Eigenfunctions of the Laplace operator
نویسنده
چکیده
The study of the Laplace operator and its corresponding eigenvalue problem is crucial to understand the foundations of 3D shape analysis. For that reason the most important mathematical properties of the Laplace operator in Euclidean spaces, its eigenvalues and eigenfunctions are summarized and explained in this report. The basic definitions and concepts of infinite dimensional function spaces, that are required are introduced beforehand. In order to motivate the study of the Laplace eigenvalue problem, the connection between the Laplace eigenvalues and the geometry of the domain, to which the operator is applied, is demonstrated by presenting Weyl’s law. Furthermore it is shown how the Fourier transform can be derived from the consequences of the spectral theorem and orthogonal basis functions of the L space. Finally some applications of the Laplace eigenfunctions in 3D shape analysis are demonstrated briefly.
منابع مشابه
On Generalization of Sturm-Liouville Theory for Fractional Bessel Operator
In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...
متن کاملHeat Kernel Smoothing of Anatomical Manifolds via Laplace-Beltrami Eigenfunctions
We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...
متن کاملSpherical Parameterization for Genus Zero Surfaces Using Laplace-Beltrami Eigenfunctions
In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-tri...
متن کاملHeat Kernel Smoothing of Anatomical Manifolds via Laplace-Beltrami Eigenfunctions Submitted to IEEE Transactions on Medical Imaging
We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...
متن کاملAsymptotic Property of Eigenvalues and Eigenfunctions of the Laplace Operator in Domain with a Perturbed Boundary
In this paper, we consider the variations of eigenvalues and eigenfunctions for the Laplace operator with homogeneous Dirichlet boundary conditions under deformation of the underlying domain of definition. We derive recursive formulas for the Taylor coefficients of the eigenvalues as functions of the shape-perturbation parameter and we establish the existence of a set of eigenfunctions that is ...
متن کامل