point. The location of the extremum, , is determined by taking the derivative of this function with respect to and setting it to zero. If the offset is larger than in any dimension, then that's an indication that the extremum lies closer to another candidate keypoint. In this case, the candidate keypoint is changed and the interpolation performed instead about that point. Otherwise the offset is added to its candidate keypoint to get the interpolated estimate for the location of the extremum. A similar subpixel determination of the locations of scale-space extrema is performed in the real-time implementation based on hybrid pyramids developed by Lindeberg and his co-workers. Discarding low-contrast keypoints To discard the keypoints with low contrast, the value of the second-order Taylor expansion is computed at the offset . If this value is less than , the candidate keypoint is discarded. Otherwise it is kept, with final scale-space location , where is the original location of the keypoint. Eliminating edge responses The DoG function will have strong responses along edges, even if the candidate keypoint is not robust to small amounts of noise. Therefore, in order to increase stability, we need to eliminate the keypoints that have poorly determined locations but have high edge responses. For poorly defined peaks in the DoG function, the principal curvature across the edge would be much larger than the principal curvature along it. Finding these principal curvatures amounts to solving for the eigenvalues of the second-order Hessian matrix, H: The eigenvalues of H are proportional to the principal curvatures of D. It turns out that the ratio of the two eigenvalues, say is the larger one, and the smaller one, with ratio , is sufficient for SIFT's purposes. The trace of H, i.e., , gives us the sum of the two eigenvalues, while its determinant, i.e., , yields the product. The ratio can be shown to be equal to , which depends only on the ratio of the eigenvalues rather than their individual values. R is minimum when the eigenvalues are equal to each other. Therefore, the higher the absolute difference between the two eigenvalues, which is equivalent to a higher absolute difference between the two principal curvatures of D, the higher the value of R. It follows that, for some threshold eigenvalue ratio , if R for a candidate keypoint is larger than , that keypoint is poorly localized and hence