Regularization preserving localization of close edges

  • The proposed approach is valid in the discrete domain as in the continous domain
  • Close edges in image induce mutual influence in the detection process
  • Mutual influence generaly leads to delocalization
  • There are two types of configuration in mutual influence :

d is the distance between two close edges (d ≥ 1)

  • To avoid delocalization of the edges, the regularization filter h must verifiy these conditions :

  • Only one known detector verifies both these conditions
filter h delocalization in opposite configuration delocalization in stair configuration
Canny or Gaussian derivative Yes Yes
Canny-Deriche Yes Yes
Shen & Castan No No
Demingy Yes No
Box Yes Yes
Triangle Yes No

Example :

(Left) Original image. (Center) Detection by Shen and Castan filter . (Right) Detection by Gaussian (derivative) filter . Same SNR, Threshold . The last image presents clearly edge delocalization.

Conclusion: considering edge localization (with or without noise) gaussian filter is not a good candidate for regularization. Canny tried to take into account the problem of close edges in his work by introducing bounded filter support

Ref: O. Laligant et al. « Regularization preserving localization of close edges ». IEEE Signal Processing Letters, Mars 2007