From the numerical point of view, it is more convenient to simulate the cracks by a small value of 
instead of considering topological perturbations of 
. The resolution of problem (2.21) with 
is an approximation of the resolution of the perturbed problem (2.22), becoming more precise as
goes to 0
.
As in the previous section (inpainting problems), our algorithm is extremely efficient as it requires only 
resolutions of a partial differential equation in
: the direct and adjoint original problems, and then the direct perturbed problem. And the complexity of this algorithm is still
(see section 2.7).
As shown in [28], the quality of the numerical results is very good. Once again, the algorithm relies on a thresholding of the topological gradient in order to define the edge set. Contrary to inpainting problems, the connexity of the edges is not crucial since it does not change significantly the quality of the restored image. However, section 2.8 presents a way to identify connex edges, with fewer badly identified edges.