# Keypoint Localization

After finding keypoint candidates, it is necessary to find stabile keypoints. It means that the points with low contrast and poor localization along the edges will be removed. It can be accomplished by using the Taylor expansion of the DoG image and the location of the extremum, x, can be determined by following formula [32]:

where *D, dD* and *j-D* are evaluated at the same selected point and x = *(x, y, a) ^{J }*is the offset from this point. It is worth noting that

*D*is the Taylor expansion up to second order of original

*D(x, y,&).*

The function value at the extremum D(x*)* is calculated by

To reject the unstable extrema with low contrast, based on the experimental results in [32], those *x* with D(x) < 0.03 is discarded.

To define the extreme points along the edges, a 2 x 2 Hessian matrix is utilized as

where *Dxx*, *Dxy* and *Dyy* is the second partial derivative of the DoG image.

To further eliminate the influence of the points localized along the edges, the candidate keypoints which are unable to satisfy following situation will be eliminated:

where *Tr(H)* and *Det(H)* is respectively the trace and the determinant of matrix *H*. *Y =* 10 is the ratio between the eigenvalue of *H* with the largest magnitude and the one with smaller magnitude.