## 4 Non-dimensional Algorithms  
In 2003 [PERSON], along with [PERSON] and [PERSON] [3], presented a new Star-ID technique that was robust to calibration errors. For flight systems in which temperature variations would cause cyclic variations in the accuracy of the calibration, the new technique would promise to eliminate the ambiguity in matching star patterns. Instead of using the inter-star angles between stars in a triangle, [PERSON] used the triangle's interior angles, the angle between two stars, with a third star as a vertex. While the inter-star angles respond linearly to changes in temperature, the triangle interior angles are invariant in the first order of the distortion, as illustrated in Figure 9. [PERSON]'s technique uses the smallest and largest of the interior angles to place stars in a catalog, so the feature extraction time is \(O(\lg b)\). The database is subsequently searched with [PERSON]'s \(k\)-vector searching technique, taking \(O(k)\) time. [PERSON]'s numerical tests found that at least five stars must be matched before the technique produces, which introduces a cross-checking routine, using \(O(b\,k^{2})\)-time. [PERSON] concludes the paper by using Star-ID to re-calibrate the camera. [PERSON] also published a method in 2005 [28], which he billed as being robust to errors introduced by new CMOS Active Pixel Sensors (APS). His metric is the sine of star-triangle interior angles, but instead of using any combination of stars, he used only the closest two stars, and used only one of the three (two independent) interior angles as a parameter. His pattern selection pattern means there is only one entry in the catalog for each star; so his catalog size is \(O(n)\). It also follows that the feature extraction time is \(O(f\lg 2)=O(f)\). Furthermore, [PERSON] does not specify a method for selecting star triangles from the catalog, but according to his published parameter distribution, the fastest method available would be a binary-tree search, taking \(O(k\lg n)\)-time. [PERSON] then actually computes the attitude for each star triangle, and finds all the stars from the catalog that should be visible, which should take no less than \(O(f)\), and more likely \(O(f\lg n)\). Each observation is then transformed into \begin{table}
\begin{tabular}{|l|c|c|c|c|c|c|} \hline Author & Year & Feature & Database & Database & Validation & Used \\  & & Extraction & Size & Search & & measurements \\  & & & & & & /Available \\ \hline \hline [PERSON] & 1981 & \(O(b)\) & \(O(nf^{3})\) & \(O(f^{3})\) & N/A & 3/3 \\ \hline [PERSON] & 1992 & \(O(f\lg b)\) & \(O(n)\) & \(O(n)\) & N/A & 3/3 \\ \hline [PERSON] & 1993 & \(O(f\lg b)\) & \(O(nb)\) & \(O(b(\Delta mn)^{2})\) & \(O(a^{b})\) & 9/12 \\ \hline [PERSON] & 1995 & \(O(b\lg b)\) & \(O(nf^{2})\) & \(O(nf^{2})\) & \(O(k)\) & 6/6 \\ \hline [PERSON] & 1996 & \(O(f\lg b)\) & \(O(n)\) & \(O(\lg n)\) & N/A & 6/6 \\ \hline [PERSON] & 1997 & \(O(f)\) & \(O(n)\) & \(O(n)\) & N/A & \(2f/2f\) \\ \hline [PERSON] & 1997 & \(O(b)\) & \(O(nf)\) & \(O(k)\) & \(O(bk^{2})\) & 6/6 \\ \hline [PERSON] & 2008 & \(O(b)\) & \(O(nf)\) & \(O(k)\) & \(O(kf^{2})\) & 6/6 \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of Advances in Asymptotic Performance. the reference frame. The observed stars are then matched up with catalog stars, and the inter-star angles compared. The process by which this is done is not described, but likely takes \(O(f\lg f)\)-time. The best of the matches of all the triangles is then selected. The final analytic time of [PERSON]'s algorithm is then \(O(kf\lg f\lg n)\). It is unclear whether [PERSON]'s performance data is on his original 45,000-star catalog, or another mentioned, reduced 1,300-star catalog, but his timed results are disappointing; all of his averages are longer than a second on a 650 MHz processor. Although the tests are performed in MATLAB, which unnecessarily increases the computation time, it is unclear why [PERSON] claims the algorithm is fast from his reported data, and without any performance comparison to any other algorithm.