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3D Shape Registration Using Spectral Graph Embedding and Probabilistic Matching
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Avinash Sharma, Radu Horaud, Diana Mateus
We can now build the concept of the graph Laplacian operator. We consider the following variants of the Laplacian matrix [41, 40, 42]: The unnormalized Laplacian, which is also referred to as the combinatorial LaplacianL,The normalized LaplacianL˜, andThe random-walk LaplacianL˜R also referred to as the discrete Laplace operator.
EEG inverse problem I
Published in Munsif Ali Jatoi, Nidal Kamel, Brain Source Localization Using EEG Signal Analysis, 2017
The discrete Laplace operator B is introduced to emphasize relationships between current densities, and thus, the spatial resolution is not taken into consideration, which results in blurred localization images. For a regular cubic grid within the brain volume, if the distance between neighboring grids is assumed to be d, then this operator is defined as: B=6d2(A−I3M)
A Primer on Laplacians
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
The fact that no discrete Laplace operator satisfies all of the desired properties simultaneously is not a coincidence. It can be shown that general simplicial meshes do not allow for discrete Laplacians that satisfy (Loc) + (Sym) + (Lin) + (Pos); see [410] for more on this topic and Figure 5.1 (right) for a simple example. This limitation provides a taxonomy on existing literature and explains the plethora of existing discrete Laplacians: Since not all desired properties can be fulfilled simultaneously, it depends on the application at hand to design discrete Laplacians that are tailored towards the specific needs of a concrete problem.
Semi-automatic 3D reconstruction of middle and inner ear structures using CBCT
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2023
Florian Beguet, Thierry Cresson, Mathieu Schmittbuhl, Cédric Doucet, David Camirand, Philippe Harris, Jean-Luc Mari, Jacques de Guise
where is the attraction strength parameter that weighs the volume energy – a squared sum between vertices and targets – weighted by . The internal energy is given by the product of the current position of the vertices and the Laplacian matrix . being the matrix representing the coordinates of obtained by applying the discrete Laplace operator on the initial mesh where represents edges.
Recent trends and advances in solving the inverse problem for EEG source localization
Published in Inverse Problems in Science and Engineering, 2019
Fahim Gohar Awan, Omer Saleem, Asima Kiran
The functioning of LORETA technique is identical to MNE, except that it uses discrete Laplace operator for regularization instead of using the identity matrix [42]. As LORETA consider the connectivity given by the discrete Laplacian (), it has been rigorously used for regularization of smoothly distributed sources. The solution is given by the following equation in this case. It is to be noted that the expression given in Equation (8) is derived from the Tinkonov regularization; such that, , the Laplacian operator. Hence, the regularization becomes an H2 semi-norm.
A polarization tensor approximation for the Hessian in iterative solvers for non-linear inverse problems
Published in Inverse Problems in Science and Engineering, 2021
F. M. Watson, M. G. Crabb, W. R. B. Lionheart
The regularization term for each was with the discrete Laplace operator in 2D, and a regularization parameter of (chosen heuristically to provide the best stable Gauss–Newton results as a benchmark). The regularization term was also included in the initial Hessian approximation for each of the l-BFGS methods. Each reconstruction was stopped when the relative change in residual met the numerical stagnation condition which is the default condition of the EIDORS Gauss–Newton solver.