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We now consider the crack detection technique, within the framework of the identification of the image edges, either in the hidden part of the image for the inpainting application, or in the whole image for the segmentation application [26]. The topological asymptotic analysis provides very quickly the location of the edges, as they are precisely defined by the most negative points of the topological gradient. The great advantage of the topological gradient in comparison with level line completion and TV-based inpainting methods (see e.g., [6, 8, 12, 13]) is that the identified edges in the unknown part of the image correspond to a regular extrapolation of the known edges, and as we will see on a numerical example, the topological gradient preserves the continuity of the edge curvature. Thus, the proposed approach is much more than simple edge detection.

If no edge is inserted in the hidden zone, then the resolution of a Poisson problem gives a blurred image, as the Laplace operator provides a smooth reconstruction between the black square and the white background, as shown in Figure 4(c). The restored image by the inpainting algorithm is represented in Figure 4(e). Using the edges identified by the topological gradient, the reconstruction by the Laplacian is much better, as there is now an insulating crack between the black and white zones.

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