Significant edges in the case of a non-stationary Gaussian noise

Isabelle Abraham (1) , Romain Abraham (2) , Agnes Desolneux (3) , Sebastien Li-Thiao-Te (4)

1 DSSI - Département des Sciences de la Simulation et de l'Information
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
4 CMLA - Centre de Mathématiques et de Leurs Applications

In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold $s(\eps)$ defined by: if the image $I$ is pure noise, then $\P(\norm{\nabla I}\geq s(\eps) \bigm| \Delta I = 0) \leq\eps$. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold $s(\eps)$ can be explicitly computed in the case of a stationary Gaussian noise. In images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. But in this case again, we will be able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold $s(\eps)$. We will end this paper with some experiments and compare the results with the ones obtained with some other methods of edge detection.

Mots clés: Edge detection; Significant edges; Inverse problem; Statistical hypothesis testing

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Isabelle Abraham, Romain Abraham, Agnes Desolneux, Sebastien Li-Thiao-Te. Significant edges in the case of a non-stationary Gaussian noise. Pattern Recognition, 2007, 40, pp.3277-3291. ⟨10.1016/j.patcog.2007.02.015⟩. ⟨hal-00079148v2⟩