Rank SIFT

Rank SIFT algorithm is the revised SIFT (Scale-invariant feature transform) algorithm which 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.

SIFT With Ranking Techniques

Ranking the Key Point

Ranking techniques can be used to keep certain number of key points which are detected by SIFT detector. ^[1]

Suppose ${\displaystyle \left\{I_{m},m=0,1,...M\right\}}$ is a training image sequence and ${\displaystyle p}$ is a key point obtained by SIFT detector. The following equation determines the rank of ${\displaystyle p}$ in the key point set. Larger value of ${\displaystyle R(p)}$ corresponds to the higher rank of ${\displaystyle p}$.

${\displaystyle R(p\in I_{0})=\sum _{m}I(\min _{q\in I_{m}}{\lVert H_{m}(p)-q\rVert }_{2}<\epsilon ),}$

where ${\displaystyle I(.)}$ is the indicator function, ${\displaystyle H_{m}}$ is the homography transformation from ${\displaystyle I_{0}}$ to ${\displaystyle I_{m}}$, and ${\displaystyle \epsilon }$ is the threshold.

Suppose ${\displaystyle x_{i}}$ is the feature descriptor of key point ${\displaystyle p_{i}}$ defined above. So ${\displaystyle x_{i}}$ can be labeled with the rank of ${\displaystyle p_{i}}$ in the feature vector space. Then the vector set ${\displaystyle X_{featurespace}=\left\{{\vec {x}}_{1},{\vec {x}}_{2},...\right\}}$ containing labeled elements can be used as a training set for the Ranking SVM ^[2] problem.

The learning process can be represented as follows:

${\displaystyle {\begin{array}{lcl}minimize:V({\vec {w}})={1 \over 2}{\vec {w}}\cdot {\vec {w}}\\s.t.\\{\begin{array}{lcl}\forall \ {\vec {x}}_{i}\ and\ {\vec {x}}_{j}\in X_{featurespace},\\{\vec {w}}^{T}({\vec {x}}_{i}-{\vec {x}}_{j})\geqq 1\quad if\ R(p_{i}\in I_{0})>R(p_{j}\in I_{0}).\end{array}}\end{array}}}$

The obtained optimal ${\displaystyle {\vec {w}}^{*}}$ can be used to order the future key points.

Ranking the Elements of Descriptor

Ranking techniques also can be used to generate the key point descriptor. ^[3]

Suppose ${\displaystyle {\vec {X}}=\left\{x_{1},...,x_{N}\right\}}$ is the feature vector of a key point and the elements of ${\displaystyle {R}=\left\{r_{1},...r_{N}\right\}}$ is the corresponding rank of ${\displaystyle x_{i}}$ in ${\displaystyle X}$. ${\displaystyle r_{i}}$ is defined as follows:

${\displaystyle r_{i}=\left\vert \left\{x_{k}:x_{k}\geqq x_{i}\right\}\right\vert .}$

After transforming original feature vector ${\displaystyle {\vec {X}}}$ to the ordinal descriptor ${\displaystyle {\vec {R}}}$, the difference between two ordinal descriptors can be evaluated in the following two measurements.

• The Spearman correlation coefficient

The Spearman correlation coefficient also refers to Spearman's rank correlation coefficient. For two ordinal descriptors ${\displaystyle {\vec {R}}}$ and ${\displaystyle {\vec {R}}^{'}}$, it can be proved that

${\displaystyle \rho ({\vec {R}},{\vec {R}}^{'})=1-{6\sum _{i=1}^{N}(r_{i}-r_{i}^{'})^{2} \over N(N^{2}-1)}}$

• The Kendall's Tau

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

${\displaystyle \tau ({\vec {R}},{\vec {R}}^{'})={2\sum _{i=1}^{N}\sum _{j=i+1}^{N}s(r_{i}-r_{j},r_{i}^{'}-r_{j}^{'}) \over N(N-1)},}$

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

References

1. Bing Li; Rong Xiao; Zhiwei Li; Rui Cai; Bao-Liang Lu; Lei Zhang; "Rank-SIFT: Learning to rank repeatable local interest points", Computer Vision and Pattern Recognition (CVPR), 2011
2. Joachims, T. (2003), "Optimizing Search Engines using Clickthrough Data", Proceedings of the ACM Conference on Knowledge Discovery and Data Mining
3. Toews, M.; Wells, W."SIFT-Rank: Ordinal Description for Invariant Feature Correspondence", Computer Vision and Pattern Recognition, 2009.