# A Method for Combating Random Geometric Attack on Image

March 9, 2017 | | By admin |

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5 Let f (x, y) be a smooth bivariate function. Function f has an Ak or Dk singularity if, up to a diffeomorphism, it can be written as: Ak : f = ±x2 ± yk+1 , k ≥ 0, Dk : f = ±yx2 ± yk−1 , k ≥ 4. 12) ± The singularity is further denoted A± k or Dk if the product of the coefficients of the monomials is ±1. As subsumed by this definition, an Ak singularity precludes an Ak+1 singularity, and similarly for Dk . An important characteristic of these normal forms is their zero level set. Those of the Ak sequence are illustrated on Fig.

5 Let f (x, y) be a smooth bivariate function. Function f has an Ak or Dk singularity if, up to a diffeomorphism, it can be written as: Ak : f = ±x2 ± yk+1 , k ≥ 0, Dk : f = ±yx2 ± yk−1 , k ≥ 4. 12) ± The singularity is further denoted A± k or Dk if the product of the coefficients of the monomials is ±1. As subsumed by this definition, an Ak singularity precludes an Ak+1 singularity, and similarly for Dk . An important characteristic of these normal forms is their zero level set. Those of the Ak sequence are illustrated on Fig.

Notice that in Eq. ) Notice that the type, elliptic or hyperbolic, is independent of the surface orientation. However the sign of b0 depends on the orientation of the principal directions. The corresponding geometric interpretation when moving along a curvature line and crossing the ridge is recalled on Fig. 6. 6: Sign changes of b0 along an oriented blue curvature line crossing a blue ridge : elliptic case (max of k1 , left), and hyperbolic case (min of k1 , right). Ridge points are on smooth curves on the surface called ridge lines and can be colored according to the color of the points.