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.

Let be an open bounded domain of , with , which is subject to a nonsmooth perturbation confined in a small region of size with an arbitrary point of and a fixed domain of . Let be a characteristic function associated to the unperturbed domain and be a characteristic function associated to the perforated domain . A given shape functional associated to the topologically perturbed domain, admits the following **topological asymptotic expansion**:

where is the shape functional associated to the reference domain, is a positive first order correction function of and is the remainder. The function is called the topological derivative of at .

The topological derivative can be applied to shape optimization problems in structural mechanics.^{ [1] } 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.^{ [2] } Shape optimization concerns itself with finding an optimal shape. That is, find to minimize some scalar-valued objective function, . The topological derivative technique can be coupled with level-set method.^{ [3] }

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.^{ [4] } 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.^{ [5] }

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.^{ [6] } Only operations are needed to detect edges, where is the number of pixels.^{ [7] } During the following years, other problems have been considered: classification, segmentation, inpainting and super-resolution.^{ [7] }^{ [8] }^{ [9] }^{ [10] }^{ [11] } This approach can be applied to gray-level or color images.^{ [12] } 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.^{ [13] }

In 2012, a general framework is presented to reconstruct an image given some noisy observations in a Hilbert space where is the domain where the image is defined.^{ [11] } The observation space depends on the specific application as well as the linear observation operator . The norm on the space is . The idea to recover the original image is to minimize the following functional for :

where is a positive definite tensor. The first term of the equation ensures that the recovered image 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^{ [11] }

- image denoising with and ,
- image denoising and deblurring with and with a motion blur or Gaussian blur,
- image inpainting with and , the subset is the region where the image has to be recovered.

In this framework, the asymptotic expansion of the cost function in the case of a crack provides the same topological derivative where is the normal to the crack and a constant diffusion coefficient. The functions and are solutions of the following direct and adjoint problems.^{ [11] }

in and on

in and on

Thanks to the topological gradient, it is possible to detect the edges and their orientation and to define an appropriate for the image reconstruction process.^{ [11] }

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.^{ [14] }

In 2009, the topological gradient method has been applied to tomographic reconstruction.^{ [15] } The coupling between the topological derivative and the level set has also been investigated in this application.^{ [16] }

In physics, the **Navier–Stokes equations** are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

In continuum mechanics, the **infinitesimal strain theory** is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

In the calculus of variations, a field of mathematical analysis, the **functional derivative** relates a change in a functional to a change in a function on which the functional depends.

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

**Geometrical optics**, or **ray optics**, is a model of optics that describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

In mathematics, the **exterior covariant derivative** is an analog of an exterior derivative that takes into account the presence of a connection.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

The following are important identities involving derivatives and integrals in vector calculus.

In mathematics, more specifically in dynamical systems, the **method of averaging** exploits systems containing time-scales separation: a *fast oscillation***versus** a *slow drift*. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale. It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution.

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

In mathematical logic and set theory, an **ordinal collapsing function** is a technique for defining certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals, and then “collapse” them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals.

In mathematics, **potential flow around a circular cylinder** is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.

In physics, **Berry connection** and **Berry curvature** are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.

In mathematics, the **Neumann–Poincaré operator** or **Poincaré–Neumann operator**, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space. Within the language of potential theory it reduces the partial differential equation to an integral equation on the boundary to which the theory of Fredholm operators can be applied. The theory is particularly simple in two dimensions—the case treated in detail in this article—where it is related to complex function theory, the conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains.

A **non-expanding horizon** (**NEH**) is an enclosed null surface whose intrinsic structure is preserved. An NEH is the geometric prototype of an isolated horizon which describes a black hole in equilibrium with its exterior from the quasilocal perspective. It is based on the concept and geometry of NEHs that the two quasilocal definitions of black holes, weakly isolated horizons and isolated horizons, are developed.

Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the **construction of a complex null tetrad**, where is a pair of *real* null vectors and is a pair of *complex* null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The **loop representation** is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.

The **electrokinematics theorem** connects the velocity and the charge of carriers moving inside an arbitrary volume to the currents, voltages and power on its surface through an arbitrary irrotational vector. Since it contains, as a particular application, the Ramo-Shockley theorem, the electrokinematics theorem is also known as **Ramo-Shockly-Pellegrini theorem**.

In probability theory, a branch of mathematics, **white noise analysis**, otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener process. It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes.

**Phase reduction** is a method used to reduce a multi-dimensional dynamical equation describing a nonlinear limit cycle oscillator into a one-dimensional phase equation. Many phenomena in our world such as chemical reactions, electric circuits, mechanical vibrations, cardiac cells, and spiking neurons are examples of rhythmic phenomena, and can be considered as nonlinear limit cycle oscillators.

- ↑ J. Sokolowski and A. Zochowski, 44On topological derivative in shape optimization44, 1997
- ↑ Topological Derivatives in Shape Optimization, Jan Sokołowski, May 28, 2012. Retrieved November 9, 2012
- ↑ G. Allaire, F. Jouve,
*Coupling the level set method and the topological gradient in structural optimization*, in IUTAM symposium on topological design optimization of structures, machines and materials, M. Bendsoe et al. eds., pp3-12, Springer (2006). - ↑ S. Amstutz, I. Horchani, and M. Masmoudi.
*Crack detection by the topological gradient method*. Control and Cybernetics, 34(1):81–101, 2005. - ↑ S. Amstutz, A.A. Novotny, Topological asymptotic analysis of the Kirchhoff plate bending problem. ESAIM: COCV 17(3), pp. 705-721, 2011
- ↑ L. J. Belaid, M. Jaoua, M. Masmoudi, and L. Siala.
*Image restoration and edge detection by topological asymptotic expansion*. CRAS Paris, 342(5):313–318, March 2006. - 1 2 D. Auroux and M. Masmoudi.
*Image processing by topological asymptotic analysis*. ESAIM: Proc. Mathematical methods for imaging and inverse problems, 26:24–44, April 2009. - ↑ D. Auroux, M. Masmoudi, and L. Jaafar Belaid.
*Image restoration and classification by topological asymptotic expansion*, pp. 23–42, Variational Formulations in Mechanics: Theory and Applications, E. Taroco, E.A. de Souza Neto and A.A. Novotny (Eds), CIMNE, Barcelona, Spain, 2007. - ↑ D. Auroux and M. Masmoudi.
*A one-shot inpainting algorithm based on the topological asymptotic analysis*. Computational and Applied Mathematics, 25(2-3):251–267, 2006. - ↑ D. Auroux and M. Masmoudi.
*Image processing by topological asymptotic expansion*. J. Math. Imaging Vision, 33(2):122–134, February 2009. - 1 2 3 4 5 S. Larnier, J. Fehrenbach and M. Masmoudi, The topological gradient method: From optimal design to image processing, Milan Journal of Mathematics, vol. 80, issue 2, pp. 411–441, December 2012.
- ↑ D. Auroux, L. Jaafar Belaid, and B. Rjaibi.
*Application of the topological gradient method to color image restoration*. SIAM J. Imaging Sci., 3(2):153–175, 2010. - ↑ S. Larnier and J. Fehrenbach.
*Edge detection and image restoration with anisotropic topological gradient*. In 2010 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pages 1362–1365, March 2010. - ↑ A. Drogoul, G. Aubert, The topological gradient method for semi-linear problems and application to edge detection and noise removal.
- ↑ D. Auroux, L. Jaafar Belaid, and B. Rjaibi.
*Application of the topological gradient method to tomography*. In ARIMA Proc. TamTam'09, 2010. - ↑ T. Rymarczyk, P. Tchórzewski, J. Sikora,
*Topological Approach to Image Reconstruction in Electrical Impedance Tomography*, ADVCOMP 2014 : The Eighth International Conference on Advanced Engineering Computing and Applications in Science

A. A. Novotny and J. Sokolowski, *Topological derivatives in shape optimization*, Springer, 2013.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.