Topological derivative
The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in higher dimensions than one, the term topological gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It has applications in shape optimization, topology optimization, image processing and mechanical modeling. Definition. Let formula_1 be an open bounded domain of formula_2, with formula_3, which is subject to a nonsmooth perturbation confined in a small region formula_4 of  size formula_5 with formula_6 an arbitrary point of formula_1 and formula_8 a fixed domain of formula_2. Let formula_10 be a characteristic function associated to the unperturbed domain and formula_11 be a characteristic function associated to the perforated domain formula_12. A given shape functional formula_13 associated to the topologically perturbed domain, admits the following topological asymptotic expansion: where formula_14 is the shape functional associated to the reference domain, formula_15 is a positive first order correction function of formula_14 and formula_17 is the remainder. The function formula_18 is called the topological derivative of formula_19 at formula_6. Applications. Structural mechanics. The topological derivative  can be applied to shape optimization problems in structural mechanics. The topological derivative can be considered as the singular limit of the shape derivative. It is a generalization of this classical tool in shape optimization. Shape optimization concerns itself with finding an optimal shape. That is, find formula_21 to minimize some scalar-valued objective function, formula_22. The topological derivative technique can be coupled with level-set method. In 2005, the topological asymptotic expansion for the Laplace equation with respect to the insertion of a short crack inside a plane domain had been found. It allows to detect and locate cracks  for a simple model problem: the steady-state heat equation with the heat flux imposed and the temperature measured on the boundary. The topological derivative had been fully developed for a wide range of second-order differential operators and in 2011, it had been applied to Kirchhoff plate bending problem with a fourth-order operator. Image processing. In the field of image processing, in 2006, the topological derivative has been used to perform edge detection and image restoration. The impact of an insulating crack in the domain is studied. The topological sensitivity gives information on the image edges. The presented  algorithm is non-iterative and thanks to the use of spectral methods has a short computing time. Only formula_23 operations are needed to detect edges, where formula_24 is the number of pixels. During the following years, other problems have been considered: classification, segmentation, inpainting and super-resolution. This approach can be applied to gray-level or color images. Until 2010, isotropic diffusion was used for image reconstructions. The topological gradient is also able to provide edge orientation and this information can be used to perform anisotropic diffusion. In 2012, a general framework is presented to reconstruct an image formula_25 given  some noisy observations formula_26 in a Hilbert space formula_27 where formula_1 is the domain where the image formula_29 is defined. The observation space formula_27 depends on the specific application as well as the linear observation operator formula_31. The norm on the space formula_27 is formula_33. The idea to recover the original image is to minimize the following functional for formula_34: where formula_35 is a positive definite tensor. The first term of the equation ensures that the recovered image formula_29 is regular, and the second term measures the discrepancy with the data. In this general framework, different types of image reconstruction  can be performed such as In this framework, the asymptotic expansion of the cost function formula_45 in the case of a crack provides the same topological derivative formula_46 where formula_47 is the normal to the crack and formula_48 a constant diffusion coefficient. The functions formula_49 and formula_50 are solutions of the following direct and adjoint problems. Thanks to the topological gradient, it is possible to detect the edges and their orientation and to define an appropriate formula_35 for the image reconstruction process. In image processing, the topological derivatives have also been studied in the case of a multiplicative noise of  gamma law or in presence of Poissonian statistics. Inverse problems. In 2009, the topological gradient method has been applied to tomographic reconstruction. The coupling between the topological derivative and the level set has also been investigated in this application. Books. A. A. Novotny and J. Sokolowski, "Topological derivatives in shape optimization", Springer, 2013.