Incorporating topological derivatives into level set methods
Journal of Computational Physics
Image Processing by Topological Asymptotic Expansion
Journal of Mathematical Imaging and Vision
Topological Sensitivity Analysis for the Location of Small Cavities in Stokes Flow
SIAM Journal on Control and Optimization
Application of the Topological Gradient Method to Color Image Restoration
SIAM Journal on Imaging Sciences
Journal of Biomedical Imaging - Special issue on Mathematical Methods for Images and Surfaces 2011
From restoration by topological gradient to medical image segmentation via an asymptotic expansion
Mathematical and Computer Modelling: An International Journal
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The topological sensitivity analysis provides an asymptotic expansion of a shape function when creating a small hole inside a domain. This expansion yields a descent direction which can be used for shape optimization if one wishes to keep a classical domain throughout the optimization process. In this paper, such an expansion is obtained for the Poisson equation for a large class of cost functions and arbitrarily shaped holes. In the three-dimensional case, this expansion depends on the shape of the hole but not on its orientation if the cost function involves only the solution u to the underlying partial differential equation, whereas it may also depend on its orientation if the cost function involves the gradient $\nabla u$. In contrast, the asymptotic expansion is independent of the shape in the two-dimensional case. A numerical example illustrates the use of the asymptotic expansion, which yields a minimizing sequence of classical domains in a case where no classical solution exists.