag{16}\]  
Figure 11: An example of the ring intensity profile at the specific radius (radius = 17). (**a**) Original image; (**b**) Rotated image (30\({}^{\circ}\), CCW); (**c**) Shift point \(k(17)\) in the profiles with rotation angles.