The eigenvalues of H are proportional to the principal curvatures of D. It turns out that the ratio of the two eigenvalues, say formula_34 is the larger one, and formula_35 the smaller one, with ratio formula_36, is sufficient for SIFT's purposes. The trace of H, i.e., formula_37, gives us the sum of the two eigenvalues, while its determinant, i.e., formula_38, yields the product. The ratio formula_39 can be shown to be equal to formula_40, 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 formula_41, if R for a candidate keypoint is larger than formula_42, that keypoint is poorly localized and hence rejected. The new approach uses formula_43.