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Rank SIFT algorithm is the revised SIFT algorithm that uses ranking techniques to improve the performance of the SIFT algorithm. In fact, ranking techniques can be used in key point localization or descriptor generation of the original SIFT algorithm.

Ranking the Key Point
Ranking techniques can be used to keep certain number of Key Point which are detected by SIFT detector.

Suppose $$\left \{ I_m, m=0,1,...M \right \}$$ is a training image sequence and $$p$$ is an key point obtained by SIFT detector. The following equation defines the rank of $$p$$ in the key point set.

$$ R( p \in I_0)=\sum_m I(\min_{ q \in I_m}{\lVert H_m(p)-q \rVert}_2 < \epsilon), $$

where $$I(.)$$ is the indicator function, $$H_m$$ is the homography transformation from $$I_0$$ to $$I_m$$, and $$\epsilon$$ is the threshold.

Suppose $$x_i$$ be the feature descriptor of key point $$p_i$$ which is defined above. And label $$x_i$$ with the rank of $$p_i$$. Then the vector set $$X=\left \{ x_1,x_2,...\right \}$$ containing labeled elements can be used as a training set for an Ranking SVM problem. The learning process can be represented as follows:

$$ \begin{array}{lcl} minimize: V(\vec w) = {1 \over 2} \vec w \cdot \vec w \\ s.t.\\ \begin{array}{lcl} \forall x_i \in I and\ \forall x_j \in I\\ \vec w^T(\vec x_i -\vec x_j)\geqq 1\quad if\ R(\vec x_i \in I )>R(\vec x_j \in I). \end{array} \end{array} $$

The obtained optimal $$\vec w^*$$ can be used to order the future key point.

Ranking the Elements of Descriptor
Ranking techniques also can be used to generate the key point descriptor.

Suppose $$ {X} = \left \{ x_1,...,x_N \right \} $$ is the feature vector of of a key point and the elements of $$ {R} = \left \{r_1,...r_N \right \} $$ is the corresponding rank of $$x_i$$ in $$X$$. $$r_i$$ is defined as follows:

$$ r_i = \left\vert \left \{x_k:x_k \geqq x_i \right \} \right \vert .$$

After transforming original feature vector $$X$$ to the ordinal descriptor $$R$$, the difference between two descriptors can be evaluated in the follow two measurements.

The spearman correlation coefficient also refers to Spearman's rank correlation coefficient. For tow ordinal descriptors $$R$$ and $$R^'$$, it can be proved that
 * The Spearman corelation coefficient

$$ \rho(R, R^') = 1- {6\sum_{i=1}^N(r_i-r_i^')^2 \over N(N^2-1)}$$

The Kedall's Tau also refers to Kendall tau rank correlation coefficient. In the above case, the Kedall's Tau between $$R$$ and $$R^'$$ is
 * The Kendall's Tau

$$\tau(\bar r, \bar r^')= {2\sum_{i=1}^N\sum_{j=i+1}^Ns(r_i-r_j, r_i^'-r_j^')\over N(N-1)},$$

$$ where \quad s(a,b) = \begin{cases} 1, & \text{if } sign(a) = sign(b) \\ -1, & o.w. \end{cases} $$