Edge detection using topological gradients: a scale-space approach

Samuel Amstutz (1) , Jérôme Fehrenbach (2, 3)

1 LANLG - Laboratoire d'Analyse non linéaire et Géométrie
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 ITAV - Institut des Technologies Avancées en sciences du Vivant

We provide in this paper a link between two methods of edge detection: edge detection using scale-space analysis, and edge detection using topological asymptotic analysis. More precisely, we show that the topological gradient associated with an image u is given by a combination of the gradients of two smoothed versions of the image u at two different scales, namely φ⋆u and (φ⋆φ)⋆u, where φ is the fundamental so- lution of the elliptic restoration equation. In the same setting we propose a new edge detector based on the maximization of the variance of the image. Then we generalize our approach to Gaussian kernels considering a topological asymptotic analysis of the parabolic heat equation. A numerical comparison of these detectors together with the Canny edge detector is presented.

Mots clés: Edge detection; Topological gradient; Heat equation; Scale space

Domaines: Equations aux dérivées partielles [math.AP]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Samuel Amstutz, Jérôme Fehrenbach. Edge detection using topological gradients: a scale-space approach. Journal of Mathematical Imaging and Vision, 2015, ⟨10.1007/s10851-015-0558-z⟩. ⟨hal-01025706⟩